diff --git "a/data/sensemath_v2_d4.json" "b/data/sensemath_v2_d4.json" new file mode 100644--- /dev/null +++ "b/data/sensemath_v2_d4.json" @@ -0,0 +1,28602 @@ +[ + { + "family_id": "magnitude_estimation_000401", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 10100?", + "options": [ + "103019990", + "103020020", + "103020010", + "103020000" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10200≈10000, 10100≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 4937", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 4937?", + "options": [ + "49863720", + "49863710", + "49863690", + "49863700" + ], + "answer": "49863700", + "correct_value": 49863700, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5835 * 6856", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5835 * 6856?", + "options": [ + "40004770", + "40004780", + "40004760", + "40004750" + ], + "answer": "40004760", + "correct_value": 40004760, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000402", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 9900?", + "options": [ + "99990000", + "99990020", + "99990010", + "99989990" + ], + "answer": "99990000", + "correct_value": 99990000, + "explanation": "Round to 10000: 10100≈10000, 9900≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 4936", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 4936?", + "options": [ + "49853600", + "49853610", + "49853590", + "49853620" + ], + "answer": "49853600", + "correct_value": 49853600, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6465 * 6928", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6465 * 6928?", + "options": [ + "44789540", + "44789530", + "44789520", + "44789510" + ], + "answer": "44789520", + "correct_value": 44789520, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000403", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 10200?", + "options": [ + "104039990", + "104040000", + "104040010", + "104040020" + ], + "answer": "104040000", + "correct_value": 104040000, + "explanation": "Round to 10000: 10200≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 5043", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 5043?", + "options": [ + "50934290", + "50934310", + "50934300", + "50934320" + ], + "answer": "50934300", + "correct_value": 50934300, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4527 * 4708", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4527 * 4708?", + "options": [ + "21313136", + "21313106", + "21313116", + "21313126" + ], + "answer": "21313116", + "correct_value": 21313116, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000404", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 10200?", + "options": [ + "99960020", + "99960000", + "99960010", + "99959990" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 9800≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4950", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4950?", + "options": [ + "49005020", + "49005010", + "49004990", + "49005000" + ], + "answer": "49005000", + "correct_value": 49005000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5067 * 5461", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5067 * 5461?", + "options": [ + "27670877", + "27670907", + "27670887", + "27670897" + ], + "answer": "27670887", + "correct_value": 27670887, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000405", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 10100?", + "options": [ + "98980010", + "98980020", + "98980000", + "98979990" + ], + "answer": "98980000", + "correct_value": 98980000, + "explanation": "Round to 10000: 9800≈10000, 10100≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4892", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4892?", + "options": [ + "48430820", + "48430800", + "48430790", + "48430810" + ], + "answer": "48430800", + "correct_value": 48430800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5392 * 5454", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5392 * 5454?", + "options": [ + "29407958", + "29407978", + "29407988", + "29407968" + ], + "answer": "29407968", + "correct_value": 29407968, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000406", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9800?", + "options": [ + "97019990", + "97020020", + "97020010", + "97020000" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9900≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5002", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5002?", + "options": [ + "49519820", + "49519800", + "49519810", + "49519790" + ], + "answer": "49519800", + "correct_value": 49519800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6302 * 4170", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6302 * 4170?", + "options": [ + "26279350", + "26279360", + "26279330", + "26279340" + ], + "answer": "26279340", + "correct_value": 26279340, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000407", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 10100?", + "options": [ + "99990000", + "99989990", + "99990020", + "99990010" + ], + "answer": "99990000", + "correct_value": 99990000, + "explanation": "Round to 10000: 9900≈10000, 10100≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 5164", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 5164?", + "options": [ + "50607220", + "50607210", + "50607190", + "50607200" + ], + "answer": "50607200", + "correct_value": 50607200, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4630 * 6649", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4630 * 6649?", + "options": [ + "30784860", + "30784870", + "30784880", + "30784890" + ], + "answer": "30784870", + "correct_value": 30784870, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000408", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9900?", + "options": [ + "97020020", + "97020000", + "97020010", + "97019990" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9800≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5003", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5003?", + "options": [ + "49529710", + "49529690", + "49529720", + "49529700" + ], + "answer": "49529700", + "correct_value": 49529700, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6067 * 4700", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6067 * 4700?", + "options": [ + "28514890", + "28514920", + "28514910", + "28514900" + ], + "answer": "28514900", + "correct_value": 28514900, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000409", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 10100?", + "options": [ + "103020020", + "103019990", + "103020000", + "103020010" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10200≈10000, 10100≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4873", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4873?", + "options": [ + "48242700", + "48242720", + "48242690", + "48242710" + ], + "answer": "48242700", + "correct_value": 48242700, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5878 * 6113", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5878 * 6113?", + "options": [ + "35932204", + "35932224", + "35932214", + "35932234" + ], + "answer": "35932214", + "correct_value": 35932214, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000410", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9800?", + "options": [ + "97020020", + "97020010", + "97019990", + "97020000" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9900≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4887", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4887?", + "options": [ + "49847410", + "49847390", + "49847420", + "49847400" + ], + "answer": "49847400", + "correct_value": 49847400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5609 * 4150", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5609 * 4150?", + "options": [ + "23277340", + "23277350", + "23277370", + "23277360" + ], + "answer": "23277350", + "correct_value": 23277350, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000411", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10200?", + "options": [ + "103020000", + "103020010", + "103020020", + "103019990" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10100≈10000, 10200≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5157", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5157?", + "options": [ + "52601410", + "52601420", + "52601400", + "52601390" + ], + "answer": "52601400", + "correct_value": 52601400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6160 * 6499", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6160 * 6499?", + "options": [ + "40033840", + "40033830", + "40033860", + "40033850" + ], + "answer": "40033840", + "correct_value": 40033840, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000412", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9900?", + "options": [ + "100980000", + "100980010", + "100980020", + "100979990" + ], + "answer": "100980000", + "correct_value": 100980000, + "explanation": "Round to 10000: 10200≈10000, 9900≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5039", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5039?", + "options": [ + "51397820", + "51397810", + "51397800", + "51397790" + ], + "answer": "51397800", + "correct_value": 51397800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4150 * 6200", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4150 * 6200?", + "options": [ + "25729990", + "25730000", + "25730010", + "25730020" + ], + "answer": "25730000", + "correct_value": 25730000, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000413", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 9900?", + "options": [ + "99989990", + "99990010", + "99990020", + "99990000" + ], + "answer": "99990000", + "correct_value": 99990000, + "explanation": "Round to 10000: 10100≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5056", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5056?", + "options": [ + "50054400", + "50054390", + "50054410", + "50054420" + ], + "answer": "50054400", + "correct_value": 50054400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6210 * 6425", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6210 * 6425?", + "options": [ + "39899240", + "39899260", + "39899250", + "39899270" + ], + "answer": "39899250", + "correct_value": 39899250, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000414", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9800?", + "options": [ + "99959990", + "99960020", + "99960010", + "99960000" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 10200≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4889", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4889?", + "options": [ + "48401090", + "48401110", + "48401100", + "48401120" + ], + "answer": "48401100", + "correct_value": 48401100, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5837 * 5219", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5837 * 5219?", + "options": [ + "30463293", + "30463323", + "30463313", + "30463303" + ], + "answer": "30463303", + "correct_value": 30463303, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000415", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9900?", + "options": [ + "98010000", + "98009990", + "98010010", + "98010020" + ], + "answer": "98010000", + "correct_value": 98010000, + "explanation": "Round to 10000: 9900≈10000, 9900≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 5115", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 5115?", + "options": [ + "50127010", + "50126990", + "50127000", + "50127020" + ], + "answer": "50127000", + "correct_value": 50127000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4430 * 5341", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4430 * 5341?", + "options": [ + "23660630", + "23660650", + "23660620", + "23660640" + ], + "answer": "23660630", + "correct_value": 23660630, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000416", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9900?", + "options": [ + "97020020", + "97020000", + "97020010", + "97019990" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9800≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5127", + "ns_shortcut": "Round 9900��10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5127?", + "options": [ + "50757320", + "50757300", + "50757310", + "50757290" + ], + "answer": "50757300", + "correct_value": 50757300, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4878 * 5826", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4878 * 5826?", + "options": [ + "28419218", + "28419248", + "28419238", + "28419228" + ], + "answer": "28419228", + "correct_value": 28419228, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000417", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 10100?", + "options": [ + "103020000", + "103020010", + "103020020", + "103019990" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10200≈10000, 10100≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5024", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5024?", + "options": [ + "49737620", + "49737590", + "49737610", + "49737600" + ], + "answer": "49737600", + "correct_value": 49737600, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4666 * 4590", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4666 * 4590?", + "options": [ + "21416940", + "21416960", + "21416930", + "21416950" + ], + "answer": "21416940", + "correct_value": 21416940, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000418", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9800?", + "options": [ + "97020000", + "97020020", + "97019990", + "97020010" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9900≈10000, 9800≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5030", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5030?", + "options": [ + "51305990", + "51306020", + "51306000", + "51306010" + ], + "answer": "51306000", + "correct_value": 51306000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4485 * 6463", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4485 * 6463?", + "options": [ + "28986565", + "28986575", + "28986555", + "28986545" + ], + "answer": "28986555", + "correct_value": 28986555, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000419", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10100?", + "options": [ + "102010010", + "102009990", + "102010020", + "102010000" + ], + "answer": "102010000", + "correct_value": 102010000, + "explanation": "Round to 10000: 10100≈10000, 10100≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 5093", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 5093?", + "options": [ + "49911390", + "49911410", + "49911420", + "49911400" + ], + "answer": "49911400", + "correct_value": 49911400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5203 * 5989", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5203 * 5989?", + "options": [ + "31160777", + "31160787", + "31160757", + "31160767" + ], + "answer": "31160767", + "correct_value": 31160767, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000420", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9800?", + "options": [ + "99959990", + "99960000", + "99960010", + "99960020" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 10200≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4932", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4932?", + "options": [ + "50306410", + "50306390", + "50306400", + "50306420" + ], + "answer": "50306400", + "correct_value": 50306400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6241 * 5427", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6241 * 5427?", + "options": [ + "33869927", + "33869917", + "33869897", + "33869907" + ], + "answer": "33869907", + "correct_value": 33869907, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000421", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 10100?", + "options": [ + "103020000", + "103020020", + "103020010", + "103019990" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10200≈10000, 10100≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4808", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4808?", + "options": [ + "47599190", + "47599210", + "47599200", + "47599220" + ], + "answer": "47599200", + "correct_value": 47599200, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5841 * 6731", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5841 * 6731?", + "options": [ + "39315791", + "39315781", + "39315771", + "39315761" + ], + "answer": "39315771", + "correct_value": 39315771, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000422", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 10200?", + "options": [ + "99960020", + "99960010", + "99960000", + "99959990" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 9800≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4800", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4800?", + "options": [ + "47520020", + "47519990", + "47520010", + "47520000" + ], + "answer": "47520000", + "correct_value": 47520000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5135 * 6644", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5135 * 6644?", + "options": [ + "34116950", + "34116960", + "34116930", + "34116940" + ], + "answer": "34116940", + "correct_value": 34116940, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000423", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 9800?", + "options": [ + "98980010", + "98980020", + "98979990", + "98980000" + ], + "answer": "98980000", + "correct_value": 98980000, + "explanation": "Round to 10000: 10100≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 4976", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 4976?", + "options": [ + "48764790", + "48764810", + "48764820", + "48764800" + ], + "answer": "48764800", + "correct_value": 48764800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4491 * 6695", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4491 * 6695?", + "options": [ + "30067265", + "30067235", + "30067245", + "30067255" + ], + "answer": "30067245", + "correct_value": 30067245, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000424", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9900?", + "options": [ + "98010020", + "98010010", + "98010000", + "98009990" + ], + "answer": "98010000", + "correct_value": 98010000, + "explanation": "Round to 10000: 9900≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 5030", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 5030?", + "options": [ + "49294020", + "49293990", + "49294010", + "49294000" + ], + "answer": "49294000", + "correct_value": 49294000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5391 * 5789", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5391 * 5789?", + "options": [ + "31208519", + "31208499", + "31208489", + "31208509" + ], + "answer": "31208499", + "correct_value": 31208499, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000425", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10200?", + "options": [ + "103020010", + "103019990", + "103020000", + "103020020" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10100≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4899", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4899?", + "options": [ + "48500120", + "48500090", + "48500110", + "48500100" + ], + "answer": "48500100", + "correct_value": 48500100, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4747 * 5045", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4747 * 5045?", + "options": [ + "23948635", + "23948625", + "23948605", + "23948615" + ], + "answer": "23948615", + "correct_value": 23948615, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000426", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9900?", + "options": [ + "100979990", + "100980010", + "100980020", + "100980000" + ], + "answer": "100980000", + "correct_value": 100980000, + "explanation": "Round to 10000: 10200≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5179", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5179?", + "options": [ + "51272090", + "51272110", + "51272100", + "51272120" + ], + "answer": "51272100", + "correct_value": 51272100, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4403 * 5309", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4403 * 5309?", + "options": [ + "23375527", + "23375547", + "23375517", + "23375537" + ], + "answer": "23375527", + "correct_value": 23375527, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000427", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10200?", + "options": [ + "103020010", + "103020020", + "103019990", + "103020000" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10100≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 4800", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 4800?", + "options": [ + "48480010", + "48480000", + "48480020", + "48479990" + ], + "answer": "48480000", + "correct_value": 48480000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4470 * 5102", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4470 * 5102?", + "options": [ + "22805930", + "22805960", + "22805950", + "22805940" + ], + "answer": "22805940", + "correct_value": 22805940, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000428", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9800?", + "options": [ + "99960020", + "99960010", + "99960000", + "99959990" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 10200≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4808", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4808?", + "options": [ + "49041610", + "49041600", + "49041620", + "49041590" + ], + "answer": "49041600", + "correct_value": 49041600, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5007 * 6271", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5007 * 6271?", + "options": [ + "31398897", + "31398917", + "31398887", + "31398907" + ], + "answer": "31398897", + "correct_value": 31398897, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000429", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9800?", + "options": [ + "99960020", + "99959990", + "99960000", + "99960010" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 10200≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 4829", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 4829?", + "options": [ + "48772910", + "48772890", + "48772920", + "48772900" + ], + "answer": "48772900", + "correct_value": 48772900, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6272 * 5849", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6272 * 5849?", + "options": [ + "36684948", + "36684938", + "36684918", + "36684928" + ], + "answer": "36684928", + "correct_value": 36684928, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000430", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 9900?", + "options": [ + "99990020", + "99990010", + "99989990", + "99990000" + ], + "answer": "99990000", + "correct_value": 99990000, + "explanation": "Round to 10000: 10100≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 5084", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 5084?", + "options": [ + "49823210", + "49823200", + "49823220", + "49823190" + ], + "answer": "49823200", + "correct_value": 49823200, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5438 * 6588", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5438 * 6588?", + "options": [ + "35825544", + "35825534", + "35825564", + "35825554" + ], + "answer": "35825544", + "correct_value": 35825544, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000431", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9900?", + "options": [ + "97020020", + "97019990", + "97020010", + "97020000" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9800≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 4988", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 4988?", + "options": [ + "50378820", + "50378810", + "50378790", + "50378800" + ], + "answer": "50378800", + "correct_value": 50378800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5768 * 4862", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5768 * 4862?", + "options": [ + "28044016", + "28044026", + "28044006", + "28044036" + ], + "answer": "28044016", + "correct_value": 28044016, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000432", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10200?", + "options": [ + "103019990", + "103020010", + "103020000", + "103020020" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10100≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5184", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5184?", + "options": [ + "52876790", + "52876810", + "52876820", + "52876800" + ], + "answer": "52876800", + "correct_value": 52876800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4064 * 6832", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4064 * 6832?", + "options": [ + "27765238", + "27765248", + "27765268", + "27765258" + ], + "answer": "27765248", + "correct_value": 27765248, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000433", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9800?", + "options": [ + "99960010", + "99959990", + "99960020", + "99960000" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 10200≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4877", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4877?", + "options": [ + "48282310", + "48282290", + "48282320", + "48282300" + ], + "answer": "48282300", + "correct_value": 48282300, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6657 * 6666", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6657 * 6666?", + "options": [ + "44375572", + "44375552", + "44375582", + "44375562" + ], + "answer": "44375562", + "correct_value": 44375562, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000434", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9800?", + "options": [ + "99960000", + "99959990", + "99960010", + "99960020" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 10200≈10000, 9800≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 4938", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 4938?", + "options": [ + "48886190", + "48886210", + "48886220", + "48886200" + ], + "answer": "48886200", + "correct_value": 48886200, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5516 * 4789", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5516 * 4789?", + "options": [ + "26416134", + "26416144", + "26416124", + "26416114" + ], + "answer": "26416124", + "correct_value": 26416124, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000435", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 10100?", + "options": [ + "98980010", + "98979990", + "98980000", + "98980020" + ], + "answer": "98980000", + "correct_value": 98980000, + "explanation": "Round to 10000: 9800≈10000, 10100≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5140", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5140?", + "options": [ + "50885990", + "50886000", + "50886020", + "50886010" + ], + "answer": "50886000", + "correct_value": 50886000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5485 * 4149", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5485 * 4149?", + "options": [ + "22757255", + "22757265", + "22757275", + "22757285" + ], + "answer": "22757265", + "correct_value": 22757265, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000436", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9900?", + "options": [ + "97020020", + "97020010", + "97020000", + "97019990" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9800≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5140", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5140?", + "options": [ + "52427990", + "52428000", + "52428010", + "52428020" + ], + "answer": "52428000", + "correct_value": 52428000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4266 * 6408", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4266 * 6408?", + "options": [ + "27336518", + "27336528", + "27336548", + "27336538" + ], + "answer": "27336528", + "correct_value": 27336528, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000437", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 10100?", + "options": [ + "99990000", + "99989990", + "99990020", + "99990010" + ], + "answer": "99990000", + "correct_value": 99990000, + "explanation": "Round to 10000: 9900≈10000, 10100≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4900", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4900?", + "options": [ + "49979990", + "49980000", + "49980010", + "49980020" + ], + "answer": "49980000", + "correct_value": 49980000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6787 * 6293", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6787 * 6293?", + "options": [ + "42710611", + "42710601", + "42710591", + "42710581" + ], + "answer": "42710591", + "correct_value": 42710591, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000438", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 10100", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 10100?", + "options": [ + "98980000", + "98980020", + "98980010", + "98979990" + ], + "answer": "98980000", + "correct_value": 98980000, + "explanation": "Round to 10000: 9800≈10000, 10100≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 5007", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 5007?", + "options": [ + "49068590", + "49068600", + "49068610", + "49068620" + ], + "answer": "49068600", + "correct_value": 49068600, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6117 * 4177", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6117 * 4177?", + "options": [ + "25550729", + "25550699", + "25550709", + "25550719" + ], + "answer": "25550709", + "correct_value": 25550709, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000439", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 10200?", + "options": [ + "104039990", + "104040010", + "104040000", + "104040020" + ], + "answer": "104040000", + "correct_value": 104040000, + "explanation": "Round to 10000: 10200≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5025", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5025?", + "options": [ + "51254990", + "51255020", + "51255010", + "51255000" + ], + "answer": "51255000", + "correct_value": 51255000, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4923 * 6823", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4923 * 6823?", + "options": [ + "33589639", + "33589649", + "33589629", + "33589619" + ], + "answer": "33589629", + "correct_value": 33589629, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000440", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9900?", + "options": [ + "98009990", + "98010000", + "98010010", + "98010020" + ], + "answer": "98010000", + "correct_value": 98010000, + "explanation": "Round to 10000: 9900≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5084", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5084?", + "options": [ + "51856800", + "51856820", + "51856810", + "51856790" + ], + "answer": "51856800", + "correct_value": 51856800, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6426 * 6564", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6426 * 6564?", + "options": [ + "42180254", + "42180264", + "42180274", + "42180284" + ], + "answer": "42180264", + "correct_value": 42180264, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000441", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 10200?", + "options": [ + "99960020", + "99960000", + "99959990", + "99960010" + ], + "answer": "99960000", + "correct_value": 99960000, + "explanation": "Round to 10000: 9800≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4948", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4948?", + "options": [ + "50469610", + "50469620", + "50469590", + "50469600" + ], + "answer": "50469600", + "correct_value": 50469600, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4390 * 6560", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4390 * 6560?", + "options": [ + "28798410", + "28798400", + "28798420", + "28798390" + ], + "answer": "28798400", + "correct_value": 28798400, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000442", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9800?", + "options": [ + "96039990", + "96040020", + "96040010", + "96040000" + ], + "answer": "96040000", + "correct_value": 96040000, + "explanation": "Round to 10000: 9800≈10000, 9800≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4812", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4812?", + "options": [ + "49082420", + "49082400", + "49082410", + "49082390" + ], + "answer": "49082400", + "correct_value": 49082400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4223 * 5233", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4223 * 5233?", + "options": [ + "22098969", + "22098979", + "22098959", + "22098949" + ], + "answer": "22098959", + "correct_value": 22098959, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000443", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9800?", + "options": [ + "96040000", + "96039990", + "96040010", + "96040020" + ], + "answer": "96040000", + "correct_value": 96040000, + "explanation": "Round to 10000: 9800≈10000, 9800≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 5197", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 5197?", + "options": [ + "53009410", + "53009390", + "53009400", + "53009420" + ], + "answer": "53009400", + "correct_value": 53009400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5856 * 5571", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5856 * 5571?", + "options": [ + "32623796", + "32623776", + "32623786", + "32623766" + ], + "answer": "32623776", + "correct_value": 32623776, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000444", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 9800", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 9800?", + "options": [ + "98980000", + "98980010", + "98980020", + "98979990" + ], + "answer": "98980000", + "correct_value": 98980000, + "explanation": "Round to 10000: 10100≈10000, 9800≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 5021", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 5021?", + "options": [ + "50712110", + "50712100", + "50712120", + "50712090" + ], + "answer": "50712100", + "correct_value": 50712100, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6427 * 4792", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6427 * 4792?", + "options": [ + "30798184", + "30798174", + "30798204", + "30798194" + ], + "answer": "30798184", + "correct_value": 30798184, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000445", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10200?", + "options": [ + "103020000", + "103020010", + "103020020", + "103019990" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10100≈10000, 10200≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5136", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5136?", + "options": [ + "50846390", + "50846420", + "50846400", + "50846410" + ], + "answer": "50846400", + "correct_value": 50846400, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4521 * 5275", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4521 * 5275?", + "options": [ + "23848275", + "23848285", + "23848295", + "23848265" + ], + "answer": "23848275", + "correct_value": 23848275, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000446", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 9900?", + "options": [ + "99990000", + "99990020", + "99989990", + "99990010" + ], + "answer": "99990000", + "correct_value": 99990000, + "explanation": "Round to 10000: 10100≈10000, 9900≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9900 * 5145", + "ns_shortcut": "Round 9900≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 5145?", + "options": [ + "50935520", + "50935500", + "50935490", + "50935510" + ], + "answer": "50935500", + "correct_value": 50935500, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4233 * 4971", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4233 * 4971?", + "options": [ + "21042233", + "21042263", + "21042253", + "21042243" + ], + "answer": "21042243", + "correct_value": 21042243, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000447", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9900 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Which of the following is the best estimate for 9900 * 9900?", + "options": [ + "98010020", + "98009990", + "98010000", + "98010010" + ], + "answer": "98010000", + "correct_value": 98010000, + "explanation": "Round to 10000: 9900≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10200 * 4961", + "ns_shortcut": "Round 10200≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 4961?", + "options": [ + "50602210", + "50602220", + "50602190", + "50602200" + ], + "answer": "50602200", + "correct_value": 50602200, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6455 * 6835", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 6455 * 6835?", + "options": [ + "44119925", + "44119935", + "44119915", + "44119945" + ], + "answer": "44119925", + "correct_value": 44119925, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000448", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10100 * 10200", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 10200?", + "options": [ + "103020010", + "103019990", + "103020000", + "103020020" + ], + "answer": "103020000", + "correct_value": 103020000, + "explanation": "Round to 10000: 10100≈10000, 10200≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 4874", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 4874?", + "options": [ + "47765220", + "47765200", + "47765190", + "47765210" + ], + "answer": "47765200", + "correct_value": 47765200, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4014 * 6615", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 4014 * 6615?", + "options": [ + "26552600", + "26552610", + "26552630", + "26552620" + ], + "answer": "26552610", + "correct_value": 26552610, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "magnitude_estimation_000449", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10200 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 10200 * 9900?", + "options": [ + "100980000", + "100980020", + "100979990", + "100980010" + ], + "answer": "100980000", + "correct_value": 100980000, + "explanation": "Round to 10000: 10200≈10000, 9900≈10000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10100 * 5179", + "ns_shortcut": "Round 10100≈10000, then adjust.", + "shortcut_strength": 0.7969999999999999, + "pure_math": { + "question": "Which of the following is the best estimate for 10100 * 5179?", + "options": [ + "52307910", + "52307900", + "52307890", + "52307920" + ], + "answer": "52307900", + "correct_value": 52307900, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5485 * 4039", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5485 * 4039?", + "options": [ + "22153935", + "22153905", + "22153925", + "22153915" + ], + "answer": "22153915", + "correct_value": 22153915, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "magnitude_estimation_000450", + "category": "magnitude_estimation", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9800 * 9900", + "ns_shortcut": "Round both to 10000 and multiply.", + "shortcut_strength": 0.985, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 9900?", + "options": [ + "97019990", + "97020000", + "97020010", + "97020020" + ], + "answer": "97020000", + "correct_value": 97020000, + "explanation": "Round to 10000: 9800≈10000, 9900≈10000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9800 * 4997", + "ns_shortcut": "Round 9800≈10000, then adjust.", + "shortcut_strength": 0.794, + "pure_math": { + "question": "Which of the following is the best estimate for 9800 * 4997?", + "options": [ + "48970620", + "48970610", + "48970600", + "48970590" + ], + "answer": "48970600", + "correct_value": 48970600, + "explanation": "One number rounds cleanly, the other less so." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5162 * 4712", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which of the following is the best estimate for 5162 * 4712?", + "options": [ + "24323364", + "24323344", + "24323334", + "24323354" + ], + "answer": "24323344", + "correct_value": 24323344, + "explanation": "Both numbers far from any round target; no clean rounding shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000451", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 5029", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 5029?", + "options": [ + "50295019", + "50295049", + "50295029", + "50295039" + ], + "answer": "50295029", + "correct_value": 50295029, + "explanation": "Distributive: (10000+1)*5029 = 10000*5029 + 1*5029." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9950 * 3967", + "ns_shortcut": "Use (10000-50)*m, correction is 50*3967.", + "shortcut_strength": 0.9, + "pure_math": { + "question": "What is 9950 * 3967?", + "options": [ + "39471640", + "39471670", + "39471650", + "39471660" + ], + "answer": "39471650", + "correct_value": 39471650, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5551 * 6681", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5551 * 6681?", + "options": [ + "37086251", + "37086221", + "37086241", + "37086231" + ], + "answer": "37086231", + "correct_value": 37086231, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000452", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 8474", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 8474?", + "options": [ + "84748474", + "84748494", + "84748464", + "84748484" + ], + "answer": "84748474", + "correct_value": 84748474, + "explanation": "Distributive: (10000+1)*8474 = 10000*8474 + 1*8474." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10031 * 4852", + "ns_shortcut": "Use (10000+31)*m, correction is 31*4852.", + "shortcut_strength": 0.938, + "pure_math": { + "question": "What is 10031 * 4852?", + "options": [ + "48670422", + "48670412", + "48670432", + "48670402" + ], + "answer": "48670412", + "correct_value": 48670412, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4484 * 4590", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4484 * 4590?", + "options": [ + "20581550", + "20581580", + "20581560", + "20581570" + ], + "answer": "20581560", + "correct_value": 20581560, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000453", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 7131", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 7131?", + "options": [ + "71324262", + "71324272", + "71324252", + "71324282" + ], + "answer": "71324262", + "correct_value": 71324262, + "explanation": "Distributive: (10000+2)*7131 = 10000*7131 + 2*7131." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10024 * 3857", + "ns_shortcut": "Use (10000+24)*m, correction is 24*3857.", + "shortcut_strength": 0.952, + "pure_math": { + "question": "What is 10024 * 3857?", + "options": [ + "38662578", + "38662588", + "38662568", + "38662558" + ], + "answer": "38662568", + "correct_value": 38662568, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6192 * 5203", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6192 * 5203?", + "options": [ + "32216996", + "32216976", + "32216966", + "32216986" + ], + "answer": "32216976", + "correct_value": 32216976, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000454", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 5588", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 5588?", + "options": [ + "55868824", + "55868814", + "55868844", + "55868834" + ], + "answer": "55868824", + "correct_value": 55868824, + "explanation": "Distributive: (10000-2)*5588 = 10000*5588 - 2*5588." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9969 * 1777", + "ns_shortcut": "Use (10000-31)*m, correction is 31*1777.", + "shortcut_strength": 0.938, + "pure_math": { + "question": "What is 9969 * 1777?", + "options": [ + "17714923", + "17714913", + "17714933", + "17714903" + ], + "answer": "17714913", + "correct_value": 17714913, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4684 * 6082", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4684 * 6082?", + "options": [ + "28488088", + "28488098", + "28488108", + "28488078" + ], + "answer": "28488088", + "correct_value": 28488088, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000455", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 9501", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 9501?", + "options": [ + "95029022", + "95029012", + "95029002", + "95028992" + ], + "answer": "95029002", + "correct_value": 95029002, + "explanation": "Distributive: (10000+2)*9501 = 10000*9501 + 2*9501." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9980 * 4023", + "ns_shortcut": "Use (10000-20)*m, correction is 20*4023.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 9980 * 4023?", + "options": [ + "40149550", + "40149540", + "40149530", + "40149560" + ], + "answer": "40149540", + "correct_value": 40149540, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5939 * 5098", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5939 * 5098?", + "options": [ + "30277032", + "30277022", + "30277042", + "30277012" + ], + "answer": "30277022", + "correct_value": 30277022, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000456", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 4596", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 4596?", + "options": [ + "45955424", + "45955404", + "45955394", + "45955414" + ], + "answer": "45955404", + "correct_value": 45955404, + "explanation": "Distributive: (10000-1)*4596 = 10000*4596 - 1*4596." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10016 * 2377", + "ns_shortcut": "Use (10000+16)*m, correction is 16*2377.", + "shortcut_strength": 0.968, + "pure_math": { + "question": "What is 10016 * 2377?", + "options": [ + "23808032", + "23808042", + "23808022", + "23808052" + ], + "answer": "23808032", + "correct_value": 23808032, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6339 * 4642", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6339 * 4642?", + "options": [ + "29425628", + "29425638", + "29425648", + "29425658" + ], + "answer": "29425638", + "correct_value": 29425638, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000457", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 2376", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 2376?", + "options": [ + "23757624", + "23757614", + "23757634", + "23757644" + ], + "answer": "23757624", + "correct_value": 23757624, + "explanation": "Distributive: (10000-1)*2376 = 10000*2376 - 1*2376." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9950 * 7257", + "ns_shortcut": "Use (10000-50)*m, correction is 50*7257.", + "shortcut_strength": 0.9, + "pure_math": { + "question": "What is 9950 * 7257?", + "options": [ + "72207140", + "72207170", + "72207150", + "72207160" + ], + "answer": "72207150", + "correct_value": 72207150, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5744 * 5846", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5744 * 5846?", + "options": [ + "33579424", + "33579444", + "33579434", + "33579414" + ], + "answer": "33579424", + "correct_value": 33579424, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000458", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 9151", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 9151?", + "options": [ + "91519151", + "91519141", + "91519161", + "91519171" + ], + "answer": "91519151", + "correct_value": 91519151, + "explanation": "Distributive: (10000+1)*9151 = 10000*9151 + 1*9151." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9979 * 5246", + "ns_shortcut": "Use (10000-21)*m, correction is 21*5246.", + "shortcut_strength": 0.958, + "pure_math": { + "question": "What is 9979 * 5246?", + "options": [ + "52349844", + "52349854", + "52349834", + "52349824" + ], + "answer": "52349834", + "correct_value": 52349834, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6642 * 4309", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6642 * 4309?", + "options": [ + "28620368", + "28620388", + "28620378", + "28620398" + ], + "answer": "28620378", + "correct_value": 28620378, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000459", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 5528", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 5528?", + "options": [ + "55268934", + "55268964", + "55268944", + "55268954" + ], + "answer": "55268944", + "correct_value": 55268944, + "explanation": "Distributive: (10000-2)*5528 = 10000*5528 - 2*5528." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10022 * 2810", + "ns_shortcut": "Use (10000+22)*m, correction is 22*2810.", + "shortcut_strength": 0.956, + "pure_math": { + "question": "What is 10022 * 2810?", + "options": [ + "28161840", + "28161830", + "28161820", + "28161810" + ], + "answer": "28161820", + "correct_value": 28161820, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5366 * 4804", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5366 * 4804?", + "options": [ + "25778264", + "25778254", + "25778274", + "25778284" + ], + "answer": "25778264", + "correct_value": 25778264, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000460", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 4123", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 4123?", + "options": [ + "41234123", + "41234113", + "41234143", + "41234133" + ], + "answer": "41234123", + "correct_value": 41234123, + "explanation": "Distributive: (10000+1)*4123 = 10000*4123 + 1*4123." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10039 * 5802", + "ns_shortcut": "Use (10000+39)*m, correction is 39*5802.", + "shortcut_strength": 0.922, + "pure_math": { + "question": "What is 10039 * 5802?", + "options": [ + "58246288", + "58246278", + "58246298", + "58246268" + ], + "answer": "58246278", + "correct_value": 58246278, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5660 * 6374", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5660 * 6374?", + "options": [ + "36076860", + "36076830", + "36076850", + "36076840" + ], + "answer": "36076840", + "correct_value": 36076840, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000461", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 4728", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 4728?", + "options": [ + "47289446", + "47289456", + "47289476", + "47289466" + ], + "answer": "47289456", + "correct_value": 47289456, + "explanation": "Distributive: (10000+2)*4728 = 10000*4728 + 2*4728." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9957 * 4749", + "ns_shortcut": "Use (10000-43)*m, correction is 43*4749.", + "shortcut_strength": 0.914, + "pure_math": { + "question": "What is 9957 * 4749?", + "options": [ + "47285813", + "47285793", + "47285803", + "47285783" + ], + "answer": "47285793", + "correct_value": 47285793, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5394 * 4350", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5394 * 4350?", + "options": [ + "23463890", + "23463910", + "23463900", + "23463920" + ], + "answer": "23463900", + "correct_value": 23463900, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000462", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 4433", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 4433?", + "options": [ + "44338886", + "44338866", + "44338876", + "44338856" + ], + "answer": "44338866", + "correct_value": 44338866, + "explanation": "Distributive: (10000+2)*4433 = 10000*4433 + 2*4433." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10025 * 6432", + "ns_shortcut": "Use (10000+25)*m, correction is 25*6432.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 10025 * 6432?", + "options": [ + "64480800", + "64480820", + "64480810", + "64480790" + ], + "answer": "64480800", + "correct_value": 64480800, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6046 * 6697", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6046 * 6697?", + "options": [ + "40490052", + "40490062", + "40490072", + "40490082" + ], + "answer": "40490062", + "correct_value": 40490062, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000463", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 7974", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 7974?", + "options": [ + "79755948", + "79755958", + "79755968", + "79755938" + ], + "answer": "79755948", + "correct_value": 79755948, + "explanation": "Distributive: (10000+2)*7974 = 10000*7974 + 2*7974." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10029 * 1340", + "ns_shortcut": "Use (10000+29)*m, correction is 29*1340.", + "shortcut_strength": 0.942, + "pure_math": { + "question": "What is 10029 * 1340?", + "options": [ + "13438870", + "13438880", + "13438850", + "13438860" + ], + "answer": "13438860", + "correct_value": 13438860, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5351 * 6326", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5351 * 6326?", + "options": [ + "33850446", + "33850426", + "33850436", + "33850416" + ], + "answer": "33850426", + "correct_value": 33850426, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000464", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 2043", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 2043?", + "options": [ + "20434106", + "20434096", + "20434086", + "20434076" + ], + "answer": "20434086", + "correct_value": 20434086, + "explanation": "Distributive: (10000+2)*2043 = 10000*2043 + 2*2043." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9962 * 4591", + "ns_shortcut": "Use (10000-38)*m, correction is 38*4591.", + "shortcut_strength": 0.924, + "pure_math": { + "question": "What is 9962 * 4591?", + "options": [ + "45735532", + "45735542", + "45735552", + "45735562" + ], + "answer": "45735542", + "correct_value": 45735542, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4380 * 4701", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4380 * 4701?", + "options": [ + "20590390", + "20590370", + "20590400", + "20590380" + ], + "answer": "20590380", + "correct_value": 20590380, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000465", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 9005", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 9005?", + "options": [ + "90032000", + "90032010", + "90031980", + "90031990" + ], + "answer": "90031990", + "correct_value": 90031990, + "explanation": "Distributive: (10000-2)*9005 = 10000*9005 - 2*9005." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10030 * 5162", + "ns_shortcut": "Use (10000+30)*m, correction is 30*5162.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "What is 10030 * 5162?", + "options": [ + "51774860", + "51774870", + "51774850", + "51774880" + ], + "answer": "51774860", + "correct_value": 51774860, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5191 * 6700", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5191 * 6700?", + "options": [ + "34779720", + "34779690", + "34779710", + "34779700" + ], + "answer": "34779700", + "correct_value": 34779700, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000466", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 9495", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 9495?", + "options": [ + "94959515", + "94959505", + "94959485", + "94959495" + ], + "answer": "94959495", + "correct_value": 94959495, + "explanation": "Distributive: (10000+1)*9495 = 10000*9495 + 1*9495." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10028 * 1577", + "ns_shortcut": "Use (10000+28)*m, correction is 28*1577.", + "shortcut_strength": 0.944, + "pure_math": { + "question": "What is 10028 * 1577?", + "options": [ + "15814156", + "15814146", + "15814166", + "15814176" + ], + "answer": "15814156", + "correct_value": 15814156, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6196 * 5712", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6196 * 5712?", + "options": [ + "35391562", + "35391552", + "35391572", + "35391542" + ], + "answer": "35391552", + "correct_value": 35391552, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000467", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 4382", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 4382?", + "options": [ + "43828784", + "43828754", + "43828764", + "43828774" + ], + "answer": "43828764", + "correct_value": 43828764, + "explanation": "Distributive: (10000+2)*4382 = 10000*4382 + 2*4382." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9959 * 8574", + "ns_shortcut": "Use (10000-41)*m, correction is 41*8574.", + "shortcut_strength": 0.918, + "pure_math": { + "question": "What is 9959 * 8574?", + "options": [ + "85388486", + "85388476", + "85388456", + "85388466" + ], + "answer": "85388466", + "correct_value": 85388466, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6405 * 4830", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6405 * 4830?", + "options": [ + "30936170", + "30936150", + "30936160", + "30936140" + ], + "answer": "30936150", + "correct_value": 30936150, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000468", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 3337", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 3337?", + "options": [ + "33363346", + "33363336", + "33363326", + "33363316" + ], + "answer": "33363326", + "correct_value": 33363326, + "explanation": "Distributive: (10000-2)*3337 = 10000*3337 - 2*3337." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9963 * 6793", + "ns_shortcut": "Use (10000-37)*m, correction is 37*6793.", + "shortcut_strength": 0.926, + "pure_math": { + "question": "What is 9963 * 6793?", + "options": [ + "67678649", + "67678679", + "67678669", + "67678659" + ], + "answer": "67678659", + "correct_value": 67678659, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5883 * 5894", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5883 * 5894?", + "options": [ + "34674412", + "34674392", + "34674422", + "34674402" + ], + "answer": "34674402", + "correct_value": 34674402, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000469", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 8476", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 8476?", + "options": [ + "84768476", + "84768486", + "84768466", + "84768496" + ], + "answer": "84768476", + "correct_value": 84768476, + "explanation": "Distributive: (10000+1)*8476 = 10000*8476 + 1*8476." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10025 * 3109", + "ns_shortcut": "Use (10000+25)*m, correction is 25*3109.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "What is 10025 * 3109?", + "options": [ + "31167715", + "31167725", + "31167735", + "31167745" + ], + "answer": "31167725", + "correct_value": 31167725, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6529 * 5768", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6529 * 5768?", + "options": [ + "37659292", + "37659272", + "37659282", + "37659262" + ], + "answer": "37659272", + "correct_value": 37659272, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000470", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 7447", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 7447?", + "options": [ + "74455116", + "74455126", + "74455096", + "74455106" + ], + "answer": "74455106", + "correct_value": 74455106, + "explanation": "Distributive: (10000-2)*7447 = 10000*7447 - 2*7447." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10023 * 7922", + "ns_shortcut": "Use (10000+23)*m, correction is 23*7922.", + "shortcut_strength": 0.954, + "pure_math": { + "question": "What is 10023 * 7922?", + "options": [ + "79402216", + "79402206", + "79402196", + "79402226" + ], + "answer": "79402206", + "correct_value": 79402206, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6858 * 6974", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6858 * 6974?", + "options": [ + "47827692", + "47827712", + "47827682", + "47827702" + ], + "answer": "47827692", + "correct_value": 47827692, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000471", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 7041", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 7041?", + "options": [ + "70395938", + "70395928", + "70395908", + "70395918" + ], + "answer": "70395918", + "correct_value": 70395918, + "explanation": "Distributive: (10000-2)*7041 = 10000*7041 - 2*7041." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9972 * 4868", + "ns_shortcut": "Use (10000-28)*m, correction is 28*4868.", + "shortcut_strength": 0.944, + "pure_math": { + "question": "What is 9972 * 4868?", + "options": [ + "48543696", + "48543716", + "48543686", + "48543706" + ], + "answer": "48543696", + "correct_value": 48543696, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6071 * 4181", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6071 * 4181?", + "options": [ + "25382861", + "25382871", + "25382851", + "25382841" + ], + "answer": "25382851", + "correct_value": 25382851, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000472", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 7244", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 7244?", + "options": [ + "72454488", + "72454498", + "72454478", + "72454508" + ], + "answer": "72454488", + "correct_value": 72454488, + "explanation": "Distributive: (10000+2)*7244 = 10000*7244 + 2*7244." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9952 * 2636", + "ns_shortcut": "Use (10000-48)*m, correction is 48*2636.", + "shortcut_strength": 0.904, + "pure_math": { + "question": "What is 9952 * 2636?", + "options": [ + "26233492", + "26233482", + "26233462", + "26233472" + ], + "answer": "26233472", + "correct_value": 26233472, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4468 * 6903", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4468 * 6903?", + "options": [ + "30842604", + "30842624", + "30842594", + "30842614" + ], + "answer": "30842604", + "correct_value": 30842604, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000473", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 8558", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 8558?", + "options": [ + "85597126", + "85597136", + "85597106", + "85597116" + ], + "answer": "85597116", + "correct_value": 85597116, + "explanation": "Distributive: (10000+2)*8558 = 10000*8558 + 2*8558." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9982 * 2677", + "ns_shortcut": "Use (10000-18)*m, correction is 18*2677.", + "shortcut_strength": 0.964, + "pure_math": { + "question": "What is 9982 * 2677?", + "options": [ + "26721814", + "26721834", + "26721824", + "26721804" + ], + "answer": "26721814", + "correct_value": 26721814, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4659 * 4697", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4659 * 4697?", + "options": [ + "21883333", + "21883313", + "21883343", + "21883323" + ], + "answer": "21883323", + "correct_value": 21883323, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000474", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 5819", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 5819?", + "options": [ + "58184171", + "58184191", + "58184201", + "58184181" + ], + "answer": "58184181", + "correct_value": 58184181, + "explanation": "Distributive: (10000-1)*5819 = 10000*5819 - 1*5819." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10016 * 1853", + "ns_shortcut": "Use (10000+16)*m, correction is 16*1853.", + "shortcut_strength": 0.968, + "pure_math": { + "question": "What is 10016 * 1853?", + "options": [ + "18559648", + "18559638", + "18559658", + "18559668" + ], + "answer": "18559648", + "correct_value": 18559648, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4627 * 5803", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4627 * 5803?", + "options": [ + "26850491", + "26850471", + "26850481", + "26850501" + ], + "answer": "26850481", + "correct_value": 26850481, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000475", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 9721", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 9721?", + "options": [ + "97190578", + "97190558", + "97190548", + "97190568" + ], + "answer": "97190558", + "correct_value": 97190558, + "explanation": "Distributive: (10000-2)*9721 = 10000*9721 - 2*9721." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9983 * 4137", + "ns_shortcut": "Use (10000-17)*m, correction is 17*4137.", + "shortcut_strength": 0.966, + "pure_math": { + "question": "What is 9983 * 4137?", + "options": [ + "41299691", + "41299671", + "41299661", + "41299681" + ], + "answer": "41299671", + "correct_value": 41299671, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5817 * 4213", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5817 * 4213?", + "options": [ + "24507041", + "24507031", + "24507011", + "24507021" + ], + "answer": "24507021", + "correct_value": 24507021, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000476", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 5091", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 5091?", + "options": [ + "50904919", + "50904899", + "50904909", + "50904929" + ], + "answer": "50904909", + "correct_value": 50904909, + "explanation": "Distributive: (10000-1)*5091 = 10000*5091 - 1*5091." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10046 * 8794", + "ns_shortcut": "Use (10000+46)*m, correction is 46*8794.", + "shortcut_strength": 0.908, + "pure_math": { + "question": "What is 10046 * 8794?", + "options": [ + "88344534", + "88344544", + "88344524", + "88344514" + ], + "answer": "88344524", + "correct_value": 88344524, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5379 * 4786", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5379 * 4786?", + "options": [ + "25743904", + "25743914", + "25743884", + "25743894" + ], + "answer": "25743894", + "correct_value": 25743894, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000477", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 3736", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 3736?", + "options": [ + "37356284", + "37356254", + "37356274", + "37356264" + ], + "answer": "37356264", + "correct_value": 37356264, + "explanation": "Distributive: (10000-1)*3736 = 10000*3736 - 1*3736." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10018 * 8713", + "ns_shortcut": "Use (10000+18)*m, correction is 18*8713.", + "shortcut_strength": 0.964, + "pure_math": { + "question": "What is 10018 * 8713?", + "options": [ + "87286844", + "87286834", + "87286824", + "87286854" + ], + "answer": "87286834", + "correct_value": 87286834, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6217 * 5617", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6217 * 5617?", + "options": [ + "34920899", + "34920909", + "34920889", + "34920879" + ], + "answer": "34920889", + "correct_value": 34920889, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000478", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 2734", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 2734?", + "options": [ + "27337256", + "27337266", + "27337276", + "27337286" + ], + "answer": "27337266", + "correct_value": 27337266, + "explanation": "Distributive: (10000-1)*2734 = 10000*2734 - 1*2734." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10047 * 3419", + "ns_shortcut": "Use (10000+47)*m, correction is 47*3419.", + "shortcut_strength": 0.906, + "pure_math": { + "question": "What is 10047 * 3419?", + "options": [ + "34350713", + "34350693", + "34350683", + "34350703" + ], + "answer": "34350693", + "correct_value": 34350693, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4870 * 5878", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4870 * 5878?", + "options": [ + "28625880", + "28625850", + "28625860", + "28625870" + ], + "answer": "28625860", + "correct_value": 28625860, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000479", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 1719", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 1719?", + "options": [ + "17191739", + "17191729", + "17191709", + "17191719" + ], + "answer": "17191719", + "correct_value": 17191719, + "explanation": "Distributive: (10000+1)*1719 = 10000*1719 + 1*1719." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9983 * 3235", + "ns_shortcut": "Use (10000-17)*m, correction is 17*3235.", + "shortcut_strength": 0.966, + "pure_math": { + "question": "What is 9983 * 3235?", + "options": [ + "32294995", + "32295025", + "32295015", + "32295005" + ], + "answer": "32295005", + "correct_value": 32295005, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5030 * 4387", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5030 * 4387?", + "options": [ + "22066610", + "22066620", + "22066600", + "22066630" + ], + "answer": "22066610", + "correct_value": 22066610, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000480", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 6440", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 6440?", + "options": [ + "64393560", + "64393550", + "64393580", + "64393570" + ], + "answer": "64393560", + "correct_value": 64393560, + "explanation": "Distributive: (10000-1)*6440 = 10000*6440 - 1*6440." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9961 * 6005", + "ns_shortcut": "Use (10000-39)*m, correction is 39*6005.", + "shortcut_strength": 0.922, + "pure_math": { + "question": "What is 9961 * 6005?", + "options": [ + "59815805", + "59815795", + "59815825", + "59815815" + ], + "answer": "59815805", + "correct_value": 59815805, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6491 * 6395", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6491 * 6395?", + "options": [ + "41509965", + "41509945", + "41509955", + "41509935" + ], + "answer": "41509945", + "correct_value": 41509945, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000481", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 4811", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 4811?", + "options": [ + "48100388", + "48100368", + "48100398", + "48100378" + ], + "answer": "48100378", + "correct_value": 48100378, + "explanation": "Distributive: (10000-2)*4811 = 10000*4811 - 2*4811." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9953 * 7657", + "ns_shortcut": "Use (10000-47)*m, correction is 47*7657.", + "shortcut_strength": 0.906, + "pure_math": { + "question": "What is 9953 * 7657?", + "options": [ + "76210121", + "76210141", + "76210111", + "76210131" + ], + "answer": "76210121", + "correct_value": 76210121, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4244 * 4562", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4244 * 4562?", + "options": [ + "19361128", + "19361118", + "19361138", + "19361148" + ], + "answer": "19361128", + "correct_value": 19361128, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000482", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 3484", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 3484?", + "options": [ + "34836526", + "34836536", + "34836516", + "34836506" + ], + "answer": "34836516", + "correct_value": 34836516, + "explanation": "Distributive: (10000-1)*3484 = 10000*3484 - 1*3484." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10042 * 2825", + "ns_shortcut": "Use (10000+42)*m, correction is 42*2825.", + "shortcut_strength": 0.916, + "pure_math": { + "question": "What is 10042 * 2825?", + "options": [ + "28368650", + "28368660", + "28368640", + "28368670" + ], + "answer": "28368650", + "correct_value": 28368650, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5299 * 6635", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5299 * 6635?", + "options": [ + "35158855", + "35158875", + "35158885", + "35158865" + ], + "answer": "35158865", + "correct_value": 35158865, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000483", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 1388", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 1388?", + "options": [ + "13881388", + "13881378", + "13881408", + "13881398" + ], + "answer": "13881388", + "correct_value": 13881388, + "explanation": "Distributive: (10000+1)*1388 = 10000*1388 + 1*1388." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9974 * 5033", + "ns_shortcut": "Use (10000-26)*m, correction is 26*5033.", + "shortcut_strength": 0.948, + "pure_math": { + "question": "What is 9974 * 5033?", + "options": [ + "50199132", + "50199162", + "50199152", + "50199142" + ], + "answer": "50199142", + "correct_value": 50199142, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6722 * 5016", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6722 * 5016?", + "options": [ + "33717552", + "33717542", + "33717572", + "33717562" + ], + "answer": "33717552", + "correct_value": 33717552, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000484", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 4007", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 4007?", + "options": [ + "40074017", + "40074007", + "40074027", + "40073997" + ], + "answer": "40074007", + "correct_value": 40074007, + "explanation": "Distributive: (10000+1)*4007 = 10000*4007 + 1*4007." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9974 * 1995", + "ns_shortcut": "Use (10000-26)*m, correction is 26*1995.", + "shortcut_strength": 0.948, + "pure_math": { + "question": "What is 9974 * 1995?", + "options": [ + "19898140", + "19898150", + "19898130", + "19898120" + ], + "answer": "19898130", + "correct_value": 19898130, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6597 * 4578", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6597 * 4578?", + "options": [ + "30201056", + "30201086", + "30201066", + "30201076" + ], + "answer": "30201066", + "correct_value": 30201066, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000485", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 9992", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 9992?", + "options": [ + "99909998", + "99910028", + "99910008", + "99910018" + ], + "answer": "99910008", + "correct_value": 99910008, + "explanation": "Distributive: (10000-1)*9992 = 10000*9992 - 1*9992." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10026 * 2592", + "ns_shortcut": "Use (10000+26)*m, correction is 26*2592.", + "shortcut_strength": 0.948, + "pure_math": { + "question": "What is 10026 * 2592?", + "options": [ + "25987412", + "25987402", + "25987392", + "25987382" + ], + "answer": "25987392", + "correct_value": 25987392, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5002 * 5735", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5002 * 5735?", + "options": [ + "28686460", + "28686470", + "28686490", + "28686480" + ], + "answer": "28686470", + "correct_value": 28686470, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000486", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 5675", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 5675?", + "options": [ + "56755685", + "56755675", + "56755695", + "56755665" + ], + "answer": "56755675", + "correct_value": 56755675, + "explanation": "Distributive: (10000+1)*5675 = 10000*5675 + 1*5675." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9980 * 1578", + "ns_shortcut": "Use (10000-20)*m, correction is 20*1578.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "What is 9980 * 1578?", + "options": [ + "15748460", + "15748430", + "15748450", + "15748440" + ], + "answer": "15748440", + "correct_value": 15748440, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5546 * 4686", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5546 * 4686?", + "options": [ + "25988556", + "25988576", + "25988566", + "25988546" + ], + "answer": "25988556", + "correct_value": 25988556, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000487", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 3553", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 3553?", + "options": [ + "35537126", + "35537116", + "35537106", + "35537096" + ], + "answer": "35537106", + "correct_value": 35537106, + "explanation": "Distributive: (10000+2)*3553 = 10000*3553 + 2*3553." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9964 * 2612", + "ns_shortcut": "Use (10000-36)*m, correction is 36*2612.", + "shortcut_strength": 0.928, + "pure_math": { + "question": "What is 9964 * 2612?", + "options": [ + "26025968", + "26025988", + "26025958", + "26025978" + ], + "answer": "26025968", + "correct_value": 26025968, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6263 * 6975", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6263 * 6975?", + "options": [ + "43684425", + "43684445", + "43684415", + "43684435" + ], + "answer": "43684425", + "correct_value": 43684425, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000488", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 6011", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 6011?", + "options": [ + "60122022", + "60122042", + "60122032", + "60122012" + ], + "answer": "60122022", + "correct_value": 60122022, + "explanation": "Distributive: (10000+2)*6011 = 10000*6011 + 2*6011." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9967 * 6423", + "ns_shortcut": "Use (10000-33)*m, correction is 33*6423.", + "shortcut_strength": 0.9339999999999999, + "pure_math": { + "question": "What is 9967 * 6423?", + "options": [ + "64018061", + "64018041", + "64018031", + "64018051" + ], + "answer": "64018041", + "correct_value": 64018041, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6927 * 5921", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6927 * 5921?", + "options": [ + "41014757", + "41014787", + "41014767", + "41014777" + ], + "answer": "41014767", + "correct_value": 41014767, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000489", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 1210", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 1210?", + "options": [ + "12097580", + "12097570", + "12097590", + "12097600" + ], + "answer": "12097580", + "correct_value": 12097580, + "explanation": "Distributive: (10000-2)*1210 = 10000*1210 - 2*1210." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10017 * 2113", + "ns_shortcut": "Use (10000+17)*m, correction is 17*2113.", + "shortcut_strength": 0.966, + "pure_math": { + "question": "What is 10017 * 2113?", + "options": [ + "21165941", + "21165911", + "21165921", + "21165931" + ], + "answer": "21165921", + "correct_value": 21165921, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5259 * 6624", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5259 * 6624?", + "options": [ + "34835636", + "34835606", + "34835626", + "34835616" + ], + "answer": "34835616", + "correct_value": 34835616, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000490", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 7304", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 7304?", + "options": [ + "73025412", + "73025382", + "73025392", + "73025402" + ], + "answer": "73025392", + "correct_value": 73025392, + "explanation": "Distributive: (10000-2)*7304 = 10000*7304 - 2*7304." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9960 * 8675", + "ns_shortcut": "Use (10000-40)*m, correction is 40*8675.", + "shortcut_strength": 0.92, + "pure_math": { + "question": "What is 9960 * 8675?", + "options": [ + "86403000", + "86403010", + "86402990", + "86403020" + ], + "answer": "86403000", + "correct_value": 86403000, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6851 * 4235", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6851 * 4235?", + "options": [ + "29014005", + "29013985", + "29013995", + "29013975" + ], + "answer": "29013985", + "correct_value": 29013985, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000491", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 8827", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 8827?", + "options": [ + "88261193", + "88261163", + "88261183", + "88261173" + ], + "answer": "88261173", + "correct_value": 88261173, + "explanation": "Distributive: (10000-1)*8827 = 10000*8827 - 1*8827." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9983 * 9935", + "ns_shortcut": "Use (10000-17)*m, correction is 17*9935.", + "shortcut_strength": 0.966, + "pure_math": { + "question": "What is 9983 * 9935?", + "options": [ + "99181115", + "99181125", + "99181105", + "99181095" + ], + "answer": "99181105", + "correct_value": 99181105, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6788 * 4602", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6788 * 4602?", + "options": [ + "31238386", + "31238366", + "31238376", + "31238396" + ], + "answer": "31238376", + "correct_value": 31238376, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000492", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 2912", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 2912?", + "options": [ + "29114166", + "29114186", + "29114196", + "29114176" + ], + "answer": "29114176", + "correct_value": 29114176, + "explanation": "Distributive: (10000-2)*2912 = 10000*2912 - 2*2912." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9971 * 3483", + "ns_shortcut": "Use (10000-29)*m, correction is 29*3483.", + "shortcut_strength": 0.942, + "pure_math": { + "question": "What is 9971 * 3483?", + "options": [ + "34729003", + "34729013", + "34728993", + "34728983" + ], + "answer": "34728993", + "correct_value": 34728993, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5747 * 5034", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5747 * 5034?", + "options": [ + "28930398", + "28930408", + "28930388", + "28930418" + ], + "answer": "28930398", + "correct_value": 28930398, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000493", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 7743", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 7743?", + "options": [ + "77437763", + "77437753", + "77437733", + "77437743" + ], + "answer": "77437743", + "correct_value": 77437743, + "explanation": "Distributive: (10000+1)*7743 = 10000*7743 + 1*7743." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9982 * 6878", + "ns_shortcut": "Use (10000-18)*m, correction is 18*6878.", + "shortcut_strength": 0.964, + "pure_math": { + "question": "What is 9982 * 6878?", + "options": [ + "68656186", + "68656196", + "68656206", + "68656216" + ], + "answer": "68656196", + "correct_value": 68656196, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6862 * 5260", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6862 * 5260?", + "options": [ + "36094140", + "36094130", + "36094110", + "36094120" + ], + "answer": "36094120", + "correct_value": 36094120, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000494", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 2587", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 2587?", + "options": [ + "25864816", + "25864836", + "25864846", + "25864826" + ], + "answer": "25864826", + "correct_value": 25864826, + "explanation": "Distributive: (10000-2)*2587 = 10000*2587 - 2*2587." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10029 * 1180", + "ns_shortcut": "Use (10000+29)*m, correction is 29*1180.", + "shortcut_strength": 0.942, + "pure_math": { + "question": "What is 10029 * 1180?", + "options": [ + "11834210", + "11834220", + "11834230", + "11834240" + ], + "answer": "11834220", + "correct_value": 11834220, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4697 * 4721", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4697 * 4721?", + "options": [ + "22174547", + "22174527", + "22174537", + "22174557" + ], + "answer": "22174537", + "correct_value": 22174537, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000495", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9999 * 8970", + "ns_shortcut": "Use (10000-1)*m = 10000*m - 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 9999 * 8970?", + "options": [ + "89691020", + "89691030", + "89691050", + "89691040" + ], + "answer": "89691030", + "correct_value": 89691030, + "explanation": "Distributive: (10000-1)*8970 = 10000*8970 - 1*8970." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10034 * 2729", + "ns_shortcut": "Use (10000+34)*m, correction is 34*2729.", + "shortcut_strength": 0.9319999999999999, + "pure_math": { + "question": "What is 10034 * 2729?", + "options": [ + "27382786", + "27382806", + "27382796", + "27382776" + ], + "answer": "27382786", + "correct_value": 27382786, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4747 * 4784", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4747 * 4784?", + "options": [ + "22709648", + "22709638", + "22709658", + "22709668" + ], + "answer": "22709648", + "correct_value": 22709648, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000496", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 3721", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 3721?", + "options": [ + "37202578", + "37202548", + "37202568", + "37202558" + ], + "answer": "37202558", + "correct_value": 37202558, + "explanation": "Distributive: (10000-2)*3721 = 10000*3721 - 2*3721." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10017 * 7017", + "ns_shortcut": "Use (10000+17)*m, correction is 17*7017.", + "shortcut_strength": 0.966, + "pure_math": { + "question": "What is 10017 * 7017?", + "options": [ + "70289299", + "70289289", + "70289279", + "70289309" + ], + "answer": "70289289", + "correct_value": 70289289, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4120 * 6397", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4120 * 6397?", + "options": [ + "26355640", + "26355660", + "26355630", + "26355650" + ], + "answer": "26355640", + "correct_value": 26355640, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000497", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 1155", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 1155?", + "options": [ + "11547710", + "11547680", + "11547700", + "11547690" + ], + "answer": "11547690", + "correct_value": 11547690, + "explanation": "Distributive: (10000-2)*1155 = 10000*1155 - 2*1155." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9954 * 7362", + "ns_shortcut": "Use (10000-46)*m, correction is 46*7362.", + "shortcut_strength": 0.908, + "pure_math": { + "question": "What is 9954 * 7362?", + "options": [ + "73281358", + "73281368", + "73281338", + "73281348" + ], + "answer": "73281348", + "correct_value": 73281348, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5537 * 6468", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5537 * 6468?", + "options": [ + "35813316", + "35813306", + "35813326", + "35813336" + ], + "answer": "35813316", + "correct_value": 35813316, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000498", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10001 * 8036", + "ns_shortcut": "Use (10000+1)*m = 10000*m + 1*m.", + "shortcut_strength": 0.9999, + "pure_math": { + "question": "What is 10001 * 8036?", + "options": [ + "80368036", + "80368026", + "80368046", + "80368056" + ], + "answer": "80368036", + "correct_value": 80368036, + "explanation": "Distributive: (10000+1)*8036 = 10000*8036 + 1*8036." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9962 * 8025", + "ns_shortcut": "Use (10000-38)*m, correction is 38*8025.", + "shortcut_strength": 0.924, + "pure_math": { + "question": "What is 9962 * 8025?", + "options": [ + "79945040", + "79945060", + "79945050", + "79945070" + ], + "answer": "79945050", + "correct_value": 79945050, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5549 * 6500", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5549 * 6500?", + "options": [ + "36068490", + "36068520", + "36068510", + "36068500" + ], + "answer": "36068500", + "correct_value": 36068500, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "structural_shortcuts_000499", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9998 * 1050", + "ns_shortcut": "Use (10000-2)*m = 10000*m - 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 9998 * 1050?", + "options": [ + "10497900", + "10497910", + "10497920", + "10497890" + ], + "answer": "10497900", + "correct_value": 10497900, + "explanation": "Distributive: (10000-2)*1050 = 10000*1050 - 2*1050." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "10016 * 7624", + "ns_shortcut": "Use (10000+16)*m, correction is 16*7624.", + "shortcut_strength": 0.968, + "pure_math": { + "question": "What is 10016 * 7624?", + "options": [ + "76361974", + "76361994", + "76361984", + "76362004" + ], + "answer": "76361984", + "correct_value": 76361984, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6172 * 5143", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6172 * 5143?", + "options": [ + "31742596", + "31742616", + "31742586", + "31742606" + ], + "answer": "31742596", + "correct_value": 31742596, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "structural_shortcuts_000500", + "category": "structural_shortcuts", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10002 * 9436", + "ns_shortcut": "Use (10000+2)*m = 10000*m + 2*m.", + "shortcut_strength": 0.9998, + "pure_math": { + "question": "What is 10002 * 9436?", + "options": [ + "94378882", + "94378872", + "94378892", + "94378862" + ], + "answer": "94378872", + "correct_value": 94378872, + "explanation": "Distributive: (10000+2)*9436 = 10000*9436 + 2*9436." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9962 * 8850", + "ns_shortcut": "Use (10000-38)*m, correction is 38*8850.", + "shortcut_strength": 0.924, + "pure_math": { + "question": "What is 9962 * 8850?", + "options": [ + "88163690", + "88163700", + "88163720", + "88163710" + ], + "answer": "88163700", + "correct_value": 88163700, + "explanation": "Distributive still works but correction term is larger." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4186 * 5332", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4186 * 5332?", + "options": [ + "22319772", + "22319762", + "22319752", + "22319742" + ], + "answer": "22319752", + "correct_value": 22319752, + "explanation": "Neither factor near a power of 10; no clean decomposition." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "relative_distance_000501", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 720/1504 or 2500/4890?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03252403950746202, + "pure_math": { + "question": "Which is larger: 720/1504 or 2500/4890?", + "options": [ + "2500/4890", + "3756/6378", + "3249/5725", + "720/1504" + ], + "answer": "2500/4890", + "correct_value": 0.5112474437627812, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2327/4686 or 2477/4858?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.013295035253877596, + "pure_math": { + "question": "Which is larger: 2327/4686 or 2477/4858?", + "options": [ + "3848/7549", + "2327/4686", + "4665/8287", + "2477/4858" + ], + "answer": "2477/4858", + "correct_value": 0.5098806093042404, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 820/1490 or 4657/8028?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 820/1490 or 4657/8028?", + "options": [ + "2501/4000", + "4657/8028", + "4142/7512", + "820/1490" + ], + "answer": "4657/8028", + "correct_value": 0.5800946686596911, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000502", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1709/3501 or 2022/3864?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.035145681535532824, + "pure_math": { + "question": "Which is larger: 1709/3501 or 2022/3864?", + "options": [ + "2022/3864", + "4500/7823", + "1709/3501", + "1295/2503" + ], + "answer": "2022/3864", + "correct_value": 0.5232919254658385, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2117/4346 or 4715/9178?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.026613893023564095, + "pure_math": { + "question": "Which is larger: 2117/4346 or 4715/9178?", + "options": [ + "2117/4346", + "2312/4284", + "4715/9178", + "4199/8519" + ], + "answer": "4715/9178", + "correct_value": 0.5137284811505775, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2156/3919 or 3410/5879?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2156/3919 or 3410/5879?", + "options": [ + "3410/5879", + "1023/1513", + "3269/5076", + "2156/3919" + ], + "answer": "3410/5879", + "correct_value": 0.5800306174519476, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000503", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3335/6848 or 2366/4567?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.0310608701902515, + "pure_math": { + "question": "Which is larger: 3335/6848 or 2366/4567?", + "options": [ + "3335/6848", + "2366/4567", + "3053/6518", + "3229/6375" + ], + "answer": "2366/4567", + "correct_value": 0.5180643748631487, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2714/5486 or 2660/5184?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.018403466961918802, + "pure_math": { + "question": "Which is larger: 2714/5486 or 2660/5184?", + "options": [ + "2714/5486", + "2377/4372", + "2660/5184", + "2683/5351" + ], + "answer": "2660/5184", + "correct_value": 0.5131172839506173, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1159/2107 or 5068/8737?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1159/2107 or 5068/8737?", + "options": [ + "1159/2107", + "4785/9541", + "5440/9912", + "5068/8737" + ], + "answer": "5068/8737", + "correct_value": 0.5800618061119377, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000504", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 937/1958 or 2145/4104?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.044111278366157136, + "pure_math": { + "question": "Which is larger: 937/1958 or 2145/4104?", + "options": [ + "937/1958", + "2965/6368", + "2145/4104", + "2325/3950" + ], + "answer": "2145/4104", + "correct_value": 0.5226608187134503, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1048/2147 or 710/1403?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017935483913803674, + "pure_math": { + "question": "Which is larger: 1048/2147 or 710/1403?", + "options": [ + "1048/2147", + "1046/2292", + "710/1403", + "4143/8110" + ], + "answer": "710/1403", + "correct_value": 0.5060584461867427, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1592/2893 or 1853/3194?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1592/2893 or 1853/3194?", + "options": [ + "2835/5799", + "1853/3194", + "1592/2893", + "1340/2595" + ], + "answer": "1853/3194", + "correct_value": 0.5801502817783344, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000505", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 678/1397 or 2610/5068?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.029670355745963717, + "pure_math": { + "question": "Which is larger: 678/1397 or 2610/5068?", + "options": [ + "1495/3298", + "2610/5068", + "678/1397", + "2402/4466" + ], + "answer": "2610/5068", + "correct_value": 0.5149960536700868, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2627/5305 or 2524/4921?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017710667375829325, + "pure_math": { + "question": "Which is larger: 2627/5305 or 2524/4921?", + "options": [ + "650/1564", + "2524/4921", + "2627/5305", + "4126/8872" + ], + "answer": "2524/4921", + "correct_value": 0.5129038813249339, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4726/8591 or 2077/3581?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4726/8591 or 2077/3581?", + "options": [ + "1895/3374", + "4726/8591", + "2197/3650", + "2077/3581" + ], + "answer": "2077/3581", + "correct_value": 0.580005585032114, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000506", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4821/9880 or 611/1163?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.037409968634795565, + "pure_math": { + "question": "Which is larger: 4821/9880 or 611/1163?", + "options": [ + "1007/2239", + "4384/9102", + "4821/9880", + "611/1163" + ], + "answer": "611/1163", + "correct_value": 0.5253654342218401, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3770/7655 or 3515/6912?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01604731006725213, + "pure_math": { + "question": "Which is larger: 3770/7655 or 3515/6912?", + "options": [ + "3515/6912", + "2243/4311", + "3770/7655", + "620/1217" + ], + "answer": "3515/6912", + "correct_value": 0.5085358796296297, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4621/8401 or 1568/2702?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4621/8401 or 1568/2702?", + "options": [ + "4633/6881", + "4621/8401", + "1568/2702", + "2521/3974" + ], + "answer": "1568/2702", + "correct_value": 0.5803108808290155, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000507", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1761/3607 or 839/1585?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04112018428939873, + "pure_math": { + "question": "Which is larger: 1761/3607 or 839/1585?", + "options": [ + "2466/4706", + "1761/3607", + "839/1585", + "1539/3361" + ], + "answer": "839/1585", + "correct_value": 0.5293375394321767, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3434/6901 or 579/1145?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.008067813727394735, + "pure_math": { + "question": "Which is larger: 3434/6901 or 579/1145?", + "options": [ + "3434/6901", + "4465/9420", + "579/1145", + "1790/3897" + ], + "answer": "579/1145", + "correct_value": 0.5056768558951965, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3367/6121 or 1402/2416?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3367/6121 or 1402/2416?", + "options": [ + "4071/8380", + "1402/2416", + "5253/8572", + "3367/6121" + ], + "answer": "1402/2416", + "correct_value": 0.5802980132450332, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000508", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4574/9563 or 1066/2067?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.037421482298455044, + "pure_math": { + "question": "Which is larger: 4574/9563 or 1066/2067?", + "options": [ + "1066/2067", + "4574/9563", + "4154/9609", + "603/1396" + ], + "answer": "1066/2067", + "correct_value": 0.5157232704402516, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2831/5778 or 3246/6351?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.021138689535159627, + "pure_math": { + "question": "Which is larger: 2831/5778 or 3246/6351?", + "options": [ + "3246/6351", + "2647/5642", + "2831/5778", + "1235/2166" + ], + "answer": "3246/6351", + "correct_value": 0.5111006140765234, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5282/9603 or 5469/9428?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5282/9603 or 5469/9428?", + "options": [ + "5026/9626", + "5282/9603", + "5469/9428", + "5102/7686" + ], + "answer": "5469/9428", + "correct_value": 0.580080610946118, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000509", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2291/4721 or 3587/6872?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.036694681998225065, + "pure_math": { + "question": "Which is larger: 2291/4721 or 3587/6872?", + "options": [ + "4242/7771", + "2210/5153", + "2291/4721", + "3587/6872" + ], + "answer": "3587/6872", + "correct_value": 0.5219732246798603, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4211/8613 or 1408/2790?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01574738860612057, + "pure_math": { + "question": "Which is larger: 4211/8613 or 1408/2790?", + "options": [ + "1408/2790", + "1357/2398", + "4881/9473", + "4211/8613" + ], + "answer": "1408/2790", + "correct_value": 0.5046594982078854, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2742/4984 or 1384/2386?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2742/4984 or 1384/2386?", + "options": [ + "3570/7126", + "2742/4984", + "1384/2386", + "4139/6960" + ], + "answer": "1384/2386", + "correct_value": 0.5800502933780386, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000510", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1151/2394 or 1112/2130?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.041280431124760264, + "pure_math": { + "question": "Which is larger: 1151/2394 or 1112/2130?", + "options": [ + "3705/7774", + "4009/7704", + "1151/2394", + "1112/2130" + ], + "answer": "1112/2130", + "correct_value": 0.5220657276995305, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3075/6212 or 4122/8130?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012001411385652805, + "pure_math": { + "question": "Which is larger: 3075/6212 or 4122/8130?", + "options": [ + "899/2130", + "4122/8130", + "3075/6212", + "4451/8441" + ], + "answer": "4122/8130", + "correct_value": 0.5070110701107011, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5441/9891 or 3274/5644?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5441/9891 or 3274/5644?", + "options": [ + "1933/3100", + "5441/9891", + "3274/5644", + "3421/6887" + ], + "answer": "3274/5644", + "correct_value": 0.5800850460666194, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000511", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1129/2359 or 2815/5467?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.036315003590466466, + "pure_math": { + "question": "Which is larger: 1129/2359 or 2815/5467?", + "options": [ + "4617/7544", + "1129/2359", + "3097/7257", + "2815/5467" + ], + "answer": "2815/5467", + "correct_value": 0.5149076275836839, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4439/9101 or 3283/6394?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.025701510422586338, + "pure_math": { + "question": "Which is larger: 4439/9101 or 3283/6394?", + "options": [ + "4478/7448", + "4439/9101", + "2316/5539", + "3283/6394" + ], + "answer": "3283/6394", + "correct_value": 0.5134501094776353, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4033/7331 or 1373/2367?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4033/7331 or 1373/2367?", + "options": [ + "3443/7076", + "4033/7331", + "2579/4224", + "1373/2367" + ], + "answer": "1373/2367", + "correct_value": 0.5800591465990705, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000512", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 876/1861 or 4150/8118?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04049498801861151, + "pure_math": { + "question": "Which is larger: 876/1861 or 4150/8118?", + "options": [ + "4150/8118", + "876/1861", + "848/1984", + "5079/8348" + ], + "answer": "4150/8118", + "correct_value": 0.5112096575511209, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2038/4095 or 3201/6326?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.00832685774196995, + "pure_math": { + "question": "Which is larger: 2038/4095 or 3201/6326?", + "options": [ + "3951/6710", + "2038/4095", + "3201/6326", + "913/1833" + ], + "answer": "3201/6326", + "correct_value": 0.5060069554220676, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1169/2124 or 2782/4796?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1169/2124 or 2782/4796?", + "options": [ + "1508/2543", + "3319/5369", + "1169/2124", + "2782/4796" + ], + "answer": "2782/4796", + "correct_value": 0.5800667222685572, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000513", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2115/4442 or 4220/8125?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.043247740103210586, + "pure_math": { + "question": "Which is larger: 2115/4442 or 4220/8125?", + "options": [ + "2988/6396", + "2115/4442", + "4220/8125", + "2510/5912" + ], + "answer": "4220/8125", + "correct_value": 0.5193846153846153, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4164/8518 or 1455/2846?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.022396703801317253, + "pure_math": { + "question": "Which is larger: 4164/8518 or 1455/2846?", + "options": [ + "1455/2846", + "4435/9722", + "4164/8518", + "1509/2827" + ], + "answer": "1455/2846", + "correct_value": 0.511243851018974, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1049/1907 or 3952/6813?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1049/1907 or 3952/6813?", + "options": [ + "3070/4539", + "3952/6813", + "1049/1907", + "2244/4008" + ], + "answer": "3952/6813", + "correct_value": 0.5800675179803317, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000514", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3836/7863 or 1240/2380?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03315389490401277, + "pure_math": { + "question": "Which is larger: 3836/7863 or 1240/2380?", + "options": [ + "4761/7977", + "1240/2380", + "3836/7863", + "2145/3881" + ], + "answer": "1240/2380", + "correct_value": 0.5210084033613446, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3773/7611 or 2090/4080?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.016525037291227063, + "pure_math": { + "question": "Which is larger: 3773/7611 or 2090/4080?", + "options": [ + "2896/5449", + "5960/9969", + "2090/4080", + "3773/7611" + ], + "answer": "2090/4080", + "correct_value": 0.5122549019607843, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4621/8401 or 1629/2807?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4621/8401 or 1629/2807?", + "options": [ + "4621/8401", + "1629/2807", + "2730/4365", + "3336/4971" + ], + "answer": "1629/2807", + "correct_value": 0.5803348770929818, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000515", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1065/2258 or 930/1801?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04472345596802091, + "pure_math": { + "question": "Which is larger: 1065/2258 or 930/1801?", + "options": [ + "576/1110", + "2301/5118", + "930/1801", + "1065/2258" + ], + "answer": "930/1801", + "correct_value": 0.5163797890061077, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4756/9664 or 3481/6855?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01566876065713141, + "pure_math": { + "question": "Which is larger: 4756/9664 or 3481/6855?", + "options": [ + "603/1004", + "2578/6064", + "4756/9664", + "3481/6855" + ], + "answer": "3481/6855", + "correct_value": 0.5078045222465354, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1169/2124 or 5737/9891?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1169/2124 or 5737/9891?", + "options": [ + "1169/2124", + "5737/9891", + "6250/9795", + "1903/3080" + ], + "answer": "5737/9891", + "correct_value": 0.5800222424426246, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000516", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3696/7723 or 5022/9842?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.031691638150813384, + "pure_math": { + "question": "Which is larger: 3696/7723 or 5022/9842?", + "options": [ + "677/1587", + "2788/5771", + "5022/9842", + "3696/7723" + ], + "answer": "5022/9842", + "correct_value": 0.5102621418410892, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2857/5805 or 3452/6799?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015559794411677541, + "pure_math": { + "question": "Which is larger: 2857/5805 or 3452/6799?", + "options": [ + "2761/6457", + "930/1802", + "2857/5805", + "3452/6799" + ], + "answer": "3452/6799", + "correct_value": 0.5077217237829093, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5275/9590 or 1576/2717?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5275/9590 or 1576/2717?", + "options": [ + "1576/2717", + "1560/2477", + "1505/2342", + "5275/9590" + ], + "answer": "1576/2717", + "correct_value": 0.5800515274199485, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000517", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2913/5963 or 1266/2469?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.024245707989874354, + "pure_math": { + "question": "Which is larger: 2913/5963 or 1266/2469?", + "options": [ + "4031/9632", + "1266/2469", + "5397/9634", + "2913/5963" + ], + "answer": "1266/2469", + "correct_value": 0.5127582017010935, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 507/1030 or 651/1296?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.010081805106076924, + "pure_math": { + "question": "Which is larger: 507/1030 or 651/1296?", + "options": [ + "1054/2234", + "2969/7192", + "651/1296", + "507/1030" + ], + "answer": "651/1296", + "correct_value": 0.5023148148148148, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3090/5617 or 2526/4354?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3090/5617 or 2526/4354?", + "options": [ + "3090/5617", + "1983/2979", + "4702/8627", + "2526/4354" + ], + "answer": "2526/4354", + "correct_value": 0.5801561782269178, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000518", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1224/2533 or 2082/4029?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033532060347714, + "pure_math": { + "question": "Which is larger: 1224/2533 or 2082/4029?", + "options": [ + "1224/2533", + "2082/4029", + "2039/4465", + "4320/8790" + ], + "answer": "2082/4029", + "correct_value": 0.5167535368577811, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3858/7926 or 2451/4774?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02665348863243905, + "pure_math": { + "question": "Which is larger: 3858/7926 or 2451/4774?", + "options": [ + "3858/7926", + "3615/8324", + "2451/4774", + "1021/1738" + ], + "answer": "2451/4774", + "correct_value": 0.5134059488898198, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1833/3332 or 1743/3005?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1833/3332 or 1743/3005?", + "options": [ + "708/1391", + "4441/7620", + "1743/3005", + "1833/3332" + ], + "answer": "1743/3005", + "correct_value": 0.5800332778702163, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000519", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4192/8743 or 3885/7521?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03708436006312599, + "pure_math": { + "question": "Which is larger: 4192/8743 or 3885/7521?", + "options": [ + "4192/8743", + "4112/6805", + "3109/7250", + "3885/7521" + ], + "answer": "3885/7521", + "correct_value": 0.5165536497806142, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2786/5607 or 823/1616?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012403276844538347, + "pure_math": { + "question": "Which is larger: 2786/5607 or 823/1616?", + "options": [ + "2786/5607", + "2908/5704", + "823/1616", + "4675/8854" + ], + "answer": "823/1616", + "correct_value": 0.5092821782178217, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2598/4723 or 2121/3656?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2598/4723 or 2121/3656?", + "options": [ + "2121/3656", + "647/1004", + "4091/6563", + "2598/4723" + ], + "answer": "2121/3656", + "correct_value": 0.5801422319474836, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000520", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4492/9276 or 3990/7700?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033921361088243396, + "pure_math": { + "question": "Which is larger: 4492/9276 or 3990/7700?", + "options": [ + "4492/9276", + "4533/7570", + "3990/7700", + "1086/2564" + ], + "answer": "3990/7700", + "correct_value": 0.5181818181818182, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 703/1440 or 3614/7049?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02450239198625498, + "pure_math": { + "question": "Which is larger: 703/1440 or 3614/7049?", + "options": [ + "1487/2679", + "703/1440", + "2909/4935", + "3614/7049" + ], + "answer": "3614/7049", + "correct_value": 0.5126968364306994, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2122/3857 or 4363/7521?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2122/3857 or 4363/7521?", + "options": [ + "4363/7521", + "1958/3581", + "2122/3857", + "1237/2406" + ], + "answer": "4363/7521", + "correct_value": 0.5801090280547799, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000521", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1636/3441 or 1135/2160?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.05001977784235845, + "pure_math": { + "question": "Which is larger: 1636/3441 or 1135/2160?", + "options": [ + "1636/3441", + "1135/2160", + "2662/4824", + "2364/4800" + ], + "answer": "1135/2160", + "correct_value": 0.5254629629629629, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1802/3676 or 4617/9102?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017044407129069628, + "pure_math": { + "question": "Which is larger: 1802/3676 or 4617/9102?", + "options": [ + "1802/3676", + "1393/2811", + "4617/9102", + "4620/9621" + ], + "answer": "4617/9102", + "correct_value": 0.507251153592617, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 570/1035 or 4022/6933?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 570/1035 or 4022/6933?", + "options": [ + "2283/4147", + "4022/6933", + "4783/9783", + "570/1035" + ], + "answer": "4022/6933", + "correct_value": 0.5801240444252127, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000522", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3196/6605 or 2062/3940?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03947440217955178, + "pure_math": { + "question": "Which is larger: 3196/6605 or 2062/3940?", + "options": [ + "4290/7361", + "3196/6605", + "2062/3940", + "1719/3834" + ], + "answer": "2062/3940", + "correct_value": 0.5233502538071066, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4405/8957 or 4579/8994?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0173230617392231, + "pure_math": { + "question": "Which is larger: 4405/8957 or 4579/8994?", + "options": [ + "4405/8957", + "885/1826", + "3186/5270", + "4579/8994" + ], + "answer": "4579/8994", + "correct_value": 0.5091171892372693, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3346/6082 or 3217/5545?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3346/6082 or 3217/5545?", + "options": [ + "1450/2374", + "3550/6876", + "3217/5545", + "3346/6082" + ], + "answer": "3217/5545", + "correct_value": 0.5801623083859333, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000523", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3232/6723 or 1308/2502?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.042044008702007585, + "pure_math": { + "question": "Which is larger: 3232/6723 or 1308/2502?", + "options": [ + "3232/6723", + "2950/5214", + "3599/8289", + "1308/2502" + ], + "answer": "1308/2502", + "correct_value": 0.5227817745803357, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1472/3012 or 4739/9407?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.015061966089788492, + "pure_math": { + "question": "Which is larger: 1472/3012 or 4739/9407?", + "options": [ + "3825/6790", + "4739/9407", + "1472/3012", + "1331/3228" + ], + "answer": "4739/9407", + "correct_value": 0.5037737854788987, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1697/3085 or 2986/5147?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1697/3085 or 2986/5147?", + "options": [ + "1697/3085", + "2986/5147", + "3697/7243", + "4936/9013" + ], + "answer": "2986/5147", + "correct_value": 0.5801437730716923, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000524", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3367/6924 or 4100/7968?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02827862576823753, + "pure_math": { + "question": "Which is larger: 3367/6924 or 4100/7968?", + "options": [ + "1602/2699", + "2116/4792", + "3367/6924", + "4100/7968" + ], + "answer": "4100/7968", + "correct_value": 0.5145582329317269, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 670/1355 or 1671/3275?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01576406298414129, + "pure_math": { + "question": "Which is larger: 670/1355 or 1671/3275?", + "options": [ + "2545/4545", + "1671/3275", + "670/1355", + "477/1119" + ], + "answer": "1671/3275", + "correct_value": 0.5102290076335878, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4186/7610 or 5563/9591?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4186/7610 or 5563/9591?", + "options": [ + "4096/6903", + "5563/9591", + "5007/9062", + "4186/7610" + ], + "answer": "5563/9591", + "correct_value": 0.5800229381712022, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000525", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4148/8645 or 1075/2070?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.039508749577399416, + "pure_math": { + "question": "Which is larger: 4148/8645 or 1075/2070?", + "options": [ + "4218/7012", + "1075/2070", + "4148/8645", + "846/1948" + ], + "answer": "1075/2070", + "correct_value": 0.5193236714975845, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 999/2025 or 3938/7760?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014140893470790339, + "pure_math": { + "question": "Which is larger: 999/2025 or 3938/7760?", + "options": [ + "3938/7760", + "999/2025", + "874/2097", + "3790/7857" + ], + "answer": "3938/7760", + "correct_value": 0.5074742268041237, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 660/1199 or 2527/4356?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 660/1199 or 2527/4356?", + "options": [ + "3858/5708", + "2184/3258", + "660/1199", + "2527/4356" + ], + "answer": "2527/4356", + "correct_value": 0.580119375573921, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000526", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2433/5025 or 5002/9775?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02753445050960035, + "pure_math": { + "question": "Which is larger: 2433/5025 or 5002/9775?", + "options": [ + "4027/7545", + "1529/2711", + "5002/9775", + "2433/5025" + ], + "answer": "5002/9775", + "correct_value": 0.5117135549872123, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2853/5863 or 3431/6764?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.0206332841553738, + "pure_math": { + "question": "Which is larger: 2853/5863 or 3431/6764?", + "options": [ + "3431/6764", + "1491/3514", + "2853/5863", + "4300/8556" + ], + "answer": "3431/6764", + "correct_value": 0.507244234180958, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2726/4955 or 1339/2308?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2726/4955 or 1339/2308?", + "options": [ + "2726/4955", + "1339/2308", + "3661/7122", + "5819/8672" + ], + "answer": "1339/2308", + "correct_value": 0.580155979202773, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000527", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 691/1467 or 3759/7322?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04235500970360029, + "pure_math": { + "question": "Which is larger: 691/1467 or 3759/7322?", + "options": [ + "3196/6626", + "2609/4346", + "691/1467", + "3759/7322" + ], + "answer": "3759/7322", + "correct_value": 0.5133843212237094, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3413/6995 or 3178/6178?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02648601380396809, + "pure_math": { + "question": "Which is larger: 3413/6995 or 3178/6178?", + "options": [ + "3178/6178", + "3413/6995", + "3545/7770", + "702/1686" + ], + "answer": "3178/6178", + "correct_value": 0.5144059566202654, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5134/9334 or 3584/6179?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5134/9334 or 3584/6179?", + "options": [ + "3584/6179", + "2141/3978", + "3798/5689", + "5134/9334" + ], + "answer": "3584/6179", + "correct_value": 0.5800291309273345, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000528", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 950/1988 or 3992/7689?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04131604557766683, + "pure_math": { + "question": "Which is larger: 950/1988 or 3992/7689?", + "options": [ + "950/1988", + "525/1012", + "3733/7602", + "3992/7689" + ], + "answer": "3992/7689", + "correct_value": 0.5191832487969827, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2300/4668 or 4241/8395?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01246528899511229, + "pure_math": { + "question": "Which is larger: 2300/4668 or 4241/8395?", + "options": [ + "2300/4668", + "770/1548", + "4241/8395", + "1911/3655" + ], + "answer": "4241/8395", + "correct_value": 0.5051816557474688, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4433/8060 or 4310/7430?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4433/8060 or 4310/7430?", + "options": [ + "4433/8060", + "4310/7430", + "2195/3231", + "1333/2004" + ], + "answer": "4310/7430", + "correct_value": 0.5800807537012113, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000529", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2291/4734 or 1541/2985?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03230198308823373, + "pure_math": { + "question": "Which is larger: 2291/4734 or 1541/2985?", + "options": [ + "1541/2985", + "2520/5980", + "5037/8247", + "2291/4734" + ], + "answer": "1541/2985", + "correct_value": 0.5162479061976549, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4263/8705 or 3604/7110?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.017173149272408428, + "pure_math": { + "question": "Which is larger: 4263/8705 or 3604/7110?", + "options": [ + "4263/8705", + "5029/8892", + "3604/7110", + "4703/9509" + ], + "answer": "3604/7110", + "correct_value": 0.5068917018284107, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3298/5995 or 4421/7622?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3298/5995 or 4421/7622?", + "options": [ + "3298/5995", + "4421/7622", + "1133/1724", + "6379/9713" + ], + "answer": "4421/7622", + "correct_value": 0.5800314877984781, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000530", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4215/8704 or 4048/7805?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03438178592625396, + "pure_math": { + "question": "Which is larger: 4215/8704 or 4048/7805?", + "options": [ + "3622/8009", + "4048/7805", + "4215/8704", + "830/1622" + ], + "answer": "4048/7805", + "correct_value": 0.5186418962203716, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3898/7908 or 522/1020?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.018846142402332677, + "pure_math": { + "question": "Which is larger: 3898/7908 or 522/1020?", + "options": [ + "4921/9155", + "522/1020", + "3133/7077", + "3898/7908" + ], + "answer": "522/1020", + "correct_value": 0.5117647058823529, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2328/4232 or 1511/2605?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2328/4232 or 1511/2605?", + "options": [ + "1727/3514", + "1511/2605", + "2328/4232", + "3487/5972" + ], + "answer": "1511/2605", + "correct_value": 0.5800383877159309, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000531", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1413/2924 or 5003/9805?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02700773845109594, + "pure_math": { + "question": "Which is larger: 1413/2924 or 5003/9805?", + "options": [ + "5003/9805", + "1413/2924", + "2111/4594", + "3981/8604" + ], + "answer": "5003/9805", + "correct_value": 0.5102498725140234, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3947/8037 or 2396/4719?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01663104391395137, + "pure_math": { + "question": "Which is larger: 3947/8037 or 2396/4719?", + "options": [ + "4878/9755", + "2396/4719", + "1425/2571", + "3947/8037" + ], + "answer": "2396/4719", + "correct_value": 0.5077346895528714, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 5331/9691 or 4663/8038?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 5331/9691 or 4663/8038?", + "options": [ + "3604/5544", + "4663/8038", + "5331/9691", + "2898/5850" + ], + "answer": "4663/8038", + "correct_value": 0.5801194326947001, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000532", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1577/3229 or 4499/8628?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03305531997129052, + "pure_math": { + "question": "Which is larger: 1577/3229 or 4499/8628?", + "options": [ + "863/1867", + "2613/5444", + "1577/3229", + "4499/8628" + ], + "answer": "4499/8628", + "correct_value": 0.5214418173388966, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 651/1321 or 4165/8224?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.013636074103747564, + "pure_math": { + "question": "Which is larger: 651/1321 or 4165/8224?", + "options": [ + "651/1321", + "1557/3547", + "4165/8224", + "1470/3174" + ], + "answer": "4165/8224", + "correct_value": 0.5064445525291829, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3126/5683 or 1149/1980?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3126/5683 or 1149/1980?", + "options": [ + "1149/1980", + "3126/5683", + "816/1677", + "4221/7622" + ], + "answer": "1149/1980", + "correct_value": 0.5803030303030303, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000533", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4239/8731 or 676/1290?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.038519611577387536, + "pure_math": { + "question": "Which is larger: 4239/8731 or 676/1290?", + "options": [ + "4239/8731", + "676/1290", + "1326/3086", + "3768/8780" + ], + "answer": "676/1290", + "correct_value": 0.524031007751938, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2400/4862 or 1575/3123?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.010698743534817423, + "pure_math": { + "question": "Which is larger: 2400/4862 or 1575/3123?", + "options": [ + "1575/3123", + "781/1843", + "513/1121", + "2400/4862" + ], + "answer": "1575/3123", + "correct_value": 0.5043227665706052, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1909/3470 or 2046/3527?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1909/3470 or 2046/3527?", + "options": [ + "1855/3573", + "1909/3470", + "4561/7723", + "2046/3527" + ], + "answer": "2046/3527", + "correct_value": 0.5800963992061242, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000534", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1687/3463 or 1082/2080?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.033042437637442024, + "pure_math": { + "question": "Which is larger: 1687/3463 or 1082/2080?", + "options": [ + "4627/8310", + "3697/6437", + "1082/2080", + "1687/3463" + ], + "answer": "1082/2080", + "correct_value": 0.5201923076923077, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3368/6863 or 5055/9916?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.019034683708002265, + "pure_math": { + "question": "Which is larger: 3368/6863 or 5055/9916?", + "options": [ + "3368/6863", + "5055/9916", + "1449/2629", + "1955/3580" + ], + "answer": "5055/9916", + "correct_value": 0.5097821702299314, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 742/1349 or 3645/6284?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 742/1349 or 3645/6284?", + "options": [ + "5006/8429", + "3645/6284", + "5631/9877", + "742/1349" + ], + "answer": "3645/6284", + "correct_value": 0.58004455760662, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000535", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4806/9994 or 4838/9350?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03654462196034203, + "pure_math": { + "question": "Which is larger: 4806/9994 or 4838/9350?", + "options": [ + "4838/9350", + "1125/2479", + "4529/7452", + "4806/9994" + ], + "answer": "4838/9350", + "correct_value": 0.5174331550802139, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2080/4249 or 1977/3874?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.020798297707511293, + "pure_math": { + "question": "Which is larger: 2080/4249 or 1977/3874?", + "options": [ + "1765/3230", + "2080/4249", + "906/1744", + "1977/3874" + ], + "answer": "1977/3874", + "correct_value": 0.5103252452245741, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 921/1674 or 2861/4932?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 921/1674 or 2861/4932?", + "options": [ + "1451/2137", + "921/1674", + "939/1743", + "2861/4932" + ], + "answer": "2861/4932", + "correct_value": 0.5800892133008921, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000536", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4227/8715 or 2578/4974?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03326931714450426, + "pure_math": { + "question": "Which is larger: 4227/8715 or 2578/4974?", + "options": [ + "4227/8715", + "2578/4974", + "3195/6116", + "3368/6660" + ], + "answer": "2578/4974", + "correct_value": 0.5182951347004423, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4531/9227 or 2577/5124?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.011868551438363406, + "pure_math": { + "question": "Which is larger: 4531/9227 or 2577/5124?", + "options": [ + "4531/9227", + "2479/4905", + "2577/5124", + "1395/2557" + ], + "answer": "2577/5124", + "correct_value": 0.502927400468384, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1803/3277 or 4410/7602?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1803/3277 or 4410/7602?", + "options": [ + "3354/5814", + "3310/4913", + "4410/7602", + "1803/3277" + ], + "answer": "4410/7602", + "correct_value": 0.580110497237569, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000537", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 589/1257 or 4520/8748?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04811355449264637, + "pure_math": { + "question": "Which is larger: 589/1257 or 4520/8748?", + "options": [ + "589/1257", + "3858/7807", + "3568/7877", + "4520/8748" + ], + "answer": "4520/8748", + "correct_value": 0.516689529035208, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1302/2616 or 3160/6252?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.007731837738529201, + "pure_math": { + "question": "Which is larger: 1302/2616 or 3160/6252?", + "options": [ + "4382/8072", + "605/1031", + "1302/2616", + "3160/6252" + ], + "answer": "3160/6252", + "correct_value": 0.5054382597568778, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4091/7437 or 917/1581?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4091/7437 or 917/1581?", + "options": [ + "5253/8011", + "4091/7437", + "3997/5920", + "917/1581" + ], + "answer": "917/1581", + "correct_value": 0.5800126502213788, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000538", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2070/4318 or 1903/3714?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.032996962284589426, + "pure_math": { + "question": "Which is larger: 2070/4318 or 1903/3714?", + "options": [ + "5389/8922", + "2070/4318", + "1903/3714", + "1294/2280" + ], + "answer": "1903/3714", + "correct_value": 0.5123855681206246, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1334/2702 or 2492/4911?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.013723930673614326, + "pure_math": { + "question": "Which is larger: 1334/2702 or 2492/4911?", + "options": [ + "5626/9330", + "3028/5443", + "2492/4911", + "1334/2702" + ], + "answer": "2492/4911", + "correct_value": 0.5074322948482998, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4519/8215 or 2258/3893?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4519/8215 or 2258/3893?", + "options": [ + "4519/8215", + "4936/7619", + "2017/3739", + "2258/3893" + ], + "answer": "2258/3893", + "correct_value": 0.5800154122784485, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000539", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 4708/9631 or 3885/7455?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03228863368143653, + "pure_math": { + "question": "Which is larger: 4708/9631 or 3885/7455?", + "options": [ + "3885/7455", + "4708/9631", + "3937/6491", + "1966/3569" + ], + "answer": "3885/7455", + "correct_value": 0.5211267605633803, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1458/3000 or 4799/9337?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.02797665202955979, + "pure_math": { + "question": "Which is larger: 1458/3000 or 4799/9337?", + "options": [ + "4663/9760", + "4799/9337", + "1458/3000", + "1721/3980" + ], + "answer": "4799/9337", + "correct_value": 0.5139766520295598, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 2251/4092 or 2314/3988?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 2251/4092 or 2314/3988?", + "options": [ + "2574/3795", + "2251/4092", + "2314/3988", + "3244/5317" + ], + "answer": "2314/3988", + "correct_value": 0.5802407221664995, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000540", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1357/2863 or 653/1232?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.05605412313847524, + "pure_math": { + "question": "Which is larger: 1357/2863 or 653/1232?", + "options": [ + "4261/7084", + "653/1232", + "1357/2863", + "3005/5434" + ], + "answer": "653/1232", + "correct_value": 0.5300324675324676, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2765/5666 or 1293/2574?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01433241426181775, + "pure_math": { + "question": "Which is larger: 2765/5666 or 1293/2574?", + "options": [ + "1378/3352", + "2765/5666", + "1293/2574", + "3794/8013" + ], + "answer": "1293/2574", + "correct_value": 0.5023310023310024, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3249/5906 or 1086/1871?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3249/5906 or 1086/1871?", + "options": [ + "3499/6743", + "1086/1871", + "2366/4789", + "3249/5906" + ], + "answer": "1086/1871", + "correct_value": 0.5804382683057189, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000541", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3455/7057 or 1296/2457?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03788771806343011, + "pure_math": { + "question": "Which is larger: 3455/7057 or 1296/2457?", + "options": [ + "1296/2457", + "3455/7057", + "1448/2762", + "2582/4176" + ], + "answer": "1296/2457", + "correct_value": 0.5274725274725275, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4288/8738 or 4278/8346?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.021850732869101497, + "pure_math": { + "question": "Which is larger: 4288/8738 or 4278/8346?", + "options": [ + "4288/8738", + "3812/6978", + "3244/6720", + "4278/8346" + ], + "answer": "4278/8346", + "correct_value": 0.5125808770668584, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1762/3203 or 2041/3518?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1762/3203 or 2041/3518?", + "options": [ + "2041/3518", + "1762/3203", + "2626/5023", + "5147/8264" + ], + "answer": "2041/3518", + "correct_value": 0.5801591813530415, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000542", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1738/3563 or 4691/9070?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.02940837178387079, + "pure_math": { + "question": "Which is larger: 1738/3563 or 4691/9070?", + "options": [ + "1738/3563", + "1308/2524", + "3653/6715", + "4691/9070" + ], + "answer": "4691/9070", + "correct_value": 0.517199558985667, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4477/9039 or 3111/6106?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.014200701136143745, + "pure_math": { + "question": "Which is larger: 4477/9039 or 3111/6106?", + "options": [ + "4477/9039", + "1527/2958", + "3842/6686", + "3111/6106" + ], + "answer": "3111/6106", + "correct_value": 0.509498853586636, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 882/1603 or 3215/5543?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 882/1603 or 3215/5543?", + "options": [ + "1650/3399", + "3215/5543", + "882/1603", + "4741/7427" + ], + "answer": "3215/5543", + "correct_value": 0.580010824463287, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000543", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1740/3636 or 935/1782?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04614350323921279, + "pure_math": { + "question": "Which is larger: 1740/3636 or 935/1782?", + "options": [ + "1740/3636", + "1028/1719", + "566/1095", + "935/1782" + ], + "answer": "935/1782", + "correct_value": 0.5246913580246914, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 576/1163 or 1307/2579?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.011514724557799838, + "pure_math": { + "question": "Which is larger: 576/1163 or 1307/2579?", + "options": [ + "3478/7986", + "576/1163", + "1307/2579", + "4082/9615" + ], + "answer": "1307/2579", + "correct_value": 0.5067855758045754, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3640/6617 or 1284/2213?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3640/6617 or 1284/2213?", + "options": [ + "2211/3638", + "4202/6334", + "3640/6617", + "1284/2213" + ], + "answer": "1284/2213", + "correct_value": 0.5802078626299142, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000544", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 564/1193 or 2777/5354?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04591987064375341, + "pure_math": { + "question": "Which is larger: 564/1193 or 2777/5354?", + "options": [ + "2777/5354", + "4021/6751", + "2002/4605", + "564/1193" + ], + "answer": "2777/5354", + "correct_value": 0.518677624206201, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 3073/6182 or 2312/4534?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01283669009605265, + "pure_math": { + "question": "Which is larger: 3073/6182 or 2312/4534?", + "options": [ + "545/1164", + "3073/6182", + "4129/9426", + "2312/4534" + ], + "answer": "2312/4534", + "correct_value": 0.50992501102779, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1836/3337 or 1582/2726?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1836/3337 or 1582/2726?", + "options": [ + "1836/3337", + "3966/6408", + "2258/4182", + "1582/2726" + ], + "answer": "1582/2726", + "correct_value": 0.5803374908290535, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000545", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 546/1163 or 4178/8181?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04122001958483801, + "pure_math": { + "question": "Which is larger: 546/1163 or 4178/8181?", + "options": [ + "3376/6472", + "546/1163", + "4178/8181", + "1362/2256" + ], + "answer": "4178/8181", + "correct_value": 0.510695513995844, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1123/2272 or 4239/8365?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.012476164518491761, + "pure_math": { + "question": "Which is larger: 1123/2272 or 4239/8365?", + "options": [ + "544/1169", + "4239/8365", + "1785/3232", + "1123/2272" + ], + "answer": "4239/8365", + "correct_value": 0.5067543335325763, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4097/7449 or 5590/9637?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4097/7449 or 5590/9637?", + "options": [ + "1595/2643", + "4097/7449", + "5575/9833", + "5590/9637" + ], + "answer": "5590/9637", + "correct_value": 0.5800560340354882, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000546", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2529/5242 or 3326/6478?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03098062423353115, + "pure_math": { + "question": "Which is larger: 2529/5242 or 3326/6478?", + "options": [ + "2061/4086", + "3187/7258", + "3326/6478", + "2529/5242" + ], + "answer": "3326/6478", + "correct_value": 0.5134300710095708, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 562/1151 or 1869/3717?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01455379012109348, + "pure_math": { + "question": "Which is larger: 562/1151 or 1869/3717?", + "options": [ + "1772/3473", + "562/1151", + "2516/4290", + "1869/3717" + ], + "answer": "1869/3717", + "correct_value": 0.5028248587570622, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 613/1114 or 5201/8966?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 613/1114 or 5201/8966?", + "options": [ + "5201/8966", + "4330/6880", + "613/1114", + "4958/9495" + ], + "answer": "5201/8966", + "correct_value": 0.5800803033682802, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000547", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 3562/7384 or 4349/8510?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.028651462239949632, + "pure_math": { + "question": "Which is larger: 3562/7384 or 4349/8510?", + "options": [ + "3562/7384", + "875/1810", + "2763/5571", + "4349/8510" + ], + "answer": "4349/8510", + "correct_value": 0.5110458284371328, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1146/2321 or 2935/5835?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.009246450297145381, + "pure_math": { + "question": "Which is larger: 1146/2321 or 2935/5835?", + "options": [ + "3371/6895", + "2935/5835", + "1146/2321", + "1888/3457" + ], + "answer": "2935/5835", + "correct_value": 0.5029991431019709, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 4914/8934 or 766/1320?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 4914/8934 or 766/1320?", + "options": [ + "4914/8934", + "2823/4203", + "766/1320", + "4605/9437" + ], + "answer": "766/1320", + "correct_value": 0.5803030303030303, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000548", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 2084/4345 or 1083/2117?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.031941219988552316, + "pure_math": { + "question": "Which is larger: 2084/4345 or 1083/2117?", + "options": [ + "803/1447", + "1083/2117", + "3159/6685", + "2084/4345" + ], + "answer": "1083/2117", + "correct_value": 0.5115729806329712, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 1462/2941 or 1754/3456?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.01041332155855279, + "pure_math": { + "question": "Which is larger: 1462/2941 or 1754/3456?", + "options": [ + "696/1626", + "1754/3456", + "1462/2941", + "3974/7902" + ], + "answer": "1754/3456", + "correct_value": 0.5075231481481481, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 977/1775 or 1171/2018?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 977/1775 or 1171/2018?", + "options": [ + "1167/2351", + "4396/9082", + "977/1775", + "1171/2018" + ], + "answer": "1171/2018", + "correct_value": 0.5802775024777007, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000549", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 561/1178 or 4700/9140?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.03799229491813816, + "pure_math": { + "question": "Which is larger: 561/1178 or 4700/9140?", + "options": [ + "2950/5741", + "4700/9140", + "561/1178", + "2881/5343" + ], + "answer": "4700/9140", + "correct_value": 0.5142231947483589, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 4719/9712 or 3339/6557?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.023333040836421404, + "pure_math": { + "question": "Which is larger: 4719/9712 or 3339/6557?", + "options": [ + "1473/3062", + "564/1322", + "4719/9712", + "3339/6557" + ], + "answer": "3339/6557", + "correct_value": 0.509226780539881, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 3431/6237 or 1152/1986?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 3431/6237 or 1152/1986?", + "options": [ + "3431/6237", + "4449/9055", + "1152/1986", + "4841/7443" + ], + "answer": "1152/1986", + "correct_value": 0.5800604229607251, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "relative_distance_000550", + "category": "relative_distance", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "Which is larger: 1790/3708 or 849/1603?", + "ns_shortcut": "One is below 1/2, the other above; no computation needed.", + "shortcut_strength": 0.04689191853731639, + "pure_math": { + "question": "Which is larger: 1790/3708 or 849/1603?", + "options": [ + "1790/3708", + "1339/2815", + "1261/2367", + "849/1603" + ], + "answer": "849/1603", + "correct_value": 0.5296319401122894, + "explanation": "Both straddle 1/2; compare to benchmark." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "Which is larger: 2504/5149 or 1447/2829?", + "ns_shortcut": "Compare to 1/2, but margin is tight.", + "shortcut_strength": 0.025180137384897938, + "pure_math": { + "question": "Which is larger: 2504/5149 or 1447/2829?", + "options": [ + "2932/6629", + "2504/5149", + "1447/2829", + "1804/3893" + ], + "answer": "1447/2829", + "correct_value": 0.5114881583598445, + "explanation": "One is very close to 1/2, shortcut harder to apply." + } + }, + "control": { + "task_type": "control", + "math_expression": "Which is larger: 1599/2906 or 1151/1983?", + "ns_shortcut": "Both on same side; must cross-multiply.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Which is larger: 1599/2906 or 1151/1983?", + "options": [ + "3013/5332", + "1599/2906", + "1151/1983", + "4961/7600" + ], + "answer": "1151/1983", + "correct_value": 0.5804336863338376, + "explanation": "Both above 1/2; benchmark comparison won't help." + } + } + }, + { + "family_id": "cancellation_identity_000551", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3639 + 3950 - 3949", + "ns_shortcut": "Near-cancellation: 3950 - 3949 = 1, so answer ≈ 3639 + 1.", + "shortcut_strength": 0.999746835443038, + "pure_math": { + "question": "Evaluate: 3639 + 3950 - 3949", + "options": [ + "3650", + "3660", + "3630", + "3640" + ], + "answer": "3640", + "correct_value": 3640, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2974 + 6072 - 6239", + "ns_shortcut": "Near-cancellation: 6072 - 6239 = -167, offset is manageable.", + "shortcut_strength": 0.9724967061923584, + "pure_math": { + "question": "Evaluate: 2974 + 6072 - 6239", + "options": [ + "2797", + "2817", + "2827", + "2807" + ], + "answer": "2807", + "correct_value": 2807, + "explanation": "B-C = -167; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3639 + 3950 - 5459", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3639 + 3950 - 5459", + "options": [ + "2140", + "2150", + "2120", + "2130" + ], + "answer": "2130", + "correct_value": 2130, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000552", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9781 + 1382 - 1383", + "ns_shortcut": "Near-cancellation: 1382 - 1383 = -1, so answer ≈ 9781 + -1.", + "shortcut_strength": 0.9992764109985528, + "pure_math": { + "question": "Evaluate: 9781 + 1382 - 1383", + "options": [ + "9800", + "9780", + "9770", + "9790" + ], + "answer": "9780", + "correct_value": 9780, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3778 + 6273 - 6422", + "ns_shortcut": "Near-cancellation: 6273 - 6422 = -149, offset is manageable.", + "shortcut_strength": 0.9762474095329189, + "pure_math": { + "question": "Evaluate: 3778 + 6273 - 6422", + "options": [ + "3629", + "3619", + "3649", + "3639" + ], + "answer": "3629", + "correct_value": 3629, + "explanation": "B-C = -149; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9781 + 1382 - 1838", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9781 + 1382 - 1838", + "options": [ + "9325", + "9315", + "9335", + "9345" + ], + "answer": "9325", + "correct_value": 9325, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000553", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4264 + 2511 - 2510", + "ns_shortcut": "Near-cancellation: 2511 - 2510 = 1, so answer ≈ 4264 + 1.", + "shortcut_strength": 0.9996017522899243, + "pure_math": { + "question": "Evaluate: 4264 + 2511 - 2510", + "options": [ + "4265", + "4255", + "4275", + "4285" + ], + "answer": "4265", + "correct_value": 4265, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1310 + 1194 - 1141", + "ns_shortcut": "Near-cancellation: 1194 - 1141 = 53, offset is manageable.", + "shortcut_strength": 0.9556113902847572, + "pure_math": { + "question": "Evaluate: 1310 + 1194 - 1141", + "options": [ + "1383", + "1373", + "1353", + "1363" + ], + "answer": "1363", + "correct_value": 1363, + "explanation": "B-C = 53; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4264 + 2511 - 1581", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4264 + 2511 - 1581", + "options": [ + "5184", + "5194", + "5204", + "5214" + ], + "answer": "5194", + "correct_value": 5194, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000554", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6105 + 3719 - 3720", + "ns_shortcut": "Near-cancellation: 3719 - 3720 = -1, so answer ≈ 6105 + -1.", + "shortcut_strength": 0.9997311105135789, + "pure_math": { + "question": "Evaluate: 6105 + 3719 - 3720", + "options": [ + "6104", + "6124", + "6114", + "6094" + ], + "answer": "6104", + "correct_value": 6104, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5249 + 8336 - 8124", + "ns_shortcut": "Near-cancellation: 8336 - 8124 = 212, offset is manageable.", + "shortcut_strength": 0.9745681381957774, + "pure_math": { + "question": "Evaluate: 5249 + 8336 - 8124", + "options": [ + "5461", + "5451", + "5481", + "5471" + ], + "answer": "5461", + "correct_value": 5461, + "explanation": "B-C = 212; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6105 + 3719 - 5618", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6105 + 3719 - 5618", + "options": [ + "4206", + "4196", + "4226", + "4216" + ], + "answer": "4206", + "correct_value": 4206, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000555", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7126 + 2057 - 2055", + "ns_shortcut": "Near-cancellation: 2057 - 2055 = 2, so answer ≈ 7126 + 2.", + "shortcut_strength": 0.9990277102576568, + "pure_math": { + "question": "Evaluate: 7126 + 2057 - 2055", + "options": [ + "7148", + "7118", + "7128", + "7138" + ], + "answer": "7128", + "correct_value": 7128, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5815 + 2681 - 2787", + "ns_shortcut": "Near-cancellation: 2681 - 2787 = -106, offset is manageable.", + "shortcut_strength": 0.9604625139873182, + "pure_math": { + "question": "Evaluate: 5815 + 2681 - 2787", + "options": [ + "5729", + "5699", + "5719", + "5709" + ], + "answer": "5709", + "correct_value": 5709, + "explanation": "B-C = -106; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7126 + 2057 - 1323", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7126 + 2057 - 1323", + "options": [ + "7850", + "7860", + "7870", + "7880" + ], + "answer": "7860", + "correct_value": 7860, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000556", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6861 + 5024 - 5026", + "ns_shortcut": "Near-cancellation: 5024 - 5026 = -2, so answer ≈ 6861 + -2.", + "shortcut_strength": 0.9996019108280255, + "pure_math": { + "question": "Evaluate: 6861 + 5024 - 5026", + "options": [ + "6859", + "6849", + "6879", + "6869" + ], + "answer": "6859", + "correct_value": 6859, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8817 + 7743 - 7445", + "ns_shortcut": "Near-cancellation: 7743 - 7445 = 298, offset is manageable.", + "shortcut_strength": 0.9615136252098669, + "pure_math": { + "question": "Evaluate: 8817 + 7743 - 7445", + "options": [ + "9115", + "9105", + "9135", + "9125" + ], + "answer": "9115", + "correct_value": 9115, + "explanation": "B-C = 298; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6861 + 5024 - 2865", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6861 + 5024 - 2865", + "options": [ + "9010", + "9030", + "9020", + "9040" + ], + "answer": "9020", + "correct_value": 9020, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000557", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6268 + 4816 - 4818", + "ns_shortcut": "Near-cancellation: 4816 - 4818 = -2, so answer ≈ 6268 + -2.", + "shortcut_strength": 0.9995847176079734, + "pure_math": { + "question": "Evaluate: 6268 + 4816 - 4818", + "options": [ + "6286", + "6266", + "6276", + "6256" + ], + "answer": "6266", + "correct_value": 6266, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7349 + 4514 - 4628", + "ns_shortcut": "Near-cancellation: 4514 - 4628 = -114, offset is manageable.", + "shortcut_strength": 0.974745237040319, + "pure_math": { + "question": "Evaluate: 7349 + 4514 - 4628", + "options": [ + "7255", + "7245", + "7225", + "7235" + ], + "answer": "7235", + "correct_value": 7235, + "explanation": "B-C = -114; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6268 + 4816 - 7435", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6268 + 4816 - 7435", + "options": [ + "3649", + "3669", + "3659", + "3639" + ], + "answer": "3649", + "correct_value": 3649, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000558", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3344 + 1236 - 1234", + "ns_shortcut": "Near-cancellation: 1236 - 1234 = 2, so answer ≈ 3344 + 2.", + "shortcut_strength": 0.9983818770226537, + "pure_math": { + "question": "Evaluate: 3344 + 1236 - 1234", + "options": [ + "3356", + "3336", + "3366", + "3346" + ], + "answer": "3346", + "correct_value": 3346, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3953 + 8998 - 9372", + "ns_shortcut": "Near-cancellation: 8998 - 9372 = -374, offset is manageable.", + "shortcut_strength": 0.9584352078239609, + "pure_math": { + "question": "Evaluate: 3953 + 8998 - 9372", + "options": [ + "3589", + "3579", + "3569", + "3599" + ], + "answer": "3579", + "correct_value": 3579, + "explanation": "B-C = -374; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3344 + 1236 - 1733", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3344 + 1236 - 1733", + "options": [ + "2867", + "2847", + "2837", + "2857" + ], + "answer": "2847", + "correct_value": 2847, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000559", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5670 + 1297 - 1299", + "ns_shortcut": "Near-cancellation: 1297 - 1299 = -2, so answer ≈ 5670 + -2.", + "shortcut_strength": 0.9984579799537394, + "pure_math": { + "question": "Evaluate: 5670 + 1297 - 1299", + "options": [ + "5688", + "5668", + "5678", + "5658" + ], + "answer": "5668", + "correct_value": 5668, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5682 + 9549 - 9196", + "ns_shortcut": "Near-cancellation: 9549 - 9196 = 353, offset is manageable.", + "shortcut_strength": 0.9630327783013928, + "pure_math": { + "question": "Evaluate: 5682 + 9549 - 9196", + "options": [ + "6035", + "6045", + "6025", + "6055" + ], + "answer": "6035", + "correct_value": 6035, + "explanation": "B-C = 353; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5670 + 1297 - 1721", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5670 + 1297 - 1721", + "options": [ + "5236", + "5266", + "5256", + "5246" + ], + "answer": "5246", + "correct_value": 5246, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000560", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5500 + 6201 - 6200", + "ns_shortcut": "Near-cancellation: 6201 - 6200 = 1, so answer ≈ 5500 + 1.", + "shortcut_strength": 0.9998387356877922, + "pure_math": { + "question": "Evaluate: 5500 + 6201 - 6200", + "options": [ + "5521", + "5511", + "5501", + "5491" + ], + "answer": "5501", + "correct_value": 5501, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8943 + 1428 - 1362", + "ns_shortcut": "Near-cancellation: 1428 - 1362 = 66, offset is manageable.", + "shortcut_strength": 0.953781512605042, + "pure_math": { + "question": "Evaluate: 8943 + 1428 - 1362", + "options": [ + "9029", + "9009", + "8999", + "9019" + ], + "answer": "9009", + "correct_value": 9009, + "explanation": "B-C = 66; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5500 + 6201 - 9922", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5500 + 6201 - 9922", + "options": [ + "1799", + "1769", + "1789", + "1779" + ], + "answer": "1779", + "correct_value": 1779, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000561", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1479 + 4675 - 4677", + "ns_shortcut": "Near-cancellation: 4675 - 4677 = -2, so answer ≈ 1479 + -2.", + "shortcut_strength": 0.9995721925133689, + "pure_math": { + "question": "Evaluate: 1479 + 4675 - 4677", + "options": [ + "1487", + "1497", + "1467", + "1477" + ], + "answer": "1477", + "correct_value": 1477, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9037 + 3428 - 3342", + "ns_shortcut": "Near-cancellation: 3428 - 3342 = 86, offset is manageable.", + "shortcut_strength": 0.9749124854142357, + "pure_math": { + "question": "Evaluate: 9037 + 3428 - 3342", + "options": [ + "9143", + "9133", + "9123", + "9113" + ], + "answer": "9123", + "correct_value": 9123, + "explanation": "B-C = 86; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1479 + 4675 - 6468", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1479 + 4675 - 6468", + "options": [ + "-306", + "-326", + "-296", + "-314" + ], + "answer": "-314", + "correct_value": -314, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000562", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5254 + 1710 - 1712", + "ns_shortcut": "Near-cancellation: 1710 - 1712 = -2, so answer ≈ 5254 + -2.", + "shortcut_strength": 0.9988304093567252, + "pure_math": { + "question": "Evaluate: 5254 + 1710 - 1712", + "options": [ + "5272", + "5242", + "5262", + "5252" + ], + "answer": "5252", + "correct_value": 5252, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6924 + 3845 - 3732", + "ns_shortcut": "Near-cancellation: 3845 - 3732 = 113, offset is manageable.", + "shortcut_strength": 0.9706111833550065, + "pure_math": { + "question": "Evaluate: 6924 + 3845 - 3732", + "options": [ + "7057", + "7027", + "7037", + "7047" + ], + "answer": "7037", + "correct_value": 7037, + "explanation": "B-C = 113; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5254 + 1710 - 2364", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5254 + 1710 - 2364", + "options": [ + "4620", + "4610", + "4590", + "4600" + ], + "answer": "4600", + "correct_value": 4600, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000563", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6144 + 3597 - 3599", + "ns_shortcut": "Near-cancellation: 3597 - 3599 = -2, so answer ≈ 6144 + -2.", + "shortcut_strength": 0.9994439810953573, + "pure_math": { + "question": "Evaluate: 6144 + 3597 - 3599", + "options": [ + "6162", + "6142", + "6152", + "6132" + ], + "answer": "6142", + "correct_value": 6142, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8514 + 3606 - 3714", + "ns_shortcut": "Near-cancellation: 3606 - 3714 = -108, offset is manageable.", + "shortcut_strength": 0.9700499168053245, + "pure_math": { + "question": "Evaluate: 8514 + 3606 - 3714", + "options": [ + "8416", + "8406", + "8396", + "8426" + ], + "answer": "8406", + "correct_value": 8406, + "explanation": "B-C = -108; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6144 + 3597 - 5284", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6144 + 3597 - 5284", + "options": [ + "4457", + "4447", + "4477", + "4467" + ], + "answer": "4457", + "correct_value": 4457, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000564", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2356 + 6788 - 6789", + "ns_shortcut": "Near-cancellation: 6788 - 6789 = -1, so answer ≈ 2356 + -1.", + "shortcut_strength": 0.9998526812021213, + "pure_math": { + "question": "Evaluate: 2356 + 6788 - 6789", + "options": [ + "2345", + "2355", + "2375", + "2365" + ], + "answer": "2355", + "correct_value": 2355, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3544 + 6192 - 6068", + "ns_shortcut": "Near-cancellation: 6192 - 6068 = 124, offset is manageable.", + "shortcut_strength": 0.9799741602067183, + "pure_math": { + "question": "Evaluate: 3544 + 6192 - 6068", + "options": [ + "3678", + "3688", + "3668", + "3658" + ], + "answer": "3668", + "correct_value": 3668, + "explanation": "B-C = 124; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2356 + 6788 - 9709", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2356 + 6788 - 9709", + "options": [ + "-545", + "-555", + "-565", + "-575" + ], + "answer": "-565", + "correct_value": -565, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000565", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2712 + 6582 - 6583", + "ns_shortcut": "Near-cancellation: 6582 - 6583 = -1, so answer ≈ 2712 + -1.", + "shortcut_strength": 0.9998480704952902, + "pure_math": { + "question": "Evaluate: 2712 + 6582 - 6583", + "options": [ + "2711", + "2701", + "2721", + "2731" + ], + "answer": "2711", + "correct_value": 2711, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1914 + 7864 - 7531", + "ns_shortcut": "Near-cancellation: 7864 - 7531 = 333, offset is manageable.", + "shortcut_strength": 0.9576551373346898, + "pure_math": { + "question": "Evaluate: 1914 + 7864 - 7531", + "options": [ + "2237", + "2267", + "2247", + "2257" + ], + "answer": "2247", + "correct_value": 2247, + "explanation": "B-C = 333; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2712 + 6582 - 10045", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2712 + 6582 - 10045", + "options": [ + "-749", + "-751", + "-739", + "-769" + ], + "answer": "-751", + "correct_value": -751, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000566", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7556 + 4084 - 4086", + "ns_shortcut": "Near-cancellation: 4084 - 4086 = -2, so answer ≈ 7556 + -2.", + "shortcut_strength": 0.9995102840352595, + "pure_math": { + "question": "Evaluate: 7556 + 4084 - 4086", + "options": [ + "7544", + "7554", + "7564", + "7574" + ], + "answer": "7554", + "correct_value": 7554, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5110 + 9694 - 9245", + "ns_shortcut": "Near-cancellation: 9694 - 9245 = 449, offset is manageable.", + "shortcut_strength": 0.9536826903239117, + "pure_math": { + "question": "Evaluate: 5110 + 9694 - 9245", + "options": [ + "5549", + "5569", + "5559", + "5579" + ], + "answer": "5559", + "correct_value": 5559, + "explanation": "B-C = 449; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7556 + 4084 - 2640", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7556 + 4084 - 2640", + "options": [ + "9020", + "8990", + "9000", + "9010" + ], + "answer": "9000", + "correct_value": 9000, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000567", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4064 + 5375 - 5377", + "ns_shortcut": "Near-cancellation: 5375 - 5377 = -2, so answer ≈ 4064 + -2.", + "shortcut_strength": 0.9996279069767442, + "pure_math": { + "question": "Evaluate: 4064 + 5375 - 5377", + "options": [ + "4062", + "4072", + "4082", + "4052" + ], + "answer": "4062", + "correct_value": 4062, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1735 + 1977 - 1923", + "ns_shortcut": "Near-cancellation: 1977 - 1923 = 54, offset is manageable.", + "shortcut_strength": 0.9726858877086495, + "pure_math": { + "question": "Evaluate: 1735 + 1977 - 1923", + "options": [ + "1799", + "1789", + "1779", + "1809" + ], + "answer": "1789", + "correct_value": 1789, + "explanation": "B-C = 54; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4064 + 5375 - 7712", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4064 + 5375 - 7712", + "options": [ + "1747", + "1717", + "1727", + "1737" + ], + "answer": "1727", + "correct_value": 1727, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000568", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9227 + 8064 - 8063", + "ns_shortcut": "Near-cancellation: 8064 - 8063 = 1, so answer ≈ 9227 + 1.", + "shortcut_strength": 0.9998759920634921, + "pure_math": { + "question": "Evaluate: 9227 + 8064 - 8063", + "options": [ + "9218", + "9248", + "9238", + "9228" + ], + "answer": "9228", + "correct_value": 9228, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3573 + 6561 - 6822", + "ns_shortcut": "Near-cancellation: 6561 - 6822 = -261, offset is manageable.", + "shortcut_strength": 0.9602194787379973, + "pure_math": { + "question": "Evaluate: 3573 + 6561 - 6822", + "options": [ + "3332", + "3312", + "3322", + "3302" + ], + "answer": "3312", + "correct_value": 3312, + "explanation": "B-C = -261; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9227 + 8064 - 3915", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9227 + 8064 - 3915", + "options": [ + "13366", + "13386", + "13376", + "13396" + ], + "answer": "13376", + "correct_value": 13376, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000569", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3894 + 5525 - 5523", + "ns_shortcut": "Near-cancellation: 5525 - 5523 = 2, so answer ≈ 3894 + 2.", + "shortcut_strength": 0.9996380090497737, + "pure_math": { + "question": "Evaluate: 3894 + 5525 - 5523", + "options": [ + "3906", + "3916", + "3896", + "3886" + ], + "answer": "3896", + "correct_value": 3896, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5760 + 6288 - 6442", + "ns_shortcut": "Near-cancellation: 6288 - 6442 = -154, offset is manageable.", + "shortcut_strength": 0.9755089058524173, + "pure_math": { + "question": "Evaluate: 5760 + 6288 - 6442", + "options": [ + "5596", + "5606", + "5616", + "5626" + ], + "answer": "5606", + "correct_value": 5606, + "explanation": "B-C = -154; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3894 + 5525 - 2035", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3894 + 5525 - 2035", + "options": [ + "7394", + "7374", + "7384", + "7404" + ], + "answer": "7384", + "correct_value": 7384, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000570", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9718 + 2660 - 2659", + "ns_shortcut": "Near-cancellation: 2660 - 2659 = 1, so answer ≈ 9718 + 1.", + "shortcut_strength": 0.999624060150376, + "pure_math": { + "question": "Evaluate: 9718 + 2660 - 2659", + "options": [ + "9719", + "9709", + "9739", + "9729" + ], + "answer": "9719", + "correct_value": 9719, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6763 + 3441 - 3518", + "ns_shortcut": "Near-cancellation: 3441 - 3518 = -77, offset is manageable.", + "shortcut_strength": 0.977622784074397, + "pure_math": { + "question": "Evaluate: 6763 + 3441 - 3518", + "options": [ + "6706", + "6686", + "6676", + "6696" + ], + "answer": "6686", + "correct_value": 6686, + "explanation": "B-C = -77; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9718 + 2660 - 1355", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9718 + 2660 - 1355", + "options": [ + "11043", + "11013", + "11023", + "11033" + ], + "answer": "11023", + "correct_value": 11023, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000571", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6789 + 9091 - 9092", + "ns_shortcut": "Near-cancellation: 9091 - 9092 = -1, so answer ≈ 6789 + -1.", + "shortcut_strength": 0.999890001099989, + "pure_math": { + "question": "Evaluate: 6789 + 9091 - 9092", + "options": [ + "6808", + "6778", + "6798", + "6788" + ], + "answer": "6788", + "correct_value": 6788, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6226 + 4859 - 5059", + "ns_shortcut": "Near-cancellation: 4859 - 5059 = -200, offset is manageable.", + "shortcut_strength": 0.9588392673389586, + "pure_math": { + "question": "Evaluate: 6226 + 4859 - 5059", + "options": [ + "6036", + "6016", + "6026", + "6046" + ], + "answer": "6026", + "correct_value": 6026, + "explanation": "B-C = -200; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6789 + 9091 - 5916", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6789 + 9091 - 5916", + "options": [ + "9954", + "9964", + "9984", + "9974" + ], + "answer": "9964", + "correct_value": 9964, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000572", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1208 + 8780 - 8778", + "ns_shortcut": "Near-cancellation: 8780 - 8778 = 2, so answer ≈ 1208 + 2.", + "shortcut_strength": 0.9997722095671981, + "pure_math": { + "question": "Evaluate: 1208 + 8780 - 8778", + "options": [ + "1220", + "1200", + "1210", + "1230" + ], + "answer": "1210", + "correct_value": 1210, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4456 + 5383 - 5218", + "ns_shortcut": "Near-cancellation: 5383 - 5218 = 165, offset is manageable.", + "shortcut_strength": 0.9693479472413152, + "pure_math": { + "question": "Evaluate: 4456 + 5383 - 5218", + "options": [ + "4611", + "4631", + "4621", + "4641" + ], + "answer": "4621", + "correct_value": 4621, + "explanation": "B-C = 165; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1208 + 8780 - 13847", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1208 + 8780 - 13847", + "options": [ + "-3851", + "-3859", + "-3841", + "-3871" + ], + "answer": "-3859", + "correct_value": -3859, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000573", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9000 + 5060 - 5058", + "ns_shortcut": "Near-cancellation: 5060 - 5058 = 2, so answer ≈ 9000 + 2.", + "shortcut_strength": 0.999604743083004, + "pure_math": { + "question": "Evaluate: 9000 + 5060 - 5058", + "options": [ + "9012", + "9022", + "8992", + "9002" + ], + "answer": "9002", + "correct_value": 9002, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1199 + 5748 - 5541", + "ns_shortcut": "Near-cancellation: 5748 - 5541 = 207, offset is manageable.", + "shortcut_strength": 0.9639874739039666, + "pure_math": { + "question": "Evaluate: 1199 + 5748 - 5541", + "options": [ + "1426", + "1406", + "1416", + "1396" + ], + "answer": "1406", + "correct_value": 1406, + "explanation": "B-C = 207; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9000 + 5060 - 6782", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9000 + 5060 - 6782", + "options": [ + "7278", + "7268", + "7288", + "7298" + ], + "answer": "7278", + "correct_value": 7278, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000574", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6747 + 9945 - 9944", + "ns_shortcut": "Near-cancellation: 9945 - 9944 = 1, so answer ≈ 6747 + 1.", + "shortcut_strength": 0.9998994469582705, + "pure_math": { + "question": "Evaluate: 6747 + 9945 - 9944", + "options": [ + "6748", + "6768", + "6758", + "6738" + ], + "answer": "6748", + "correct_value": 6748, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7254 + 6211 - 5963", + "ns_shortcut": "Near-cancellation: 6211 - 5963 = 248, offset is manageable.", + "shortcut_strength": 0.9600708420544196, + "pure_math": { + "question": "Evaluate: 7254 + 6211 - 5963", + "options": [ + "7512", + "7502", + "7522", + "7492" + ], + "answer": "7502", + "correct_value": 7502, + "explanation": "B-C = 248; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6747 + 9945 - 15049", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6747 + 9945 - 15049", + "options": [ + "1663", + "1633", + "1653", + "1643" + ], + "answer": "1643", + "correct_value": 1643, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000575", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3423 + 7727 - 7728", + "ns_shortcut": "Near-cancellation: 7727 - 7728 = -1, so answer ≈ 3423 + -1.", + "shortcut_strength": 0.9998705836676589, + "pure_math": { + "question": "Evaluate: 3423 + 7727 - 7728", + "options": [ + "3422", + "3442", + "3412", + "3432" + ], + "answer": "3422", + "correct_value": 3422, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8125 + 7636 - 7301", + "ns_shortcut": "Near-cancellation: 7636 - 7301 = 335, offset is manageable.", + "shortcut_strength": 0.9561288632792038, + "pure_math": { + "question": "Evaluate: 8125 + 7636 - 7301", + "options": [ + "8470", + "8460", + "8450", + "8480" + ], + "answer": "8460", + "correct_value": 8460, + "explanation": "B-C = 335; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3423 + 7727 - 5044", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3423 + 7727 - 5044", + "options": [ + "6116", + "6106", + "6126", + "6096" + ], + "answer": "6106", + "correct_value": 6106, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000576", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5487 + 9932 - 9930", + "ns_shortcut": "Near-cancellation: 9932 - 9930 = 2, so answer ≈ 5487 + 2.", + "shortcut_strength": 0.999798630688683, + "pure_math": { + "question": "Evaluate: 5487 + 9932 - 9930", + "options": [ + "5489", + "5509", + "5479", + "5499" + ], + "answer": "5489", + "correct_value": 5489, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7897 + 1444 - 1480", + "ns_shortcut": "Near-cancellation: 1444 - 1480 = -36, offset is manageable.", + "shortcut_strength": 0.9750692520775623, + "pure_math": { + "question": "Evaluate: 7897 + 1444 - 1480", + "options": [ + "7881", + "7861", + "7871", + "7851" + ], + "answer": "7861", + "correct_value": 7861, + "explanation": "B-C = -36; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5487 + 9932 - 6173", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5487 + 9932 - 6173", + "options": [ + "9256", + "9246", + "9236", + "9266" + ], + "answer": "9246", + "correct_value": 9246, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000577", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8101 + 8460 - 8459", + "ns_shortcut": "Near-cancellation: 8460 - 8459 = 1, so answer ≈ 8101 + 1.", + "shortcut_strength": 0.9998817966903073, + "pure_math": { + "question": "Evaluate: 8101 + 8460 - 8459", + "options": [ + "8112", + "8092", + "8102", + "8122" + ], + "answer": "8102", + "correct_value": 8102, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9745 + 4635 - 4420", + "ns_shortcut": "Near-cancellation: 4635 - 4420 = 215, offset is manageable.", + "shortcut_strength": 0.95361380798274, + "pure_math": { + "question": "Evaluate: 9745 + 4635 - 4420", + "options": [ + "9950", + "9970", + "9960", + "9980" + ], + "answer": "9960", + "correct_value": 9960, + "explanation": "B-C = 215; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8101 + 8460 - 13140", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8101 + 8460 - 13140", + "options": [ + "3421", + "3441", + "3411", + "3431" + ], + "answer": "3421", + "correct_value": 3421, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000578", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5722 + 2063 - 2062", + "ns_shortcut": "Near-cancellation: 2063 - 2062 = 1, so answer ≈ 5722 + 1.", + "shortcut_strength": 0.9995152690256908, + "pure_math": { + "question": "Evaluate: 5722 + 2063 - 2062", + "options": [ + "5743", + "5713", + "5733", + "5723" + ], + "answer": "5723", + "correct_value": 5723, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2763 + 4914 - 5056", + "ns_shortcut": "Near-cancellation: 4914 - 5056 = -142, offset is manageable.", + "shortcut_strength": 0.9711029711029711, + "pure_math": { + "question": "Evaluate: 2763 + 4914 - 5056", + "options": [ + "2611", + "2621", + "2641", + "2631" + ], + "answer": "2621", + "correct_value": 2621, + "explanation": "B-C = -142; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5722 + 2063 - 1132", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5722 + 2063 - 1132", + "options": [ + "6673", + "6653", + "6663", + "6643" + ], + "answer": "6653", + "correct_value": 6653, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000579", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1817 + 5124 - 5125", + "ns_shortcut": "Near-cancellation: 5124 - 5125 = -1, so answer ≈ 1817 + -1.", + "shortcut_strength": 0.9998048399687745, + "pure_math": { + "question": "Evaluate: 1817 + 5124 - 5125", + "options": [ + "1806", + "1816", + "1826", + "1836" + ], + "answer": "1816", + "correct_value": 1816, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2992 + 7169 - 6938", + "ns_shortcut": "Near-cancellation: 7169 - 6938 = 231, offset is manageable.", + "shortcut_strength": 0.9677779327660762, + "pure_math": { + "question": "Evaluate: 2992 + 7169 - 6938", + "options": [ + "3233", + "3243", + "3223", + "3213" + ], + "answer": "3223", + "correct_value": 3223, + "explanation": "B-C = 231; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1817 + 5124 - 2703", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1817 + 5124 - 2703", + "options": [ + "4248", + "4228", + "4238", + "4258" + ], + "answer": "4238", + "correct_value": 4238, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000580", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8752 + 6815 - 6816", + "ns_shortcut": "Near-cancellation: 6815 - 6816 = -1, so answer ≈ 8752 + -1.", + "shortcut_strength": 0.9998532648569333, + "pure_math": { + "question": "Evaluate: 8752 + 6815 - 6816", + "options": [ + "8751", + "8771", + "8761", + "8741" + ], + "answer": "8751", + "correct_value": 8751, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9730 + 2090 - 2170", + "ns_shortcut": "Near-cancellation: 2090 - 2170 = -80, offset is manageable.", + "shortcut_strength": 0.9617224880382775, + "pure_math": { + "question": "Evaluate: 9730 + 2090 - 2170", + "options": [ + "9660", + "9670", + "9650", + "9640" + ], + "answer": "9650", + "correct_value": 9650, + "explanation": "B-C = -80; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8752 + 6815 - 9568", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8752 + 6815 - 9568", + "options": [ + "6009", + "5999", + "5989", + "6019" + ], + "answer": "5999", + "correct_value": 5999, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000581", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4268 + 5101 - 5102", + "ns_shortcut": "Near-cancellation: 5101 - 5102 = -1, so answer ≈ 4268 + -1.", + "shortcut_strength": 0.9998039600078416, + "pure_math": { + "question": "Evaluate: 4268 + 5101 - 5102", + "options": [ + "4277", + "4267", + "4287", + "4257" + ], + "answer": "4267", + "correct_value": 4267, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1511 + 7086 - 6939", + "ns_shortcut": "Near-cancellation: 7086 - 6939 = 147, offset is manageable.", + "shortcut_strength": 0.9792548687552921, + "pure_math": { + "question": "Evaluate: 1511 + 7086 - 6939", + "options": [ + "1658", + "1678", + "1668", + "1648" + ], + "answer": "1658", + "correct_value": 1658, + "explanation": "B-C = 147; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4268 + 5101 - 2454", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4268 + 5101 - 2454", + "options": [ + "6935", + "6915", + "6925", + "6905" + ], + "answer": "6915", + "correct_value": 6915, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000582", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3270 + 9632 - 9630", + "ns_shortcut": "Near-cancellation: 9632 - 9630 = 2, so answer ≈ 3270 + 2.", + "shortcut_strength": 0.9997923588039868, + "pure_math": { + "question": "Evaluate: 3270 + 9632 - 9630", + "options": [ + "3272", + "3282", + "3262", + "3292" + ], + "answer": "3272", + "correct_value": 3272, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3565 + 7213 - 7558", + "ns_shortcut": "Near-cancellation: 7213 - 7558 = -345, offset is manageable.", + "shortcut_strength": 0.9521696936087619, + "pure_math": { + "question": "Evaluate: 3565 + 7213 - 7558", + "options": [ + "3210", + "3240", + "3220", + "3230" + ], + "answer": "3220", + "correct_value": 3220, + "explanation": "B-C = -345; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3270 + 9632 - 14201", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3270 + 9632 - 14201", + "options": [ + "-1291", + "-1311", + "-1281", + "-1299" + ], + "answer": "-1299", + "correct_value": -1299, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000583", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7896 + 2453 - 2454", + "ns_shortcut": "Near-cancellation: 2453 - 2454 = -1, so answer ≈ 7896 + -1.", + "shortcut_strength": 0.9995923359152059, + "pure_math": { + "question": "Evaluate: 7896 + 2453 - 2454", + "options": [ + "7885", + "7905", + "7895", + "7915" + ], + "answer": "7895", + "correct_value": 7895, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7339 + 7247 - 7064", + "ns_shortcut": "Near-cancellation: 7247 - 7064 = 183, offset is manageable.", + "shortcut_strength": 0.9747481716572375, + "pure_math": { + "question": "Evaluate: 7339 + 7247 - 7064", + "options": [ + "7532", + "7522", + "7542", + "7512" + ], + "answer": "7522", + "correct_value": 7522, + "explanation": "B-C = 183; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7896 + 2453 - 3316", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7896 + 2453 - 3316", + "options": [ + "7023", + "7053", + "7043", + "7033" + ], + "answer": "7033", + "correct_value": 7033, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000584", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1890 + 2392 - 2394", + "ns_shortcut": "Near-cancellation: 2392 - 2394 = -2, so answer ≈ 1890 + -2.", + "shortcut_strength": 0.9991638795986622, + "pure_math": { + "question": "Evaluate: 1890 + 2392 - 2394", + "options": [ + "1898", + "1888", + "1878", + "1908" + ], + "answer": "1888", + "correct_value": 1888, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5180 + 1094 - 1046", + "ns_shortcut": "Near-cancellation: 1094 - 1046 = 48, offset is manageable.", + "shortcut_strength": 0.9561243144424132, + "pure_math": { + "question": "Evaluate: 5180 + 1094 - 1046", + "options": [ + "5228", + "5248", + "5218", + "5238" + ], + "answer": "5228", + "correct_value": 5228, + "explanation": "B-C = 48; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1890 + 2392 - 3816", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 1890 + 2392 - 3816", + "options": [ + "476", + "486", + "456", + "466" + ], + "answer": "466", + "correct_value": 466, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000585", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4359 + 9463 - 9464", + "ns_shortcut": "Near-cancellation: 9463 - 9464 = -1, so answer ≈ 4359 + -1.", + "shortcut_strength": 0.9998943252668288, + "pure_math": { + "question": "Evaluate: 4359 + 9463 - 9464", + "options": [ + "4368", + "4378", + "4358", + "4348" + ], + "answer": "4358", + "correct_value": 4358, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8361 + 9361 - 8894", + "ns_shortcut": "Near-cancellation: 9361 - 8894 = 467, offset is manageable.", + "shortcut_strength": 0.9501121675034718, + "pure_math": { + "question": "Evaluate: 8361 + 9361 - 8894", + "options": [ + "8818", + "8828", + "8848", + "8838" + ], + "answer": "8828", + "correct_value": 8828, + "explanation": "B-C = 467; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4359 + 9463 - 4639", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4359 + 9463 - 4639", + "options": [ + "9193", + "9203", + "9173", + "9183" + ], + "answer": "9183", + "correct_value": 9183, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000586", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3507 + 2988 - 2990", + "ns_shortcut": "Near-cancellation: 2988 - 2990 = -2, so answer ≈ 3507 + -2.", + "shortcut_strength": 0.999330655957162, + "pure_math": { + "question": "Evaluate: 3507 + 2988 - 2990", + "options": [ + "3495", + "3525", + "3505", + "3515" + ], + "answer": "3505", + "correct_value": 3505, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7739 + 4460 - 4580", + "ns_shortcut": "Near-cancellation: 4460 - 4580 = -120, offset is manageable.", + "shortcut_strength": 0.9730941704035875, + "pure_math": { + "question": "Evaluate: 7739 + 4460 - 4580", + "options": [ + "7619", + "7609", + "7639", + "7629" + ], + "answer": "7619", + "correct_value": 7619, + "explanation": "B-C = -120; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3507 + 2988 - 1144", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3507 + 2988 - 1144", + "options": [ + "5351", + "5341", + "5361", + "5371" + ], + "answer": "5351", + "correct_value": 5351, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000587", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4015 + 4854 - 4853", + "ns_shortcut": "Near-cancellation: 4854 - 4853 = 1, so answer ≈ 4015 + 1.", + "shortcut_strength": 0.9997939843428101, + "pure_math": { + "question": "Evaluate: 4015 + 4854 - 4853", + "options": [ + "4026", + "4016", + "4036", + "4006" + ], + "answer": "4016", + "correct_value": 4016, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5324 + 9400 - 9137", + "ns_shortcut": "Near-cancellation: 9400 - 9137 = 263, offset is manageable.", + "shortcut_strength": 0.9720212765957447, + "pure_math": { + "question": "Evaluate: 5324 + 9400 - 9137", + "options": [ + "5577", + "5587", + "5607", + "5597" + ], + "answer": "5587", + "correct_value": 5587, + "explanation": "B-C = 263; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4015 + 4854 - 1874", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4015 + 4854 - 1874", + "options": [ + "6985", + "6995", + "7015", + "7005" + ], + "answer": "6995", + "correct_value": 6995, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000588", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9092 + 3990 - 3991", + "ns_shortcut": "Near-cancellation: 3990 - 3991 = -1, so answer ≈ 9092 + -1.", + "shortcut_strength": 0.9997493734335839, + "pure_math": { + "question": "Evaluate: 9092 + 3990 - 3991", + "options": [ + "9111", + "9101", + "9081", + "9091" + ], + "answer": "9091", + "correct_value": 9091, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3999 + 1475 - 1546", + "ns_shortcut": "Near-cancellation: 1475 - 1546 = -71, offset is manageable.", + "shortcut_strength": 0.951864406779661, + "pure_math": { + "question": "Evaluate: 3999 + 1475 - 1546", + "options": [ + "3938", + "3918", + "3928", + "3948" + ], + "answer": "3928", + "correct_value": 3928, + "explanation": "B-C = -71; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9092 + 3990 - 6160", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9092 + 3990 - 6160", + "options": [ + "6922", + "6912", + "6942", + "6932" + ], + "answer": "6922", + "correct_value": 6922, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000589", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3427 + 1463 - 1462", + "ns_shortcut": "Near-cancellation: 1463 - 1462 = 1, so answer ≈ 3427 + 1.", + "shortcut_strength": 0.9993164730006835, + "pure_math": { + "question": "Evaluate: 3427 + 1463 - 1462", + "options": [ + "3448", + "3438", + "3428", + "3418" + ], + "answer": "3428", + "correct_value": 3428, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1490 + 8682 - 8264", + "ns_shortcut": "Near-cancellation: 8682 - 8264 = 418, offset is manageable.", + "shortcut_strength": 0.9518544114259387, + "pure_math": { + "question": "Evaluate: 1490 + 8682 - 8264", + "options": [ + "1898", + "1918", + "1928", + "1908" + ], + "answer": "1908", + "correct_value": 1908, + "explanation": "B-C = 418; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3427 + 1463 - 1922", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 3427 + 1463 - 1922", + "options": [ + "2968", + "2988", + "2958", + "2978" + ], + "answer": "2968", + "correct_value": 2968, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000590", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5536 + 1154 - 1155", + "ns_shortcut": "Near-cancellation: 1154 - 1155 = -1, so answer ≈ 5536 + -1.", + "shortcut_strength": 0.9991334488734835, + "pure_math": { + "question": "Evaluate: 5536 + 1154 - 1155", + "options": [ + "5545", + "5535", + "5525", + "5555" + ], + "answer": "5535", + "correct_value": 5535, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8316 + 7043 - 6769", + "ns_shortcut": "Near-cancellation: 7043 - 6769 = 274, offset is manageable.", + "shortcut_strength": 0.9610961238108761, + "pure_math": { + "question": "Evaluate: 8316 + 7043 - 6769", + "options": [ + "8590", + "8580", + "8610", + "8600" + ], + "answer": "8590", + "correct_value": 8590, + "explanation": "B-C = 274; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5536 + 1154 - 1589", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 5536 + 1154 - 1589", + "options": [ + "5101", + "5111", + "5091", + "5121" + ], + "answer": "5101", + "correct_value": 5101, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000591", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8528 + 3299 - 3298", + "ns_shortcut": "Near-cancellation: 3299 - 3298 = 1, so answer ≈ 8528 + 1.", + "shortcut_strength": 0.9996968778417702, + "pure_math": { + "question": "Evaluate: 8528 + 3299 - 3298", + "options": [ + "8519", + "8529", + "8539", + "8549" + ], + "answer": "8529", + "correct_value": 8529, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8085 + 6684 - 6834", + "ns_shortcut": "Near-cancellation: 6684 - 6834 = -150, offset is manageable.", + "shortcut_strength": 0.9775583482944344, + "pure_math": { + "question": "Evaluate: 8085 + 6684 - 6834", + "options": [ + "7955", + "7935", + "7925", + "7945" + ], + "answer": "7935", + "correct_value": 7935, + "explanation": "B-C = -150; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8528 + 3299 - 5326", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8528 + 3299 - 5326", + "options": [ + "6501", + "6491", + "6521", + "6511" + ], + "answer": "6501", + "correct_value": 6501, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000592", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6703 + 7620 - 7622", + "ns_shortcut": "Near-cancellation: 7620 - 7622 = -2, so answer ≈ 6703 + -2.", + "shortcut_strength": 0.9997375328083989, + "pure_math": { + "question": "Evaluate: 6703 + 7620 - 7622", + "options": [ + "6711", + "6721", + "6691", + "6701" + ], + "answer": "6701", + "correct_value": 6701, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4842 + 2423 - 2489", + "ns_shortcut": "Near-cancellation: 2423 - 2489 = -66, offset is manageable.", + "shortcut_strength": 0.9727610400330169, + "pure_math": { + "question": "Evaluate: 4842 + 2423 - 2489", + "options": [ + "4776", + "4766", + "4796", + "4786" + ], + "answer": "4776", + "correct_value": 4776, + "explanation": "B-C = -66; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6703 + 7620 - 4009", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 6703 + 7620 - 4009", + "options": [ + "10304", + "10334", + "10314", + "10324" + ], + "answer": "10314", + "correct_value": 10314, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000593", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7292 + 7887 - 7889", + "ns_shortcut": "Near-cancellation: 7887 - 7889 = -2, so answer ≈ 7292 + -2.", + "shortcut_strength": 0.99974641815646, + "pure_math": { + "question": "Evaluate: 7292 + 7887 - 7889", + "options": [ + "7280", + "7300", + "7310", + "7290" + ], + "answer": "7290", + "correct_value": 7290, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5325 + 4060 - 4169", + "ns_shortcut": "Near-cancellation: 4060 - 4169 = -109, offset is manageable.", + "shortcut_strength": 0.973152709359606, + "pure_math": { + "question": "Evaluate: 5325 + 4060 - 4169", + "options": [ + "5236", + "5206", + "5226", + "5216" + ], + "answer": "5216", + "correct_value": 5216, + "explanation": "B-C = -109; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7292 + 7887 - 3037", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7292 + 7887 - 3037", + "options": [ + "12162", + "12142", + "12132", + "12152" + ], + "answer": "12142", + "correct_value": 12142, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000594", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8364 + 7192 - 7193", + "ns_shortcut": "Near-cancellation: 7192 - 7193 = -1, so answer ≈ 8364 + -1.", + "shortcut_strength": 0.999860956618465, + "pure_math": { + "question": "Evaluate: 8364 + 7192 - 7193", + "options": [ + "8383", + "8353", + "8363", + "8373" + ], + "answer": "8363", + "correct_value": 8363, + "explanation": "B-C = -1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8255 + 6767 - 6544", + "ns_shortcut": "Near-cancellation: 6767 - 6544 = 223, offset is manageable.", + "shortcut_strength": 0.967045958327176, + "pure_math": { + "question": "Evaluate: 8255 + 6767 - 6544", + "options": [ + "8488", + "8498", + "8468", + "8478" + ], + "answer": "8478", + "correct_value": 8478, + "explanation": "B-C = 223; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8364 + 7192 - 10012", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 8364 + 7192 - 10012", + "options": [ + "5534", + "5564", + "5544", + "5554" + ], + "answer": "5544", + "correct_value": 5544, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000595", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2576 + 4339 - 4341", + "ns_shortcut": "Near-cancellation: 4339 - 4341 = -2, so answer ≈ 2576 + -2.", + "shortcut_strength": 0.9995390643005301, + "pure_math": { + "question": "Evaluate: 2576 + 4339 - 4341", + "options": [ + "2594", + "2564", + "2574", + "2584" + ], + "answer": "2574", + "correct_value": 2574, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8238 + 6243 - 6371", + "ns_shortcut": "Near-cancellation: 6243 - 6371 = -128, offset is manageable.", + "shortcut_strength": 0.9794970366810828, + "pure_math": { + "question": "Evaluate: 8238 + 6243 - 6371", + "options": [ + "8120", + "8100", + "8110", + "8130" + ], + "answer": "8110", + "correct_value": 8110, + "explanation": "B-C = -128; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2576 + 4339 - 7037", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2576 + 4339 - 7037", + "options": [ + "-108", + "-138", + "-122", + "-118" + ], + "answer": "-122", + "correct_value": -122, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000596", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9205 + 6773 - 6772", + "ns_shortcut": "Near-cancellation: 6773 - 6772 = 1, so answer ≈ 9205 + 1.", + "shortcut_strength": 0.9998523549387273, + "pure_math": { + "question": "Evaluate: 9205 + 6773 - 6772", + "options": [ + "9226", + "9196", + "9206", + "9216" + ], + "answer": "9206", + "correct_value": 9206, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8386 + 6423 - 6198", + "ns_shortcut": "Near-cancellation: 6423 - 6198 = 225, offset is manageable.", + "shortcut_strength": 0.9649696403549743, + "pure_math": { + "question": "Evaluate: 8386 + 6423 - 6198", + "options": [ + "8611", + "8631", + "8601", + "8621" + ], + "answer": "8611", + "correct_value": 8611, + "explanation": "B-C = 225; still a moderate shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9205 + 6773 - 9324", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 9205 + 6773 - 9324", + "options": [ + "6674", + "6664", + "6644", + "6654" + ], + "answer": "6654", + "correct_value": 6654, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "cancellation_identity_000597", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4777 + 1848 - 1847", + "ns_shortcut": "Near-cancellation: 1848 - 1847 = 1, so answer ≈ 4777 + 1.", + "shortcut_strength": 0.9994588744588745, + "pure_math": { + "question": "Evaluate: 4777 + 1848 - 1847", + "options": [ + "4798", + "4788", + "4768", + "4778" + ], + "answer": "4778", + "correct_value": 4778, + "explanation": "B-C = 1; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4607 + 4868 - 4673", + "ns_shortcut": "Near-cancellation: 4868 - 4673 = 195, offset is manageable.", + "shortcut_strength": 0.9599424815119145, + "pure_math": { + "question": "Evaluate: 4607 + 4868 - 4673", + "options": [ + "4792", + "4802", + "4812", + "4822" + ], + "answer": "4802", + "correct_value": 4802, + "explanation": "B-C = 195; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4777 + 1848 - 1000", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 4777 + 1848 - 1000", + "options": [ + "5625", + "5615", + "5645", + "5635" + ], + "answer": "5625", + "correct_value": 5625, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000598", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7540 + 1933 - 1935", + "ns_shortcut": "Near-cancellation: 1933 - 1935 = -2, so answer ≈ 7540 + -2.", + "shortcut_strength": 0.9989653388515262, + "pure_math": { + "question": "Evaluate: 7540 + 1933 - 1935", + "options": [ + "7558", + "7538", + "7548", + "7528" + ], + "answer": "7538", + "correct_value": 7538, + "explanation": "B-C = -2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7167 + 4130 - 4255", + "ns_shortcut": "Near-cancellation: 4130 - 4255 = -125, offset is manageable.", + "shortcut_strength": 0.9697336561743342, + "pure_math": { + "question": "Evaluate: 7167 + 4130 - 4255", + "options": [ + "7032", + "7062", + "7042", + "7052" + ], + "answer": "7042", + "correct_value": 7042, + "explanation": "B-C = -125; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7540 + 1933 - 1012", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 7540 + 1933 - 1012", + "options": [ + "8481", + "8451", + "8461", + "8471" + ], + "answer": "8461", + "correct_value": 8461, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000599", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2021 + 4865 - 4863", + "ns_shortcut": "Near-cancellation: 4865 - 4863 = 2, so answer ≈ 2021 + 2.", + "shortcut_strength": 0.9995889003083248, + "pure_math": { + "question": "Evaluate: 2021 + 4865 - 4863", + "options": [ + "2033", + "2013", + "2043", + "2023" + ], + "answer": "2023", + "correct_value": 2023, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7016 + 6649 - 6426", + "ns_shortcut": "Near-cancellation: 6649 - 6426 = 223, offset is manageable.", + "shortcut_strength": 0.9664611219732291, + "pure_math": { + "question": "Evaluate: 7016 + 6649 - 6426", + "options": [ + "7249", + "7259", + "7229", + "7239" + ], + "answer": "7239", + "correct_value": 7239, + "explanation": "B-C = 223; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2021 + 4865 - 7453", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2021 + 4865 - 7453", + "options": [ + "-553", + "-567", + "-583", + "-563" + ], + "answer": "-567", + "correct_value": -567, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "cancellation_identity_000600", + "category": "cancellation_identity", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2704 + 1588 - 1586", + "ns_shortcut": "Near-cancellation: 1588 - 1586 = 2, so answer ≈ 2704 + 2.", + "shortcut_strength": 0.9987405541561712, + "pure_math": { + "question": "Evaluate: 2704 + 1588 - 1586", + "options": [ + "2706", + "2696", + "2726", + "2716" + ], + "answer": "2706", + "correct_value": 2706, + "explanation": "B-C = 2; result ≈ A + offset." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7101 + 8902 - 9105", + "ns_shortcut": "Near-cancellation: 8902 - 9105 = -203, offset is manageable.", + "shortcut_strength": 0.9771961356998428, + "pure_math": { + "question": "Evaluate: 7101 + 8902 - 9105", + "options": [ + "6908", + "6898", + "6888", + "6918" + ], + "answer": "6898", + "correct_value": 6898, + "explanation": "B-C = -203; still a moderate shortcut." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2704 + 1588 - 1056", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Evaluate: 2704 + 1588 - 1056", + "options": [ + "3236", + "3226", + "3256", + "3246" + ], + "answer": "3236", + "correct_value": 3236, + "explanation": "Large gap between B and C; no cancellation shortcut." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000602", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2520 * 3783", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.968875, + "pure_math": { + "question": "Estimate 2520 × 3783.", + "options": [ + "9533170", + "9533150", + "9533180", + "9533160" + ], + "answer": "9533160", + "correct_value": 9533160, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2497 * 3111", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.888275, + "pure_math": { + "question": "Estimate 2497 × 3111.", + "options": [ + "7768177", + "7768167", + "7768157", + "7768187" + ], + "answer": "7768167", + "correct_value": 7768167, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9467 * 3526", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9467 × 3526.", + "options": [ + "33380642", + "33380662", + "33380652", + "33380632" + ], + "answer": "33380642", + "correct_value": 33380642, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000603", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2520 * 3783", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.968875, + "pure_math": { + "question": "Estimate 2520 × 3783.", + "options": [ + "9533170", + "9533150", + "9533180", + "9533160" + ], + "answer": "9533160", + "correct_value": 9533160, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2497 * 3111", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.888275, + "pure_math": { + "question": "Estimate 2497 × 3111.", + "options": [ + "7768177", + "7768167", + "7768157", + "7768187" + ], + "answer": "7768167", + "correct_value": 7768167, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9467 * 3526", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9467 × 3526.", + "options": [ + "33380642", + "33380662", + "33380652", + "33380632" + ], + "answer": "33380642", + "correct_value": 33380642, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000604", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5081 * 2047", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.98015, + "pure_math": { + "question": "Estimate 5081 × 2047.", + "options": [ + "10400807", + "10400817", + "10400827", + "10400797" + ], + "answer": "10400807", + "correct_value": 10400807, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5087 * 1499", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.86605, + "pure_math": { + "question": "Estimate 5087 × 1499.", + "options": [ + "7625403", + "7625433", + "7625423", + "7625413" + ], + "answer": "7625413", + "correct_value": 7625413, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5858 * 5856", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5858 × 5856.", + "options": [ + "34304448", + "34304438", + "34304458", + "34304468" + ], + "answer": "34304448", + "correct_value": 34304448, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000605", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1274 * 8785", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9413375, + "pure_math": { + "question": "Estimate 1274 × 8785.", + "options": [ + "11192110", + "11192090", + "11192080", + "11192100" + ], + "answer": "11192090", + "correct_value": 11192090, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1273 * 5384", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8272999999999999, + "pure_math": { + "question": "Estimate 1273 × 5384.", + "options": [ + "6853832", + "6853842", + "6853852", + "6853822" + ], + "answer": "6853832", + "correct_value": 6853832, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3736 * 9032", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3736 × 9032.", + "options": [ + "33743572", + "33743562", + "33743542", + "33743552" + ], + "answer": "33743552", + "correct_value": 33743552, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000606", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5081 * 2047", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.98015, + "pure_math": { + "question": "Estimate 5081 × 2047.", + "options": [ + "10400807", + "10400817", + "10400827", + "10400797" + ], + "answer": "10400807", + "correct_value": 10400807, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5087 * 1499", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.86605, + "pure_math": { + "question": "Estimate 5087 × 1499.", + "options": [ + "7625403", + "7625433", + "7625423", + "7625413" + ], + "answer": "7625413", + "correct_value": 7625413, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5858 * 5856", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5858 × 5856.", + "options": [ + "34304448", + "34304438", + "34304458", + "34304468" + ], + "answer": "34304448", + "correct_value": 34304448, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000607", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1274 * 8785", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9413375, + "pure_math": { + "question": "Estimate 1274 × 8785.", + "options": [ + "11192110", + "11192090", + "11192080", + "11192100" + ], + "answer": "11192090", + "correct_value": 11192090, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1273 * 5384", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8272999999999999, + "pure_math": { + "question": "Estimate 1273 × 5384.", + "options": [ + "6853832", + "6853842", + "6853852", + "6853822" + ], + "answer": "6853832", + "correct_value": 6853832, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3736 * 9032", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3736 × 9032.", + "options": [ + "33743572", + "33743562", + "33743542", + "33743552" + ], + "answer": "33743552", + "correct_value": 33743552, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000608", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5009 * 1968", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.9911, + "pure_math": { + "question": "Estimate 5009 × 1968.", + "options": [ + "9857722", + "9857702", + "9857732", + "9857712" + ], + "answer": "9857712", + "correct_value": 9857712, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5065 * 1359", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83325, + "pure_math": { + "question": "Estimate 5065 × 1359.", + "options": [ + "6883335", + "6883325", + "6883345", + "6883355" + ], + "answer": "6883335", + "correct_value": 6883335, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9859 * 8865", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9859 × 8865.", + "options": [ + "87400035", + "87400045", + "87400055", + "87400025" + ], + "answer": "87400035", + "correct_value": 87400035, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000609", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5009 * 1968", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.9911, + "pure_math": { + "question": "Estimate 5009 × 1968.", + "options": [ + "9857722", + "9857702", + "9857732", + "9857712" + ], + "answer": "9857712", + "correct_value": 9857712, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5065 * 1359", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83325, + "pure_math": { + "question": "Estimate 5065 × 1359.", + "options": [ + "6883335", + "6883325", + "6883345", + "6883355" + ], + "answer": "6883335", + "correct_value": 6883335, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9859 * 8865", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9859 × 8865.", + "options": [ + "87400035", + "87400045", + "87400055", + "87400025" + ], + "answer": "87400035", + "correct_value": 87400035, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000610", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1273 * 7425", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9548625, + "pure_math": { + "question": "Estimate 1273 × 7425.", + "options": [ + "9452045", + "9452015", + "9452025", + "9452035" + ], + "answer": "9452025", + "correct_value": 9452025, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1226 * 9734", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8820250000000001, + "pure_math": { + "question": "Estimate 1226 × 9734.", + "options": [ + "11933904", + "11933894", + "11933874", + "11933884" + ], + "answer": "11933884", + "correct_value": 11933884, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8575 * 1527", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8575 × 1527.", + "options": [ + "13094025", + "13094015", + "13094035", + "13094045" + ], + "answer": "13094025", + "correct_value": 13094025, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000611", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1273 * 7425", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9548625, + "pure_math": { + "question": "Estimate 1273 × 7425.", + "options": [ + "9452045", + "9452015", + "9452025", + "9452035" + ], + "answer": "9452025", + "correct_value": 9452025, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1226 * 9734", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8820250000000001, + "pure_math": { + "question": "Estimate 1226 × 9734.", + "options": [ + "11933904", + "11933894", + "11933874", + "11933884" + ], + "answer": "11933884", + "correct_value": 11933884, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8575 * 1527", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8575 × 1527.", + "options": [ + "13094025", + "13094015", + "13094035", + "13094045" + ], + "answer": "13094025", + "correct_value": 13094025, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000612", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1233 * 7868", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.98495, + "pure_math": { + "question": "Estimate 1233 × 7868.", + "options": [ + "9701264", + "9701244", + "9701254", + "9701234" + ], + "answer": "9701244", + "correct_value": 9701244, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1258 * 5343", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8307375, + "pure_math": { + "question": "Estimate 1258 × 5343.", + "options": [ + "6721484", + "6721504", + "6721494", + "6721514" + ], + "answer": "6721494", + "correct_value": 6721494, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8725 * 6136", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8725 × 6136.", + "options": [ + "53536610", + "53536620", + "53536590", + "53536600" + ], + "answer": "53536600", + "correct_value": 53536600, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000613", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2540 * 3676", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.9515, + "pure_math": { + "question": "Estimate 2540 × 3676.", + "options": [ + "9337060", + "9337040", + "9337050", + "9337030" + ], + "answer": "9337040", + "correct_value": 9337040, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2508 * 3039", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.878275, + "pure_math": { + "question": "Estimate 2508 × 3039.", + "options": [ + "7621822", + "7621832", + "7621802", + "7621812" + ], + "answer": "7621812", + "correct_value": 7621812, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5945 * 2628", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5945 × 2628.", + "options": [ + "15623450", + "15623480", + "15623470", + "15623460" + ], + "answer": "15623460", + "correct_value": 15623460, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000614", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1274 * 8677", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9480875, + "pure_math": { + "question": "Estimate 1274 × 8677.", + "options": [ + "11054488", + "11054518", + "11054498", + "11054508" + ], + "answer": "11054498", + "correct_value": 11054498, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1239 * 9766", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.885225, + "pure_math": { + "question": "Estimate 1239 × 9766.", + "options": [ + "12100084", + "12100094", + "12100074", + "12100064" + ], + "answer": "12100074", + "correct_value": 12100074, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7143 * 4661", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 7143 × 4661.", + "options": [ + "33293513", + "33293543", + "33293523", + "33293533" + ], + "answer": "33293523", + "correct_value": 33293523, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000615", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2539 * 3729", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.958325, + "pure_math": { + "question": "Estimate 2539 × 3729.", + "options": [ + "9467951", + "9467941", + "9467921", + "9467931" + ], + "answer": "9467931", + "correct_value": 9467931, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2518 * 5231", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.842525, + "pure_math": { + "question": "Estimate 2518 × 5231.", + "options": [ + "13171668", + "13171648", + "13171678", + "13171658" + ], + "answer": "13171658", + "correct_value": 13171658, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5669 * 4738", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5669 × 4738.", + "options": [ + "26859742", + "26859722", + "26859712", + "26859732" + ], + "answer": "26859722", + "correct_value": 26859722, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000616", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1233 * 7868", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.98495, + "pure_math": { + "question": "Estimate 1233 × 7868.", + "options": [ + "9701264", + "9701244", + "9701254", + "9701234" + ], + "answer": "9701244", + "correct_value": 9701244, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1258 * 5343", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8307375, + "pure_math": { + "question": "Estimate 1258 × 5343.", + "options": [ + "6721484", + "6721504", + "6721494", + "6721514" + ], + "answer": "6721494", + "correct_value": 6721494, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8725 * 6136", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8725 × 6136.", + "options": [ + "53536610", + "53536620", + "53536590", + "53536600" + ], + "answer": "53536600", + "correct_value": 53536600, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000617", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4966 * 1988", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.9936, + "pure_math": { + "question": "Estimate 4966 × 1988.", + "options": [ + "9872408", + "9872398", + "9872428", + "9872418" + ], + "answer": "9872408", + "correct_value": 9872408, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4954 * 1446", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8569, + "pure_math": { + "question": "Estimate 4954 × 1446.", + "options": [ + "7163504", + "7163494", + "7163474", + "7163484" + ], + "answer": "7163484", + "correct_value": 7163484, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6863 * 3034", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6863 × 3034.", + "options": [ + "20822342", + "20822362", + "20822352", + "20822332" + ], + "answer": "20822342", + "correct_value": 20822342, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000618", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2540 * 3676", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.9515, + "pure_math": { + "question": "Estimate 2540 × 3676.", + "options": [ + "9337060", + "9337040", + "9337050", + "9337030" + ], + "answer": "9337040", + "correct_value": 9337040, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2508 * 3039", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.878275, + "pure_math": { + "question": "Estimate 2508 × 3039.", + "options": [ + "7621822", + "7621832", + "7621802", + "7621812" + ], + "answer": "7621812", + "correct_value": 7621812, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5945 * 2628", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5945 × 2628.", + "options": [ + "15623450", + "15623480", + "15623470", + "15623460" + ], + "answer": "15623460", + "correct_value": 15623460, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000619", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1274 * 8677", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9480875, + "pure_math": { + "question": "Estimate 1274 × 8677.", + "options": [ + "11054488", + "11054518", + "11054498", + "11054508" + ], + "answer": "11054498", + "correct_value": 11054498, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1239 * 9766", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.885225, + "pure_math": { + "question": "Estimate 1239 × 9766.", + "options": [ + "12100084", + "12100094", + "12100074", + "12100064" + ], + "answer": "12100074", + "correct_value": 12100074, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7143 * 4661", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 7143 × 4661.", + "options": [ + "33293513", + "33293543", + "33293523", + "33293533" + ], + "answer": "33293523", + "correct_value": 33293523, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000620", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2539 * 3729", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.958325, + "pure_math": { + "question": "Estimate 2539 × 3729.", + "options": [ + "9467951", + "9467941", + "9467921", + "9467931" + ], + "answer": "9467931", + "correct_value": 9467931, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2518 * 5231", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.842525, + "pure_math": { + "question": "Estimate 2518 × 5231.", + "options": [ + "13171668", + "13171648", + "13171678", + "13171658" + ], + "answer": "13171658", + "correct_value": 13171658, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5669 * 4738", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5669 × 4738.", + "options": [ + "26859742", + "26859722", + "26859712", + "26859732" + ], + "answer": "26859722", + "correct_value": 26859722, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000621", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2510 * 4386", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.94975, + "pure_math": { + "question": "Estimate 2510 × 4386.", + "options": [ + "11008860", + "11008850", + "11008870", + "11008880" + ], + "answer": "11008860", + "correct_value": 11008860, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2463 * 5175", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8457250000000001, + "pure_math": { + "question": "Estimate 2463 × 5175.", + "options": [ + "12746045", + "12746035", + "12746015", + "12746025" + ], + "answer": "12746025", + "correct_value": 12746025, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2846 * 5764", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2846 × 5764.", + "options": [ + "16404344", + "16404364", + "16404334", + "16404354" + ], + "answer": "16404344", + "correct_value": 16404344, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000622", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4910 * 1977", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.98525, + "pure_math": { + "question": "Estimate 4910 × 1977.", + "options": [ + "9707060", + "9707080", + "9707090", + "9707070" + ], + "answer": "9707070", + "correct_value": 9707070, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4955 * 2641", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83525, + "pure_math": { + "question": "Estimate 4955 × 2641.", + "options": [ + "13086175", + "13086165", + "13086155", + "13086145" + ], + "answer": "13086155", + "correct_value": 13086155, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1168 * 6839", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1168 × 6839.", + "options": [ + "7987952", + "7987942", + "7987972", + "7987962" + ], + "answer": "7987952", + "correct_value": 7987952, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000623", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4966 * 1988", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.9936, + "pure_math": { + "question": "Estimate 4966 × 1988.", + "options": [ + "9872408", + "9872398", + "9872428", + "9872418" + ], + "answer": "9872408", + "correct_value": 9872408, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4954 * 1446", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8569, + "pure_math": { + "question": "Estimate 4954 × 1446.", + "options": [ + "7163504", + "7163494", + "7163474", + "7163484" + ], + "answer": "7163484", + "correct_value": 7163484, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6863 * 3034", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6863 × 3034.", + "options": [ + "20822342", + "20822362", + "20822352", + "20822332" + ], + "answer": "20822342", + "correct_value": 20822342, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000624", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2510 * 4386", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.94975, + "pure_math": { + "question": "Estimate 2510 × 4386.", + "options": [ + "11008860", + "11008850", + "11008870", + "11008880" + ], + "answer": "11008860", + "correct_value": 11008860, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2463 * 5175", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8457250000000001, + "pure_math": { + "question": "Estimate 2463 × 5175.", + "options": [ + "12746045", + "12746035", + "12746015", + "12746025" + ], + "answer": "12746025", + "correct_value": 12746025, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2846 * 5764", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2846 × 5764.", + "options": [ + "16404344", + "16404364", + "16404334", + "16404354" + ], + "answer": "16404344", + "correct_value": 16404344, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000625", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4910 * 1977", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.98525, + "pure_math": { + "question": "Estimate 4910 × 1977.", + "options": [ + "9707060", + "9707080", + "9707090", + "9707070" + ], + "answer": "9707070", + "correct_value": 9707070, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4955 * 2641", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83525, + "pure_math": { + "question": "Estimate 4955 × 2641.", + "options": [ + "13086175", + "13086165", + "13086155", + "13086145" + ], + "answer": "13086155", + "correct_value": 13086155, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1168 * 6839", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1168 × 6839.", + "options": [ + "7987952", + "7987942", + "7987972", + "7987962" + ], + "answer": "7987952", + "correct_value": 7987952, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000626", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1227 * 7737", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9743625, + "pure_math": { + "question": "Estimate 1227 × 7737.", + "options": [ + "9493309", + "9493299", + "9493319", + "9493289" + ], + "answer": "9493299", + "correct_value": 9493299, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1229 * 10560", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8316, + "pure_math": { + "question": "Estimate 1229 × 10560.", + "options": [ + "12978250", + "12978240", + "12978230", + "12978260" + ], + "answer": "12978240", + "correct_value": 12978240, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9233 * 1353", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9233 × 1353.", + "options": [ + "12492239", + "12492259", + "12492269", + "12492249" + ], + "answer": "12492249", + "correct_value": 12492249, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000627", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1227 * 7737", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9743625, + "pure_math": { + "question": "Estimate 1227 × 7737.", + "options": [ + "9493309", + "9493299", + "9493319", + "9493289" + ], + "answer": "9493299", + "correct_value": 9493299, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1229 * 10560", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8316, + "pure_math": { + "question": "Estimate 1229 × 10560.", + "options": [ + "12978250", + "12978240", + "12978230", + "12978260" + ], + "answer": "12978240", + "correct_value": 12978240, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9233 * 1353", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9233 × 1353.", + "options": [ + "12492239", + "12492259", + "12492269", + "12492249" + ], + "answer": "12492249", + "correct_value": 12492249, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000628", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1248 * 7302", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.955575, + "pure_math": { + "question": "Estimate 1248 × 7302.", + "options": [ + "9112886", + "9112916", + "9112896", + "9112906" + ], + "answer": "9112896", + "correct_value": 9112896, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1262 * 10295", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8517625, + "pure_math": { + "question": "Estimate 1262 × 10295.", + "options": [ + "12992300", + "12992280", + "12992290", + "12992310" + ], + "answer": "12992290", + "correct_value": 12992290, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6542 * 1175", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6542 × 1175.", + "options": [ + "7686860", + "7686840", + "7686850", + "7686870" + ], + "answer": "7686850", + "correct_value": 7686850, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000629", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1248 * 7302", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.955575, + "pure_math": { + "question": "Estimate 1248 × 7302.", + "options": [ + "9112886", + "9112916", + "9112896", + "9112906" + ], + "answer": "9112896", + "correct_value": 9112896, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1262 * 10295", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8517625, + "pure_math": { + "question": "Estimate 1262 × 10295.", + "options": [ + "12992300", + "12992280", + "12992290", + "12992310" + ], + "answer": "12992290", + "correct_value": 12992290, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6542 * 1175", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 6542 × 1175.", + "options": [ + "7686860", + "7686840", + "7686850", + "7686870" + ], + "answer": "7686850", + "correct_value": 7686850, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000630", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5090 * 1947", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.97775, + "pure_math": { + "question": "Estimate 5090 × 1947.", + "options": [ + "9910230", + "9910250", + "9910220", + "9910240" + ], + "answer": "9910230", + "correct_value": 9910230, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4989 * 1489", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87115, + "pure_math": { + "question": "Estimate 4989 × 1489.", + "options": [ + "7428621", + "7428631", + "7428611", + "7428641" + ], + "answer": "7428621", + "correct_value": 7428621, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4544 * 1768", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 4544 × 1768.", + "options": [ + "8033802", + "8033812", + "8033782", + "8033792" + ], + "answer": "8033792", + "correct_value": 8033792, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000631", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5090 * 1947", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.97775, + "pure_math": { + "question": "Estimate 5090 × 1947.", + "options": [ + "9910230", + "9910250", + "9910220", + "9910240" + ], + "answer": "9910230", + "correct_value": 9910230, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4989 * 1489", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87115, + "pure_math": { + "question": "Estimate 4989 × 1489.", + "options": [ + "7428621", + "7428631", + "7428611", + "7428641" + ], + "answer": "7428621", + "correct_value": 7428621, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4544 * 1768", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 4544 × 1768.", + "options": [ + "8033802", + "8033812", + "8033782", + "8033792" + ], + "answer": "8033792", + "correct_value": 8033792, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000632", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2472 * 3887", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.980275, + "pure_math": { + "question": "Estimate 2472 × 3887.", + "options": [ + "9608664", + "9608684", + "9608674", + "9608654" + ], + "answer": "9608664", + "correct_value": 9608664, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2504 * 5135", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.857325, + "pure_math": { + "question": "Estimate 2504 × 5135.", + "options": [ + "12858040", + "12858050", + "12858060", + "12858030" + ], + "answer": "12858040", + "correct_value": 12858040, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1859 * 8461", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1859 × 8461.", + "options": [ + "15728989", + "15729009", + "15729019", + "15728999" + ], + "answer": "15728999", + "correct_value": 15728999, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000633", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4960 * 2196", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.947, + "pure_math": { + "question": "Estimate 4960 × 2196.", + "options": [ + "10892170", + "10892180", + "10892160", + "10892150" + ], + "answer": "10892160", + "correct_value": 10892160, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4999 * 1587", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.89665, + "pure_math": { + "question": "Estimate 4999 × 1587.", + "options": [ + "7933413", + "7933423", + "7933403", + "7933433" + ], + "answer": "7933413", + "correct_value": 7933413, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9047 * 9028", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9047 × 9028.", + "options": [ + "81676306", + "81676316", + "81676336", + "81676326" + ], + "answer": "81676316", + "correct_value": 81676316, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000634", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1228 * 7243", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9438875, + "pure_math": { + "question": "Estimate 1228 × 7243.", + "options": [ + "8894414", + "8894404", + "8894394", + "8894424" + ], + "answer": "8894404", + "correct_value": 8894404, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1250 * 10446", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.847125, + "pure_math": { + "question": "Estimate 1250 × 10446.", + "options": [ + "13057500", + "13057490", + "13057520", + "13057510" + ], + "answer": "13057500", + "correct_value": 13057500, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1554 * 1462", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1554 × 1462.", + "options": [ + "2271938", + "2271948", + "2271958", + "2271968" + ], + "answer": "2271948", + "correct_value": 2271948, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000635", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5060 * 1868", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.961, + "pure_math": { + "question": "Estimate 5060 × 1868.", + "options": [ + "9452090", + "9452070", + "9452080", + "9452100" + ], + "answer": "9452080", + "correct_value": 9452080, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5046 * 1382", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8409, + "pure_math": { + "question": "Estimate 5046 × 1382.", + "options": [ + "6973572", + "6973562", + "6973592", + "6973582" + ], + "answer": "6973572", + "correct_value": 6973572, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2928 * 5764", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2928 × 5764.", + "options": [ + "16877012", + "16876982", + "16876992", + "16877002" + ], + "answer": "16876992", + "correct_value": 16876992, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000636", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1274 * 7201", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9404625, + "pure_math": { + "question": "Estimate 1274 × 7201.", + "options": [ + "9174094", + "9174074", + "9174084", + "9174064" + ], + "answer": "9174074", + "correct_value": 9174074, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1262 * 9859", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8790125, + "pure_math": { + "question": "Estimate 1262 × 9859.", + "options": [ + "12442068", + "12442048", + "12442058", + "12442078" + ], + "answer": "12442058", + "correct_value": 12442058, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3753 * 5259", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3753 × 5259.", + "options": [ + "19737037", + "19737027", + "19737017", + "19737047" + ], + "answer": "19737027", + "correct_value": 19737027, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000637", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2472 * 3887", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.980275, + "pure_math": { + "question": "Estimate 2472 × 3887.", + "options": [ + "9608664", + "9608684", + "9608674", + "9608654" + ], + "answer": "9608664", + "correct_value": 9608664, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2504 * 5135", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.857325, + "pure_math": { + "question": "Estimate 2504 × 5135.", + "options": [ + "12858040", + "12858050", + "12858060", + "12858030" + ], + "answer": "12858040", + "correct_value": 12858040, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1859 * 8461", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1859 × 8461.", + "options": [ + "15728989", + "15729009", + "15729019", + "15728999" + ], + "answer": "15728999", + "correct_value": 15728999, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000638", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2548 * 3929", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.981525, + "pure_math": { + "question": "Estimate 2548 × 3929.", + "options": [ + "10011112", + "10011092", + "10011082", + "10011102" + ], + "answer": "10011092", + "correct_value": 10011092, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2472 * 2853", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.851025, + "pure_math": { + "question": "Estimate 2472 × 2853.", + "options": [ + "7052626", + "7052616", + "7052606", + "7052636" + ], + "answer": "7052616", + "correct_value": 7052616, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8668 * 1774", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8668 × 1774.", + "options": [ + "15377052", + "15377032", + "15377042", + "15377022" + ], + "answer": "15377032", + "correct_value": 15377032, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000639", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4960 * 2196", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.947, + "pure_math": { + "question": "Estimate 4960 × 2196.", + "options": [ + "10892170", + "10892180", + "10892160", + "10892150" + ], + "answer": "10892160", + "correct_value": 10892160, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4999 * 1587", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.89665, + "pure_math": { + "question": "Estimate 4999 × 1587.", + "options": [ + "7933413", + "7933423", + "7933403", + "7933433" + ], + "answer": "7933413", + "correct_value": 7933413, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9047 * 9028", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9047 × 9028.", + "options": [ + "81676306", + "81676316", + "81676336", + "81676326" + ], + "answer": "81676316", + "correct_value": 81676316, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000640", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1228 * 7243", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9438875, + "pure_math": { + "question": "Estimate 1228 × 7243.", + "options": [ + "8894414", + "8894404", + "8894394", + "8894424" + ], + "answer": "8894404", + "correct_value": 8894404, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1250 * 10446", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.847125, + "pure_math": { + "question": "Estimate 1250 × 10446.", + "options": [ + "13057500", + "13057490", + "13057520", + "13057510" + ], + "answer": "13057500", + "correct_value": 13057500, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1554 * 1462", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1554 × 1462.", + "options": [ + "2271938", + "2271948", + "2271958", + "2271968" + ], + "answer": "2271948", + "correct_value": 2271948, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000641", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1231 * 8094", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.986525, + "pure_math": { + "question": "Estimate 1231 × 8094.", + "options": [ + "9963724", + "9963714", + "9963704", + "9963734" + ], + "answer": "9963714", + "correct_value": 9963714, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1258 * 5897", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8653625, + "pure_math": { + "question": "Estimate 1258 × 5897.", + "options": [ + "7418436", + "7418426", + "7418446", + "7418416" + ], + "answer": "7418426", + "correct_value": 7418426, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7031 * 2366", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 7031 × 2366.", + "options": [ + "16635356", + "16635336", + "16635346", + "16635366" + ], + "answer": "16635346", + "correct_value": 16635346, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000642", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5082 * 1800", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.9418, + "pure_math": { + "question": "Estimate 5082 × 1800.", + "options": [ + "9147610", + "9147590", + "9147620", + "9147600" + ], + "answer": "9147600", + "correct_value": 9147600, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4938 * 2655", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83005, + "pure_math": { + "question": "Estimate 4938 × 2655.", + "options": [ + "13110400", + "13110380", + "13110390", + "13110410" + ], + "answer": "13110390", + "correct_value": 13110390, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4534 * 9665", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 4534 × 9665.", + "options": [ + "43821110", + "43821120", + "43821100", + "43821130" + ], + "answer": "43821110", + "correct_value": 43821110, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000643", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5060 * 1868", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.961, + "pure_math": { + "question": "Estimate 5060 × 1868.", + "options": [ + "9452090", + "9452070", + "9452080", + "9452100" + ], + "answer": "9452080", + "correct_value": 9452080, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5046 * 1382", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8409, + "pure_math": { + "question": "Estimate 5046 × 1382.", + "options": [ + "6973572", + "6973562", + "6973592", + "6973582" + ], + "answer": "6973572", + "correct_value": 6973572, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2928 * 5764", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 2928 × 5764.", + "options": [ + "16877012", + "16876982", + "16876992", + "16877002" + ], + "answer": "16876992", + "correct_value": 16876992, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000644", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2510 * 3724", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.9635, + "pure_math": { + "question": "Estimate 2510 × 3724.", + "options": [ + "9347260", + "9347250", + "9347230", + "9347240" + ], + "answer": "9347240", + "correct_value": 9347240, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2451 * 2846", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.84595, + "pure_math": { + "question": "Estimate 2451 × 2846.", + "options": [ + "6975546", + "6975556", + "6975566", + "6975536" + ], + "answer": "6975546", + "correct_value": 6975546, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4235 * 1437", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 4235 × 1437.", + "options": [ + "6085685", + "6085715", + "6085695", + "6085705" + ], + "answer": "6085695", + "correct_value": 6085695, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000645", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1274 * 7201", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.9404625, + "pure_math": { + "question": "Estimate 1274 × 7201.", + "options": [ + "9174094", + "9174074", + "9174084", + "9174064" + ], + "answer": "9174074", + "correct_value": 9174074, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1262 * 9859", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8790125, + "pure_math": { + "question": "Estimate 1262 × 9859.", + "options": [ + "12442068", + "12442048", + "12442058", + "12442078" + ], + "answer": "12442058", + "correct_value": 12442058, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3753 * 5259", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 3753 × 5259.", + "options": [ + "19737037", + "19737027", + "19737017", + "19737047" + ], + "answer": "19737027", + "correct_value": 19737027, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000646", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2473 * 4227", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.966225, + "pure_math": { + "question": "Estimate 2473 × 4227.", + "options": [ + "10453371", + "10453391", + "10453381", + "10453361" + ], + "answer": "10453371", + "correct_value": 10453371, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2526 * 2722", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83505, + "pure_math": { + "question": "Estimate 2526 × 2722.", + "options": [ + "6875782", + "6875762", + "6875792", + "6875772" + ], + "answer": "6875772", + "correct_value": 6875772, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5568 * 1443", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5568 × 1443.", + "options": [ + "8034634", + "8034644", + "8034624", + "8034614" + ], + "answer": "8034624", + "correct_value": 8034624, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000647", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2530 * 3992", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.993, + "pure_math": { + "question": "Estimate 2530 × 3992.", + "options": [ + "10099760", + "10099770", + "10099750", + "10099780" + ], + "answer": "10099760", + "correct_value": 10099760, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2470 * 5025", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.865875, + "pure_math": { + "question": "Estimate 2470 × 5025.", + "options": [ + "12411760", + "12411750", + "12411770", + "12411740" + ], + "answer": "12411750", + "correct_value": 12411750, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4536 * 7228", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 4536 × 7228.", + "options": [ + "32786218", + "32786228", + "32786198", + "32786208" + ], + "answer": "32786208", + "correct_value": 32786208, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000648", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2520 * 3623", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.948875, + "pure_math": { + "question": "Estimate 2520 × 3623.", + "options": [ + "9129950", + "9129980", + "9129970", + "9129960" + ], + "answer": "9129960", + "correct_value": 9129960, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2530 * 4978", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.87175, + "pure_math": { + "question": "Estimate 2530 × 4978.", + "options": [ + "12594340", + "12594330", + "12594360", + "12594350" + ], + "answer": "12594340", + "correct_value": 12594340, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1446 * 1354", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 1446 × 1354.", + "options": [ + "1957884", + "1957904", + "1957894", + "1957874" + ], + "answer": "1957884", + "correct_value": 1957884, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000649", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1266 * 7268", + "ns_shortcut": "Round to 1250×8000 for easy mental computation.", + "shortcut_strength": 0.94785, + "pure_math": { + "question": "Estimate 1266 × 7268.", + "options": [ + "9201308", + "9201288", + "9201278", + "9201298" + ], + "answer": "9201288", + "correct_value": 9201288, + "explanation": "Round to compatible pair 1250×8000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1255 * 10479", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.8430625, + "pure_math": { + "question": "Estimate 1255 × 10479.", + "options": [ + "13151155", + "13151145", + "13151165", + "13151135" + ], + "answer": "13151145", + "correct_value": 13151145, + "explanation": "One number rounds to 1250, other is farther from 8000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5352 * 4434", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 5352 × 4434.", + "options": [ + "23730758", + "23730788", + "23730778", + "23730768" + ], + "answer": "23730768", + "correct_value": 23730768, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "compatible_numbers_000650", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4925 * 2150", + "ns_shortcut": "Round to 5000×2000 for easy mental computation.", + "shortcut_strength": 0.955, + "pure_math": { + "question": "Estimate 4925 × 2150.", + "options": [ + "10588770", + "10588760", + "10588740", + "10588750" + ], + "answer": "10588750", + "correct_value": 10588750, + "explanation": "Round to compatible pair 5000×2000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5002 * 2659", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.83505, + "pure_math": { + "question": "Estimate 5002 × 2659.", + "options": [ + "13300338", + "13300328", + "13300318", + "13300308" + ], + "answer": "13300318", + "correct_value": 13300318, + "explanation": "One number rounds to 5000, other is farther from 2000." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9874 * 9535", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 9874 × 9535.", + "options": [ + "94148600", + "94148590", + "94148580", + "94148610" + ], + "answer": "94148590", + "correct_value": 94148590, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "compatible_numbers_000651", + "category": "compatible_numbers", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2548 * 3929", + "ns_shortcut": "Round to 2500×4000 for easy mental computation.", + "shortcut_strength": 0.981525, + "pure_math": { + "question": "Estimate 2548 × 3929.", + "options": [ + "10011112", + "10011092", + "10011082", + "10011102" + ], + "answer": "10011092", + "correct_value": 10011092, + "explanation": "Round to compatible pair 2500×4000 = 10000000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2472 * 2853", + "ns_shortcut": "Compatible pair shortcut works but less precise.", + "shortcut_strength": 0.851025, + "pure_math": { + "question": "Estimate 2472 × 2853.", + "options": [ + "7052626", + "7052616", + "7052606", + "7052636" + ], + "answer": "7052616", + "correct_value": 7052616, + "explanation": "One number rounds to 2500, other is farther from 4000." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8668 * 1774", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Estimate 8668 × 1774.", + "options": [ + "15377052", + "15377032", + "15377042", + "15377022" + ], + "answer": "15377032", + "correct_value": 15377032, + "explanation": "Not near any compatible pair; full multiplication needed." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "landmark_comparison_000652", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 48380 vs 22179", + "ns_shortcut": "Use 10% as landmark: 10% of 48380 = 4838.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 48380 greater than 22179?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4354.2, + "explanation": "9% ≈ 10%. 10% of 48380 = 4838. Compare to 22179." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "15% of 48380 vs 22179", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 15% of 48380 greater than 22179?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7257.0, + "explanation": "15% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 48380 vs 22179", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 48380 greater than 22179?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 40155.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000653", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 75650 vs 34563", + "ns_shortcut": "Use 50% as landmark: 50% of 75650 = 37825.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 75650 greater than 34563?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 36312.0, + "explanation": "48% ≈ 50%. 50% of 75650 = 37825. Compare to 34563." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 75650 vs 34563", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 75650 greater than 34563?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 34042.5, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 75650 vs 34563", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 75650 greater than 34563?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 62789.5, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000654", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 95860 vs 25675", + "ns_shortcut": "Use 10% as landmark: 10% of 95860 = 9586.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 95860 greater than 25675?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9586.0, + "explanation": "10% ≈ 10%. 10% of 95860 = 9586. Compare to 25675." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "13% of 95860 vs 25675", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 13% of 95860 greater than 25675?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 12461.8, + "explanation": "13% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 95860 vs 25675", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 95860 greater than 25675?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 35468.2, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000655", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 60670 vs 16769", + "ns_shortcut": "Use 25% as landmark: 25% of 60670 = 15167.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 60670 greater than 16769?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15774.2, + "explanation": "26% ≈ 25%. 25% of 60670 = 15167. Compare to 16769." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 60670 vs 16769", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 60670 greater than 16769?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 11527.3, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 60670 vs 16769", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 60670 greater than 16769?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 22447.9, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000656", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 38290 vs 9069", + "ns_shortcut": "Use 25% as landmark: 25% of 38290 = 9572.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 38290 greater than 9069?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 9572.5, + "explanation": "25% ≈ 25%. 25% of 38290 = 9572. Compare to 9069." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "19% of 38290 vs 9069", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 19% of 38290 greater than 9069?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7275.1, + "explanation": "19% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "43% of 38290 vs 9069", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 43% of 38290 greater than 9069?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 16464.7, + "explanation": "43% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000657", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 43310 vs 15814", + "ns_shortcut": "Use 50% as landmark: 50% of 43310 = 21655.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 43310 greater than 15814?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 21655.0, + "explanation": "50% ≈ 50%. 50% of 43310 = 21655. Compare to 15814." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53% of 43310 vs 15814", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 53% of 43310 greater than 15814?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 22954.3, + "explanation": "53% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 43310 vs 15814", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 43310 greater than 15814?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 31616.3, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000658", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 36100 vs 12867", + "ns_shortcut": "Use 50% as landmark: 50% of 36100 = 18050.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 36100 greater than 12867?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 18772.0, + "explanation": "52% ≈ 50%. 50% of 36100 = 18050. Compare to 12867." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "54% of 36100 vs 12867", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 54% of 36100 greater than 12867?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 19494.0, + "explanation": "54% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 36100 vs 12867", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 36100 greater than 12867?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 26353.0, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000659", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 71650 vs 29517", + "ns_shortcut": "Use 10% as landmark: 10% of 71650 = 7165.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 71650 greater than 29517?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7165.0, + "explanation": "10% ≈ 10%. 10% of 71650 = 7165. Compare to 29517." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 71650 vs 29517", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 71650 greater than 29517?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10031.0, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 71650 vs 29517", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 71650 greater than 29517?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 48005.5, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000660", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 31870 vs 12925", + "ns_shortcut": "Use 25% as landmark: 25% of 31870 = 7967.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 31870 greater than 12925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8604.9, + "explanation": "27% ≈ 25%. 25% of 31870 = 7967. Compare to 12925." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32% of 31870 vs 12925", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 32% of 31870 greater than 12925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10198.4, + "explanation": "32% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 31870 vs 12925", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 31870 greater than 12925?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 23265.1, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000661", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 97500 vs 21848", + "ns_shortcut": "Use 25% as landmark: 25% of 97500 = 24375.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 97500 greater than 21848?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 26325.0, + "explanation": "27% ≈ 25%. 25% of 97500 = 24375. Compare to 21848." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "21% of 97500 vs 21848", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 21% of 97500 greater than 21848?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 20475.0, + "explanation": "21% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 97500 vs 21848", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 97500 greater than 21848?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 80925.0, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000662", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 39180 vs 11688", + "ns_shortcut": "Use 25% as landmark: 25% of 39180 = 9795.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 39180 greater than 11688?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10186.8, + "explanation": "26% ≈ 25%. 25% of 39180 = 9795. Compare to 11688." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 39180 vs 11688", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 39180 greater than 11688?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7052.4, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 39180 vs 11688", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 39180 greater than 11688?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 32519.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000663", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 75230 vs 17577", + "ns_shortcut": "Use 10% as landmark: 10% of 75230 = 7523.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 75230 greater than 17577?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7523.0, + "explanation": "10% ≈ 10%. 10% of 75230 = 7523. Compare to 17577." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 75230 vs 17577", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 75230 greater than 17577?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10532.2, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 75230 vs 17577", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 75230 greater than 17577?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 27835.1, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000664", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 44600 vs 13866", + "ns_shortcut": "Use 25% as landmark: 25% of 44600 = 11150.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 44600 greater than 13866?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 10704.0, + "explanation": "24% ≈ 25%. 25% of 44600 = 11150. Compare to 13866." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 44600 vs 13866", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 44600 greater than 13866?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 12934.0, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 44600 vs 13866", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 44600 greater than 13866?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 32558.0, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000665", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 34690 vs 11606", + "ns_shortcut": "Use 25% as landmark: 25% of 34690 = 8672.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 34690 greater than 11606?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9366.3, + "explanation": "27% ≈ 25%. 25% of 34690 = 8672. Compare to 11606." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "28% of 34690 vs 11606", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 28% of 34690 greater than 11606?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9713.2, + "explanation": "28% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 34690 vs 11606", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 34690 greater than 11606?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 23242.3, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000666", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 79090 vs 21700", + "ns_shortcut": "Use 50% as landmark: 50% of 79090 = 39545.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 79090 greater than 21700?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 41126.8, + "explanation": "52% ≈ 50%. 50% of 79090 = 39545. Compare to 21700." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44% of 79090 vs 21700", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 44% of 79090 greater than 21700?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 34799.6, + "explanation": "44% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 79090 vs 21700", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 79090 greater than 21700?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 29263.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000667", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 36720 vs 8924", + "ns_shortcut": "Use 50% as landmark: 50% of 36720 = 18360.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 36720 greater than 8924?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 19094.4, + "explanation": "52% ≈ 50%. 50% of 36720 = 18360. Compare to 8924." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53% of 36720 vs 8924", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 53% of 36720 greater than 8924?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 19461.6, + "explanation": "53% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 36720 vs 8924", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 36720 greater than 8924?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 23133.6, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000668", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 93060 vs 31194", + "ns_shortcut": "Use 10% as landmark: 10% of 93060 = 9306.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 93060 greater than 31194?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9306.0, + "explanation": "10% ≈ 10%. 10% of 93060 = 9306. Compare to 31194." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "16% of 93060 vs 31194", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 16% of 93060 greater than 31194?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 14889.6, + "explanation": "16% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 93060 vs 31194", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 93060 greater than 31194?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 67933.8, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000669", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 18930 vs 6267", + "ns_shortcut": "Use 25% as landmark: 25% of 18930 = 4732.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 18930 greater than 6267?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5111.1, + "explanation": "27% ≈ 25%. 25% of 18930 = 4732. Compare to 6267." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 18930 vs 6267", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 18930 greater than 6267?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3407.4, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 18930 vs 6267", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 18930 greater than 6267?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 13818.9, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000670", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 29950 vs 7166", + "ns_shortcut": "Use 10% as landmark: 10% of 29950 = 2995.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 29950 greater than 7166?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3294.5, + "explanation": "11% ≈ 10%. 10% of 29950 = 2995. Compare to 7166." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 29950 vs 7166", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 29950 greater than 7166?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4193.0, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "87% of 29950 vs 7166", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 87% of 29950 greater than 7166?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 26056.5, + "explanation": "87% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000671", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 79470 vs 33317", + "ns_shortcut": "Use 25% as landmark: 25% of 79470 = 19867.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 79470 greater than 33317?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 19867.5, + "explanation": "25% ≈ 25%. 25% of 79470 = 19867. Compare to 33317." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 79470 vs 33317", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 79470 greater than 33317?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15894.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 79470 vs 33317", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 79470 greater than 33317?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 65960.1, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000672", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 52240 vs 13685", + "ns_shortcut": "Use 10% as landmark: 10% of 52240 = 5224.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 52240 greater than 13685?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5746.4, + "explanation": "11% ≈ 10%. 10% of 52240 = 5224. Compare to 13685." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4% of 52240 vs 13685", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 4% of 52240 greater than 13685?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2089.6, + "explanation": "4% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 52240 vs 13685", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 52240 greater than 13685?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 35000.8, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000673", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 16520 vs 4299", + "ns_shortcut": "Use 25% as landmark: 25% of 16520 = 4130.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 16520 greater than 4299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4295.2, + "explanation": "26% ≈ 25%. 25% of 16520 = 4130. Compare to 4299." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 16520 vs 4299", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 16520 greater than 4299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2973.6, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 16520 vs 4299", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 16520 greater than 4299?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 11068.4, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000674", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 51340 vs 17242", + "ns_shortcut": "Use 25% as landmark: 25% of 51340 = 12835.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 51340 greater than 17242?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 12835.0, + "explanation": "25% ≈ 25%. 25% of 51340 = 12835. Compare to 17242." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 51340 vs 17242", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 51340 greater than 17242?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 14888.6, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 51340 vs 17242", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 51340 greater than 17242?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 34397.8, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000675", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "51% of 68470 vs 27360", + "ns_shortcut": "Use 50% as landmark: 50% of 68470 = 34235.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 51% of 68470 greater than 27360?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 34919.7, + "explanation": "51% ≈ 50%. 50% of 68470 = 34235. Compare to 27360." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "53% of 68470 vs 27360", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 53% of 68470 greater than 27360?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 36289.1, + "explanation": "53% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 68470 vs 27360", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 68470 greater than 27360?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 56830.1, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000676", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 41260 vs 11171", + "ns_shortcut": "Use 25% as landmark: 25% of 41260 = 10315.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 41260 greater than 11171?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 11140.2, + "explanation": "27% ≈ 25%. 25% of 41260 = 10315. Compare to 11171." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 41260 vs 11171", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 41260 greater than 11171?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8252.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 41260 vs 11171", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 41260 greater than 11171?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 34245.8, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000677", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 23770 vs 9136", + "ns_shortcut": "Use 50% as landmark: 50% of 23770 = 11885.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 23770 greater than 9136?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 11409.6, + "explanation": "48% ≈ 50%. 50% of 23770 = 11885. Compare to 9136." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44% of 23770 vs 9136", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 44% of 23770 greater than 9136?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 10458.8, + "explanation": "44% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 23770 vs 9136", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 23770 greater than 9136?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 17352.1, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000678", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 55000 vs 22317", + "ns_shortcut": "Use 25% as landmark: 25% of 55000 = 13750.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 55000 greater than 22317?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 12650.0, + "explanation": "23% ≈ 25%. 25% of 55000 = 13750. Compare to 22317." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "28% of 55000 vs 22317", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 28% of 55000 greater than 22317?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 15400.0, + "explanation": "28% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 55000 vs 22317", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 55000 greater than 22317?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 20350.0, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000679", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "48% of 63500 vs 24915", + "ns_shortcut": "Use 50% as landmark: 50% of 63500 = 31750.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 48% of 63500 greater than 24915?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 30480.0, + "explanation": "48% ≈ 50%. 50% of 63500 = 31750. Compare to 24915." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44% of 63500 vs 24915", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 44% of 63500 greater than 24915?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 27940.0, + "explanation": "44% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 63500 vs 24915", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 63500 greater than 24915?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 52705.0, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000680", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 72010 vs 15760", + "ns_shortcut": "Use 25% as landmark: 25% of 72010 = 18002.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 72010 greater than 15760?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 16562.3, + "explanation": "23% ≈ 25%. 25% of 72010 = 18002. Compare to 15760." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "32% of 72010 vs 15760", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 32% of 72010 greater than 15760?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 23043.2, + "explanation": "32% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "63% of 72010 vs 15760", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 63% of 72010 greater than 15760?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 45366.3, + "explanation": "63% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000681", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9% of 52080 vs 22238", + "ns_shortcut": "Use 10% as landmark: 10% of 52080 = 5208.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 9% of 52080 greater than 22238?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4687.2, + "explanation": "9% ≈ 10%. 10% of 52080 = 5208. Compare to 22238." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 52080 vs 22238", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 52080 greater than 22238?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3124.8, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 52080 vs 22238", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 52080 greater than 22238?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 43226.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000682", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8% of 52750 vs 15517", + "ns_shortcut": "Use 10% as landmark: 10% of 52750 = 5275.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 8% of 52750 greater than 15517?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4220.0, + "explanation": "8% ≈ 10%. 10% of 52750 = 5275. Compare to 15517." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5% of 52750 vs 15517", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 5% of 52750 greater than 15517?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 2637.5, + "explanation": "5% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 52750 vs 15517", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 52750 greater than 15517?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 43782.5, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000683", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 87480 vs 24547", + "ns_shortcut": "Use 25% as landmark: 25% of 87480 = 21870.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 87480 greater than 24547?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 23619.6, + "explanation": "27% ≈ 25%. 25% of 87480 = 21870. Compare to 24547." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 87480 vs 24547", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 87480 greater than 24547?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 17496.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 87480 vs 24547", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 87480 greater than 24547?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 72608.4, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000684", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "23% of 17490 vs 8451", + "ns_shortcut": "Use 25% as landmark: 25% of 17490 = 4372.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 23% of 17490 greater than 8451?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4022.7, + "explanation": "23% ≈ 25%. 25% of 17490 = 4372. Compare to 8451." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 17490 vs 8451", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 17490 greater than 8451?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5247.0, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 17490 vs 8451", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 17490 greater than 8451?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6471.3, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000685", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 76710 vs 22129", + "ns_shortcut": "Use 50% as landmark: 50% of 76710 = 38355.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 76710 greater than 22129?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 39889.2, + "explanation": "52% ≈ 50%. 50% of 76710 = 38355. Compare to 22129." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "45% of 76710 vs 22129", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 45% of 76710 greater than 22129?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 34519.5, + "explanation": "45% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 76710 vs 22129", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 76710 greater than 22129?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 51395.7, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000686", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 85760 vs 36782", + "ns_shortcut": "Use 10% as landmark: 10% of 85760 = 8576.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 85760 greater than 36782?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 8576.0, + "explanation": "10% ≈ 10%. 10% of 85760 = 8576. Compare to 36782." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6% of 85760 vs 36782", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 6% of 85760 greater than 36782?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5145.6, + "explanation": "6% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 85760 vs 36782", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 85760 greater than 36782?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 31731.2, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000687", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 63180 vs 19879", + "ns_shortcut": "Use 50% as landmark: 50% of 63180 = 31590.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 63180 greater than 19879?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 32853.6, + "explanation": "52% ≈ 50%. 50% of 63180 = 31590. Compare to 19879." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "57% of 63180 vs 19879", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 57% of 63180 greater than 19879?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 36012.6, + "explanation": "57% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 63180 vs 19879", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 63180 greater than 19879?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 46121.4, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000688", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "27% of 63860 vs 30514", + "ns_shortcut": "Use 25% as landmark: 25% of 63860 = 15965.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 27% of 63860 greater than 30514?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 17242.2, + "explanation": "27% ≈ 25%. 25% of 63860 = 15965. Compare to 30514." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 63860 vs 30514", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 63860 greater than 30514?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 12772.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 63860 vs 30514", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 63860 greater than 30514?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 42786.2, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000689", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 62370 vs 23823", + "ns_shortcut": "Use 25% as landmark: 25% of 62370 = 15592.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 62370 greater than 23823?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 14968.8, + "explanation": "24% ≈ 25%. 25% of 62370 = 15592. Compare to 23823." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "29% of 62370 vs 23823", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 29% of 62370 greater than 23823?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 18087.3, + "explanation": "29% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 62370 vs 23823", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 62370 greater than 23823?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 51767.1, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000690", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 37850 vs 18596", + "ns_shortcut": "Use 25% as landmark: 25% of 37850 = 9462.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 37850 greater than 18596?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9841.0, + "explanation": "26% ≈ 25%. 25% of 37850 = 9462. Compare to 18596." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "18% of 37850 vs 18596", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 18% of 37850 greater than 18596?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6813.0, + "explanation": "18% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 37850 vs 18596", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 37850 greater than 18596?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 14004.5, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000691", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 89710 vs 36335", + "ns_shortcut": "Use 25% as landmark: 25% of 89710 = 22427.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 89710 greater than 36335?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 22427.5, + "explanation": "25% ≈ 25%. 25% of 89710 = 22427. Compare to 36335." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "30% of 89710 vs 36335", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 30% of 89710 greater than 36335?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 26913.0, + "explanation": "30% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 89710 vs 36335", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 89710 greater than 36335?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 60105.7, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000692", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "10% of 61660 vs 27126", + "ns_shortcut": "Use 10% as landmark: 10% of 61660 = 6166.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 10% of 61660 greater than 27126?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6166.0, + "explanation": "10% ≈ 10%. 10% of 61660 = 6166. Compare to 27126." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3% of 61660 vs 27126", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 3% of 61660 greater than 27126?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1849.8, + "explanation": "3% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 61660 vs 27126", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 61660 greater than 27126?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 51177.8, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000693", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 37310 vs 17689", + "ns_shortcut": "Use 25% as landmark: 25% of 37310 = 9327.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 37310 greater than 17689?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9700.6, + "explanation": "26% ≈ 25%. 25% of 37310 = 9327. Compare to 17689." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 37310 vs 17689", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 37310 greater than 17689?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 7462.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 37310 vs 17689", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 37310 greater than 17689?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 30967.3, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000694", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "24% of 90400 vs 25771", + "ns_shortcut": "Use 25% as landmark: 25% of 90400 = 22600.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 24% of 90400 greater than 25771?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 21696.0, + "explanation": "24% ≈ 25%. 25% of 90400 = 22600. Compare to 25771." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "22% of 90400 vs 25771", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 22% of 90400 greater than 25771?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 19888.0, + "explanation": "22% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 90400 vs 25771", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 90400 greater than 25771?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 65992.0, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000695", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "50% of 45760 vs 12614", + "ns_shortcut": "Use 50% as landmark: 50% of 45760 = 22880.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 50% of 45760 greater than 12614?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 22880.0, + "explanation": "50% ≈ 50%. 50% of 45760 = 22880. Compare to 12614." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "44% of 45760 vs 12614", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.94, + "pure_math": { + "question": "Is 44% of 45760 greater than 12614?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 20134.4, + "explanation": "44% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 45760 vs 12614", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 45760 greater than 12614?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 37980.8, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000696", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "52% of 76930 vs 26921", + "ns_shortcut": "Use 50% as landmark: 50% of 76930 = 38465.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 52% of 76930 greater than 26921?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 40003.6, + "explanation": "52% ≈ 50%. 50% of 76930 = 38465. Compare to 26921." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "55% of 76930 vs 26921", + "ns_shortcut": "Use 50% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 55% of 76930 greater than 26921?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 42311.5, + "explanation": "55% is near 50% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "67% of 76930 vs 26921", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 67% of 76930 greater than 26921?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 51543.1, + "explanation": "67% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000697", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "12% of 27300 vs 10901", + "ns_shortcut": "Use 10% as landmark: 10% of 27300 = 2730.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 12% of 27300 greater than 10901?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3276.0, + "explanation": "12% ≈ 10%. 10% of 27300 = 2730. Compare to 10901." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "14% of 27300 vs 10901", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 14% of 27300 greater than 10901?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3822.0, + "explanation": "14% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 27300 vs 10901", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 27300 greater than 10901?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 19929.0, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000698", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "26% of 91070 vs 43013", + "ns_shortcut": "Use 25% as landmark: 25% of 91070 = 22767.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 26% of 91070 greater than 43013?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 23678.2, + "explanation": "26% ≈ 25%. 25% of 91070 = 22767. Compare to 43013." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "21% of 91070 vs 43013", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.96, + "pure_math": { + "question": "Is 21% of 91070 greater than 43013?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 19124.7, + "explanation": "21% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 91070 vs 43013", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 91070 greater than 43013?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 66481.1, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000699", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "11% of 57250 vs 11559", + "ns_shortcut": "Use 10% as landmark: 10% of 57250 = 5725.", + "shortcut_strength": 0.99, + "pure_math": { + "question": "Is 11% of 57250 greater than 11559?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6297.5, + "explanation": "11% ≈ 10%. 10% of 57250 = 5725. Compare to 11559." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7% of 57250 vs 11559", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.97, + "pure_math": { + "question": "Is 7% of 57250 greater than 11559?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 4007.5, + "explanation": "7% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "73% of 57250 vs 11559", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 73% of 57250 greater than 11559?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 41792.5, + "explanation": "73% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000700", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "25% of 26260 vs 11086", + "ns_shortcut": "Use 25% as landmark: 25% of 26260 = 6565.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "Is 25% of 26260 greater than 11086?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 6565.0, + "explanation": "25% ≈ 25%. 25% of 26260 = 6565. Compare to 11086." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "20% of 26260 vs 11086", + "ns_shortcut": "Use 25% as approximate anchor, then adjust.", + "shortcut_strength": 0.95, + "pure_math": { + "question": "Is 20% of 26260 greater than 11086?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 5252.0, + "explanation": "20% is near 25% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "37% of 26260 vs 11086", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 37% of 26260 greater than 11086?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 9716.2, + "explanation": "37% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "landmark_comparison_000701", + "category": "landmark_comparison", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8% of 21520 vs 10615", + "ns_shortcut": "Use 10% as landmark: 10% of 21520 = 2152.", + "shortcut_strength": 0.98, + "pure_math": { + "question": "Is 8% of 21520 greater than 10615?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 1721.6, + "explanation": "8% ≈ 10%. 10% of 21520 = 2152. Compare to 10615." + } + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "17% of 21520 vs 10615", + "ns_shortcut": "Use 10% as approximate anchor, then adjust.", + "shortcut_strength": 0.9299999999999999, + "pure_math": { + "question": "Is 17% of 21520 greater than 10615?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "No", + "correct_value": 3658.4, + "explanation": "17% is near 10% but requires adjustment." + } + }, + "control": { + "task_type": "control", + "math_expression": "83% of 21520 vs 10615", + "ns_shortcut": "No clean shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "Is 83% of 21520 greater than 10615?", + "options": [ + "Yes", + "No", + "Cannot be determined", + "Need more information" + ], + "answer": "Yes", + "correct_value": 17861.6, + "explanation": "83% is not near any standard landmark; compute directly." + } + } + }, + { + "family_id": "equation_reasoning_000702", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8731 × 2490 × 7066 = ___ × 7066 × 8731", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8731 × 2490 × 7066 = ___ × 7066 × 8731", + "options": [ + "2480", + "2500", + "2510", + "2490" + ], + "answer": "2490", + "correct_value": 2490, + "explanation": "A and C appear on both sides; blank = B = 2490." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9377 + 4162 + ___ = 4162 + 3644 + 8500", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9377 + 4162 + ___ = 4162 + 3644 + 8500", + "options": [ + "2777", + "2767", + "2757", + "2787" + ], + "answer": "2767", + "correct_value": 2767, + "explanation": "B cancels; blank = C + D − A = 3644 + 8500 − 9377 = 2767." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3870 + 8736 + ___ = 4273 + 6287", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3870 + 8736 + ___ = 4273 + 6287", + "options": [ + "-2034", + "-2046", + "-2064", + "-2044" + ], + "answer": "-2046", + "correct_value": -2046, + "explanation": "No matching terms; blank = 4273 + 6287 − 3870 − 8736 = -2046." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000703", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4391 + 8555 + ___ = 8555 + 8523 + 4391", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4391 + 8555 + ___ = 8555 + 8523 + 4391", + "options": [ + "8523", + "8513", + "8543", + "8533" + ], + "answer": "8523", + "correct_value": 8523, + "explanation": "A and B appear on both sides and cancel; blank = C = 8523." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4963 + 3897 + ___ = 3897 + 8559 + 9195", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4963 + 3897 + ___ = 3897 + 8559 + 9195", + "options": [ + "12781", + "12801", + "12811", + "12791" + ], + "answer": "12791", + "correct_value": 12791, + "explanation": "B cancels; blank = C + D − A = 8559 + 9195 − 4963 = 12791." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3609 + 5011 + ___ = 8572 + 2144", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3609 + 5011 + ___ = 8572 + 2144", + "options": [ + "2106", + "2086", + "2116", + "2096" + ], + "answer": "2096", + "correct_value": 2096, + "explanation": "No matching terms; blank = 8572 + 2144 − 3609 − 5011 = 2096." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000704", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9515 + 8703 = ___ + 7313", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9515 + 8703 = ___ + 7313", + "options": [ + "10895", + "10925", + "10915", + "10905" + ], + "answer": "10905", + "correct_value": 10905, + "explanation": "Blank = A + B − C = 9515 + 8703 − 7313 = 10905." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9803 + 2982 + ___ = 2982 + 9751 + 1388", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9803 + 2982 + ___ = 2982 + 9751 + 1388", + "options": [ + "1356", + "1336", + "1346", + "1326" + ], + "answer": "1336", + "correct_value": 1336, + "explanation": "B cancels; blank = C + D − A = 9751 + 1388 − 9803 = 1336." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5799 + 7527 + ___ = 1653 + 6953", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5799 + 7527 + ___ = 1653 + 6953", + "options": [ + "-4720", + "-4710", + "-4700", + "-4730" + ], + "answer": "-4720", + "correct_value": -4720, + "explanation": "No matching terms; blank = 1653 + 6953 − 5799 − 7527 = -4720." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000705", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3032 + 2697 = ___ + 3391", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3032 + 2697 = ___ + 3391", + "options": [ + "2348", + "2338", + "2328", + "2358" + ], + "answer": "2338", + "correct_value": 2338, + "explanation": "Blank = A + B − C = 3032 + 2697 − 3391 = 2338." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3768 + 7004 + ___ = 7004 + 7764 + 6567", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3768 + 7004 + ___ = 7004 + 7764 + 6567", + "options": [ + "10583", + "10553", + "10563", + "10573" + ], + "answer": "10563", + "correct_value": 10563, + "explanation": "B cancels; blank = C + D − A = 7764 + 6567 − 3768 = 10563." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4442 + 5333 + ___ = 2591 + 6407", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4442 + 5333 + ___ = 2591 + 6407", + "options": [ + "-793", + "-763", + "-773", + "-777" + ], + "answer": "-777", + "correct_value": -777, + "explanation": "No matching terms; blank = 2591 + 6407 − 4442 − 5333 = -777." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000706", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5259 × 6610 × 5838 = ___ × 5838 × 5259", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5259 × 6610 × 5838 = ___ × 5838 × 5259", + "options": [ + "6610", + "6630", + "6620", + "6600" + ], + "answer": "6610", + "correct_value": 6610, + "explanation": "A and C appear on both sides; blank = B = 6610." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5225 + 5510 + ___ = 5510 + 1585 + 3294", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5225 + 5510 + ___ = 5510 + 1585 + 3294", + "options": [ + "-346", + "-364", + "-334", + "-344" + ], + "answer": "-346", + "correct_value": -346, + "explanation": "B cancels; blank = C + D − A = 1585 + 3294 − 5225 = -346." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1425 + 4621 + ___ = 7076 + 6942", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1425 + 4621 + ___ = 7076 + 6942", + "options": [ + "7962", + "7972", + "7982", + "7992" + ], + "answer": "7972", + "correct_value": 7972, + "explanation": "No matching terms; blank = 7076 + 6942 − 1425 − 4621 = 7972." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000707", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8787 × (8206 + 5831) = 8787 × ___ + 8787 × 5831", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8787 × (8206 + 5831) = 8787 × ___ + 8787 × 5831", + "options": [ + "8196", + "8206", + "8226", + "8216" + ], + "answer": "8206", + "correct_value": 8206, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 8206." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3400 + 5789 + ___ = 5789 + 7295 + 5215", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3400 + 5789 + ___ = 5789 + 7295 + 5215", + "options": [ + "9100", + "9130", + "9110", + "9120" + ], + "answer": "9110", + "correct_value": 9110, + "explanation": "B cancels; blank = C + D − A = 7295 + 5215 − 3400 = 9110." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5963 + 3430 + ___ = 1197 + 5250", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5963 + 3430 + ___ = 1197 + 5250", + "options": [ + "-2946", + "-2944", + "-2964", + "-2934" + ], + "answer": "-2946", + "correct_value": -2946, + "explanation": "No matching terms; blank = 1197 + 5250 − 5963 − 3430 = -2946." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000708", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3791 + 6639 = ___ + 3553", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3791 + 6639 = ___ + 3553", + "options": [ + "6877", + "6887", + "6867", + "6897" + ], + "answer": "6877", + "correct_value": 6877, + "explanation": "Blank = A + B − C = 3791 + 6639 − 3553 = 6877." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7192 + 8950 + ___ = 8950 + 3770 + 9605", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7192 + 8950 + ___ = 8950 + 3770 + 9605", + "options": [ + "6203", + "6193", + "6183", + "6173" + ], + "answer": "6183", + "correct_value": 6183, + "explanation": "B cancels; blank = C + D − A = 3770 + 9605 − 7192 = 6183." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7223 + 1233 + ___ = 2060 + 6609", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7223 + 1233 + ___ = 2060 + 6609", + "options": [ + "233", + "213", + "203", + "223" + ], + "answer": "213", + "correct_value": 213, + "explanation": "No matching terms; blank = 2060 + 6609 − 7223 − 1233 = 213." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000709", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3459 × 5559 × 2959 = ___ × 2959 × 3459", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3459 × 5559 × 2959 = ___ × 2959 × 3459", + "options": [ + "5559", + "5579", + "5549", + "5569" + ], + "answer": "5559", + "correct_value": 5559, + "explanation": "A and C appear on both sides; blank = B = 5559." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8525 + 1199 + ___ = 1199 + 9587 + 4741", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8525 + 1199 + ___ = 1199 + 9587 + 4741", + "options": [ + "5823", + "5803", + "5793", + "5813" + ], + "answer": "5803", + "correct_value": 5803, + "explanation": "B cancels; blank = C + D − A = 9587 + 4741 − 8525 = 5803." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7528 + 2619 + ___ = 6398 + 9313", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7528 + 2619 + ___ = 6398 + 9313", + "options": [ + "5584", + "5554", + "5574", + "5564" + ], + "answer": "5564", + "correct_value": 5564, + "explanation": "No matching terms; blank = 6398 + 9313 − 7528 − 2619 = 5564." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000710", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8317 + 9567 + ___ = 9567 + 5062 + 8317", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8317 + 9567 + ___ = 9567 + 5062 + 8317", + "options": [ + "5082", + "5062", + "5052", + "5072" + ], + "answer": "5062", + "correct_value": 5062, + "explanation": "A and B appear on both sides and cancel; blank = C = 5062." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5015 + 4741 + ___ = 4741 + 8472 + 3907", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5015 + 4741 + ___ = 4741 + 8472 + 3907", + "options": [ + "7354", + "7374", + "7384", + "7364" + ], + "answer": "7364", + "correct_value": 7364, + "explanation": "B cancels; blank = C + D − A = 8472 + 3907 − 5015 = 7364." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5546 + 2496 + ___ = 4113 + 2043", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 5546 + 2496 + ___ = 4113 + 2043", + "options": [ + "-1886", + "-1904", + "-1874", + "-1884" + ], + "answer": "-1886", + "correct_value": -1886, + "explanation": "No matching terms; blank = 4113 + 2043 − 5546 − 2496 = -1886." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000711", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4403 + 4728 + ___ = 4728 + 4996 + 4403", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4403 + 4728 + ___ = 4728 + 4996 + 4403", + "options": [ + "5016", + "4996", + "5006", + "4986" + ], + "answer": "4996", + "correct_value": 4996, + "explanation": "A and B appear on both sides and cancel; blank = C = 4996." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9797 + 1836 + ___ = 1836 + 6224 + 2196", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9797 + 1836 + ___ = 1836 + 6224 + 2196", + "options": [ + "-1363", + "-1377", + "-1393", + "-1373" + ], + "answer": "-1377", + "correct_value": -1377, + "explanation": "B cancels; blank = C + D − A = 6224 + 2196 − 9797 = -1377." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8240 + 2366 + ___ = 8799 + 4234", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8240 + 2366 + ___ = 8799 + 4234", + "options": [ + "2437", + "2417", + "2447", + "2427" + ], + "answer": "2427", + "correct_value": 2427, + "explanation": "No matching terms; blank = 8799 + 4234 − 8240 − 2366 = 2427." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000712", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5914 + 1157 = ___ + 8041", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5914 + 1157 = ___ + 8041", + "options": [ + "-980", + "-960", + "-950", + "-970" + ], + "answer": "-970", + "correct_value": -970, + "explanation": "Blank = A + B − C = 5914 + 1157 − 8041 = -970." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4191 + 8475 + ___ = 8475 + 1008 + 5124", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4191 + 8475 + ___ = 8475 + 1008 + 5124", + "options": [ + "1941", + "1951", + "1961", + "1931" + ], + "answer": "1941", + "correct_value": 1941, + "explanation": "B cancels; blank = C + D − A = 1008 + 5124 − 4191 = 1941." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9070 + 3892 + ___ = 6384 + 9311", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9070 + 3892 + ___ = 6384 + 9311", + "options": [ + "2733", + "2743", + "2753", + "2723" + ], + "answer": "2733", + "correct_value": 2733, + "explanation": "No matching terms; blank = 6384 + 9311 − 9070 − 3892 = 2733." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000713", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5328 × (9719 + 5079) = 5328 × ___ + 5328 × 5079", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5328 × (9719 + 5079) = 5328 × ___ + 5328 × 5079", + "options": [ + "9719", + "9709", + "9729", + "9739" + ], + "answer": "9719", + "correct_value": 9719, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 9719." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8209 + 2826 + ___ = 2826 + 3984 + 8988", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8209 + 2826 + ___ = 2826 + 3984 + 8988", + "options": [ + "4753", + "4783", + "4773", + "4763" + ], + "answer": "4763", + "correct_value": 4763, + "explanation": "B cancels; blank = C + D − A = 3984 + 8988 − 8209 = 4763." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9122 + 3672 + ___ = 3929 + 2074", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9122 + 3672 + ___ = 3929 + 2074", + "options": [ + "-6791", + "-6789", + "-6779", + "-6809" + ], + "answer": "-6791", + "correct_value": -6791, + "explanation": "No matching terms; blank = 3929 + 2074 − 9122 − 3672 = -6791." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000714", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8261 + 6610 + ___ = 6610 + 9824 + 8261", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8261 + 6610 + ___ = 6610 + 9824 + 8261", + "options": [ + "9814", + "9844", + "9824", + "9834" + ], + "answer": "9824", + "correct_value": 9824, + "explanation": "A and B appear on both sides and cancel; blank = C = 9824." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4797 + 4493 + ___ = 4493 + 6033 + 9221", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4797 + 4493 + ___ = 4493 + 6033 + 9221", + "options": [ + "10457", + "10447", + "10467", + "10477" + ], + "answer": "10457", + "correct_value": 10457, + "explanation": "B cancels; blank = C + D − A = 6033 + 9221 − 4797 = 10457." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2686 + 5916 + ___ = 4445 + 3532", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2686 + 5916 + ___ = 4445 + 3532", + "options": [ + "-615", + "-635", + "-625", + "-605" + ], + "answer": "-625", + "correct_value": -625, + "explanation": "No matching terms; blank = 4445 + 3532 − 2686 − 5916 = -625." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000715", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3814 + 3587 = ___ + 3574", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3814 + 3587 = ___ + 3574", + "options": [ + "3827", + "3837", + "3817", + "3847" + ], + "answer": "3827", + "correct_value": 3827, + "explanation": "Blank = A + B − C = 3814 + 3587 − 3574 = 3827." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7830 + 5413 + ___ = 5413 + 4747 + 5306", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7830 + 5413 + ___ = 5413 + 4747 + 5306", + "options": [ + "2223", + "2243", + "2233", + "2213" + ], + "answer": "2223", + "correct_value": 2223, + "explanation": "B cancels; blank = C + D − A = 4747 + 5306 − 7830 = 2223." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3375 + 6017 + ___ = 6467 + 6798", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3375 + 6017 + ___ = 6467 + 6798", + "options": [ + "3873", + "3863", + "3883", + "3893" + ], + "answer": "3873", + "correct_value": 3873, + "explanation": "No matching terms; blank = 6467 + 6798 − 3375 − 6017 = 3873." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000716", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7447 + 7834 = ___ + 1092", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7447 + 7834 = ___ + 1092", + "options": [ + "14209", + "14199", + "14189", + "14179" + ], + "answer": "14189", + "correct_value": 14189, + "explanation": "Blank = A + B − C = 7447 + 7834 − 1092 = 14189." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6793 + 3292 + ___ = 3292 + 8485 + 5492", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6793 + 3292 + ___ = 3292 + 8485 + 5492", + "options": [ + "7194", + "7174", + "7204", + "7184" + ], + "answer": "7184", + "correct_value": 7184, + "explanation": "B cancels; blank = C + D − A = 8485 + 5492 − 6793 = 7184." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4417 + 5193 + ___ = 5829 + 7119", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4417 + 5193 + ___ = 5829 + 7119", + "options": [ + "3328", + "3358", + "3348", + "3338" + ], + "answer": "3338", + "correct_value": 3338, + "explanation": "No matching terms; blank = 5829 + 7119 − 4417 − 5193 = 3338." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000717", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6336 + 4267 = ___ + 4522", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6336 + 4267 = ___ + 4522", + "options": [ + "6101", + "6081", + "6091", + "6071" + ], + "answer": "6081", + "correct_value": 6081, + "explanation": "Blank = A + B − C = 6336 + 4267 − 4522 = 6081." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2219 + 4983 + ___ = 4983 + 3628 + 6332", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2219 + 4983 + ___ = 4983 + 3628 + 6332", + "options": [ + "7741", + "7751", + "7761", + "7731" + ], + "answer": "7741", + "correct_value": 7741, + "explanation": "B cancels; blank = C + D − A = 3628 + 6332 − 2219 = 7741." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1753 + 7390 + ___ = 8979 + 5093", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1753 + 7390 + ___ = 8979 + 5093", + "options": [ + "4929", + "4939", + "4919", + "4949" + ], + "answer": "4929", + "correct_value": 4929, + "explanation": "No matching terms; blank = 8979 + 5093 − 1753 − 7390 = 4929." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000718", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2661 + 1919 + ___ = 1919 + 6614 + 2661", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2661 + 1919 + ___ = 1919 + 6614 + 2661", + "options": [ + "6614", + "6624", + "6604", + "6634" + ], + "answer": "6614", + "correct_value": 6614, + "explanation": "A and B appear on both sides and cancel; blank = C = 6614." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8178 + 5656 + ___ = 5656 + 2819 + 8730", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8178 + 5656 + ___ = 5656 + 2819 + 8730", + "options": [ + "3361", + "3371", + "3391", + "3381" + ], + "answer": "3371", + "correct_value": 3371, + "explanation": "B cancels; blank = C + D − A = 2819 + 8730 − 8178 = 3371." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6317 + 7779 + ___ = 3486 + 3464", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6317 + 7779 + ___ = 3486 + 3464", + "options": [ + "-7144", + "-7146", + "-7164", + "-7134" + ], + "answer": "-7146", + "correct_value": -7146, + "explanation": "No matching terms; blank = 3486 + 3464 − 6317 − 7779 = -7146." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000719", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8180 + 5017 + ___ = 5017 + 6371 + 8180", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8180 + 5017 + ___ = 5017 + 6371 + 8180", + "options": [ + "6371", + "6381", + "6361", + "6391" + ], + "answer": "6371", + "correct_value": 6371, + "explanation": "A and B appear on both sides and cancel; blank = C = 6371." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1358 + 3301 + ___ = 3301 + 2114 + 6143", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1358 + 3301 + ___ = 3301 + 2114 + 6143", + "options": [ + "6889", + "6899", + "6909", + "6919" + ], + "answer": "6899", + "correct_value": 6899, + "explanation": "B cancels; blank = C + D − A = 2114 + 6143 − 1358 = 6899." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4016 + 6263 + ___ = 8673 + 4730", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4016 + 6263 + ___ = 8673 + 4730", + "options": [ + "3124", + "3144", + "3134", + "3114" + ], + "answer": "3124", + "correct_value": 3124, + "explanation": "No matching terms; blank = 8673 + 4730 − 4016 − 6263 = 3124." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000720", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5243 × 5183 × 1379 = ___ × 1379 × 5243", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5243 × 5183 × 1379 = ___ × 1379 × 5243", + "options": [ + "5203", + "5173", + "5183", + "5193" + ], + "answer": "5183", + "correct_value": 5183, + "explanation": "A and C appear on both sides; blank = B = 5183." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8001 + 7493 + ___ = 7493 + 8968 + 1786", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8001 + 7493 + ___ = 7493 + 8968 + 1786", + "options": [ + "2743", + "2763", + "2773", + "2753" + ], + "answer": "2753", + "correct_value": 2753, + "explanation": "B cancels; blank = C + D − A = 8968 + 1786 − 8001 = 2753." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1265 + 8163 + ___ = 1525 + 9261", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1265 + 8163 + ___ = 1525 + 9261", + "options": [ + "1358", + "1368", + "1348", + "1378" + ], + "answer": "1358", + "correct_value": 1358, + "explanation": "No matching terms; blank = 1525 + 9261 − 1265 − 8163 = 1358." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000721", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6468 × (2099 + 2237) = 6468 × ___ + 6468 × 2237", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6468 × (2099 + 2237) = 6468 × ___ + 6468 × 2237", + "options": [ + "2109", + "2089", + "2119", + "2099" + ], + "answer": "2099", + "correct_value": 2099, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 2099." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9712 + 3592 + ___ = 3592 + 9413 + 2437", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9712 + 3592 + ___ = 3592 + 9413 + 2437", + "options": [ + "2148", + "2138", + "2158", + "2128" + ], + "answer": "2138", + "correct_value": 2138, + "explanation": "B cancels; blank = C + D − A = 9413 + 2437 − 9712 = 2138." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2197 + 8060 + ___ = 5536 + 8630", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2197 + 8060 + ___ = 5536 + 8630", + "options": [ + "3919", + "3909", + "3929", + "3899" + ], + "answer": "3909", + "correct_value": 3909, + "explanation": "No matching terms; blank = 5536 + 8630 − 2197 − 8060 = 3909." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000722", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2196 × (3976 + 6567) = 2196 × ___ + 2196 × 6567", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2196 × (3976 + 6567) = 2196 × ___ + 2196 × 6567", + "options": [ + "3976", + "3986", + "3996", + "3966" + ], + "answer": "3976", + "correct_value": 3976, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 3976." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1351 + 7577 + ___ = 7577 + 5813 + 1104", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1351 + 7577 + ___ = 7577 + 5813 + 1104", + "options": [ + "5576", + "5586", + "5566", + "5556" + ], + "answer": "5566", + "correct_value": 5566, + "explanation": "B cancels; blank = C + D − A = 5813 + 1104 − 1351 = 5566." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4690 + 6281 + ___ = 7998 + 3465", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4690 + 6281 + ___ = 7998 + 3465", + "options": [ + "482", + "502", + "492", + "512" + ], + "answer": "492", + "correct_value": 492, + "explanation": "No matching terms; blank = 7998 + 3465 − 4690 − 6281 = 492." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000723", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4282 + 6810 = ___ + 3500", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4282 + 6810 = ___ + 3500", + "options": [ + "7582", + "7612", + "7592", + "7602" + ], + "answer": "7592", + "correct_value": 7592, + "explanation": "Blank = A + B − C = 4282 + 6810 − 3500 = 7592." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6054 + 3975 + ___ = 3975 + 2653 + 1429", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6054 + 3975 + ___ = 3975 + 2653 + 1429", + "options": [ + "-1958", + "-1968", + "-1988", + "-1972" + ], + "answer": "-1972", + "correct_value": -1972, + "explanation": "B cancels; blank = C + D − A = 2653 + 1429 − 6054 = -1972." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2134 + 6365 + ___ = 6166 + 4770", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2134 + 6365 + ___ = 6166 + 4770", + "options": [ + "2457", + "2427", + "2437", + "2447" + ], + "answer": "2437", + "correct_value": 2437, + "explanation": "No matching terms; blank = 6166 + 4770 − 2134 − 6365 = 2437." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000724", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7951 × 5441 × 4817 = ___ × 4817 × 7951", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 7951 × 5441 × 4817 = ___ × 4817 × 7951", + "options": [ + "5461", + "5441", + "5431", + "5451" + ], + "answer": "5441", + "correct_value": 5441, + "explanation": "A and C appear on both sides; blank = B = 5441." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3249 + 8203 + ___ = 8203 + 3017 + 7145", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3249 + 8203 + ___ = 8203 + 3017 + 7145", + "options": [ + "6933", + "6903", + "6923", + "6913" + ], + "answer": "6913", + "correct_value": 6913, + "explanation": "B cancels; blank = C + D − A = 3017 + 7145 − 3249 = 6913." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7232 + 1806 + ___ = 8320 + 8185", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7232 + 1806 + ___ = 8320 + 8185", + "options": [ + "7477", + "7487", + "7467", + "7457" + ], + "answer": "7467", + "correct_value": 7467, + "explanation": "No matching terms; blank = 8320 + 8185 − 7232 − 1806 = 7467." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000725", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3042 + 2521 = ___ + 3868", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3042 + 2521 = ___ + 3868", + "options": [ + "1705", + "1715", + "1685", + "1695" + ], + "answer": "1695", + "correct_value": 1695, + "explanation": "Blank = A + B − C = 3042 + 2521 − 3868 = 1695." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2971 + 9013 + ___ = 9013 + 6871 + 6191", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2971 + 9013 + ___ = 9013 + 6871 + 6191", + "options": [ + "10101", + "10111", + "10091", + "10081" + ], + "answer": "10091", + "correct_value": 10091, + "explanation": "B cancels; blank = C + D − A = 6871 + 6191 − 2971 = 10091." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2487 + 9305 + ___ = 6895 + 8831", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2487 + 9305 + ___ = 6895 + 8831", + "options": [ + "3944", + "3934", + "3924", + "3954" + ], + "answer": "3934", + "correct_value": 3934, + "explanation": "No matching terms; blank = 6895 + 8831 − 2487 − 9305 = 3934." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000726", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1264 + 1118 = ___ + 7226", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1264 + 1118 = ___ + 7226", + "options": [ + "-4826", + "-4844", + "-4836", + "-4856" + ], + "answer": "-4844", + "correct_value": -4844, + "explanation": "Blank = A + B − C = 1264 + 1118 − 7226 = -4844." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6811 + 6904 + ___ = 6904 + 6078 + 2069", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6811 + 6904 + ___ = 6904 + 6078 + 2069", + "options": [ + "1336", + "1346", + "1356", + "1326" + ], + "answer": "1336", + "correct_value": 1336, + "explanation": "B cancels; blank = C + D − A = 6078 + 2069 − 6811 = 1336." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4757 + 2383 + ___ = 7384 + 8722", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4757 + 2383 + ___ = 7384 + 8722", + "options": [ + "8956", + "8986", + "8976", + "8966" + ], + "answer": "8966", + "correct_value": 8966, + "explanation": "No matching terms; blank = 7384 + 8722 − 4757 − 2383 = 8966." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000727", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3305 + 5079 + ___ = 5079 + 3905 + 3305", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3305 + 5079 + ___ = 5079 + 3905 + 3305", + "options": [ + "3915", + "3925", + "3895", + "3905" + ], + "answer": "3905", + "correct_value": 3905, + "explanation": "A and B appear on both sides and cancel; blank = C = 3905." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3297 + 8697 + ___ = 8697 + 1638 + 2002", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3297 + 8697 + ___ = 8697 + 1638 + 2002", + "options": [ + "343", + "333", + "363", + "353" + ], + "answer": "343", + "correct_value": 343, + "explanation": "B cancels; blank = C + D − A = 1638 + 2002 − 3297 = 343." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7089 + 7797 + ___ = 3580 + 7868", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7089 + 7797 + ___ = 3580 + 7868", + "options": [ + "-3452", + "-3422", + "-3438", + "-3432" + ], + "answer": "-3438", + "correct_value": -3438, + "explanation": "No matching terms; blank = 3580 + 7868 − 7089 − 7797 = -3438." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000728", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6981 × 2792 × 3660 = ___ × 3660 × 6981", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6981 × 2792 × 3660 = ___ × 3660 × 6981", + "options": [ + "2812", + "2782", + "2802", + "2792" + ], + "answer": "2792", + "correct_value": 2792, + "explanation": "A and C appear on both sides; blank = B = 2792." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6014 + 7613 + ___ = 7613 + 7114 + 5344", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 6014 + 7613 + ___ = 7613 + 7114 + 5344", + "options": [ + "6454", + "6434", + "6444", + "6464" + ], + "answer": "6444", + "correct_value": 6444, + "explanation": "B cancels; blank = C + D − A = 7114 + 5344 − 6014 = 6444." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6432 + 1466 + ___ = 8708 + 1560", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6432 + 1466 + ___ = 8708 + 1560", + "options": [ + "2390", + "2370", + "2360", + "2380" + ], + "answer": "2370", + "correct_value": 2370, + "explanation": "No matching terms; blank = 8708 + 1560 − 6432 − 1466 = 2370." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000729", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9604 + 3781 = ___ + 3591", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9604 + 3781 = ___ + 3591", + "options": [ + "9814", + "9804", + "9784", + "9794" + ], + "answer": "9794", + "correct_value": 9794, + "explanation": "Blank = A + B − C = 9604 + 3781 − 3591 = 9794." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4648 + 2509 + ___ = 2509 + 7879 + 2086", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4648 + 2509 + ___ = 2509 + 7879 + 2086", + "options": [ + "5307", + "5337", + "5327", + "5317" + ], + "answer": "5317", + "correct_value": 5317, + "explanation": "B cancels; blank = C + D − A = 7879 + 2086 − 4648 = 5317." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7800 + 9793 + ___ = 8215 + 3202", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7800 + 9793 + ___ = 8215 + 3202", + "options": [ + "-6194", + "-6164", + "-6174", + "-6176" + ], + "answer": "-6176", + "correct_value": -6176, + "explanation": "No matching terms; blank = 8215 + 3202 − 7800 − 9793 = -6176." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000730", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5208 + 3100 = ___ + 6290", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5208 + 3100 = ___ + 6290", + "options": [ + "2028", + "2018", + "2008", + "2038" + ], + "answer": "2018", + "correct_value": 2018, + "explanation": "Blank = A + B − C = 5208 + 3100 − 6290 = 2018." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8828 + 9074 + ___ = 9074 + 4215 + 3061", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8828 + 9074 + ___ = 9074 + 4215 + 3061", + "options": [ + "-1568", + "-1552", + "-1538", + "-1548" + ], + "answer": "-1552", + "correct_value": -1552, + "explanation": "B cancels; blank = C + D − A = 4215 + 3061 − 8828 = -1552." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3336 + 4695 + ___ = 2346 + 2553", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3336 + 4695 + ___ = 2346 + 2553", + "options": [ + "-3118", + "-3132", + "-3128", + "-3148" + ], + "answer": "-3132", + "correct_value": -3132, + "explanation": "No matching terms; blank = 2346 + 2553 − 3336 − 4695 = -3132." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000731", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4706 × 5106 × 4024 = ___ × 4024 × 4706", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4706 × 5106 × 4024 = ___ × 4024 × 4706", + "options": [ + "5106", + "5096", + "5116", + "5126" + ], + "answer": "5106", + "correct_value": 5106, + "explanation": "A and C appear on both sides; blank = B = 5106." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9518 + 9468 + ___ = 9468 + 5370 + 3157", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9518 + 9468 + ___ = 9468 + 5370 + 3157", + "options": [ + "-991", + "-1009", + "-989", + "-979" + ], + "answer": "-991", + "correct_value": -991, + "explanation": "B cancels; blank = C + D − A = 5370 + 3157 − 9518 = -991." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3379 + 1560 + ___ = 7308 + 1786", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3379 + 1560 + ___ = 7308 + 1786", + "options": [ + "4145", + "4175", + "4155", + "4165" + ], + "answer": "4155", + "correct_value": 4155, + "explanation": "No matching terms; blank = 7308 + 1786 − 3379 − 1560 = 4155." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000732", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1558 × 4896 × 3652 = ___ × 3652 × 1558", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 1558 × 4896 × 3652 = ___ × 3652 × 1558", + "options": [ + "4916", + "4896", + "4906", + "4886" + ], + "answer": "4896", + "correct_value": 4896, + "explanation": "A and C appear on both sides; blank = B = 4896." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8877 + 8927 + ___ = 8927 + 6634 + 4076", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 8877 + 8927 + ___ = 8927 + 6634 + 4076", + "options": [ + "1853", + "1833", + "1843", + "1823" + ], + "answer": "1833", + "correct_value": 1833, + "explanation": "B cancels; blank = C + D − A = 6634 + 4076 − 8877 = 1833." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8425 + 5305 + ___ = 9943 + 8206", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8425 + 5305 + ___ = 9943 + 8206", + "options": [ + "4409", + "4429", + "4439", + "4419" + ], + "answer": "4419", + "correct_value": 4419, + "explanation": "No matching terms; blank = 9943 + 8206 − 8425 − 5305 = 4419." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000733", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6525 × (4052 + 5182) = 6525 × ___ + 6525 × 5182", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6525 × (4052 + 5182) = 6525 × ___ + 6525 × 5182", + "options": [ + "4052", + "4042", + "4062", + "4072" + ], + "answer": "4052", + "correct_value": 4052, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 4052." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2581 + 6609 + ___ = 6609 + 9504 + 5306", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2581 + 6609 + ___ = 6609 + 9504 + 5306", + "options": [ + "12219", + "12249", + "12239", + "12229" + ], + "answer": "12229", + "correct_value": 12229, + "explanation": "B cancels; blank = C + D − A = 9504 + 5306 − 2581 = 12229." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1416 + 9901 + ___ = 3068 + 2549", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1416 + 9901 + ___ = 3068 + 2549", + "options": [ + "-5680", + "-5700", + "-5710", + "-5690" + ], + "answer": "-5700", + "correct_value": -5700, + "explanation": "No matching terms; blank = 3068 + 2549 − 1416 − 9901 = -5700." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000734", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5844 + 3069 + ___ = 3069 + 6695 + 5844", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5844 + 3069 + ___ = 3069 + 6695 + 5844", + "options": [ + "6695", + "6685", + "6705", + "6715" + ], + "answer": "6695", + "correct_value": 6695, + "explanation": "A and B appear on both sides and cancel; blank = C = 6695." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2359 + 9269 + ___ = 9269 + 3991 + 7513", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2359 + 9269 + ___ = 9269 + 3991 + 7513", + "options": [ + "9135", + "9165", + "9145", + "9155" + ], + "answer": "9145", + "correct_value": 9145, + "explanation": "B cancels; blank = C + D − A = 3991 + 7513 − 2359 = 9145." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1273 + 4255 + ___ = 6798 + 9611", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1273 + 4255 + ___ = 6798 + 9611", + "options": [ + "10871", + "10881", + "10901", + "10891" + ], + "answer": "10881", + "correct_value": 10881, + "explanation": "No matching terms; blank = 6798 + 9611 − 1273 − 4255 = 10881." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000735", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9838 × (4775 + 9697) = 9838 × ___ + 9838 × 9697", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9838 × (4775 + 9697) = 9838 × ___ + 9838 × 9697", + "options": [ + "4795", + "4775", + "4765", + "4785" + ], + "answer": "4775", + "correct_value": 4775, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 4775." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2640 + 3713 + ___ = 3713 + 4707 + 3491", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2640 + 3713 + ___ = 3713 + 4707 + 3491", + "options": [ + "5578", + "5568", + "5558", + "5548" + ], + "answer": "5558", + "correct_value": 5558, + "explanation": "B cancels; blank = C + D − A = 4707 + 3491 − 2640 = 5558." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1801 + 7567 + ___ = 5857 + 7386", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1801 + 7567 + ___ = 5857 + 7386", + "options": [ + "3875", + "3895", + "3885", + "3865" + ], + "answer": "3875", + "correct_value": 3875, + "explanation": "No matching terms; blank = 5857 + 7386 − 1801 − 7567 = 3875." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000736", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3230 × (5915 + 1005) = 3230 × ___ + 3230 × 1005", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3230 × (5915 + 1005) = 3230 × ___ + 3230 × 1005", + "options": [ + "5905", + "5935", + "5925", + "5915" + ], + "answer": "5915", + "correct_value": 5915, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5915." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4420 + 2094 + ___ = 2094 + 1970 + 7723", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4420 + 2094 + ___ = 2094 + 1970 + 7723", + "options": [ + "5293", + "5263", + "5273", + "5283" + ], + "answer": "5273", + "correct_value": 5273, + "explanation": "B cancels; blank = C + D − A = 1970 + 7723 − 4420 = 5273." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3625 + 4972 + ___ = 7410 + 3612", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 3625 + 4972 + ___ = 7410 + 3612", + "options": [ + "2445", + "2425", + "2435", + "2415" + ], + "answer": "2425", + "correct_value": 2425, + "explanation": "No matching terms; blank = 7410 + 3612 − 3625 − 4972 = 2425." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000737", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6741 × 2155 × 4707 = ___ × 4707 × 6741", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6741 × 2155 × 4707 = ___ × 4707 × 6741", + "options": [ + "2145", + "2165", + "2155", + "2175" + ], + "answer": "2155", + "correct_value": 2155, + "explanation": "A and C appear on both sides; blank = B = 2155." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3022 + 3487 + ___ = 3487 + 4372 + 8264", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3022 + 3487 + ___ = 3487 + 4372 + 8264", + "options": [ + "9614", + "9604", + "9634", + "9624" + ], + "answer": "9614", + "correct_value": 9614, + "explanation": "B cancels; blank = C + D − A = 4372 + 8264 − 3022 = 9614." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6365 + 1176 + ___ = 2921 + 9529", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6365 + 1176 + ___ = 2921 + 9529", + "options": [ + "4919", + "4909", + "4899", + "4929" + ], + "answer": "4909", + "correct_value": 4909, + "explanation": "No matching terms; blank = 2921 + 9529 − 6365 − 1176 = 4909." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000738", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4804 + 8180 = ___ + 5895", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 4804 + 8180 = ___ + 5895", + "options": [ + "7079", + "7099", + "7109", + "7089" + ], + "answer": "7089", + "correct_value": 7089, + "explanation": "Blank = A + B − C = 4804 + 8180 − 5895 = 7089." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3517 + 2153 + ___ = 2153 + 6593 + 5458", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3517 + 2153 + ___ = 2153 + 6593 + 5458", + "options": [ + "8524", + "8554", + "8544", + "8534" + ], + "answer": "8534", + "correct_value": 8534, + "explanation": "B cancels; blank = C + D − A = 6593 + 5458 − 3517 = 8534." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2848 + 8643 + ___ = 7978 + 3759", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2848 + 8643 + ___ = 7978 + 3759", + "options": [ + "266", + "246", + "256", + "236" + ], + "answer": "246", + "correct_value": 246, + "explanation": "No matching terms; blank = 7978 + 3759 − 2848 − 8643 = 246." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000739", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2451 + 6940 + ___ = 6940 + 4953 + 2451", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 2451 + 6940 + ___ = 6940 + 4953 + 2451", + "options": [ + "4953", + "4963", + "4973", + "4943" + ], + "answer": "4953", + "correct_value": 4953, + "explanation": "A and B appear on both sides and cancel; blank = C = 4953." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7908 + 1928 + ___ = 1928 + 2723 + 5590", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7908 + 1928 + ___ = 1928 + 2723 + 5590", + "options": [ + "405", + "395", + "415", + "425" + ], + "answer": "405", + "correct_value": 405, + "explanation": "B cancels; blank = C + D − A = 2723 + 5590 − 7908 = 405." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7875 + 5116 + ___ = 6843 + 2361", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7875 + 5116 + ___ = 6843 + 2361", + "options": [ + "-3773", + "-3783", + "-3803", + "-3787" + ], + "answer": "-3787", + "correct_value": -3787, + "explanation": "No matching terms; blank = 6843 + 2361 − 7875 − 5116 = -3787." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000740", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9109 × 4627 × 8106 = ___ × 8106 × 9109", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9109 × 4627 × 8106 = ___ × 8106 × 9109", + "options": [ + "4617", + "4637", + "4627", + "4647" + ], + "answer": "4627", + "correct_value": 4627, + "explanation": "A and C appear on both sides; blank = B = 4627." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7157 + 9502 + ___ = 9502 + 7268 + 1689", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7157 + 9502 + ___ = 9502 + 7268 + 1689", + "options": [ + "1810", + "1790", + "1820", + "1800" + ], + "answer": "1800", + "correct_value": 1800, + "explanation": "B cancels; blank = C + D − A = 7268 + 1689 − 7157 = 1800." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2555 + 2924 + ___ = 4909 + 6909", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2555 + 2924 + ___ = 4909 + 6909", + "options": [ + "6359", + "6339", + "6349", + "6329" + ], + "answer": "6339", + "correct_value": 6339, + "explanation": "No matching terms; blank = 4909 + 6909 − 2555 − 2924 = 6339." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000741", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8212 × (8548 + 9564) = 8212 × ___ + 8212 × 9564", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8212 × (8548 + 9564) = 8212 × ___ + 8212 × 9564", + "options": [ + "8538", + "8568", + "8558", + "8548" + ], + "answer": "8548", + "correct_value": 8548, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 8548." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1420 + 4539 + ___ = 4539 + 3961 + 1395", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 1420 + 4539 + ___ = 4539 + 3961 + 1395", + "options": [ + "3926", + "3946", + "3936", + "3956" + ], + "answer": "3936", + "correct_value": 3936, + "explanation": "B cancels; blank = C + D − A = 3961 + 1395 − 1420 = 3936." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7752 + 1464 + ___ = 8481 + 5153", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7752 + 1464 + ___ = 8481 + 5153", + "options": [ + "4418", + "4428", + "4408", + "4438" + ], + "answer": "4418", + "correct_value": 4418, + "explanation": "No matching terms; blank = 8481 + 5153 − 7752 − 1464 = 4418." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000742", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5074 × (5532 + 8307) = 5074 × ___ + 5074 × 8307", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5074 × (5532 + 8307) = 5074 × ___ + 5074 × 8307", + "options": [ + "5522", + "5552", + "5542", + "5532" + ], + "answer": "5532", + "correct_value": 5532, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5532." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5311 + 3631 + ___ = 3631 + 4430 + 9966", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5311 + 3631 + ___ = 3631 + 4430 + 9966", + "options": [ + "9085", + "9095", + "9075", + "9105" + ], + "answer": "9085", + "correct_value": 9085, + "explanation": "B cancels; blank = C + D − A = 4430 + 9966 − 5311 = 9085." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6710 + 3818 + ___ = 9189 + 2456", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6710 + 3818 + ___ = 9189 + 2456", + "options": [ + "1107", + "1117", + "1127", + "1137" + ], + "answer": "1117", + "correct_value": 1117, + "explanation": "No matching terms; blank = 9189 + 2456 − 6710 − 3818 = 1117." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000743", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6010 × (2659 + 2481) = 6010 × ___ + 6010 × 2481", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6010 × (2659 + 2481) = 6010 × ___ + 6010 × 2481", + "options": [ + "2659", + "2679", + "2669", + "2649" + ], + "answer": "2659", + "correct_value": 2659, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 2659." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4050 + 7270 + ___ = 7270 + 4312 + 5336", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4050 + 7270 + ___ = 7270 + 4312 + 5336", + "options": [ + "5598", + "5618", + "5608", + "5588" + ], + "answer": "5598", + "correct_value": 5598, + "explanation": "B cancels; blank = C + D − A = 4312 + 5336 − 4050 = 5598." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8172 + 6922 + ___ = 3608 + 3761", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8172 + 6922 + ___ = 3608 + 3761", + "options": [ + "-7715", + "-7735", + "-7725", + "-7705" + ], + "answer": "-7725", + "correct_value": -7725, + "explanation": "No matching terms; blank = 3608 + 3761 − 8172 − 6922 = -7725." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000744", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6087 × 9292 × 5853 = ___ × 5853 × 6087", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6087 × 9292 × 5853 = ___ × 5853 × 6087", + "options": [ + "9312", + "9292", + "9282", + "9302" + ], + "answer": "9292", + "correct_value": 9292, + "explanation": "A and C appear on both sides; blank = B = 9292." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4254 + 2529 + ___ = 2529 + 2483 + 7629", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4254 + 2529 + ___ = 2529 + 2483 + 7629", + "options": [ + "5848", + "5868", + "5858", + "5878" + ], + "answer": "5858", + "correct_value": 5858, + "explanation": "B cancels; blank = C + D − A = 2483 + 7629 − 4254 = 5858." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9729 + 6796 + ___ = 7500 + 3179", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 9729 + 6796 + ___ = 7500 + 3179", + "options": [ + "-5834", + "-5844", + "-5846", + "-5864" + ], + "answer": "-5846", + "correct_value": -5846, + "explanation": "No matching terms; blank = 7500 + 3179 − 9729 − 6796 = -5846." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000745", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6305 + 4683 + ___ = 4683 + 8250 + 6305", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6305 + 4683 + ___ = 4683 + 8250 + 6305", + "options": [ + "8260", + "8270", + "8240", + "8250" + ], + "answer": "8250", + "correct_value": 8250, + "explanation": "A and B appear on both sides and cancel; blank = C = 8250." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5485 + 1431 + ___ = 1431 + 9101 + 4677", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 5485 + 1431 + ___ = 1431 + 9101 + 4677", + "options": [ + "8283", + "8313", + "8293", + "8303" + ], + "answer": "8293", + "correct_value": 8293, + "explanation": "B cancels; blank = C + D − A = 9101 + 4677 − 5485 = 8293." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8214 + 4538 + ___ = 3381 + 1203", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8214 + 4538 + ___ = 3381 + 1203", + "options": [ + "-8162", + "-8182", + "-8168", + "-8152" + ], + "answer": "-8168", + "correct_value": -8168, + "explanation": "No matching terms; blank = 3381 + 1203 − 8214 − 4538 = -8168." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000746", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8191 + 7791 = ___ + 4149", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 8191 + 7791 = ___ + 4149", + "options": [ + "11853", + "11843", + "11833", + "11823" + ], + "answer": "11833", + "correct_value": 11833, + "explanation": "Blank = A + B − C = 8191 + 7791 − 4149 = 11833." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7342 + 5284 + ___ = 5284 + 6963 + 5318", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 7342 + 5284 + ___ = 5284 + 6963 + 5318", + "options": [ + "4939", + "4959", + "4949", + "4929" + ], + "answer": "4939", + "correct_value": 4939, + "explanation": "B cancels; blank = C + D − A = 6963 + 5318 − 7342 = 4939." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4845 + 6245 + ___ = 9853 + 6850", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 4845 + 6245 + ___ = 9853 + 6850", + "options": [ + "5623", + "5603", + "5633", + "5613" + ], + "answer": "5613", + "correct_value": 5613, + "explanation": "No matching terms; blank = 9853 + 6850 − 4845 − 6245 = 5613." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + } + }, + { + "family_id": "equation_reasoning_000747", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5045 × (9819 + 3109) = 5045 × ___ + 5045 × 3109", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5045 × (9819 + 3109) = 5045 × ___ + 5045 × 3109", + "options": [ + "9809", + "9829", + "9839", + "9819" + ], + "answer": "9819", + "correct_value": 9819, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 9819." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9771 + 6136 + ___ = 6136 + 2717 + 4732", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 9771 + 6136 + ___ = 6136 + 2717 + 4732", + "options": [ + "-2318", + "-2322", + "-2338", + "-2308" + ], + "answer": "-2322", + "correct_value": -2322, + "explanation": "B cancels; blank = C + D − A = 2717 + 4732 − 9771 = -2322." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8623 + 5409 + ___ = 1871 + 8540", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 8623 + 5409 + ___ = 1871 + 8540", + "options": [ + "-3621", + "-3609", + "-3619", + "-3639" + ], + "answer": "-3621", + "correct_value": -3621, + "explanation": "No matching terms; blank = 1871 + 8540 − 8623 − 5409 = -3621." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000748", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3807 + 1362 + ___ = 1362 + 8347 + 3807", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 3807 + 1362 + ___ = 1362 + 8347 + 3807", + "options": [ + "8337", + "8367", + "8357", + "8347" + ], + "answer": "8347", + "correct_value": 8347, + "explanation": "A and B appear on both sides and cancel; blank = C = 8347." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3422 + 2044 + ___ = 2044 + 6806 + 2071", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3422 + 2044 + ___ = 2044 + 6806 + 2071", + "options": [ + "5455", + "5465", + "5445", + "5475" + ], + "answer": "5455", + "correct_value": 5455, + "explanation": "B cancels; blank = C + D − A = 6806 + 2071 − 3422 = 5455." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1451 + 9041 + ___ = 8058 + 3184", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 1451 + 9041 + ___ = 8058 + 3184", + "options": [ + "770", + "760", + "750", + "740" + ], + "answer": "750", + "correct_value": 750, + "explanation": "No matching terms; blank = 8058 + 3184 − 1451 − 9041 = 750." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000749", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9821 × (5991 + 6415) = 9821 × ___ + 9821 × 6415", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 9821 × (5991 + 6415) = 9821 × ___ + 9821 × 6415", + "options": [ + "5991", + "5981", + "6011", + "6001" + ], + "answer": "5991", + "correct_value": 5991, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 5991." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3363 + 9841 + ___ = 9841 + 6152 + 1111", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 3363 + 9841 + ___ = 9841 + 6152 + 1111", + "options": [ + "3890", + "3910", + "3920", + "3900" + ], + "answer": "3900", + "correct_value": 3900, + "explanation": "B cancels; blank = C + D − A = 6152 + 1111 − 3363 = 3900." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2960 + 2163 + ___ = 2042 + 9679", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 2960 + 2163 + ___ = 2042 + 9679", + "options": [ + "6608", + "6598", + "6618", + "6588" + ], + "answer": "6598", + "correct_value": 6598, + "explanation": "No matching terms; blank = 2042 + 9679 − 2960 − 2163 = 6598." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000750", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5076 × 3938 × 3969 = ___ × 3969 × 5076", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 5076 × 3938 × 3969 = ___ × 3969 × 5076", + "options": [ + "3928", + "3948", + "3958", + "3938" + ], + "answer": "3938", + "correct_value": 3938, + "explanation": "A and C appear on both sides; blank = B = 3938." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2957 + 6436 + ___ = 6436 + 2459 + 2542", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 2957 + 6436 + ___ = 6436 + 2459 + 2542", + "options": [ + "2044", + "2054", + "2064", + "2034" + ], + "answer": "2044", + "correct_value": 2044, + "explanation": "B cancels; blank = C + D − A = 2459 + 2542 − 2957 = 2044." + }, + "distractor_types": [ + "correct", + "nearby_same_digit", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7677 + 7247 + ___ = 8772 + 3269", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 7677 + 7247 + ___ = 8772 + 3269", + "options": [ + "-2877", + "-2897", + "-2883", + "-2867" + ], + "answer": "-2883", + "correct_value": -2883, + "explanation": "No matching terms; blank = 8772 + 3269 − 7677 − 7247 = -2883." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "equation_reasoning_000751", + "category": "equation_reasoning", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6818 × (3906 + 8051) = 6818 × ___ + 6818 × 8051", + "ns_shortcut": "Recognize commutativity/distributive law — matching terms cancel.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What value fills the blank? 6818 × (3906 + 8051) = 6818 × ___ + 6818 × 8051", + "options": [ + "3896", + "3906", + "3916", + "3926" + ], + "answer": "3906", + "correct_value": 3906, + "explanation": "Distributive law: A×(B+C) = A×B + A×C; blank = B = 3906." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4023 + 5268 + ___ = 5268 + 7576 + 4286", + "ns_shortcut": "Partial cancellation: B cancels but remaining terms must be computed.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What value fills the blank? 4023 + 5268 + ___ = 5268 + 7576 + 4286", + "options": [ + "7859", + "7839", + "7829", + "7849" + ], + "answer": "7839", + "correct_value": 7839, + "explanation": "B cancels; blank = C + D − A = 7576 + 4286 − 4023 = 7839." + }, + "distractor_types": [ + "nearby_same_digit", + "correct", + "nearby_same_digit", + "nearby_same_digit" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6565 + 8401 + ___ = 2540 + 9583", + "ns_shortcut": "No algebraic shortcut; compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What value fills the blank? 6565 + 8401 + ___ = 2540 + 9583", + "options": [ + "-2837", + "-2857", + "-2843", + "-2827" + ], + "answer": "-2843", + "correct_value": -2843, + "explanation": "No matching terms; blank = 2540 + 9583 − 6565 − 8401 = -2843." + }, + "distractor_types": [ + "nearby_same_digit", + "nearby_same_digit", + "correct", + "nearby_same_digit" + ] + } + }, + { + "family_id": "option_elimination_000752", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4077 × 6150", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4077 × 6150?", + "options": [ + "25073551", + "25073552", + "25073550", + "25073549" + ], + "answer": "25073550", + "correct_value": 25073550, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8240 × 8382", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8240 × 8382?", + "options": [ + "69067681", + "69067679", + "69067680", + "69067670" + ], + "answer": "69067680", + "correct_value": 69067680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6329 × 7169", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6329 × 7169?", + "options": [ + "53862601", + "53092601", + "44992601", + "45372601" + ], + "answer": "45372601", + "correct_value": 45372601, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000753", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6918 × 8652", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6918 × 8652?", + "options": [ + "598545432", + "125694519", + "5985450", + "59854536" + ], + "answer": "59854536", + "correct_value": 59854536, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7335 × 4896", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7335 × 4896?", + "options": [ + "35912161", + "35912140", + "35912159", + "35912160" + ], + "answer": "35912160", + "correct_value": 35912160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6129 × 3929", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6129 × 3929?", + "options": [ + "22120841", + "24080841", + "22320841", + "27220841" + ], + "answer": "24080841", + "correct_value": 24080841, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000754", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1220 × 5134", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1220 × 5134?", + "options": [ + "6263483", + "6263479", + "6263481", + "6263480" + ], + "answer": "6263480", + "correct_value": 6263480, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7810 × 8136", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7810 × 8136?", + "options": [ + "63542159", + "63542160", + "63542170", + "63542161" + ], + "answer": "63542160", + "correct_value": 63542160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3947 × 4463", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3947 × 4463?", + "options": [ + "18675461", + "17615461", + "14985461", + "17575461" + ], + "answer": "17615461", + "correct_value": 17615461, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000755", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4893 × 1523", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4893 × 1523?", + "options": [ + "7452040", + "7452042", + "7452038", + "7452039" + ], + "answer": "7452039", + "correct_value": 7452039, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3210 × 3880", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3210 × 3880?", + "options": [ + "12454780", + "12454799", + "12454801", + "12454800" + ], + "answer": "12454800", + "correct_value": 12454800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9357 × 5973", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9357 × 5973?", + "options": [ + "63209361", + "55889361", + "57259361", + "63039361" + ], + "answer": "55889361", + "correct_value": 55889361, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000756", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1864 × 4190", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1864 × 4190?", + "options": [ + "7810160", + "781016", + "14839304", + "14058288" + ], + "answer": "7810160", + "correct_value": 7810160, + "explanation": "Product has 7 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2190 × 7082", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2190 × 7082?", + "options": [ + "15509570", + "15509580", + "15509581", + "15509579" + ], + "answer": "15509580", + "correct_value": 15509580, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2745 × 9369", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2745 × 9369?", + "options": [ + "24647905", + "24967905", + "30457905", + "25717905" + ], + "answer": "25717905", + "correct_value": 25717905, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000757", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3083 × 4139", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3083 × 4139?", + "options": [ + "12760537", + "12760540", + "12760536", + "12760538" + ], + "answer": "12760537", + "correct_value": 12760537, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2635 × 7916", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2635 × 7916?", + "options": [ + "20858661", + "20858659", + "20858660", + "20858680" + ], + "answer": "20858660", + "correct_value": 20858660, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8927 × 2827", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8927 × 2827?", + "options": [ + "21436629", + "29656629", + "25236629", + "26726629" + ], + "answer": "25236629", + "correct_value": 25236629, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000758", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7525 × 5474", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7525 × 5474?", + "options": [ + "41191853", + "41191849", + "41191851", + "41191850" + ], + "answer": "41191850", + "correct_value": 41191850, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5660 × 3744", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5660 × 3744?", + "options": [ + "21191020", + "21191041", + "21191039", + "21191040" + ], + "answer": "21191040", + "correct_value": 21191040, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1649 × 9833", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1649 × 9833?", + "options": [ + "16214617", + "17424617", + "14514617", + "16024617" + ], + "answer": "16214617", + "correct_value": 16214617, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000759", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7167 × 1638", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7167 × 1638?", + "options": [ + "11739545", + "11739546", + "11739548", + "11739547" + ], + "answer": "11739546", + "correct_value": 11739546, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4760 × 9768", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4760 × 9768?", + "options": [ + "46495681", + "46495679", + "46495690", + "46495680" + ], + "answer": "46495680", + "correct_value": 46495680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7949 × 7647", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7949 × 7647?", + "options": [ + "60786003", + "68796003", + "53716003", + "57036003" + ], + "answer": "60786003", + "correct_value": 60786003, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000760", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5927 × 1249", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5927 × 1249?", + "options": [ + "7402822", + "7402826", + "7402823", + "7402824" + ], + "answer": "7402823", + "correct_value": 7402823, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3475 × 1180", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3475 × 1180?", + "options": [ + "4100499", + "4100510", + "4100501", + "4100500" + ], + "answer": "4100500", + "correct_value": 4100500, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6167 × 3645", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6167 × 3645?", + "options": [ + "20748715", + "22478715", + "20618715", + "23088715" + ], + "answer": "22478715", + "correct_value": 22478715, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000761", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4770 × 3207", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4770 × 3207?", + "options": [ + "15297389", + "15297391", + "15297390", + "15297392" + ], + "answer": "15297390", + "correct_value": 15297390, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1810 × 3370", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1810 × 3370?", + "options": [ + "6099701", + "6099700", + "6099680", + "6099699" + ], + "answer": "6099700", + "correct_value": 6099700, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1433 × 1427", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1433 × 1427?", + "options": [ + "2303891", + "2213891", + "1839891", + "2044891" + ], + "answer": "2044891", + "correct_value": 2044891, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000762", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8898 × 4584", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 8898 × 4584?", + "options": [ + "40788431", + "40788432", + "40788434", + "40788433" + ], + "answer": "40788432", + "correct_value": 40788432, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8310 × 9680", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8310 × 9680?", + "options": [ + "80440800", + "80440801", + "80440799", + "80440820" + ], + "answer": "80440800", + "correct_value": 80440800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9873 × 5635", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9873 × 5635?", + "options": [ + "55634355", + "48734355", + "66614355", + "61604355" + ], + "answer": "55634355", + "correct_value": 55634355, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000763", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1621 × 2615", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1621 × 2615?", + "options": [ + "4238918", + "4238914", + "4238916", + "4238915" + ], + "answer": "4238915", + "correct_value": 4238915, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6955 × 7216", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6955 × 7216?", + "options": [ + "50187290", + "50187279", + "50187281", + "50187280" + ], + "answer": "50187280", + "correct_value": 50187280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5867 × 4971", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5867 × 4971?", + "options": [ + "25344857", + "24414857", + "26274857", + "29164857" + ], + "answer": "29164857", + "correct_value": 29164857, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000764", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8805 × 4756", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 8805 × 4756?", + "options": [ + "41876583", + "41876579", + "41876581", + "41876580" + ], + "answer": "41876580", + "correct_value": 41876580, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1555 × 8380", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1555 × 8380?", + "options": [ + "13030899", + "13030901", + "13030900", + "13030880" + ], + "answer": "13030900", + "correct_value": 13030900, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1461 × 9049", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1461 × 9049?", + "options": [ + "14200589", + "13220589", + "15160589", + "13640589" + ], + "answer": "13220589", + "correct_value": 13220589, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000765", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1104 × 5487", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1104 × 5487?", + "options": [ + "6057649", + "6057647", + "6057650", + "6057648" + ], + "answer": "6057648", + "correct_value": 6057648, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8530 × 8644", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8530 × 8644?", + "options": [ + "73733320", + "73733319", + "73733300", + "73733321" + ], + "answer": "73733320", + "correct_value": 73733320, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9071 × 8363", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9071 × 8363?", + "options": [ + "88380773", + "86000773", + "80270773", + "75860773" + ], + "answer": "75860773", + "correct_value": 75860773, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000766", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4077 × 6150", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4077 × 6150?", + "options": [ + "25073551", + "25073552", + "25073550", + "25073549" + ], + "answer": "25073550", + "correct_value": 25073550, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "8240 × 8382", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 8240 × 8382?", + "options": [ + "69067681", + "69067679", + "69067680", + "69067670" + ], + "answer": "69067680", + "correct_value": 69067680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6329 × 7169", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6329 × 7169?", + "options": [ + "53862601", + "53092601", + "44992601", + "45372601" + ], + "answer": "45372601", + "correct_value": 45372601, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000767", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6918 × 8652", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6918 × 8652?", + "options": [ + "598545432", + "125694519", + "5985450", + "59854536" + ], + "answer": "59854536", + "correct_value": 59854536, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7335 × 4896", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7335 × 4896?", + "options": [ + "35912161", + "35912140", + "35912159", + "35912160" + ], + "answer": "35912160", + "correct_value": 35912160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6129 × 3929", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6129 × 3929?", + "options": [ + "22120841", + "24080841", + "22320841", + "27220841" + ], + "answer": "24080841", + "correct_value": 24080841, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000768", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4950 × 8282", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4950 × 8282?", + "options": [ + "40995903", + "40995901", + "40995899", + "40995900" + ], + "answer": "40995900", + "correct_value": 40995900, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6270 × 3586", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6270 × 3586?", + "options": [ + "22484219", + "22484221", + "22484220", + "22484210" + ], + "answer": "22484220", + "correct_value": 22484220, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1629 × 2035", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1629 × 2035?", + "options": [ + "3323015", + "3315015", + "3335015", + "3670015" + ], + "answer": "3315015", + "correct_value": 3315015, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000769", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4071 × 6447", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4071 × 6447?", + "options": [ + "2624571", + "26245737", + "262457376", + "262457440" + ], + "answer": "26245737", + "correct_value": 26245737, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7835 × 7120", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7835 × 7120?", + "options": [ + "55785200", + "55785199", + "55785201", + "55785220" + ], + "answer": "55785200", + "correct_value": 55785200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3949 × 6659", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3949 × 6659?", + "options": [ + "25336391", + "27986391", + "23706391", + "26296391" + ], + "answer": "26296391", + "correct_value": 26296391, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000770", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3282 × 8889", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3282 × 8889?", + "options": [ + "29173700", + "29173697", + "29173698", + "29173699" + ], + "answer": "29173698", + "correct_value": 29173698, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6300 × 3428", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6300 × 3428?", + "options": [ + "21596399", + "21596410", + "21596401", + "21596400" + ], + "answer": "21596400", + "correct_value": 21596400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8573 × 3255", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8573 × 3255?", + "options": [ + "27905115", + "28025115", + "23195115", + "30245115" + ], + "answer": "27905115", + "correct_value": 27905115, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000771", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5562 × 7803", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5562 × 7803?", + "options": [ + "43400288", + "43400285", + "43400286", + "43400287" + ], + "answer": "43400286", + "correct_value": 43400286, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3220 × 4794", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3220 × 4794?", + "options": [ + "15436679", + "15436670", + "15436680", + "15436681" + ], + "answer": "15436680", + "correct_value": 15436680, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9029 × 4643", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9029 × 4643?", + "options": [ + "34681647", + "41921647", + "47521647", + "44771647" + ], + "answer": "41921647", + "correct_value": 41921647, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000772", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2726 × 5890", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2726 × 5890?", + "options": [ + "3211228", + "16056140", + "1605614", + "1605614062" + ], + "answer": "16056140", + "correct_value": 16056140, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7015 × 8432", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7015 × 8432?", + "options": [ + "59150480", + "59150481", + "59150460", + "59150479" + ], + "answer": "59150480", + "correct_value": 59150480, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8137 × 6445", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8137 × 6445?", + "options": [ + "51942965", + "54592965", + "55742965", + "52442965" + ], + "answer": "52442965", + "correct_value": 52442965, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000773", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5656 × 1959", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5656 × 1959?", + "options": [ + "9972084", + "1108005", + "110801089", + "11080104" + ], + "answer": "11080104", + "correct_value": 11080104, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9530 × 7204", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9530 × 7204?", + "options": [ + "68654100", + "68654120", + "68654121", + "68654119" + ], + "answer": "68654120", + "correct_value": 68654120, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8843 × 3025", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8843 × 3025?", + "options": [ + "26750075", + "31650075", + "28600075", + "31740075" + ], + "answer": "26750075", + "correct_value": 26750075, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000774", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8968 × 8292", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 8968 × 8292?", + "options": [ + "148725306", + "156161571", + "141289041", + "74362656" + ], + "answer": "74362656", + "correct_value": 74362656, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2985 × 6260", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2985 × 6260?", + "options": [ + "18686101", + "18686099", + "18686090", + "18686100" + ], + "answer": "18686100", + "correct_value": 18686100, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3359 × 3871", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3359 × 3871?", + "options": [ + "11432689", + "13002689", + "12312689", + "14772689" + ], + "answer": "13002689", + "correct_value": 13002689, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000775", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "5559 × 9837", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 5559 × 9837?", + "options": [ + "54683883", + "54683882", + "54683885", + "54683884" + ], + "answer": "54683883", + "correct_value": 54683883, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7515 × 6352", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7515 × 6352?", + "options": [ + "47735280", + "47735279", + "47735281", + "47735300" + ], + "answer": "47735280", + "correct_value": 47735280, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6763 × 9965", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6763 × 9965?", + "options": [ + "67833295", + "67393295", + "66413295", + "64953295" + ], + "answer": "67393295", + "correct_value": 67393295, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000776", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6135 × 7717", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6135 × 7717?", + "options": [ + "47343795", + "47343796", + "47343798", + "47343794" + ], + "answer": "47343795", + "correct_value": 47343795, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3570 × 1412", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3570 × 1412?", + "options": [ + "5040839", + "5040840", + "5040850", + "5040841" + ], + "answer": "5040840", + "correct_value": 5040840, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4761 × 9455", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4761 × 9455?", + "options": [ + "39715255", + "37465255", + "42125255", + "45015255" + ], + "answer": "45015255", + "correct_value": 45015255, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000777", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "7521 × 4423", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 7521 × 4423?", + "options": [ + "33265382", + "33265386", + "33265383", + "33265384" + ], + "answer": "33265383", + "correct_value": 33265383, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2895 × 9642", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2895 × 9642?", + "options": [ + "27913591", + "27913600", + "27913590", + "27913589" + ], + "answer": "27913590", + "correct_value": 27913590, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8473 × 6827", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8473 × 6827?", + "options": [ + "63685171", + "58625171", + "47495171", + "57845171" + ], + "answer": "57845171", + "correct_value": 57845171, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000778", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1220 × 5134", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1220 × 5134?", + "options": [ + "6263483", + "6263479", + "6263481", + "6263480" + ], + "answer": "6263480", + "correct_value": 6263480, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "7810 × 8136", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 7810 × 8136?", + "options": [ + "63542159", + "63542160", + "63542170", + "63542161" + ], + "answer": "63542160", + "correct_value": 63542160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3947 × 4463", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3947 × 4463?", + "options": [ + "18675461", + "17615461", + "14985461", + "17575461" + ], + "answer": "17615461", + "correct_value": 17615461, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000779", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6325 × 3054", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6325 × 3054?", + "options": [ + "19316551", + "19316550", + "19316549", + "19316553" + ], + "answer": "19316550", + "correct_value": 19316550, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5240 × 2594", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5240 × 2594?", + "options": [ + "13592550", + "13592561", + "13592560", + "13592559" + ], + "answer": "13592560", + "correct_value": 13592560, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6361 × 8057", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6361 × 8057?", + "options": [ + "45500577", + "47680577", + "55870577", + "51250577" + ], + "answer": "51250577", + "correct_value": 51250577, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000780", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "8020 × 2628", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 8020 × 2628?", + "options": [ + "21076560", + "210765613", + "2107656056", + "2107656" + ], + "answer": "21076560", + "correct_value": 21076560, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6365 × 6990", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6365 × 6990?", + "options": [ + "44491349", + "44491360", + "44491351", + "44491350" + ], + "answer": "44491350", + "correct_value": 44491350, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1475 × 9751", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1475 × 9751?", + "options": [ + "15392725", + "16922725", + "11932725", + "14382725" + ], + "answer": "14382725", + "correct_value": 14382725, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000781", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1459 × 1977", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1459 × 1977?", + "options": [ + "2884443", + "2884444", + "2884442", + "2884446" + ], + "answer": "2884443", + "correct_value": 2884443, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2115 × 1132", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2115 × 1132?", + "options": [ + "2394181", + "2394179", + "2394180", + "2394190" + ], + "answer": "2394180", + "correct_value": 2394180, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1529 × 5743", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1529 × 5743?", + "options": [ + "9260047", + "8781047", + "9884047", + "9469047" + ], + "answer": "8781047", + "correct_value": 8781047, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000782", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1133 × 1067", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1133 × 1067?", + "options": [ + "241775", + "1208911", + "120883", + "12089157" + ], + "answer": "1208911", + "correct_value": 1208911, + "explanation": "Product has 7 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2730 × 9160", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2730 × 9160?", + "options": [ + "25006801", + "25006799", + "25006800", + "25006780" + ], + "answer": "25006800", + "correct_value": 25006800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7655 × 6267", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7655 × 6267?", + "options": [ + "42983885", + "57303885", + "45823885", + "47973885" + ], + "answer": "47973885", + "correct_value": 47973885, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000783", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2598 × 7416", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2598 × 7416?", + "options": [ + "19266770", + "19266768", + "19266767", + "19266769" + ], + "answer": "19266768", + "correct_value": 19266768, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3345 × 2034", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3345 × 2034?", + "options": [ + "6803750", + "6803730", + "6803731", + "6803729" + ], + "answer": "6803730", + "correct_value": 6803730, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2571 × 6841", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2571 × 6841?", + "options": [ + "17588211", + "17758211", + "15968211", + "15858211" + ], + "answer": "17588211", + "correct_value": 17588211, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000784", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6660 × 8455", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6660 × 8455?", + "options": [ + "101358540", + "112620600", + "56310300", + "118251630" + ], + "answer": "56310300", + "correct_value": 56310300, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2330 × 7234", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2330 × 7234?", + "options": [ + "16855230", + "16855221", + "16855219", + "16855220" + ], + "answer": "16855220", + "correct_value": 16855220, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3761 × 1059", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3761 × 1059?", + "options": [ + "3284899", + "4549899", + "3908899", + "3982899" + ], + "answer": "3982899", + "correct_value": 3982899, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000785", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6056 × 4809", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6056 × 4809?", + "options": [ + "2912330471", + "2912325", + "2912330441", + "29123304" + ], + "answer": "29123304", + "correct_value": 29123304, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "6210 × 8004", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 6210 × 8004?", + "options": [ + "49704850", + "49704839", + "49704841", + "49704840" + ], + "answer": "49704840", + "correct_value": 49704840, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7127 × 7331", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7127 × 7331?", + "options": [ + "59388037", + "51908037", + "52248037", + "58928037" + ], + "answer": "52248037", + "correct_value": 52248037, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000786", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3972 × 8996", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3972 × 8996?", + "options": [ + "35732112", + "3573211248", + "7146416", + "357321137" + ], + "answer": "35732112", + "correct_value": 35732112, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3600 × 5672", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3600 × 5672?", + "options": [ + "20419190", + "20419199", + "20419201", + "20419200" + ], + "answer": "20419200", + "correct_value": 20419200, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "6925 × 7753", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 6925 × 7753?", + "options": [ + "53689525", + "58779525", + "60889525", + "54819525" + ], + "answer": "53689525", + "correct_value": 53689525, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000787", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3567 × 7987", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3567 × 7987?", + "options": [ + "28489629", + "28489630", + "28489628", + "28489632" + ], + "answer": "28489629", + "correct_value": 28489629, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4010 × 6726", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4010 × 6726?", + "options": [ + "26971261", + "26971259", + "26971260", + "26971240" + ], + "answer": "26971260", + "correct_value": 26971260, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7953 × 6161", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7953 × 6161?", + "options": [ + "39698433", + "47548433", + "48998433", + "39668433" + ], + "answer": "48998433", + "correct_value": 48998433, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000788", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2243 × 8255", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2243 × 8255?", + "options": [ + "18515964", + "18515965", + "18515966", + "18515968" + ], + "answer": "18515965", + "correct_value": 18515965, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1875 × 4468", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1875 × 4468?", + "options": [ + "8377501", + "8377520", + "8377499", + "8377500" + ], + "answer": "8377500", + "correct_value": 8377500, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7461 × 1355", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7461 × 1355?", + "options": [ + "10699655", + "10109655", + "11959655", + "11729655" + ], + "answer": "10109655", + "correct_value": 10109655, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000789", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3121 × 1321", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3121 × 1321?", + "options": [ + "4122842", + "4122840", + "4122841", + "4122844" + ], + "answer": "4122841", + "correct_value": 4122841, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9430 × 6180", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9430 × 6180?", + "options": [ + "58277400", + "58277399", + "58277401", + "58277390" + ], + "answer": "58277400", + "correct_value": 58277400, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "7373 × 4227", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 7373 × 4227?", + "options": [ + "30475671", + "31455671", + "32725671", + "31165671" + ], + "answer": "31165671", + "correct_value": 31165671, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000790", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4420 × 5326", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4420 × 5326?", + "options": [ + "23540919", + "23540920", + "23540921", + "23540923" + ], + "answer": "23540920", + "correct_value": 23540920, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1610 × 9938", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1610 × 9938?", + "options": [ + "16000179", + "16000181", + "16000180", + "16000200" + ], + "answer": "16000180", + "correct_value": 16000180, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3365 × 2025", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3365 × 2025?", + "options": [ + "7273125", + "6677125", + "6814125", + "6075125" + ], + "answer": "6814125", + "correct_value": 6814125, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000791", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4610 × 7330", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4610 × 7330?", + "options": [ + "3379130046", + "33791300", + "6758260", + "3379130077" + ], + "answer": "33791300", + "correct_value": 33791300, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2605 × 5854", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2605 × 5854?", + "options": [ + "15249650", + "15249670", + "15249669", + "15249671" + ], + "answer": "15249670", + "correct_value": 15249670, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "3765 × 6065", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 3765 × 6065?", + "options": [ + "26494725", + "22834725", + "26064725", + "24244725" + ], + "answer": "22834725", + "correct_value": 22834725, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000792", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "4893 × 1523", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 4893 × 1523?", + "options": [ + "7452040", + "7452042", + "7452038", + "7452039" + ], + "answer": "7452039", + "correct_value": 7452039, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3210 × 3880", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3210 × 3880?", + "options": [ + "12454780", + "12454799", + "12454801", + "12454800" + ], + "answer": "12454800", + "correct_value": 12454800, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "9357 × 5973", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 9357 × 5973?", + "options": [ + "63209361", + "55889361", + "57259361", + "63039361" + ], + "answer": "55889361", + "correct_value": 55889361, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000793", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "3395 × 6722", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 3395 × 6722?", + "options": [ + "22821191", + "22821190", + "22821189", + "22821193" + ], + "answer": "22821190", + "correct_value": 22821190, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9985 × 9804", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9985 × 9804?", + "options": [ + "97892941", + "97892960", + "97892940", + "97892939" + ], + "answer": "97892940", + "correct_value": 97892940, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2251 × 9459", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2251 × 9459?", + "options": [ + "25332209", + "23342209", + "21292209", + "23812209" + ], + "answer": "21292209", + "correct_value": 21292209, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000794", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2277 × 4068", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2277 × 4068?", + "options": [ + "9262835", + "9262837", + "9262836", + "9262838" + ], + "answer": "9262836", + "correct_value": 9262836, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "5415 × 7006", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 5415 × 7006?", + "options": [ + "37937491", + "37937490", + "37937510", + "37937489" + ], + "answer": "37937490", + "correct_value": 37937490, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "2459 × 1543", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 2459 × 1543?", + "options": [ + "3907237", + "3794237", + "3474237", + "3604237" + ], + "answer": "3794237", + "correct_value": 3794237, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000795", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "9405 × 8910", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 9405 × 8910?", + "options": [ + "83798551", + "83798549", + "83798550", + "83798552" + ], + "answer": "83798550", + "correct_value": 83798550, + "explanation": "Product divisible by 9 (both factors divisible by 3); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "4260 × 6314", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 4260 × 6314?", + "options": [ + "26897620", + "26897641", + "26897640", + "26897639" + ], + "answer": "26897640", + "correct_value": 26897640, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8371 × 2153", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8371 × 2153?", + "options": [ + "15482763", + "16052763", + "18712763", + "18022763" + ], + "answer": "18022763", + "correct_value": 18022763, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + } + }, + { + "family_id": "option_elimination_000796", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6125 × 6048", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6125 × 6048?", + "options": [ + "37044000", + "37043999", + "37044003", + "37044001" + ], + "answer": "37044000", + "correct_value": 37044000, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3560 × 8686", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3560 × 8686?", + "options": [ + "30922160", + "30922159", + "30922170", + "30922161" + ], + "answer": "30922160", + "correct_value": 30922160, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8169 × 2525", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8169 × 2525?", + "options": [ + "20626725", + "23906725", + "23006725", + "22406725" + ], + "answer": "20626725", + "correct_value": 20626725, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000797", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6777 × 6015", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6777 × 6015?", + "options": [ + "4076361", + "4076365567", + "40763655", + "8152727" + ], + "answer": "40763655", + "correct_value": 40763655, + "explanation": "Product has 8 digits; only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "3925 × 5618", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 3925 × 5618?", + "options": [ + "22050670", + "22050651", + "22050649", + "22050650" + ], + "answer": "22050650", + "correct_value": 22050650, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "5275 × 3233", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 5275 × 3233?", + "options": [ + "17054075", + "20244075", + "20144075", + "16344075" + ], + "answer": "17054075", + "correct_value": 17054075, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000798", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1760 × 7480", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1760 × 7480?", + "options": [ + "13164803", + "13164800", + "13164801", + "13164799" + ], + "answer": "13164800", + "correct_value": 13164800, + "explanation": "Product ends in 0 (5 × even); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "2760 × 5986", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 2760 × 5986?", + "options": [ + "16521360", + "16521380", + "16521361", + "16521359" + ], + "answer": "16521360", + "correct_value": 16521360, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "8329 × 7465", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 8329 × 7465?", + "options": [ + "62175985", + "61435985", + "68395985", + "52275985" + ], + "answer": "62175985", + "correct_value": 62175985, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000799", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "1077 × 7895", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 1077 × 7895?", + "options": [ + "8502918", + "8502915", + "8502914", + "8502916" + ], + "answer": "8502915", + "correct_value": 8502915, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9135 × 5574", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9135 × 5574?", + "options": [ + "50918470", + "50918489", + "50918490", + "50918491" + ], + "answer": "50918490", + "correct_value": 50918490, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4975 × 7349", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4975 × 7349?", + "options": [ + "36561275", + "38921275", + "39561275", + "34621275" + ], + "answer": "36561275", + "correct_value": 36561275, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000800", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "6809 × 9143", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 6809 × 9143?", + "options": [ + "62254688", + "62254686", + "62254687", + "62254690" + ], + "answer": "62254687", + "correct_value": 62254687, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "1090 × 5974", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 1090 × 5974?", + "options": [ + "6511661", + "6511660", + "6511640", + "6511659" + ], + "answer": "6511660", + "correct_value": 6511660, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "control": { + "task_type": "control", + "math_expression": "4073 × 6239", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 4073 × 6239?", + "options": [ + "25411447", + "28471447", + "23491447", + "30431447" + ], + "answer": "25411447", + "correct_value": 25411447, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "correct", + "distractor", + "distractor", + "distractor" + ] + } + }, + { + "family_id": "option_elimination_000801", + "category": "option_elimination", + "strong_shortcut": { + "task_type": "strong_shortcut", + "math_expression": "2531 × 7515", + "ns_shortcut": "Check trailing digit / parity / magnitude — only one option matches.", + "shortcut_strength": 1.0, + "pure_math": { + "question": "What is 2531 × 7515?", + "options": [ + "19020468", + "19020465", + "19020466", + "19020464" + ], + "answer": "19020465", + "correct_value": 19020465, + "explanation": "Product odd (odd × odd); only one option matches." + }, + "distractor_types": [ + "distractor", + "correct", + "distractor", + "distractor" + ] + }, + "weak_shortcut": { + "task_type": "weak_shortcut", + "math_expression": "9035 × 8546", + "ns_shortcut": "Trailing digit narrows options but does not uniquely identify the answer.", + "shortcut_strength": 0.5, + "pure_math": { + "question": "What is 9035 × 8546?", + "options": [ + "77213111", + "77213100", + "77213109", + "77213110" + ], + "answer": "77213110", + "correct_value": 77213110, + "explanation": "Feature narrows to 2 options; eliminates some but not all distractors." + }, + "distractor_types": [ + "distractor", + "distractor", + "distractor", + "correct" + ] + }, + "control": { + "task_type": "control", + "math_expression": "1451 × 9671", + "ns_shortcut": "All options have same structural features; must compute precisely.", + "shortcut_strength": 0.0, + "pure_math": { + "question": "What is 1451 × 9671?", + "options": [ + "16752621", + "11452621", + "14032621", + "12302621" + ], + "answer": "14032621", + "correct_value": 14032621, + "explanation": "All options have same structural features; must compute precisely." + }, + "distractor_types": [ + "distractor", + "distractor", + "correct", + "distractor" + ] + } + } +] \ No newline at end of file