The Larger-Minus-Smaller Error
Mr. Delgado's third graders are subtracting three-digit numbers. On an exit ticket, 8 of 24 students answer 402 − 178 = 336. Their work shows they subtracted the smaller digit from the larger in each column: 2−8 becomes 8−2, 0−7 becomes 7−0, and 4−1 = 3. Six more students regrouped but forgot to reduce the hundreds. The class has base-ten blocks, place-value mats, and a number line. Mr. Delgado has 15 minutes tomorrow before the unit test. He wants to address the misconception without simply reteaching the algorithm.
1. Diagnose the misconception and explain why reteaching the steps alone is unlikely to work.
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The 336 responses show a consistent 'larger minus smaller' digit rule applied without place value. Students are treating digits as independent numbers, not as 4 hundreds, 0 tens, and 2 ones. The six others have a partial algorithm: they regroup but lose track of the reduced hundred. The misconception is conceptual, so reteaching steps will not fix it; students need to see the quantities and the need to decompose.
2. Choose a representation sequence for the 15 minutes and justify the order.
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Start with base-ten blocks for 402: students must trade 1 hundred for 10 tens, then 1 ten for 10 ones before subtracting 178. Record each trade on a place-value mat. Then connect to a number line: start at 402 and subtract 100, 70, and 8, or count up from 178. Finally, compare the block model to the written algorithm, labeling the regrouped values. Use the wrong answer 336 and ask students to prove it with blocks. Sequence: 5 min blocks, 5 min number line, 5 min written connection.
3. Write an assessment prompt that reveals whether students can transfer the idea, and state success criteria.
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Give two new problems and one error-analysis prompt: 'A student says 503 − 267 = 344. Use a model to show why this is wrong.' Success criteria: correct answer, a labeled trade, and an explanation that a hundred was decomposed. Listen for 'I can't subtract 7 from 3, so I trade a ten.' If students still use larger-minus-smaller, reteach in a small group with blocks before the test.