503 − 268 = 365: a rule, not a slip
You teach Class V (34 children). A ten-item diagnostic on subtraction shows a clear pattern: 19 children answered 503 − 268 as 365 and 400 − 137 as 337; the same 19 answered 62 − 47 as 25. When you ask one of them, Sonam, to explain, he says, 'Eight minus three is five, six minus zero is six, five minus two is three.' He gets every addition item right and can name the place value of digits up to thousands. In a follow-up interview, given bundles of ten sticks, he solves 62 − 47 correctly by opening a bundle. Your textbook's next chapter is multiplication, and the half-yearly examination is six weeks away.
1. Explain what this pattern is and why it is not carelessness.
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It is a consistent, correctly applied rule: 'in every column, subtract the smaller digit from the larger'. Sonam states the rule aloud and applies it without exception to every column, across three different items, including 62 − 47 = 25. Carelessness produces varied answers; a rule produces the same answer every time.
What has gone wrong is not arithmetic but the concept of exchange. The written algorithm is a recording system for something that happens with quantities: when there are not enough ones, a ten is opened into ten ones. If the algorithm is taught as column instructions ('if the top is smaller, borrow'), children who never saw the exchange behind it invent a rule that needs no exchange - because swapping is easier, always works arithmetically, and was never contradicted by anything they could see.
The interview is the key evidence: with bundles he is correct. So the concept is available in the concrete, and what is missing is the mapping between the concrete action and the written notation. That is a teaching gap, not an ability gap, and it is fixable in weeks - which matters with the examination in six weeks.
2. Design a three-lesson sequence that moves these children from bundles to the written algorithm, including the zero in the minuend.
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Lesson 1 - exchange, recorded. Pairs work 62 − 47 with bundles and a place-value mat. Rule stated aloud: 'I cannot take seven ones from two ones, so I open one ten.' Each move is recorded in a two-column table (tens / ones) beside the bundles. Then the same problem is drawn: cross out a ten, draw ten ones. Finish with three items done as drawings, no digits.
Lesson 2 - drawing beside the digits. Now write 62 − 47 in columns next to the drawing. The crossing-out becomes the small 'borrowed' notation: cross the 6, write 5; write 1 in front of the 2. Then introduce the zero case: 503 − 268. With bundles: there are no tens to open, so open a hundred into ten tens, then open one ten into ten ones. Do it physically first, then as a drawing, then as notation. This is the step textbooks skip and where most of the 19 will fail again.
Lesson 3 - algorithm plus a sense check. Mixed practice, five items, two of them with a zero in the minuend. Add estimation: '503 − 268 is about 500 − 270, which is 230, so 365 cannot be right.' Keep one chart on the wall: 'Never swap. Open a ten.'
3. What will you do with the 15 children who are already correct, and how will you assess the whole class after three lessons?
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The 15 correct children are not given more of the same. First, check they are correct for the right reason: a five-item check including two zero-in-minuend items (503 − 268, 700 − 246) and one item asking them to explain the crossed-out digit in their own words. Some will be procedurally correct but unable to explain - those need Lesson 2's drawing work, not enrichment.
Those who pass become explainers: paired with a child who holds the swap rule, with the instruction to ask 'why did you cross out the five?' rather than to give the answer. Then give them error-analysis work - three wrong worked examples (365, 337, 25) with the task 'find the rule behind the mistake and write the correct solution'. This deepens their place-value understanding and keeps them productively busy while you work with the 19.
Assessment after three lessons: an eight-item written test, of which three items have a zero in the minuend, plus one interview with six children: 'Explain why you crossed out the 5.' Success criterion: zero instances of smaller-from-larger across the class, and every interviewed child can describe the exchange, not just perform it. Score the explanation separately from the answer.