Sixteen square units, two pieces: when area checking stops being enough
A table holds flat plastic pieces cut on a 1 cm grid. The task card shows a 4 cm by 4 cm square target - 16 square units - and says: fill it exactly with two pieces, no gaps and nothing hanging over the edge. Five pieces are offered. A is a 2-by-4 rectangle, 8 units. B is a 4-by-2 rectangle, 8 units. C is an L made from a 3-by-3 square with one corner unit cut away, 8 units. D is a 2-by-3 rectangle, 6 units. E is a 4-by-4 square with a six-unit L removed, 10 units. The child counts each piece correctly, announces that A and B are the only two with 8 units, then sets A and C together because 'C is also eight.' The pair overlaps along one row and leaves two units of the target bare. She retries A with C in a different rotation for four minutes, then asks whether the pieces are broken.
1. Show why C cannot pair with A to fill the square, without relying on trial and error.
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Area is necessary and not sufficient. A is 8 and C is 8, so the areas sum to 16 and pass the first filter - which is exactly why the child's reasoning stalled there.
The second filter is complement shape: place one piece and ask whether the leftover is the other piece. Put A, a 2-by-4 rectangle, against the left edge of the 4-by-4 target. The leftover is a clean 2-by-4 rectangle, which is B. Now put C into a corner of the target instead. C's footprint is not a rectangle, so the leftover region is 8 units with a notch in it: its outline is the mirror of C's outline, an L, and no rotation of a 2-by-4 rectangle becomes an L. The pair fails however it is turned.
The general rule to teach: two equal-area pieces tile a square only when the leftover after placing the first is congruent to the second. The area test narrows the field; the shape test decides it.
2. The child's first move was to sort by size. Is that a strength or a weakness?
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Both, and it is worth naming precisely which.
Sorting by area shows she has conserved the target's area and understands that the pieces must total 16 under unit counting - a real insight at this age, and one many children miss by judging pieces purely by how big they look. That should be credited.
The weakness appears the moment area is treated as a decision rule rather than a filter. When she said 'C is also eight' she stopped testing shape and spent four minutes rotating a piece that could never work.
The fix is one added question, asked before any piece is placed: 'if this one goes here, what shape is left?' With A on the table she can trace the leftover with a finger and see a rectangle; with C she sees a notch. One prompt converts area-sorting from a stopping rule into a first step.
For diagnostic purposes, note that her four minutes were not wasted: she was systematically varying rotation, which is the right search behaviour aimed at the wrong candidate.
3. The correct pair is two congruent rectangles, one merely rotated. What misconception could make a child reject it, and how should the teacher respond?
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A child who assumes the two pieces in a puzzle must be different shapes will reject A with B on sight, even after passing the area check - the pair looks like cheating, like using the same piece twice. That assumption is imported from jigsaw puzzles, where duplicate pieces never occur; the task card says nothing about the pieces needing to be distinct.
The response is physical and quick. Have her stack A on B and see that they match, then place one and rotate the other into position while saying the arithmetic aloud: eight plus eight is sixteen, and the region left after A is exactly B's shape. Then generalise the rule: what matters is that the outlines complement one another, not that the pieces differ.
This distinction matters for interpretation. A child who rejects the correct pair for this reason and then settles on A with C has not made an area error at all - she has made an assumption about puzzles. Reading it as a spatial error would send the remediation in the wrong direction.