Row Constancy vs. Cancellation in a 3-by-3 Matrix
During a fall practice block, a coordinator at Riverside Elementary watches two students disagree over a 3-by-3 matrix item with the bottom-right cell blank. Every cell is built from at most three parts: an outline shape, one straight line, and nothing else. Row 1 holds a square outline crossed by a diagonal line, a square outline crossed by a horizontal line, and then a diagonal line plus a horizontal line with no square. Row 2 holds a circle outline with a vertical line, a circle outline with a slanted line, and then a vertical line plus a slanted line with no circle. Row 3 holds a triangle outline with a horizontal line, a triangle outline with a vertical line, and the blank. The options: A is a triangle with both lines inside, B is a bare triangle, C is a horizontal line crossing a vertical line with no outline shape, and D is a triangle containing one line. Student 1 argues for A: "Every row keeps the same shape, so the last cell has to be a triangle." Student 2 argues for C but cannot explain why. The block ends in four minutes.
1. Which option is correct, and what rule generates the third cell of each row? Show the check that settles the disagreement.
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The rule is cancellation, not constancy: whatever the first two cells share disappears, and whatever appears in only one of them survives into the third cell.
Row 1: the square outline is in both cells, so it cancels; the diagonal appears only in cell 1 and the horizontal only in cell 2, so both survive. The printed cell 3 is exactly a diagonal plus a horizontal with no square, which confirms the rule.
Row 2: the circle cancels; the vertical and the slanted line survive, matching the printed cell 3.
Row 3: the triangle outline is in both cells, so it cancels; the horizontal and vertical lines survive. The missing cell is a horizontal line crossing a vertical line with no outline shape, which is Option C.
Column check: every third-column cell in rows 1 and 2 contains only line combinations and no outline shape, so C fits the column as well. Options A, B and D all retain the triangle, the one element the rule removes.
2. Student 1's rule, "every row keeps the same shape," is not arbitrary. Explain why it looks right, and identify the evidence that defeats it.
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The observation behind it is real: in every visible row, the first two cells share an outline shape (two squares, two circles, two triangles), so the row reads as having a theme. A solver who scans left to right and stops after two cells has a strong, simple prediction, and Option A is drawn to match it exactly.
The defeat is structural. The row-constant-shape pattern describes the inputs, not the output. In rows 1 and 2 the third cell drops the outline shape entirely, so the column, not the row, controls the form of the output. Student 1 is generalizing from two cells and never testing the rule against a cell that is already printed.
The transferable coaching point: once you have formed a rule, apply it to the visible cell in the same row or column before choosing. A rule that fits the first two cells but contradicts the third is not a rule, it is a guess.
3. Write a two-sentence prompt that redirects Student 1 without naming the answer or the rule.
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Cover the blank with your finger, then look at the two cells directly above it. What do all three cells in that column have in common, and does your option keep that?
A second sentence that works as a follow-up: "Now look at what happened to the square in row 1's third cell, and ask whether your triangle survives the same treatment."
Both sentences point at evidence rather than at the answer. They force a column check and a comparison with a completed row, which are the two moves Student 1 skipped. Neither reveals that the shared element cancels, so the student still has to derive the relation, which is the reasoning the item is testing.