The Rotating Notch Matrix
A 3×3 matrix of geometric figures is incomplete. Each cell contains a shape (triangle, square, or circle), a shading pattern (unshaded, left-half shaded, or fully shaded), and a small notch on one side. The completed cells are: Row 1-triangle, unshaded, notch top; square, left-half shaded, notch right; circle, fully shaded, notch bottom. Row 2-square, left-half shaded, notch top; circle, fully shaded, notch right; triangle, unshaded, notch bottom. Row 3-circle, fully shaded, notch top; triangle, unshaded, notch right; [missing]. A student says the missing cell must be a square, left-half shaded, with notch bottom because each row contains one of each shape and one of each shading. Another student says it must be a square, left-half shaded, with notch top because the notch rotates clockwise down each column. The coach asks which proposal is correct and why the other reasoning is incomplete.
1. Identify the missing figure and justify it with at least two patterns.
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Missing figure: square, left-half shaded, notch on the bottom. Shape: each row contains triangle, square, and circle once; row 3 already has circle and triangle, so it needs square. Shading: each row contains unshaded, left-half shaded, and fully shaded once; row 3 already has fully shaded and unshaded, so it needs left-half shaded. Notch: each column has a constant notch side-column 1 top, column 2 right, column 3 bottom-so the missing cell takes the bottom notch. The two row patterns and the column pattern agree.
2. The second student says the notch rotates clockwise down each column. Test that claim against the given cells.
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The claim fails immediately. Down column 1 the notches are top, top, top-no rotation. Down column 2 they are right, right, right-also no rotation. Down column 3 they are bottom, bottom, and therefore bottom. The notch is constant within a column, not rotating down it. The student may have confused “down each column” with “across each row”: across rows 1 and 2 the notch does go top → right → bottom, but that is a left-to-right rotation, not a top-to-bottom one.
3. What extra cell would you need in order to distinguish “constant by column” from “rotates left-to-right within a row”?
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You would need a row whose first notch is not top. In the given rows, both rules happen to produce top → right → bottom, so they are observationally equivalent. If a new row began with a right notch, “rotates left-to-right” would predict right → bottom → left, while “constant by column” would keep the column’s existing notch (top, right, or bottom). A single such cell in a fourth row would separate the two rules.