The Grid With a Blank Bottom-Right Cell
Mr. Bell draws a three-by-three grid and fills in eight cells. Top row: a triangle with one dot (unshaded), a triangle with two dots (shaded), a triangle with three dots (unshaded). Middle row: a square with two dots (shaded), a square with three dots (unshaded), a square with four dots (shaded). Bottom row: a pentagon with three dots (unshaded), a pentagon with four dots (shaded), and one blank cell at the bottom right. Mr. Bell asks what belongs in the blank. Ana says, 'A pentagon with five dots, unshaded.' Dev says, 'A hexagon with five dots, unshaded - the number of sides goes up by one each time you move right, so the third column needs six sides.' Before anyone votes, Mr. Bell asks the class to test each claim against every figure already on the board, not just the ones nearest the blank.
1. Test Dev's rule - 'sides go up by one moving right' - against the whole grid. What happens?
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Dev's rule fails on the first row it touches.
Moving right along the top row, the sides go 3, 3, 3 - not 3, 4, 5. Along the middle row they go 4, 4, 4, and along the bottom row 5, 5, and then the blank. The side count never changes as you move right; it changes as you move down: 3, 4, 5 down the first column, and the same down the second and third.
Dev looked only at the cells closest to the blank, which is exactly the trap Mr. Bell warned about. A rule that is invented at the edge of a grid has to be carried back to the far side and checked there. One failed check is enough to reject it - you do not average the evidence, you look for a counterexample and stop.
2. Give the figure that belongs in the blank, and defend it with two independent checks.
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A pentagon with five dots, unshaded.
Row check: in the bottom row the outer shape stays a pentagon, the dots go 3, 4, so the next is 5, and the shading runs unshaded, shaded, unshaded.
Column check: in the third column the sides go 3, 4, 5, the dots go 3, 4, 5, and the shading runs unshaded, shaded, unshaded. Same answer.
The two checks matter because they are not the same reasoning. A grid question can be solved by reading rows or by reading columns, and a correct cell satisfies both. If your row rule gives five dots and your column rule gives something else, you have misread a cell, and the fix is to re-read the board rather than to pick the answer you like better.
3. A third student says the answer is 'a pentagon with five dots, but shaded.' Which rule does that break, and where on the board is the proof?
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It breaks the shading pattern, and the proof is the right-hand column already on the board.
The third figure in the top row is unshaded (the triangle with three dots) and the third figure in the middle row is unshaded (the square with four dots). So the third column runs unshaded, shaded, unshaded, and the blank must be unshaded. The same pattern shows up along every row: first unshaded, second shaded, third unshaded.
This is the most common way to lose a mark on a figure grid. The student got the shape right, the dot count right, and threw the answer away on shading because shading was checked last, in a hurry. A reliable method is to name all three attributes - shape, count, fill - and confirm each one separately before writing the answer down.