Matrix Fill and a Punched, Folded Square: Naming the Rule Behind Each Choice
Two nonverbal items are printed on a card. Item one is a three-by-three matrix. In the first row an outlined square, a half-shaded square and a solid square contain 1, 2 and 3 small dots respectively. The second row repeats the same fills and dot counts, but the shape is a triangle in all three cells. The third row begins with an outlined circle holding 1 dot and a half-shaded circle holding 2 dots; the bottom-right cell is blank. Item two reads: a square sheet of paper is folded in half top to bottom, then in half left to right, and one hole is punched all the way through at a point that lies on no fold. Amara answers item one with “a solid circle with one dot,” tracking the fills across the row and stopping there. Ben answers item two with “four holes, one near each corner.” Cass answers “two holes, both on the centre line.” The teacher asks the group to justify each answer by naming the rule that produces it, and to say which parts of the answers are right and which are not.
1. Give the correct bottom-right cell and identify what Amara's answer leaves out.
Read the worked answer
A solid circle containing 3 dots.
The matrix changes attributes along two different directions:
- Down each column the shape stays fixed: column one is squares, column two triangles, column three circles.
- Across each row two attributes advance together: the fill runs outlined, then half-shaded, then solid, and the dot count runs 1, then 2, then 3.
The missing cell sits in the third column, so it must be a circle, and in the third row, so it must carry the third fill and the third dot count.
Amara tracked the fill cycle correctly and chose the right shape, but she carried the dot count from column one instead of advancing it across the row. Her cell would fit the third column only. The check to apply is that the chosen cell must satisfy its row rule and its column rule at the same time. When two attributes vary, a rule that explains one of them is only half a rule, and partial explanations are the most common way a matrix answer goes wrong.
2. Evaluate Ben's and Cass's folding answers. How many holes are there, and where do they fall?
Read the worked answer
Four holes, placed as mirror images of the punch point - but Ben's reasoning about corners is wrong.
Each fold doubles the layers. The top-to-bottom fold gives 2 layers; the left-to-right fold gives 4. One punch through 4 layers makes 4 holes when the paper is opened.
The placement comes from reflection, not from convenience. Unfolding the second fold reflects the punch across the vertical centre line; unfolding the first reflects both of those across the horizontal centre line. The four holes therefore sit one in each quadrant, matching the punch's distance from each fold line. They land near the corners only if the punch was near a corner, so Ben has the count right and the location unjustified.
Cass's count is wrong for the stated punch. Two holes would require the punch to sit exactly on a fold line, so that pairs of holes merge into one. Her answer describes a different, equally valid version of the item, not this one.
3. Suppose the punch had been made exactly on the second fold line. What changes, and why does the layer count stay the same while the hole count drops?
Read the worked answer
The paper is still 4 layers thick, but the opened sheet shows 2 holes.
Layers and holes are separate quantities. The two folds still stack 4 layers, so the punch still passes through 4 thicknesses of paper. But when a punch is centred on a fold line, the two layers joined at that line are mirror halves of the same hole rather than two separate holes. Opening that fold merges the pair into one hole straddling the crease.
With two folds, a punch on the second fold line lies on 2 of the 4 layers in matching pairs, so 4 layer punctures become 2 visible holes, arranged as mirror images across the remaining horizontal fold line.
The general principle: count layers to get the maximum number of holes, then subtract for every punch that lies on a crease, because creases turn pairs of holes into one. Students who memorize “folds double the holes” without the crease exception will get every on-the-fold variant wrong.