Writing Read-Aloud Scripts for Figure Matrices and Paper Folding
Mr. Whitfield is preparing read-aloud scripts for four nonverbal practice sets, because one student in his group has a read-aloud accommodation and two others are still learning English. Each script must describe a figure precisely enough that only one answer is possible. Script 1, a figure matrix with two rows and two columns: the top-left cell is a square containing one small solid dot at its center; the top-right cell is the same square with the dot touching the middle of the top edge; the bottom-left cell is a square containing two small solid dots, side by side, at its center; the bottom-right cell is missing. Script 2, paper folding: a square sheet is folded in half left to right, then in half top to bottom, then one hole is punched through all layers. Script 3 repeats Script 2, but the hole is punched exactly on the crease made by the second fold. Script 4 repeats Script 2, with the hole punched at the exact center of the folded packet.
1. Supply the missing cell for Script 1, state the rules the matrix runs on, and explain what must be added for the answer to be strictly unique.
Read the worked answer
Answer: a square with two small solid dots, side by side, touching the middle of the top edge.
Rules. Across a row, dot position changes and dot count does not: one dot at center becomes one dot at the top edge. Down a column, dot count doubles and position does not: the centered single dot becomes two centered dots. Applying both to the bottom-right cell gives two dots at the top edge.
Why it is not strictly unique as written. Only three cells are filled. A solver could propose 'the number of dots equals the column plus the row' and put three dots in the bottom-right, and nothing in the three given cells refutes that. The matrix convention - the same transformation along every row and down every column - is what forces the answer, and Mr. Whitfield has to state or imply that convention in the script. Practical fixes: add a third row or column so the doubling rule is confirmed twice, or open the script with 'the same change happens along every row and down every column.' He should also hold the other attributes fixed explicitly ('same size square, dots the same size in every cell'), otherwise the answer could legitimately vary in dot size or spacing.
2. For Scripts 2, 3, and 4, state how many holes appear when the paper is fully unfolded and justify each count from the layer structure.
Read the worked answer
Script 2 - 4 holes. Start with 1 layer; the left-right fold gives 2 layers; the top-bottom fold gives 4. A punch away from every crease passes through 4 distinct points, and unfolding reflects them into a pattern symmetric about both creases: four holes, one in each quadrant.
Script 3 - 2 holes. The punch sits on the crease made by the second fold. Those two layers are mirror images about that crease, so the two punched points coincide when the crease is opened flat. The remaining doubling from the first fold still separates the holes sideways, so 4 layers yield 2 distinct holes.
Script 4 - 1 hole. The exact center of the folded packet is the center of the original square, which lies on both creases. Every layer's punched point maps onto that same point when unfolded, so all four coincide.
The general rule: a punch through n layers gives n holes, minus one coincidence for each crease it lies on. Script 2 has 4 layers and 0 creases; Script 3 has 4 layers and 1 crease; Script 4 has 4 layers and 2 creases. Note that Script 2 would also give 2 holes if the punch happened to land on the first crease - which is the flaw addressed next.
3. Identify which script is genuinely ambiguous for a listener, and rewrite it so the answer cannot vary with an unstated detail.
Read the worked answer
Script 2 is ambiguous, and its ambiguity is fatal because the expected answer of 4 depends on where the hole goes. A listener is told only that 'one hole is punched through all layers.' If the punch lands on either crease, the count drops to 2 or 1. The listener has no way to know which case is intended, so a correct answer is impossible to distinguish from a wrong one.
A secondary ambiguity: 'folded in half left to right' does not say which half goes over which. For a hole count that is irrelevant, but if a later question asks where a hole sits relative to a corner, the direction of each fold changes the answer.
Rewrite: 'Start with a square sheet. Fold the left edge over to meet the right edge and press the crease. Keep it folded, then fold the top edge down to meet the bottom edge and press the crease. You now have a small square with four layers. Punch one hole straight through all four layers at a point about one centimetre in from each edge of the packet and not on any crease. Then open the sheet completely. How many holes are there?'
The rewrite fixes the layer count, fixes the hole's location relative to every crease, and fixes the fold direction - the three details the count depends on.