Punch, Fold, Count
A spatial-visualisation station gives each student one sheet: a 16 cm by 16 cm square. Students fold the top edge down onto the bottom edge, crease, then fold the left edge across onto the right edge, crease, and hold the packet flat. The packet is now an 8 cm by 8 cm square sitting in the bottom-right quarter of the original sheet, with four layers.
A single circular punch, 1 cm across, is then pushed through all layers at a marked point. Two different points are used in successive rounds. Point P is 2 cm from the packet's right edge and 2 cm from its bottom edge. Point Q is on the packet's left edge, 2 cm above its bottom edge - that edge is a crease, not a cut edge. Students then unfold completely and count separate holes. The most common answers are 2, 4 and 8. The station sheet says only 'count the holes you actually see', which is the source of the argument.
1. For punch P, give the number of holes and their positions relative to the original square.
Read the worked answer
Four holes, at the corners of a rectangle inset 2 cm from every edge of the 16 cm sheet.
Take the bottom-left corner of the original sheet as (0,0). The punch sits at (14,2) in the packet, which is the bottom-right quarter. Unfold the left-right crease at x = 8 and the hole mirrors to (2,2). Unfold the top-bottom crease at y = 8 and both holes mirror to (14,14) and (2,14).
The count is neither two nor eight. Each fold doubles the stack, so two folds give four layers, and a punch through the stack makes one hole per layer: four. Eight would require three folds. The mirror lines are the creases, at x = 8 and y = 8, not the paper's outer edges - that substitution is the usual error, and it is what produces answers scattered across the sheet rather than clustered near the corners.
2. For punch Q, how many separate holes does the unfolded sheet show? Explain the difference from P.
Read the worked answer
Two holes. Point Q lies on the left edge of the packet, and that edge is the crease made by the second fold, at x = 8. A punch centred on a crease does not produce an independent hole per layer: it removes a half-disc from the layers on each side of the crease, and when those layers unfold the two half-discs lie edge to edge and form one complete hole centred on x = 8 at y = 2. The same happens in the top half, giving a second hole at (8,14).
The layer count is still four, so nothing about the folding has changed; the holes merge in pairs, so the visible count halves. Keep the two questions apart: how many layers were pierced (four), and how many separate holes the unfolded sheet shows (two). The station's instruction to 'count the holes you actually see' is fair, but only if students first decide whether the punch sits on a crease.
3. A student argues: 'The punch is 1 cm across, so a hole formed on a crease should be 2 cm across.' Evaluate that claim.
Read the worked answer
Wrong, and the error is worth naming precisely. The punch is a 1 cm circle, and the material removed from all four layers together is that same 1 cm circle, redistributed among the layers.
On the crease, one layer keeps the part of the disc on one side of the crease, and the facing layer contributes the part on the other side. Those two parts are the two halves of the original circle, so they reassemble into a hole 1 cm across, not 2 cm.
A 2 cm hole would require two separate punches about 1 cm apart, one on each side of the crease, leaving a strip of paper between them that then separates; or two half-holes deliberately placed side by side rather than mirror-imaged.
A useful check the student can run: folding and unfolding never changes the total area of paper removed, only how that area is arranged across the sheet. Doubling the diameter would quadruple the area, which folding cannot do.