Three Folds, Three Miscounts: Paper Folding Without the Paper
Nia, a tenth grader, works through six paper-folding items at the kitchen table using real paper, because the actual test shows only drawings of a folded sheet and four answer choices. Trial 1: she folds a square sheet in half top to bottom, creases it, folds it in half again left to right, creases it, then punches one hole straight through the packet at a point near the corner where the two open edges meet. She predicts 8 holes and counts 4. Trial 2: she repeats the folds and punches on the folded edge instead, 2 cm from the corner where the two creases meet; she predicts 4 and counts 2. Trial 3: she punches exactly at that crease corner; she predicts 2 and counts 1. All three predictions are exactly twice what the paper shows, and she concludes that 'the real test must count holes differently.' She has 20 minutes and wants a rule she can apply to drawings alone.
1. Explain the correct count for trial 1 from the layer structure, and diagnose Nia's consistent doubling.
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Two folds, each halving the sheet, produce 2 × 2 = 4 layers in the packet. A punch near an open corner passes through all four layers at four different places on the original sheet, so unfolding gives 4 separate holes, one in each corner region, arranged symmetrically about both midlines.
Her 8 is exactly double, and the same factor appears in all three trials (8 vs 4, 4 vs 2, 2 vs 1), so this is systematic, not a miscount. The usual cause is counting a hole on each surface of each layer: four layers means eight faces, so the punch seems to make eight holes. But the two faces of one layer are the two sides of a single hole - you count holes in the sheet, not in the stack.
The second common cause is doubling once for each crease on the assumption that a crease always splits a hole. That assumption is only true when the punch misses the crease; trial 2 shows what happens when it does not.
2. Explain trials 2 and 3, then state one rule that covers all three trials.
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Trial 2: the punch is centred on one crease. Unfolding along that crease opens the hole onto itself - the two half-holes on either side of the crease are mirror images that join into one - so the four layer positions collapse in pairs and the count is 2, not 4.
Trial 3: the punch sits at the corner where both creases meet, so unfolding along either crease joins halves twice over. All four positions coincide and the result is 1 hole, at the centre of the sheet.
One rule covers all three: the fold count tells you how many layer positions the punch passes through (2 per halving fold), not how many holes result. Count distinct positions, then halve the total for each crease the punch lies on, because a hole centred on a crease is its own mirror image.
One extra case to keep separate: a punch on an open edge makes a notch on the boundary rather than an interior hole, and answer choices draw it as a partial shape at the paper's edge.
3. Nia wants to memorise 'double for every fold.' Give her the single question to ask on every item and the cases where the rule fails.
Read the worked answer
The question is: does the punch touch a fold line or a folded corner? If it touches nothing, multiply by 2 for each halving fold. If it touches one crease, halve; if it sits at a crease corner, halve again. That question is what makes the count correct in all three trials, and it is answerable from the drawing alone, which is the point.
The memorised rule fails in three situations. First, punches on a crease or crease corner, as in trials 2 and 3 - the rule overcounts. Second, folds that are not halves: folding one corner to the centre produces a triangular flap, so the layers under the punch are not uniform and the punch may pass through only some of them; multiplying by a power of two is then meaningless. Third, diagonal folds, where the creases are diagonals of the square and the mirror symmetry is a reflection across a diagonal, so the holes land in positions a student expecting midline symmetry will not predict.
A good use of her remaining 20 minutes is to draw, not fold: sketch the unfolded square and mark where the punch would land on each layer, then count distinct marks.