Paper Folding: Layer Count and Fold-Line Symmetry
A square sheet of paper is 20 cm on each side. It is folded exactly in half by bringing the left edge to the right edge and creasing the vertical midline; then it is folded in half again by bringing the top edge to the bottom edge and creasing the horizontal midline. With the packet still folded, a single circular hole is punched through all layers. The punch center is 2 cm from the left folded edge and 3 cm from the top folded edge. A student says two holes will appear when the paper is unfolded because there were two folds. Another says four holes because 'each fold doubles the layers.' The teacher asks the group to track the number of layers after each fold, to state whether the punch lies on a crease, and to predict what would change if the punch were centered exactly on the left folded edge but kept 3 cm from the top folded edge.
1. How many holes will the unfolded sheet have, and where will they be? Explain the layer count and symmetry.
Read the worked answer
Four holes. After the first fold the sheet has 2 layers; after the second fold it has 4 layers. A punch through all layers makes one hole per layer if it does not lie on a crease. The punch is 2 cm from the vertical folded edge and 3 cm from the horizontal folded edge, so it is not on either crease. Unfolding reflects the hole across the vertical midline and the horizontal midline, giving four distinct holes at positions 2 cm from the vertical center and 3 cm from the horizontal center, one in each quadrant. The folds do not simply add two holes; they multiply the layers, and the creases determine the symmetry of the copies.
2. Diagnose the 'two folds, two holes' reasoning. What is the correct relationship between folds and holes?
Read the worked answer
The student is treating each fold as adding one hole. In fact, each fold doubles the number of layers in the region being punched. Starting with 1 layer, one fold gives 2 layers; a second gives 4. A punch through all 4 layers therefore makes 4 holes, provided it misses the creases. If the punch hit a crease, two half-holes would join into one hole on unfolding, so the count would drop. The correct relationship is: holes = number of layers penetrated, adjusted for any punch centered on a fold line.
3. If the punch were centered exactly on the left folded edge but still 3 cm from the top folded edge, how many holes would result? Explain.
Read the worked answer
Two holes. The punch now lies on the vertical crease, so the two half-holes on either side of that crease join into one hole when the first fold is opened. The horizontal fold still copies that hole to the other side of the horizontal midline, giving two holes. If the punch were centered on both folded edges, the result would be one hole. This question tests whether the solver is tracking crease positions, not just layer count; a layer-count-only rule would wrongly predict four.