Reasoning · Grade 2-1 Making Shapes

Problem

Fold and cut paper, predict shapes

A square sheet is folded in half so the fold becomes its left edge. A shape is cut from the folded sheet, always touching the fold. In the example a triangle cut opened into a diamond hole. Find the hole that appears for cuts (1), (2) and (3).
Your answer
How to solve
Strategy Visualize Spatial Relationships — Fold-and-cut is a mirror problem: the fold line is a line of symmetry, so the hole is the cut piece plus its mirror image across the fold. The cleanest way to predict it is to mentally (or physically) unfold and reflect each cut across the fold (tool 17, helped by actually folding scrap paper, tool 10). We first re-read the given Example (tool 9, an easier solved case) to confirm the rule 'point on the fold becomes the far tip of the doubled shape', then draw each reflected result (tool 1).
1STEP 1

Read the rule from the worked Example

The fold is a mirror, so reflecting the cut across it shows the hole.

2STEP 2

Cut (1): triangle on the fold

Cut (1) has one corner off the fold, which mirrors into two.

1 off-fold corner → 2 after mirroring = 3-sided shape
3STEP 3

Cut (1) result: a downward-pointing triangle

The top edges join up, so (1) opens into a downward triangle.

4STEP 4

Cut (2): four-cornered piece on the fold

Cut (2) has two corners off the fold, mirroring into four.

2 off-fold corners → 4, plus the bottom tip = 5 sides
5STEP 5

Cut (2) result: a pentagon

Adding the tip on the fold makes (2) a pentagon.

6STEP 6

Cut (3): stepped plus/T piece on the fold

In (3) the right tab gains a mirror tab, forming a horizontal bar.

vertical bar + horizontal bar through the middle = plus sign
7STEP 7

Cut (3) result: a plus (cross) shape

Crossing the centre bar, (3) opens into a plus shape.

Answer
(1) triangle (2) pentagon (3) plus shape
Each result is symmetric about the vertical fold line, which must be true for any fold-and-cut hole, and matches the Example's rule (the on-fold V-point doubled into a diamond). The side counts are consistent with doubling: cut (1) had 1 off-fold corner - ＞ a 3-sided triangle; cut (2) had 2 off-fold corners plus a bottom tip - ＞ a 5-sided pentagon; cut (3)'s stepped right-angle piece - ＞ a right-angled plus sign. All three holes sit centered on the fold, as expected.
Takeaway

When you fold then cut, the fold is a mirror, so just flip the cut to the other side and you can see the hole before you even open the paper!

  • Read the rule from the worked Example
  • Cut (1): triangle on the fold
  • Cut (1) result: a downward-pointing triangle
  • Cut (2): four-cornered piece on the fold
  • Cut (2) result: a pentagon
  • Cut (3): stepped plus/T piece on the fold
  • Cut (3) result: a plus (cross) shape