Symmetry · Usually met around P3
Folding paper, cutting and punching holes
Each fold doubles the layers; opening flips the holes across the crease.
How to do it
Method
Each fold doubles the layers; opening flips the holes across the crease
Folded twice, 2 holes: 2 × 4 = 8 holes
Another way
Open it one fold at a time
2 × 2 = 4, 4 × 2 = 8
In full
A crease is a line of symmetry: opened out, the pattern is always symmetric about the crease. Punching: a hole goes through every layer of the folded paper. Folded once there are 2 layers, folded twice 4, and each fold doubles the layers. So 1 hole makes 2 when the paper is opened from one fold, and 4 from two folds. As it opens, each hole flips across the crease like a mirror image, just as far from it on the other side; it does not slide across. A hole punched right on the crease only cuts half a hole, and the two halves open into 1 hole. Cutting works the same way: half a shape cut along the crease opens into a whole shape symmetric about the crease. Another way: undo it one step at a time; each time a fold opens, mirror every hole across that crease.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A square of paper is folded in half and holes are punched in it as shown (the dashed line is the crease). Opened out, how many holes does it have?
Show the answer and working
Answer: 2 holes
Folded once, the paper has 2 layers and each hole goes through both: 1 × 2 = 2.
Example 2
A square of paper is folded in half and in half again, then punched as shown (the dashed lines are creases). Opened out, how many holes does it have?
Show the answer and working
Answer: 4 holes
Folded twice, the paper has 4 layers and each hole goes through all 4: 1 × 4 = 4.
Example 3
A piece of paper is folded and the shaded part is cut off along the crease, as shown (the dashed line is the crease). Opened out, what shape is the piece?
- A. A rectangle
- B. A right-angled triangle
- C. A trapezium
- D. An isosceles triangle
Choose the right answer
Show the answer and working
Answer: D. An isosceles triangle
The crease becomes the shape's line of symmetry: flip the half you cut over the crease, and the two halves together make the whole shape.
Practise it in the game →Symmetry · Free to start, unlimited questions, no download