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Non-Verbal Reasoning · Spatial transformations

NVR: paper folding and hole punching

A free preview from the Athena Learning Library — the same lesson registered students learn from.

In this lesson: Predict the hole pattern when folded, punched paper is opened out.

Before you start

Cut a folded paper snowflake and one snip appears again and again when you open it. Punching folded paper follows the same mirroring rule.

Watch it happen

Fold, punch, unfold

A square sheet is folded twice (the right half is folded over onto the left half; then the top half is folded down over the bottom half), punched once at the centre, and opened out to show four holes.

Start with one square sheet of paper, flat.

Step 1 of 4

The idea

Think of the holes backwards. Instead of guessing forwards, work in reverse: start with the punched, folded paper, then undo one fold at a time. Every time you undo a fold, each hole is copied across that fold line, like a reflection in a mirror.

Follow this routine. First, count the layers the punch went through: a punch through four layers makes up to four holes. Next, undo the last fold you made and mirror every hole to the other side of that fold line, keeping the same distance from the line. Then undo the earlier fold and mirror all the holes again.

Watch for the special case: a hole punched exactly on a fold line is not doubled by that fold, because its mirror copy lands on top of itself. Holes on two fold lines, such as the exact centre, stay as one hole. Distance matters too: a hole near a fold opens out near the middle, while a hole near an edge stays near an edge.

Worked example

A square sheet is folded twice and one hole is punched through the folded packet, as the pictures show. When the sheet is opened out, how many holes are there, and where?

A square sheet folded twice (the right half is folded over onto the left half; then the bottom half is folded up over the top half) with one punched hole at the bottom left of the folded packet.
  1. Start flat, then fold 1: the right half is folded over onto the left half. The sheet is now two layers thick.
    A square sheet folded twice (the right half is folded over onto the left half; then the bottom half is folded up over the top half) with one punched hole at the bottom left of the folded packet.
  2. Fold 2: the bottom half is folded up over the top half. Now the folded packet is four layers thick, and one hole is punched through all four layers, at the bottom left of the packet.
    A square sheet folded twice (the right half is folded over onto the left half; then the bottom half is folded up over the top half) with one punched hole at the bottom left of the folded packet.
  3. Unfold in your mind, one fold at a time. Opening the last fold mirrors the hole across that fold line: 2 holes. Opening the first fold mirrors both of those: 4 holes.
    A square sheet folded twice (the right half is folded over onto the left half; then the bottom half is folded up over the top half) with one punched hole at the bottom left of the folded packet.
  4. The opened sheet shows four holes, one in each quarter of the sheet — each fold line acts like a mirror for every hole.
    A square sheet folded twice (the right half is folded over onto the left half; then the bottom half is folded up over the top half) with one punched hole at the bottom left of the folded packet.

Answer: 4 holes — one in each quarter of the sheet.

Try one yourself

This sheet is folded just once (the top half is folded down over the bottom half) and punched once. How many holes when it is opened out?

A square sheet folded once (the top half is folded down over the bottom half) with one punched hole.

Watch out

Quick check

Try these on your own first — then reveal each answer.

1. This sheet is folded three times and punched once. How many holes when it is opened out?

A square sheet folded three times with one punched hole.

2. This sheet is folded twice and punched twice. How many holes when it is opened out?

A square sheet folded twice with punched holes.

Practise this concept

Reading is the first step — the concept sticks when your child answers questions on it. Registered students practise graded questions for this lesson, with instant marking and worked explanations, and the lesson is marked complete in their progress.

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