Unfold backwards, one crease at a time

Two formats, one skill

Fold and punch gives you a square being folded, a hole punched through the folded paper, and asks for the unfolded square.

Fold along the line gives you a figure with a dotted crease and asks for the folded result.

Different pictures, and exactly the same question underneath: can you apply a reflection accurately, in your head, without slipping into a rotation?

Reflection. Not rotation, not translation. That one distinction decides most of these.

Unfold backwards

Start from the folded state and reverse the last fold first. Reflect every hole across that crease, so each hole gains a mirror partner. Then reverse the crease before it, and do the same again.

With a true half fold you get a clean doubling each time: one hole becomes two, two become four, four become eight.

The important words there are true half fold.

When the holes do not double

This is the part that catches people.

If a fold is not exactly in half, the paper is not doubled everywhere. It is two layers thick only where the flap actually lands. Everywhere else it is still a single layer.

So a hole punched in the covered part comes out as two holes, and a hole punched outside that region comes out as one.

Fold along the line: flip the flap

The dotted line is the crease, and the smaller part is the flap. The flap reflects across the crease; the part that stays put does not move.

Then eliminate on three checks:

Is it genuinely reflected, rather than slid across or turned round? Has the crease stayed exactly on the dotted line? Is the flap the same shape and the same size?

In the exam, eliminate rather than construct. It is faster and it produces fewer mistakes.

Where do pupils lose marks?

Partial unfolding — reversing one crease and stopping. That option is genuinely half right, which is exactly why it is tempting.

Rotation instead of reflection. On a symmetrical figure those give the same answer, so the question will use an asymmetric one.

A drifting crease — the right shape reflected across the wrong line.

Assuming everything doubles when the fold was not a true half fold.

How can we practise this at home?

Paper and a hole punch, and nothing else. Fold, punch, and — before opening it — ask how many holes there will be and roughly where. Then open it.

The prediction is the practice. Opening the paper without a prediction first teaches nothing, because the answer arrives before the thinking does.

Then the version that teaches the most: fold it unevenly on purpose, punch through, and ask again. The moment a child sees three holes instead of four is the moment the layer rule stops being a rule and starts being obvious.