This session for the Mathematical Association Annual Conference 2021 presents some ideas for using origami for teaching and learning mathematics.

It covers some mathematics arising from a single fold, fractions, how to trisect a right angle by folding paper and therefore make an equilateral triangle, a six-pointed star, a tetrahedron and an approximation of a regular icosahedron.

**Two basic folds**

A valley fold is concave. A mountain fold is convex. Most origami diagrams used a dashed line for valleys. Mountain fold is usually dash-dot-dot.

**One fold**

Haga’s Turned Up Part (TUP)

https://www.geogebra.org/m/undpadhp

What happens when the bottom right corner goes to the midpoint of the top edge? (Haga theorem).

**Fractions**

Folding in half, quarters, eighths, sixteenths, etc is easy.

How about dividing the edge of a square into thirds?

Crease an A4 sheet into a 4 by 4 grid.

Make a magazine box. What size paper do you need to make two smaller magazine boxes fit inside? What are the ratios of lengths, surface areas and volumes?

Dave Mitchell’s Two-way Tube – which configuration has the greater volume, or are they the same?

__https://www.atm.org.uk/Shop/Learning-Mathematics-with-Origami-book-and-pdf/act0101pk__

**60 degree fold**

Fold the lower right corner to the centre line and *at the same time* make the fold go through the lower left corner.

How could you prove this? What kinds of proof are convincing and give insight? You might find adding some extra lines helpful: an answer is in Why does the “60 degree fold” work?

This is a special case of trisecting an acute angle.

Here are some more tasks using the 60 degree fold taken from LEARNING MATHEMATICS WITH ORIGAMI by Tung Ken Lam and Sue Pope, Association of Teachers of Mathematics, ISBN 9781898611950