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Red paper boat being followed by a flotilla of smaller white paper boats

Origami (from the Japanese for ‘to fold’ + ‘paper’) is used the world over to produce beautiful creations that sit in the fascinating intersection of mathematics and art. I’ve been in a multitude of sessions and workshops where I am in awe of not only the beautiful constructions made possible but also the wonderful maths behind them.

The recent British Congress of Mathematics Educators (BCME 9) was no different. Amongst the sessions I attended I was able to take part in a variety of beautiful tile constructions with Dr Jennie Golding.

One of the shapes we produced was a regular hexagon folded from a piece of A4. It’s a simple process but opens up a world of what ifs? and whys?

Folding the shapes is great for dexterity and hand-eye co-ordination, and then you can investigate the properties of the created shapes and experience the mathematics enacted in a creative activity. Recognising that 3D shapes (and the boundaries of those that are possible) can be created from a 2D material is a useful idea, particularly when you start to consider nets. Visualisation skills are enhanced as you start to predict what is being constructed. Shapes are met in a wide range of orientations and transformations; corresponding vertices, angles and edges can be identified.

As pupils meet various angle properties, congruency, similarity, trigonometry and different-but-equally-valid constructions can be investigated and justified. Initial constructions can be communicated in a variety of ways: written or spoken instructions, diagrams, a video or a combination. Getting pupils to communicate their own constructions in any format also gives them an opportunity to experience the challenges of designing clear instructions for others.

Now there are lots of ways to construct a regular hexagon, but I particularly liked this one as it didn’t rely on any use of trigonometry or the ratio of side of A4 in the explanation of why it worked. In fact, I’m going to go several steps back and show how to construct an angle of 60º using any rectangular sheet of paper.

Here’s how to do it…

Getting someone to explain why they know this angle has to be 60º just by looking at the final construction starts an important mathematical conversation. You can see that the angle constructed has been made with three layers of paper overlapping exactly and therefore each angle is 1/3 of 180º. But why does this work? How does this work? Do you think it will be challenging to prove?

Well actually – no. Surprisingly, the only facts that the explanation requires are that the interior angles in a triangle sum to 180º and side-angle-side congruency. If you’d like to pause at this point and see if you can construct the proof yourself, have a go!

Here’s my explanation – in diagrams, with as few words as possible:

In fact I would hypothesise (I haven’t tried it out yet – but would love to hear from someone if they do) that a surprising number of Key Stage 3 pupils (and maybe even younger) would be able to show why this works. Some of those might then be on a journey to folding a rhombus (repeat this process on the left hand side), an equilateral triangle (fold the rhombus in half; there’s a great discussion here whether all rhombi fold in half to give equilateral triangles, if not what do they fold in half to give - and why does this one result in an equilateral triangle?) and a regular hexagon (fold the two acute angles of the rhombus to the centre of the original rectangle).

A final thing to think about with this construction is: what are the ratios of paper for which this wouldn’t work? What about the ratios for which a rhombus is not constructed by repeating the construction on the right? Or the ratio of paper for which the second fold meets the centre line (the final picture in the initial instructions) and hence the middle fold in the sheet is the length of one diagonal?

These type of paper folding activities are an ideal environment in which to practice and develop spatial reasoning; the ability to see, to inspect and reflect on spatial objects, images, relationships and transformations. It is the act of generating images, and inspecting images to answer questions, that provides the “input” and critical cognitive tools for pupils to carry out formal geometric analysis (Battista, M, 2007).

Battista, M.T., 2007. The Development of Geometric and Spatial Thinking, in: Lester, F.K. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning: A Project of the National Council of Teachers of Mathematics. Information Age, pp. 843–908.

SOMETHING TO TRY:

KS1: Give pupils lots of paper circles to work with. Ask “How can you fold a circle to find it’s centre? What’s the minimum number of folds you must make? Why does this work?”

KS2: Give pupils instructions in a variety of formats explaining how to fold a square from a rectangular piece of paper.

Discuss which instructions were easiest to follow and why? Ask for an explanation as to why this produces a square? Would it work if you started with a different shape, such as a kite?

Check out the Origami Resource Centre for some ideas.

KS3: Use the construction of a 60° angle described above. Pupils can either try to explain why the angle folded is 60° or they could be given the series of photos above and asked either to put them in order and/or to write the mathematical explanation for each picture.

KS4/5: A size paper is in the ratio of 1:√2. Explain why the following folding results in a kite that is a cyclic quadrilateral.

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Articles: mathematics, teacher-authored, outside USA

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