Symmetry
Trihexaflexagon
Activity Guide
OVERVIEW
The trihexaflexagon activity is about starting conversations about maths. Students can learn to
make one at school and share it with their friends and family/carers.
It can be easily tailored to suit a range of year groups and abilities. You might like to present simple
concepts of equilateral triangles, hexagons, or focus on symmetry. The document ‘The symmetry
of the Trihexaflexagon illustrated’ provides details of how the triangles move when flexed.
This document outlines one way of presenting the trihexaflexagon and follows the video script for
the lesson presented on the National Literacy and Numeracy Website – Trihexaflexagon page.
PREPARATION
 Decide how you will tailor this activity to the class, will it focus on shapes and patterns, or
symmetry.
 Decide and print off enough templates for each student. The A4 pages have two templates
per page.
EQUIPMENT
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One template for each student plus spare ones to cover any mishaps or extension
activities.
Dry glue.
Pencils, as required.
Lesson plan – Introduction to students (Year 4-5)
“We are going to look at some very weird maths today - weird because it doesn’t look like maths.
Let me present to you - a flexagon!”
[Reveal the face of a trihexaflexagon]
“So can you see the face, how many sides does it have? What do we call a six-sided shape?
A hexagon because hexa is a very old word for six. So this is called a hexaflexagon, hexa as in
six and the flexagon because we can bend it or flex it.
So if we flex this shape in a special way we can get it to a point and if we have a look it pops open
to ANOTHER FACE!”
[Demonstrate how it flexes to another face and continue working through the faces - HAPPY
MONSTER, CAT until back to the start]
“So it has three faces and if we have three of something - like a shape with three sides it’s called a
triangle or if it’s an athletic contest with three events we call it a triathlon… so tri is an old word
meaning “three” and hexa is an old word meaning “six” so this is called a trihexaflexagon.”
[Demonstrate the three different faces moving back and forth so it actually shows six different
faces]
“OK so when I am flexing my face to reveal the happy monster, where is my face going? Around
the back!”
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© Simon Pampena - free for educational use within Australia
“Yes. But what has happened to the serious face of the boy? It is gone – replaced by this happy
face! So what do you expect is behind the monster face? A SAD MONSTER!
And if we go to our last picture, Mr Cat. What happens if we flex it around, what do we see?
IT’S A BIRD!”
So what is going on? How does this happen?
“OK so now let’s look at how the faces change around.
If we start with the serious face of the boy, when we pinch the sides together in a flex we can see
that three diamond shapes come together. And, then, when we turn it around, each of the
diamonds have turned upside down.
What do you think is going on? What happens to the mouth? What happens to the left eye? What
happens to the right eye?
Let’s try moving around the one of the faces on the whiteboard. So where does the mouth go? So
where does the left eye go? So where does the right eye go?
It works because our faces have mirror-symmetry or are bi-symmetrical. That means if you look in
the mirror, you can recognise yourself. That’s because the left half of your face is the mirror image
of the right half of your face.
Now let’s look at the monster face. Draw in some eyebrows. So if you do some upward strokes
you’ll get a happy face. What happens if you flex it? What happens when it is the unhappy monster
face; where are the eyebrows?
What about glasses? How would that work?”
WHERE’S THE MATHS?
Through activities and discoveries such as the flexagon mathematicians are working out how and
why this works which leads to all sorts of discoveries. For example, they came up with clever ways
to describe this type of movement that ended up being useful describing other things, like DNA. So
the students may not think that there’s any maths here but in actual fact, this is what
mathematicians love to do!
DESIGN YOUR OWN
Students may like to personalise their trihexafexagons by colouring them in, or use one of the
blank templates to see what they can create! It works best if they make it before they colour it,
otherwise they need to work out how the triangles relate to each other when it is made so their
patterns work.
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© Simon Pampena - free for educational use within Australia