Triangle Animals

I can join two equilateral triangles together along their edges like this:

How many different ways are there to join three equilateral triangles together? (You must match an edge to an edge.)
How do you know you have got them all?

How many different ways are there to join four equilateral triangles together?
Have you got a system for checking that you have them all?

You might like to use this interactivity to try out your ideas. Click the red triangle to make a new triangle appear. You can move the triangles around by dragging with your mouse. If you hover your mouse over a corner, you will see a white circle where you can click to rotate the triangle.

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For a further challenge, you might like to look at Tri-Five .

Why do this problem?

There are two main reasons for using this problem in the classroom. The first is to encourage children to develop a systematic approach to solving problems. The second is to enable them to understand that rotating a shape does not change the shape itself, it just puts it in a different place (or orientation). Both these ideas can be touched upon as you introduce the problem.

Possible approach

To begin with, explore just two triangles. This might be best done by having cut out triangles on an OHP, or by using the interactivity. Suggest to the group that one of the triangles could be kept still while you look at the different positions of the second triangle. This will encourage them to move the second triangle around the first in a particular direction i.e. having a system so that no possibilities are left out. Then, by looking at each of the different shapes and rotating them, children can be asked what they notice. (They are all the same.)
In this way, the class can conclude that in fact there is only one way to put two triangles together. This initial exploration and discussion will equip them for tackling the problem itself.

Key questions

Have you decided what is the same and what is different?
How do you know that you have them all?

Possible extension

Children could have a go at Tri-Five. They could also see what happens if they use two or three isoceles triangles. Are their results different? Why? Again the focus should be on developing the use of accruate descriptive language and beginning to introduce the correct mathematical vocabulary of vertices, sides and angles.

Possible support

Having lots of cut-out triangles for children to stick down will help them access this challenge.  You could use this sheet of equilateral triangles for printing and cutting (and possibly laminating too).