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Putting Two and Two Together

Age 7 to 11 Challenge Level:

Putting Two and Two Together

Two equilateral triangles can be put together in one way only, if they must touch all along a side:

two equilateral triangles

These isosceles triangles of the same area can be put together in more than one way.

Remember that reflection can be used as well as rotation when finding the ways.

two isosceles triangles

Can you find any other ways?

You could cut out triangles from these sheets or you might want to use the interactivity below:

Full Screen Version

These triangles are half of the above equilateral and isosceles triangles:

two smaller triangles

In how many ways do you think two of these can be put together using, reflection as well as rotation, if same-sized sides must touch the full length of the side?

Can you give any reasons for this?

Full Screen Version

You could cut some of your other triangles in half to test out your prediction.

Can you find a triangle such that two identical copies of it can be put together in more ways keeping to the rule that same-sized sides must touch the full length of the side?

Why do this problem?

This problem encourages children to work in a systematic way to find all possibilities and it gives a context for analysing properties of triangles.

It is essential to have cut-out triangles available for the children to manipulate while working on this problem. You can print off these sheets, if possible on coloured card.

Possible approach

A good way to introduce the problem would be for children themselves to investigate the different ways (in fact only one) you can put two yellow (equilateral) triangles together. This will allow you to emphasise what we mean by "different". Using the interactivity on the interactive whiteboard will allow the children to show you the ways they have found and then you can look at them all together as a group, focusing on whether they really are different from each other. When looking at two green triangles, you could ask children to work in pairs. How do they know they have all the possibilities? You could invite some pairs to share the ways that they had worked systematically in order to know they hadn't missed any out.

Then give everyone an opportunity to predict how many ways two blue triangles can be fitted together - ask them to think carefully on their own, then chat to a partner, then talk about their predictions as a class. In discussion, encourage pupils to justify their predictions and then invite them to test them out.

Key questions

How many ways have you found for putting together the yellow/ green/ blue triangles?
How do you know you have found all the possible ones?
Can you think of a reason why this is so?
How many sides of different lengths does that triangle have?

Possible extension

Learners could try using three copies of each triangle instead of two.

Possible support

Suggest using cut-out triangles to manipulate and then draw round. These can be printed off on coloured card.