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Four Triangles Puzzle

Age 5 to 11
Challenge Level

Four Triangles Puzzle

If you cut a square diagonally from corner to corner you get four right-angled isosceles triangles.

How many different shapes can you make by fitting the four triangles back together?

You may only fit long sides to long sides and short sides to short sides.

The whole length of the side must be joined.

You might like to record what you do.

You could use this interactivity to try out your ideas:

Click on the grey triangles to make new pieces.
You can rotate them by dragging the corners.

Printable NRICH Roadshow resource.

Why do this problem?

This low threshold high ceiling task invites learners to be creative with a familiar triangle in a systematic and logical way. It can develop understanding of angles and right angles, and encourages children to visualise.

Possible approach

You could use large triangles to introduce this activity on the board. Alternatively, the simple interactivity would be a good place to start. Pose the problem for the children and perhaps invite one or two to come to the board/interactivity to manipulate the triangles. This will help establish the "rules". Leave learners to try out more ideas in pairs, using cut-out triangles. This sheet could be printed out on coloured paper for children to use. It will make 48 small triangles. Recording could be done on square dotty paper or cm squared paper.

After a little while, bring the group together to share ways of working. Invite pairs to describe how they are going about finding the triangles. This will lead into a discussion of finding a system for making successive arrangements, which means it is less likely any will be left out. Some children might have looked at just two triangle arrangements to start with, then those with three triangles. Simplifying in this way can be very powerful in mathematics and you may want to suggest this even if none of the children have tried it. At this point, it may also be appropriate to talk about how they are checking that each arrangement is in fact different to the others.

The patterns produced during this investigation, along with descriptions about how they were found, would make a lovely classroom display.

Key questions

Does it make a difference if you turn one triangle over in this arrangement?
How are you making sure all your patterns are different?

Possible extension

Some children will be able to justify that they have found all the possible variations. You could challenge them to predict and explain how many different arrangements there will be for other numbers of triangles.

Possible support

Working in pairs would offer support for learners, and the opportunity to talk about their thinking as they work.  It might be helpful for some children to be able to stick their arrangements of triangles onto paper as they go along rather than having to record them separately.  Photographs could be taken and then shared as a class to discuss duplicates or systematic ways of convincing others that all the possible solutions have been found.