Tessellating Triangles
Can you make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
triangles?
Equilateral triangles have three sides the same length and three angles the same. Can you make them fit together to cover the paper without any gaps between them? This is called 'tessellating'.  Image

What about triangles with two equal sides? These are isosceles triangles.
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Can you tessellate all isosceles triangles?
Now try with right angled triangles. These have one right angle or 90 degree angle.
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Some triangles have sides that are all different. Can you tessellate these?
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Could you tessellate any triangle?
Make plenty of triangles before you start (in different colours if
possible).
There are lots of interesting designs to be made by tessellating triangles. Have a go yourself, and if you discover anything interesting, email primary.nrich@maths.org to tell us what you've done!
Please don't worry that your solution is not "complete"  we'd like to hear about anything you have tried.
Teachers  you might like to send in a summary of your children's work.
Why do this problem?
This problem encourages children to use the right vocabulary when talking about shape properties. They will begin to understand that, for a shape to tessellate, the angles where they come together are important.Possible approach
The most useful resource for this investigation would be a large number of cutout triangles, either paper/card or plastic. Children may also find dotty/squared paper useful.
Encourage the pupils to talk about what they are doing, perhaps with a partner, and report their findings back to the class frequently. Try to question the children in such a way so as to lead them to explore this, perhaps by drawing their attention to a particular part of the tessellation. For example, use lots of coloured tiles to build a pattern like this:
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At the point marked by the arrow you could try asking questions like:
 If I took out one of the triangles, how do I know which way it fits in to make the tessellation?
 Why won't it fit if the triangle is rotated?
This should prompt your pupils into considering the angles within the individual shape itself, which can be extended to discussion about the sum of the angles at the point shown. This work can make a lovely display!