### Why do this problem

This
problem involves a simple construction which gives a surprising
and useful result. Having confirmed the sizes of angles this
knowledge can be applied to make images and polyhedra. Different
routes to solution can also lead to useful discussions, including
how well learners explain their reasoning and the elegance of their
methods.

### Possible approach

Produce some equilateral triangles.

Ask the group to consider how they know they are
equilateral?

Building on the idea that angles must be $60^o$, ask them to
prove that this is the case.

If learners are inexperienced with proof, after lots of
discussion, you may wish to use

these proof cards
(illustrated below) to aid in structuring a logical argument.

You may wish to give out all sixteen cards and ask learners to
arrange them to form a logical argument. Alternatively, you might
for example, give learners the images and ask them to write the
text, or the text in order and ask them to find the pairs. The aim
is to extend discussion in terms of the structure of the argument,
its strengths and weaknesses or how they might have done it
differently.

Learners can make triangles of different sizes by halving and
quartering various sheets of paper in order to demonstrate that the
paper does not have to be A4 size.

Follow on by utilising the ability to make equilateral
triangles in different contexts.

### Key questions

What do you know?

Can you see and use any symmetry?

### Possible extension

See:

Paper
Folding - Models of the Platonic Solids.
Can learners justify all the results used as they work?

### Possible support

Spending time making triangles and feeling confident about their
properties is a useful starting point. Finding triangles that might
be congruent, cutting them out and testing the congruency by
matching them can then lead to identifying why sides and angles
might be equal.