### Why do this problem?

This
problem involves using practical equipment to approach a
mathematical problem.

It challenges the usual misconception
that all tetrahedra are regular.

It needs systematic thinking and
visualisation and has some surprises in it - there are a few
examples that are quite unexpected.

It is hard to be convinced that you have
found all the possibilities and difficult to make the distinction
between the two tetrahedra that are mirror images of one
another.

### Possible approach

Hand out one of each type of triangle to students working in
pairs or small groups. Invite them to make a list of four things
that they think are mathematically most important about the
triangles - either by considering each triangle individually or
when compared to each other.

Share ideas, making sure these points are covered:

- two are isoceles and the other two are equilateral
- one triangle has a right angle
- the triangles have sides of only two possible lengths.

Spend some time comparing the triangles to establish which
sides are "short" and which "long".

Hand out further triangles and invite the students to create a
tetrahedron with some of the triangles. This task may result in the
need to discuss what a tetrahedron is and that a tetrahedron can be
made from triangles that are not equilateral. You may wish to have
some examples ready to illustrate these points.

Present the problem.

Whilst the class works on the problem it may be useful to stop
to discuss progress and approaches that will enable them to
convince themselves and each other that they have all the
possibilities.

### Key questions

If you take one of each of the triangles is it possible to
make a tetrahedron and how do you know?

Have you got a systematic approach for finding all the
different tetrahedra?

How are you recording your findings?

How do you know that you have tried all possible ways of
putting the same four triangles together?

### Possible extension

Students who have met Pythagoras' Theorem may like to quantify
the relationship between long and short sides.

### Possible support

Work with two and then three different types of triangle to
establish a systematic approach.

See also:

Paper
Folding - Models of the Platonic Solids