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