### Why do this problem?

This
problem provides the basis for a proof of Pythagoras' theorem
using a dissection. In constructing a proof there is the
opportunity for rich discussion about what constitutes a proof and
why it is not enough to say that the pieces seem to form two
squares, it has to be proved that lines match up and corners are
right angles.

### Possible approach

In pairs, students construct a square and choose a point on
one side. Make sure everyone understands that they need to mark
each point the same distance from each corner as shown in the
diagram. Different pairs could choose different sized squares and
mark at different distances from the edge to give a variety of
different dissections.

Cutting out and reassembling the pieces is really only the first
part of this challenge. If anyone really struggles to reassemble
it, there's a diagram in the Hint which might help. Once the two
joined squares are formed, pairs should come up with arguments to
convince themselves that the angles which appear to be $90^{\circ}$
actually are, and that lines which match up are equal in
length.

Pairs could produce a poster or a presentation to show the stages
of their proof, and then present their proofs to the rest of the
class. Encourage students to be critical of each step in their own
and others' proofs to make sure they are mathematically
sound.

### Key questions

Which angles are $90^{\circ}$? How can you be sure?

How do the side lengths of the squares formed relate to the
side lengths of the right angled triangles?

Is this enough to prove Pythagoras'
Theorem? What do we need to do to make it a proof rather
than a demonstration?

### Possible extension

Explore other proofs of Pythagoras' Theorem in this

problem.
### Possible support

Tilted
squares offers another way to think about proving Pythagoras'
Theorem.