### Matter of Scale

Prove Pythagoras Theorem using enlargements and scale factors.

### Dodecawhat

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

### Trapezium Four

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

# Pythagoras Puzzler

### 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.