### Sine Problem

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

### Frieze Patterns in Cast Iron

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

### The Frieze Tree

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

# Squaring the Rectangle

### Why do this problem?

This problem encourages students to think deeply about area and length in order to prove a theorem. The suggestions and interactivity in the task provide some scaffolding to help students to investigate, make conjectures, and hopefully prove some of their conclusions. To get to a complete proof will require some perseverance.

### Possible approach

You may wish to spend some time working on Triangle Transformations before starting on this problem.

"Draw a rectangle with any dimensions that you like. Your challenge is to find a way to cut it up and reassemble it to make a square."

Students may start by picking particular rectangles that can be easily rearranged. After they have had some time to explore, you may wish to share the interactivity below:

Invite students to work out how the square has been cut up, and how they could reverse-engineer this to start with a rectangle and finish up with a square.

Alternatively, you might wish to share the image below, and invite them to consider how the pieces could be rearranged to make a square.

It might be fruitful to discuss the side length of a square which has the same area as a rectangle with sides of length $a$ and $b$: "If the top right corner of the rectangle has coordinates (a, b), how could you calculate the coordinates of the other dots?"

### Key questions

How do the pieces move to turn a square into a rectangle and back again?
How could you identify the points which are needed to make the pieces?
Can every rectangle be cut up and reassembled to make a square?

### Possible extension

Students could consider other polygons and how to dissect them and reassemble them to make squares. They could find out about the Wallace-Bolyai-Gerwein Theorem, and perhaps the analogous problem in three dimensions, which is Hilbert's Third Problem.

### Possible support

Students could use squared paper and construct squares, and then find rectangles with the same area - for example, turning a $6 \times 6$ square into a $9 \times 4$ rectangle, using the dissection shown above.