### Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

### The Old Goats

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't fight each other but can reach every corner of the field?

### Rolling Triangle

The triangle ABC is equilateral. The arc AB has centre C, the arc BC has centre A and the arc CA has centre B. Explain how and why this shape can roll along between two parallel tracks.

# An Unusual Shape

### Why do this problem?

This problem offers the opportunity to practise calculating areas of circles and fractions of a circle in the context of an optimisation task.

### Possible approach

This printable worksheet may be useful: An Unusual Shape.

Start by showing the diagram from the problem, and ask learners to think on their own for a few moments about where the goat might have been tethered to yield the area shown.
Next, allow them to discuss their ideas with their partner, and finally share convincing arguments with the whole class.

Having determined where the hook is, the challenge is to work out the area available to the goat and then to consider different positions of the hook in order to find the maximum possible area.

This is a fantastic opportunity to talk about the benefits of factorising, working in terms of $\pi$, and only using the calculator at the very end of the computation.

Once learners have established where to fix the hook for maximum goat nutrition (!), move on to other lengths of rope. Suggest that pairs of learners work with different lengths of rope, and finish by sharing their findings with the rest of the class.

### Key questions

What can you say about the radius of each part of a circle?
How does this help you to pinpoint where the hook must be?
How does the space available to the goat change if the hook is moved?

### Possible extension

What happens if the rope is longer than the sum of the sides of the shed?
Investigate what happens to the area available for sheds of different dimensions, or sheds which are not rectangular.

### Possible support

The activity can be modelled by building the frame of the shed from multilink cubes, and learners could use string to work out the shapes of the regions that could be made available to the goat, when fixing the hook at different points.