### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

# Round and Round a Circle

### Why do this problem?

This problem encourages learners to examine the relationships between the ratios of sides of right angled triangles, and how those ratios vary over changing angles. The aim is to look in detail at the under-pinning structure of the animation.

### Possible approach

This takes the problem "Where is the Dot? " slightly further.
Ask the group to watch the animation carefully, explaining that you will ask them to recall in detail what they saw. They should not make notes as they need to watch the animation carefully - seeing it as a whole will help them make connections.
Spend some time trying to recreate the animation - checking by running the animation and pausing the screen after major steps.
Ask the learners to work in pairs "making sense" of the data that the animation produces. Trying to think about connections? As you move around the group encourage recording of findings and ideas to share. You may even like them to produce a display of what they discover. It may be useful to ask them to think about mathematics they have met before that might be useful (e.g. Pythaogoras theorem) or encourage them to think about drawing some graphs (what would the independent and dependent variables be?).

There may be points in this part of the lesson when you will want to stop and share some of the ideas learners have and if others agree or have found similar or conflicting things.

### Key questions

• What do the numbers refer to?
• Are they the same value at different points during the dot's motion around the circle?
• Are the three values connected and if so how?
• What can graphs of the values tell you?

### Possible support

Spending some time talking together about the animation and what it is showing and creating a story board of key moments will help to unpick the underpinning structure.

You should also look at "Where is the dot" and related problems.