### Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

### Doodles

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

### Russian Cubes

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

# Why 24?

### Why do this problem?

This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights. It is well suited for students who are working on the difference of two squares.

Initially working on the building blocks then gives students the opportunity to work on harder mathematical challenges than they might otherwise attempt.

The problem is structured in a way that makes it ideal for students to work on in small groups.

### Possible approach

This task might ideally be completed in groups of three or four.
Hand out a set of building block cards (Word, PDF) to each group. (The final challenge will need to be removed to be handed out later.) Within groups, there are several ways of structuring the task, depending on how experienced the students are at working together.

Each student, or pair of students, could be given their own building block to work on. After they have had an opportunity to make progress on their question, encourage them to share their findings with each other and work together on each other's tasks.

Alternatively, the whole group could work together on all the building blocks, ensuring that the group doesn't move on until everyone understands.

When everyone in the group is satisfied that they have explored in detail the challenges in the building blocks, hand out the final challenge.

The teacher's role is to challenge groups to explain and justify their mathematical thinking, so that all members of the group are in a position to contribute to the solution of the challenge.

It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.

### Key questions

What important mathematical insights does my building block give me?
How can these insights help the group tackle the final challenge?

### Possible extension

Of course, students could be offered the Final Challenge without seeing any of the building blocks.

### Possible support

Encourage groups not to move on until everyone in the group understands. The building blocks could be distributed within groups in a way that plays to the strengths of particular students.
The final challenge could also be done as a proof sorter activity using this set of cards (Word, PDF).