### At a Glance

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

### Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they thus became owners of stock of precisely equal value. Tell me the prices of their gems and the values of their stocks. (Editors note: In this problem you are only given enough information to find the values of rubies, saphires and diamonds relative to the price of a pearl.)

# Areas and Ratios

### Why do this problem?

This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a pair of related 'building blocks' designed to lead students to helpful insights. It requires students to apply their understanding of area and proportionality.

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

These printable cards for sorting may be useful: Area and Ratio Cards

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 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 both of 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.