Copyright © University of Cambridge. All rights reserved.

### 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 circle
theorems, or for applying basic understanding of angles in
triangles.

Initially working on the building blocks gives students the
opportunity to then 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

Hand out a set of building block cards (

Word,

PDF) to each group of
three or four students. (The final challenge will need to be
removed to be handed out later.)

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.
As the four introductory tasks are very similar, anything that one
student finds useful can be shared with the rest of the group to
help them to make progress on their own building block.

When everyone in the group is satisfied that they have
explored the challenges in the building blocks, encourage them to
discuss the similarities between their findings. Before giving the
group the final challenge, ask if they can predict what they will
be asked to do. Then set them 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?

What would happen if I tried the same process with 10- 11- 13-
or 200-point circles?

### Possible extension

Other circle theorems can be explored in a similar way in the
problems

Subtended
Angles and

Right Angles.

### 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.