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'Cyclic Quadrilaterals' printed from http://nrich.maths.org/

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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.