### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

# CD Heaven

### Why do this problem :

This problem creates a context where students are challenged to pick out detail from a large block of text, represent that data and the relationships between the data as a set of equations, which need solving and interpreting back into the practical context.

### Possible approach :

This kind of work can be hard and not always rewarding for individuals not motivated just because it's a puzzle. It may be that the problem achieves most to develop a problem solving culture in the classroom if it is approached with plenty of discussion between students and in the spirit of a group effort. Less motivated students can still verify and critique the proposals of others.

Pair students to extract the relevant data and relationships from the text. Check orally that all the group has all the needed data and debate items that are not universally agreed upon.

Ask pairs to express these relationships as equations, again pooling all and discussing elements of contention.

Set students to work looking for a solution, which is then written up poster-size for display around the room.

Students peruse each others results and presentation of method, before a whole group discussion of the routes followed and the comparative efficiency of the methods deployed. Finally, ask different students to explain how the algebraic results work as a meaningful solution to the practical problem context.

### Key questions :

• Pick out all the data that you are going to need from this text.
• Try to get clear about exactly how these values relate to each other and express those relationships as algebra (equation)
• What does it mean to solve those equations and can you see how to manage that ?

### Possible support :

The style of approach suggested above should allow a wide range of ability within a group to function effectively as a problem-solving culture - where abler students explain more, and less able students providing verification of results and feedback on effectiveness of communication.

### Possible extension :

Matchless is an interesting challenge which pushes students into a deeper understanding of simultaneous equations.