### Arranging Squares

A task which depends on members of the group working collaboratively to reach a single goal.

### Triangles in the Middle

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

# Fitting Flat Shapes

### Why do this problem ?

Approximating physical quantities by idealised mathematical shapes is a commonly used tool in mathematical biology. Working with these shapes requires a good degree of skill at geometrical visualisation. By consider packing problems, students will develop this skill and see how important packing is in nature.

### Possible approach

This question could be posed individually or for group discussion. This problem also works effectively when students are given time to reflect on the question and look for packing in nature. Ask the question and let students consider it over, say, a week. What shapes and packings have they noticed in nature? Could they find any images to share? Then consider the questions of efficient packings. This results might make an effective display.

### Key questions

• How reasonable is the mathematical idealisation that you make?
• Are there any objects which are particularly well represented by a certain shape?
• Do any sorts of packing occur particularly often?

### Possible extension

Can students think of good evolutionary or chemical reasons for the shapes and packings that certain organisms take?

Are there situations in which efficient packings might be particularly helpful or particularly unhelpful?

How is symmetry important in packings?

### Possible support

Some students might need cut-outs in order to experiment with the packing possibilities. Some students might also struggle with the 'open' nature of the question, as there is no 'complete' answer. To begin, they might like to read the Student Guide to Getting Started with Rich Tasks.