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Fitting Flat Shapes
Age 16 to 18
Challenge Level 
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.
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