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'Polydron' printed from http://nrich.maths.org/

Do you have any Polydron in your school?
Here are some questions about the square Polydrons.
You can see in the picture that a square can be made in two different ways.
 How much bigger is the one made from 4 right angled isosceles triangles than the one made from just one square Polydron?

Polydron is great for connecting and folding pieces together.
Using only square Polydron you can can easily click them together to make other shapes.
If you connect five squares together we call it a pentomino.
There are 12 different ones.

Can you find them all?
 Do all your pentominoes have the same perimeter length?
 How many pentominoes have line symmetry?

Rotational symmetry?
What if you could fold them up?
 How many of the pentominoes will fold up and clip together to make 'lidless' boxes? Why not discuss first which will fold up and which won't, before trying to fold them?

Why do this problem?
This
activity can be a good start for exploring the special properties of shapes. Children benefit from lots of informal play with linking shapes such as Polydron, during which time they conceptualise the different characteristics of 2d shapes and visualise how they might fold up to make 3d shapes. This visualising
is a very important aspect of being a mathematician.
Possible approach
Ask the children to make some squares using Polydron.
What is the same/different about them?
You could bring in the language of similarity  all squares are the same shape but may be different sizes. In what way might we say one square is 'bigger ' than another?
The pentomino activity is not a new one, but using Polydron allows children to try lots of examples. If they keep to the same colours then they can be encouraged to work systematically. They could record their work on squared paper, or you could take photographs of the pentominoes and make a display which could be sorted according to whether they fold up into a lidless box or not.
Alternatively, other criteria could be used to sort.
Key questions
What can you tell me about the square polydrons and the triangular polydrons?
What is the same about them? What is different?
What could 'bigger' mean?
Before you fold them up, can you tell what 3d shape they will make?
Possible extension
What if you had six squares joined together (hexominoes)?
Can you use what you found out about pentominoes to find some hexominoes that fold up into a box with a lid?
Possible support
This is a 'low threshold high ceiling' task in that all of the children in a class will be able to begin the activities, provising they have some moderate degree of fine motor skill.