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This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.


This activity investigates how you might make squares and pentominoes from Polydron.

Multilink Cubes

If you had 36 cubes, what different cuboids could you make?

Count the Trapeziums

Age 7 to 11
Challenge Level

Why do this problem?

It's quite easy to hazard a guess about how many trapezia there are in this problem but to be absolutely certain (and convince someone else that you have all the possible solutions) requires some really systematic work.

Possible approach

You could begin with a whole class challenge of a similar but simpler kind - for example how many trapezia in this shape?

Check that all the children know what a trapezium is and ask for a system for finding all possibilities in this diagram. Emphasise working systematically and what this means in practice - for example starting at the top and working clockwise.

Offer the recording sheet for those who want it - and scissors for them to cut out the variations and re-order them to check for missing diagrams. Working systematically does not come naturally to young children so being able to impose a structure onto randomly generated pictures can be a valuable step in learning how to be systematic.

Key questions

Where will we start?
What story can we tell which will convince a friend that we have them all? How do we know?
Is there another way of arranging them in a pattern?

Possible extension

Adding another layer of triangles to the bottom of the diagram increases the complexity, but for children who are already working systematically this will be only slightly more challenging.

There is a collection of similar style problems here.

Possible support

Reducing the picture by one layer can beĀ helpful for children who find visualising difficult. Provide a recording sheet so that they can model the same ordering strategy as in the main activity.