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Shape Times Shape

The coloured shapes stand for eleven of the numbers from $0$ to $12$. Each shape is a different number.

Can you work out what they are from the multiplications below?

multiplications using shapes

This activity is featured in the hands-on Brain Buster Maths Boxes, developed by members of the NRICH Team and produced by BEAM. For more details see here .


Click here for a poster of this problem.

Why do this problem?

This problem helps children become familiar with the idea of a symbol (in this case a shape) representing a number. It is also an opportunity for pupils to practise using multiplication and division in a challenging context.

Possible approach

In order to introduce the idea of a shape representing a number, you could start the lesson by having some shapes representing numbers in addition sums. For example:
square + rectangle=5, circle + square = 11, rectangle + rectangle = 8
Ask partners to talk to each other about how they would work out what each shape stands for in the calculations above and share their ideas amongst the whole group. In this case the last sum is actually the most helpful to start with.

You could then introduce the Shape Times Shape problem by displaying it on the whiteboard and explaining the task. (It might also be useful for pairs of children to have a paper copy.) Ask the children to think on their own about where they might start. Invite them to share their ideas with a partner, then discuss the options amongst the whole group. Look out for good reasoning, based on their knowledge of number properties, and encourage everyone to strive for this level of explanation.

Set them off in pairs to tackle the problem indicating that the plenary will focus on how they went about solving it. You may want to stop them after a few minutes to find out how they are recording their work and to share some efficient ways.

Key questions

Which sum is useful to start with? Why?
Now that we know that shape, which sum could we look at next? What does that tell us? How do we know?
How are you keeping track of what you have done?

Possible extension

Learners could make up their own problem using shapes as symbols and test it on a friend. The problem What's it Worth? requires similar thinking processes to this problem and would be a good one for pupils to try next.

Possible support

Children might find it easiest to have numbered counters or cards available so that they can physically form the sums to check their reasoning. You might want to support their recording by giving out a this sheet with each of the shapes drawn on. Alternatively, this sheet has a copy of the question and space for recording (thank you to Rose Prentice for sharing it with us).