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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.

Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?


Age 5 to 11
Challenge Level



The tangram is based on the dissection of a square into seven pieces.
Can you make other squares using some, not all, of the pieces?
Can you make five different squares?
What is the smallest square you can make?
What is the largest?
Tangram Image
You might find it helpful to use the interactivity below to try out your ideas, or to print and cut out one of the tangrams on this sheet.


Why do this problem?

Not only will children be using their knowledge of properties of squares as they try this activity, but they will also be putting into practice their visualising skills. Tangrams can be great to work on in pairs and this will encourage the pupils to talk together about what they are doing - a great opportunity for you to listen!

One of the main benefits of tangrams is the ability to manipulate the pieces; to "play" with the shapes and get a feel for the challenge. For this reason, it would be a good idea to encourage pupils to print off and cut out the shapes for themselves from this sheet. Alternatively, if you have access to tablets or computers, learners could use the interactivity in pairs.
You may like to read our Let's Get Flexible with Geometry article to find out more about developing learners' mathematical flexibility through geometry.

Possible approach

You could display the interactivity on the board and start by asking children to look at the image on the left (of the tangram pieces). Invite them to consider what they see and what they might like to ask. Give them time to think on their own, then talk to a partner about their 'noticings'. Encourage some pairs to share their observations and questions with the whole group. Try to take a step back and allow the class to respond to the comments and questions themselves, rather than you validating all contributions. Some children might talk about the different shapes they can see (and colours), some might describe the outline shape of the square. Build on their thoughts to introduce the idea of a tangram.

You can then set the class off in pairs on the challenges. After some time, invite them to offer 'hints' for the rest of the group, without giving away a solution. Perhaps they can offer advice about what they are finding helpful to focus on as they tackle the challenges, perhaps they can suggest a starting point for one of the squares... These challenges are not straightforward and learners are likely to get stuck. Talk about this with the class. How does it feel to be stuck? How are they trying to get unstuck? How does it feel when they make a breakthrough? You can stop the class as appropriate to share experiences.

The completed squares could make a lovely classroom display, accompanied by some brief notes written by the class, focusing on how they went about finding solutions.


Key questions

How many pieces have you got altogether?
What could you put with this piece to make a square?
Are all the pieces different?
What's the smallest square you can make?


Possible extension

To find more tangrams on the site (many of which have interactivities), enter 'tangram' in the top right-hand search box. Our interactive Tangram Browser contains a whole collection of tangrams.


Possible support

You may have to encourage some children to experiment and 'have a go'.