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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?

Possible Pieces

Age 5 to 11
Challenge Level

Possible Pieces

We are going to look at possible jigsaw pieces.
The kind of jigsaw we're looking at is rectangular, with straight edges.

Some of the most common jigsaw shapes are a bit like these three pieces below:


So, for each jigsaw piece we start with a square template, then some sides have a peg. The edge pieces of the jigsaw have one or two straight sides.

We will only use pieces that have at least one peg and one hole.

Challenge 1

Using pieces that have at least one peg and one hole, find all the possible ways of making a rectangular jigsaw three pieces wide and two pieces deep, with straight edges all the way around. All six pieces should be different.

Challenge 2

Find all the possible pieces that have at least one peg and one hole.

Challenge 3

Read all of this one before starting!
Again, only use pieces that have at least one peg and one hole and none the same.
Find as many possible ways of making a two by four rectangular jigsaw, starting with this piece in the top left-hand corner.

Before you start making/drawing/constructing pieces, think carefully and then estimate how many possible arrangements there will be and be able to explain your reasoning to others.

Why do this problem?

This activity requires little mathematics 'curriculum' knowledge (in terms of number/geometry etc), so learners can be 'freed up' to focus on their problem-solving and mathematical thinking skills. A certain degree of resilience and perseverance will be needed.  It is a great context in which to give learners the experience of working on a challenge that takes a longer time than many and you may wish to offer them the opportunity to return to the task at a later date. It is likely that pupils will take up the chance to develop a systematic way of getting a solution.

Possible approach

The pupils could start off by doing some quick jigsaws, or you could have a box containing jigsaw pieces out for them to see. Ask them what they notice about the pieces and encourage them to share their thoughts with a partner. Gather the whole group together to exchange ideas, listening out for those children who identify similarities and differences, such as "some pieces have bits cut out" or "this one has some straight edges" etc.  You could use this opportunity to introduce the language of 'peg' and 'hole' (as used in the task) to the group.

You can then present the challenge to the children. You may like to say very little else at that stage and let learners work in pairs or small groups on the task. (You may like to give out printed copies of the task.) After some time, bring them together again to share what they have done so far and to clarify the task. You could invite some pairs/groups to talk about the way they are working so that a variety of approaches is highlighted. Some children may be drawing shapes on whiteboards or paper, others may be making them from card, some may have found more abstract ways to record what they are doing. You could discuss the advantages and disadvantages of each method. (It may be useful for learners to have some squared paper but try to provide any resources which they request, even if you haven't anticipated them!)

Invite learners to explain how they are making sure their jigsaw pieces are all different from one another. If the children haven't had much experience of working in a systematic way, you could ask each pair/group to make the pieces out of card, then after a longer period of time, display the pieces somewhere easy to see. With the help of the children, group the pieces together, for example all those with exactly one straight side; all those with just one 'hole'. In this way, the class will be able to identify pieces that are missing from the set.

You could leave this as a 'simmering activity' for children to contribute to during the week and then devote time at a later date to drawing their ideas together.

Key questions

Tell me how you're finding a solution.
How are you making sure you do not make any the same?
Tell me how you're trying to get solutions to challenge 2 ( or 3).

Possible extension

As mentioned above, because this activity requires little formal mathematics in terms of number/geometry etc, pupils will have the opportunity to really focus on their approach. You can encourage learners to reflect on the way/s they tackled the challenges and how they overcame any moments of being stuck.

Possible support

It may be appropriate only to share the first challenge initially so that learners have chance to focus on that without distractions.