# Possible Pieces

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

## Problem

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:

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

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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.

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## Getting Started

How are you making sure you do not make any the same?

Here is a picture of the kind of jigsaw the problem is based on, in case you are not familiar with jigsaws. (This is just an example, this particular picture will not necessarily help you answer the challenges!)

Here is a picture of the kind of jigsaw the problem is based on, in case you are not familiar with jigsaws. (This is just an example, this particular picture will not necessarily help you answer the challenges!)

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## Student Solutions

Yvonne from KÃ¼geliloo School in Zurich Switzerland wrote to say they found eight solutions to challenge 1 and 32 solutions to challenge 3. It would have been good to see pictures of them.

Millie from Nene Valley Primary sent in her work with a picture:

First I tried to make one puzzle that worked, and then I realised that there was a pattern of pegs and holes. The pegs went round in a circle and so did the holes, so only the middle peg or hole could change, or the circle could go round the other way.

A larger more legible version is available here.pdf

Well done Millie, and thanks to Yvonne for her two answers. If you have a go at this challenge, let us know what you find out.

Millie from Nene Valley Primary sent in her work with a picture:

First I tried to make one puzzle that worked, and then I realised that there was a pattern of pegs and holes. The pegs went round in a circle and so did the holes, so only the middle peg or hole could change, or the circle could go round the other way.

A larger more legible version is available here.pdf

Image

Well done Millie, and thanks to Yvonne for her two answers. If you have a go at this challenge, let us know what you find out.

## Teachers' Resources

### 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.

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).