### Homes

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

### Train Carriages

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

### Let's Investigate Triangles

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

# Jigsaw Pieces

## Jigsaw Pieces

Jigsaw puzzles have lots of different shaped pieces that all fit together without gaps. Most puzzle pieces have four sides with buttons added on to them or cut out of them similar to these:

Imagine that the puzzle pieces are roughly a rectangular shape and all the same size. They can have a button added on or taken out of the middle of any side.

How many different puzzle pieces could there be?

Have you got them all? Have you got any that are really the same?

Have you remembered the straight edges?

### Why do this problem?

This problem can be intriguing for children, since it is somewhat unusual by drawing attention to jigsaw pieces. It offers experiences in spatial awareness, as well as a starting point for being systematic in the way the shapes are found. It also presents opportunties for learners to find their own way of representing their solutions.

### 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 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 on the task. After some time, bring them together again to share what they have done so far. You could invite some pairs 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.

Invite pairs 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 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 at least 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. Finding all the possibilities is quite a challenge so you could leave this as a 'simmering activity' for children to contribute to during the week.

### Key questions

What pieces have you found?
Do you think there are more to be found, and if so why?
How do you know all your pieces are different from each other?

### Possible extension

Can they put together some of the different shapes that they have found to make a rectangular shape with straight edges?

### Possible support

You could have lots of rectangles already cut out of card for children to make into pieces. Having plenty of random jigsaw pieces around for learners to scrutinise will also help.