### 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?

### Stairs

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

### 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?

# Chain of Changes

## Chain of Changes

Here is a set of shapes. For this problem we will call them pieces .

Each of these pieces is either a different colour or a different shape from all the others.

These problems ask you to arrange the pieces in a line so that you change either colour or shape in the next piece along. If we start with a blue triangle the next shape has to be either another triangle or another blue shape.

The first puzzle is to arrange all the shapes in such a line starting with the blue triangle and ending with the red circle. There are lots of different ways of doing it!

The second problem is to arrange the pieces in the same way, starting with the blue triangle and ending with the red circle, but to change first colour, then shape or vice-versa.

If you put a yellow triangle after the blue one and so change the colour, next you must put another yellow piece and so change the shape.

You will not be able to use all the pieces in this way but the problem is to see how many you can use.

Why do you think you cannot use all the pieces?

You might like to try using this interactivity.

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### Why do this problem?

This problem will help children to refer to the shapes by name and visualise the next shape to place in the pattern. They will need to look carefully at the properties of each shape. It will also encourage them to use a trial and improvement approach in solving problems.

### Possible approach

You will need plenty of Logic Blocks or of shapes cut out from this sheet which has two copies of each of the shapes. A full set of Logic Blocks can provide enough for four children, pairs or small groups depending on how the children are working. (One group has the large, thick pieces, one the large, thin pieces and so on).

You could start with one of the pieces and ask children to describe it. Ask if they can find one which is the same shape but a different colour. Then you could go on either using the large, thick blocks or this interactivity to make sure that all in the group really understand the problems to be solved. After this the children could work in pairs or threes on the actual problem so that they are able to talk through their ideas with the others.

At the end you could discuss the different ways that the children found of doing the first part of the problem. Then discuss why there was not a way of doing the second part using all the pieces. Children could also discuss the pattern that results in changing shape and colour alternately.

The work makes a good display using either the children's own drawings or paper copies of the pieces using this sheet.

### Key questions

Are you going to change the colour or the shape this time?
Which shape are you going to use next?
Can you find another way of doing it?
Why can't you use all the shapes this time?

### Possible extension

Those who find these tasks straightforward could use a full set of Logic Blocks and also change the size and thickness of the pieces.

### Possible support

It might help some children to make their own chains which started with the blue triangle, but without specifying the end point.