### 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?

# Cuisenaire Counting

## Cuisenaire Counting

I have some rods that are different colours and different lengths.

I could make a yellow rod using white and red rods in many different ways. Here are some of them:

Can you find all the ways to match the yellow rod using reds and whites?

How will you know that you haven't missed any out?

You could use the interactivity below to try out your ideas, or you could use our Cuisenaire Environment.

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The dark green rod is longer than the yellow rod.
Can you find all the ways of matching the dark green rod using white and red rods?
How will you know that you have found them all?

Why not use the interactivity below to find all the ways of matching the dark green rod, or again, you could use our Cuisenaire Environment.

Full screen version

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### Why do this problem?

In this activity, children have an environment for exploring partitioning of numbers. Of course, it is essential to have cuisenaire rods available for them, so that they can solve this problem practically too.  These notes are written to suggest that the activity could be used to focus on sharing good strategies to find all possibilities. These strategies can then be applied to further challenges.

### Possible approach

If your children are not already familiar with Cuisenaire rods, it would be good to give them time to 'play' with the rods before having a go at this activity.

If you are working with a small group, it might be appropriate to introduce the first challenge (making a yellow with reds and whites) using the rods themselves.  If you are working with the whole class, then it might be better to use the interactivity or Cuisenaire Environment on the interactive whiteboard.   Challenge the children to find a way of making the yellow from red and/or white rods, and ask a pupil to make an arrangement.  Ask for another, different, arrangement and invite someone else to make that next to the first. Then set them off to see whether they can find all the different ways, perhaps working in pairs.

After a suitable length of time, draw everyone together again to talk about what they have done so far.  At this stage, you may need to have a discussion about what 'different' means.  Ask the group how they are making sure they don't leave out any possibilities. It could be that some learners have developed a system for creating or ordering the different ways, for example by starting with all whites, then putting in one red etc.  If this isn't the case, ask pairs to draw one arrangement on a strip of paper and pin as many different arrangements as they have found on the board.  You can then ask the children to order the arrangements by moving the strips.  This way, it will be easier to see if they have missed any out.  You could complete the solution of this part of the problem all together and then invite pupils to comment on what they might do to solve the problem if they are given a differently coloured rod (e.g. dark green) to make from reds and whites.  Do they have a sense of the need for a system?

As they work on the second part of the challenge in their pairs, you will be able to listen and observe.  Have they got the idea of creating possibilities in a particular order?  Are they using the same order that the whole group used, or have they developed their own?

Allow time at the end of the session for children to look at what other pairs have done.  It might be that this activity (and any extensions you suggest - see below) form a 'simmering' task over a period of a few days or more.  Allocate a space on the wall for children to add to the possible arrangements and then come back to their contributions at a later date.

### Key questions

Can you use just white rods?
Can you use just red rods?
Can you find another way to make the yellow rod?
How could you put those rods in a different order to make the yellow?
How will you know that you have found all the possiblities?
How could you use what you've done to help find all the possible ways for making the dark green rod?

### Possible extension

Some children may like to continue to work in a systematic way to find all possible ways of making a black, then brown, then blue, then orange rod from reds and whites.  Alternatively, they may like to apply their method to finding all the ways of making different rods using light green in addition to red and white rods.  This interactivity could be useful for the latter.

### Possible support

Children might find it helpful to use cm squared paper with the cuisenaire rods for recording purposes.