### Three Squares

What is the greatest number of squares you can make by overlapping three squares?

### Two Dice

Find all the numbers that can be made by adding the dots on two dice.

### Biscuit Decorations

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

# Same Length Trains

## Same Length Trains

Matt and Katie were using Cuisenaire rods. They placed rods end to end to make 'trains'.
Here are their two trains:

What is the same about the two trains?
What is different?

You might have noticed:

• the trains are the same length as each other
• the top train uses rods that are all different (light green, black, pink and dark green)
• the bottom train uses rods that are all the same (white)
• there are four rods in the top train
• there are 20 rods in the bottom train
• the white rods are the shortest
• the white rods are all the same length
• each colour rod is a different length

Can you make some more trains the same length as Matt's and Katie's trains, each using rods of just one colour?
How many different trains can you make?

If you haven't got any Cuisenaire rods, you may like to use the interactivity below to try out your ideas.

### Why do this problem?

This problem is a good opportunity to encourage children to have a system for finding all possible solutions. It is also an ideal context in which to help children deepen their understanding of factors and multiples in a playful environment.

### Possible approach

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

Introduce the task by telling the 'story' of Matt and Katie creating 'trains'. Show the picture of their two trains and invite children to talk in pairs about what is the same and what is different about them. After a short time, draw the whole group together and share their noticings, writing them up on the board for all to see. Encourage everyone to comment on these noticings, rather than you being the one to answer questions or validate the contributions. Having listened to the class' observations, you may wish to reveal the bullet points in the problem too.

You can then introduce Cuisenaire rods and set up the task for children to work on in pairs, either using real rods or the interactivity on a tablet or computer. As they work, look out for the different ways that they are approaching the task.  Some children might be choosing a rod at random to see whether it works, others may decide to start with the next shortest rod (red), or the longest rod (orange) and work up/down in order.  You could bring everyone together for a mini plenary to share your observations and to encourage systematic working. Remember that there are lots of ways to work systematically, so try to take time to understand a pair's system, even if it is one that you wouldn't choose to adopt.

You may like to make an area of the board or wall or flipchart available for children to come up and record when they have found a train that works, or have the interactivity on the whiteboard so that they can create a train for all to see. Encourage the rest of the class to be checking the solutions that are put up, rather than you yourself being the 'checker'. In the plenary, you could use the solutions that have been contributed by the class to model one way of working systematically (or ask a pair to share their way) and that will help reveal whether all the possible solutions have been found, or not.

If you are keen to use this as an opportunity to focus on factors and multiples, look out for pairs who are working in a numerical way and share their approach.  Alternatively, you could draw attention to the numerical representation of the rods by reminding the class, or asking them, how many white rods were used in the pictured train. If we say a white rod is 1 unit long, how long are the other rods, comparing them to white? What colour trains have the children been able to make that are 20 white rods long?  Why has no-one been able to make a train which is made only of light green rods, for example?

### Key questions

What have you tried so far?
How will you remember your solutions?
How can you record what you have done?
Which colour rods fit in exactly?
Which colour rods cannot be fitted in exactly?
How will you know that you have found them all?

### Possible support

Try to offer real Cuisenaire rods if at all possible, otherwise invite pairs of children to use the interactivity on a tablet or computer. Some children might find it helpful to record on squared paper.

### Possible extension

Learners could try using different numbers of white rods to make 'same length trains' with rods of just one colour. Using 21, 22, 23 and 24 could prove interesting. When is it impossible to make trains using rods of just one colour other than white?
In addition, children might like to investigate making some 'same length trains' using rods of all different colours.