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.

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? 
 

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?