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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?

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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?

Red Express Train

Age 5 to 7
Challenge Level

Why do this problem?

This activity can be used as a first entry for young pupils in exploring a somewhat open problem in which ordinary addition or subtraction play no apparent part. I have found that it is also useful in helping pupils develop some perseverance skills.

Possible approach

You may stick to the train situation but it may be that in the classroom environment there may be another parallel story. You may decide on setting out five small chairs of one colour and use two of a different colour in a straight line. Or, have five children of one gender and two of the other sitting behind each other. There may be other reources in which two can be added to five. The pupils that I have worked with have used two sets of cubes - $5$ of one colour and $2$ of another. There will likely be some opportunities of seeing different ways of approaching the challenge, there may also be different systems used for ensuring that all the possibilities are found.

Key questions

How are you finding ways of having the carriages?
Have you got the correct number of carriages?
What could you do next?
Do you thinkthere are other ways of putting them?

Possible extension

Try extending the numbers used $5$ & $3$, $6$ & $3$ etc.
Go to two on five to explore a similar activity but now in more dimensions.

Possible support

When the pupils are at their tables some may find it helpful to have coloured cubes or something similar to represent the carriages.