Challenge Level

This problem encourages learners to connect geometrical and numerical ideas, which will help them to generalise and prove. The task makes a very good introduction to algebraic thinking, however, it also has a very accessible starting point so everyone can begin.

*This problem featured in an NRICH Primary webinar in March 2022.*

You could start by giving everyone in the group some interlocking cubes (or Lego blocks) and asking them to make a stick (or tower) of seven. Ask them to break their stick into two pieces and hold one piece in each hand. Ask how they have done this and record the various ways on the board. (Try to be guided by the group in terms of how you record. Perhaps learners would like you to sketch the two resulting sticks; perhaps they are happy for numbers to be recorded; some may wish to include an addition sign...) Then enquire if there are any other ways that it could have been done. How will they know that they have found all the different ways?

At this point, it might be useful to record each different solution on a different piece of paper, as you can then move them around the board (or table, or floor) to impose some sort of order on them. (Alternatively, you could gather the sticks/towers themselves.) This will help reveal which, if any, have been omitted. This is the basis of working systematically.

Some learners may suggest that 2+5 is the same as 5+2. Numerically they are identical, but the context is also important. In this case, because we are using seven **differently coloured** cubes or Lego blocks, splitting into 2 and 5 will *look different* from 5 and 2.

After this introduction, learners could work in pairs on finding the number of ways that sticks of six and eight cubes can be broken into two pieces. Look out for pairs who record their findings clearly, keeping track of what they have tried, and using their recording to help them know whether they have found all possibilities. You could share some of these recordings in a mini
plenary if appropriate. Try to encourage learners to use any resources they choose.

After a suitable length of time, gather everyone together again. Encourage learners to explain how they knew they had found all the ways in each case. Make a note on the board of the number of ways for six cubes, seven cubes and eight cubes. What do they notice? Can they use what they notice to predict the number of ways of breaking up twelve cubes into two pieces, for example? Work together
as a whole group to find the ways practically, to verify their conjecture.

Is the number of ways of breaking a stick of cubes into two pieces always going to be one less than the number of cubes? Explain that mathematicians aren't usually satisfied with finding a few examples, they want to build a convincing argument that something is always true. Can the class help create such an argument?

How are you going to record what you have done?

How do you know you have found all the ways of breaking it into two pieces?

Can you see a connection between the total number of cubes and the number of ways you can break the stick into two pieces?

How many ways do you think you can break 20 cubes?

How could you express that "for any number"?

How would you convince a mathematician that this is always going to be true?

Learners might find it helpful to use interlocking cubes or Lego blocks and to record on squared paper with coloured pencils/pens.

If all the cubes are the same colour, a split of 4 and 2 will look the same as a split of 2 and 4. Challenge learners to consider how many ways there are of splitting 6 cubes now. Can they predict how may ways there will be with any number of cubes?

Then they could try a different problem in which generalisation is required such as Sticky Triangles.