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Break it Up!

Age 5 to 11
Challenge Level

Break it Up!

You have a stick of 7 interlocking cubes (or a tower of 7 Lego blocks). You cannot change the order of the cubes.

 

stick of seven differently coloured cubes

 

You break off a bit of it leaving it in two pieces.

Here are three of the ways in which you can do it:

2 interlocking cubes and 5 interlocking cubes

3 interlocking cubes and 4 interlocking cubes

5 interlocking cubes and 2 interlocking cubes

 

In how many different ways can it be done?

Now try with a stick of 8 cubes and a stick of 6 cubes:

eight interlocking cubes of different colours six interlocking cubes of different colours

 

Now predict how many ways there will be with 5 cubes.

Try it! Were you right?

How many ways with 20 cubes? 50 cubes? 100 cubes?

ANY number of cubes?

 

Why do this problem?

This problem makes a very good introduction to algebraic thinking, however, it also has a very accessible starting point so everyone can begin.

Possible approach

You could start by giving all the group some interlocking cubes (or Lego blocks) and asking them to make a stick (or tower) of seven. Then tell them to break their stick into two pieces and hold one piece in each hand. Ask how they have done this and write the various ways on the board and 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. This will help reveal which, if any, have been omitted. This is the basis of working systematically.

 
Some learners may claim that 2+5 is the same as 5+2. "Not always!" is the answer to this. 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 the introduction, learners could work in pairs on finding the number of ways that sticks of six, eight and nine cubes can be broken into two pieces. Look out for ways that pairs 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 pairs to use any resources they choose.
 
When learners feel ready to generalise they can go on to working out the number of ways with $20$, $50$ and $100$ cubes and then ANY number of cubes.
 
At the end of the lesson all can come together to discuss their findings. They can be asked how they knew they had found all the ways of breaking a stick into two pieces. Those who know a good way of expressing "any number" can explain their reasoning.

 

Key questions

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?
If you break it into 0+7, does this give you 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 20/50/100 cubes?
How could you express that "for any number"?
 
 

Possible support

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

Possible extension

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.