Break it up!

In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

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

 

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Break it up!

 

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

Here is one of the ways in which you can do it:

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Break it up!

 

Here is another way you can do it:

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Break it up!

 

In how many different ways can it be done?

Now try with a stick of eight cubes:

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Break it up!

What about with a stick of six cubes?

 

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Break it up!

 

What do you notice?

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

Try it! Were you right?

How many ways with 20 cubes? 

Will your noticing always be true? Can you create an argument that would convince mathematicians?