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Button-up Some More

Age 7 to 11
Challenge Level

Button-up Some More

You might like to have a look at Button-up before trying this problem.
I have a jacket which has four buttons.
Sometimes, I do the buttons up starting with the top button. Sometimes, I start somewhere else.  
How many different ways of buttoning it up can you find?
Look back at the number of different ways you found for buttoning up three buttons and four buttons.
Can you predict the number of ways of buttoning up a coat with five buttons?
Six buttons ...?  

This problem was inspired by an idea from Bernard Murphy.

Why do this problem?

In Button Up, the focus was on working in a systematic way.  This follow-up problem will allow children to consolidate their understanding of working systematically, but the main objective is to encourage them to identify and explain patterns, which will lead to  generalisations.
Here you can see a response from a school in Canada together with some useful notes that might be helpful to read before watching the video.

Possible approach

It would be a good idea to introduce the problem in a similar way to that suggested in the teachers' notes of Button Up.
The focus for some pairs as they work on this activity will be on developing a system so that they know they have found all the possible ways.  However,  you might expect that learners working at a higher level won't need to write out all $120$ ways for five buttons (and perhaps not all $24$ ways for four buttons).  Encourage these children to explain how they know that the total is right, even though they haven't listed all the possibilities.
For example, for four buttons, they might find that starting with the top button, there are six different ways, so this means there will be six different ways starting with the second button, six ways starting with the third and six with the fourth, making $24$ ways in total. Alternatively, they could argue that if the first way for three buttons is ABC, you could add in a fourth button in four different ways i.e. DABC, ADBC, ABDC and ABCD. You can add a fourth button into all six ways, giving $6 \times 4=24$ ways for four buttons.
You can then encourage children to be able to predict the number of ways of buttoning up a jacket  with any number of buttons.     Can they convince themselves why their method works?  Can they convince  another pair why it works?  
This activity might make a good 'simmering' task so that it is worked on over a period of a few days or weeks before you bring all the children's ideas together.

Key questions

How do you know you have found all the ways?
How could you use the number of ways to button up three buttons to help you work out the number of ways for four buttons?
How will you record what you're doing?

Possible extension

Rather than being interested in the order of buttoning, invite the children to investigate the total distance that their hands have to travel to do up all the buttons. For example, for three buttons, what is the greatest distance that their hands can travel? What is the least distance? How about for four buttons?  Five buttons? 
Can learners make any generalisations about the distances travelled? For example, how would they achieve the shortest distance for any number of buttons? How would they achieve the longest distance for any number?

Possible support

Some learners might benefit from writing each order on a separate strip of paper.  The strips can then be ordered so that any missing possibilities might be identified more easily.