7350
/www/nrich/html/content/id/7350/
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2011-02-01T00:00:01
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You might like to have a look at <a href="http://nrich.maths.org/7227&part=">Button-up</a> before trying this problem.<br></br>
<br></br>
I have a jacket which has four buttons.<br></br>
Sometimes, I do the buttons up starting with the top button. Sometimes, I start somewhere else. <br></br>
How many different ways of buttoning it up can you find?<br></br>
<br></br>
Look back at the number of different ways you found for buttoning up three buttons and four buttons.<br></br>
Can you predict the number of ways of buttoning up a coat with five buttons?<br></br>
Six buttons ...? <br></br>
<br></br>
<p><span style="font-style: italic;">This problem was inspired by an idea from Bernard Murphy.</span></p>
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<p class="editorial">Madeleine and Luke at Greystoke Primary worked
carefully on four buttons:</p>
<div>We decided that we needed an organised way of working to make
sure that we found all the different ways.</div>
<div>So we started with all the ways when you start with the first
button, then all the ways you start with the second button
etc.</div>
<div>In total we found $6$ ways for each so $6 \times 4 =
24$.</div>
<div>So we think that there are $24$ ways of buttoning up the
coat.</div>
<div>Here are all the ways:</div>
<div>$1234$, $1243$, $1324$, $1342$, $1423$, $1432$, $2134$,
$2143$, $2431$, $2413$, $2314$, $1341$, $3124$, $3142$, $3412$,
$3421$, $3241$, $3214$, $4321$, $4312$, $4213$, $4231$, $4123$,
$4132$</div>
<div>$6 \times 4 = 24$ ways.</div>
<br></br>
<br></br>
<p><span class="editorial">Jordan from Birds Bush Primary gave some
helpful detail about how we can make sure we don't miss any
possibilities:</span></p>
<br></br>
At the start of a solution start with the basic solution of
'$1234$'.<br></br>
Then switch the last two numbers,$1243$.<br></br>
As you can't swap them again, swap the second number in the
solution to another but not the first number. In this case
it's one eg: $1324$.<br></br>
Now swap the last two numbers' places - $1342$.<br></br>
As before you swap the second number in the solution with the only
number which hasn't been second: $1432$.<br></br>
And again swap the last two numbers' places - $1423$.<br></br>
To get all solutions just use the same method but change the first
number until all numbers have been in first place and you should
end up with the $24$ different combinations as follows;<br></br>
$1234$, $1243$, $1324$, $1342$, $1432$, $1423$, $2341$, $2314$,
$2413$, $2431$, $2134$, $2143$, $3214$, $3241$, $3412$, $3421$,
$3124$, $3142$, $4321$, $4312$, $4231$, $4213$, $4123$, $4132$.
<br></br>
<br></br>
<p><span class="editorial">Children from La Mare De Carteret
Primary school agreed with 24 being the total number of ways for
four buttons.</span></p>
<br></br>
Scarlet, Will and Oliver, also from Greystoke Primary, investigate
more buttons:<br></br>
<br></br>
We looked at what happens when there are $5$ buttons on the
coat.<br></br>
Again we had an organised method and started by looking at all the
ways when starting with the first button.<br></br>
We found $24$ ways when starting with the first button, so we
didn't need to do any more working, we just calculated $24
\times 5$ which is $120$ ways of buttoning up the coat with
$5$ buttons. <br></br>
<br></br>
<p><span class="editorial">I like the way you decided you didn't
need to work out the number of ways for each button being first as
it would be the same for all buttons. Noticing short-cuts
like that is a great skill for a mathematician.</span></p>
<br></br>
<p><span class="editorial">Ella who goes to North Molton Primary
looked at the number of ways to do up the buttons in a more general
way. Here is what she wrote: </span></p>
<div>First of all I got a piece of paper and drew all four buttons,
naming them one, two, three and four.</div>
<div>Then, I experimented with the numbers, seeing how many
different ways there were to order them.</div>
<div>I started of by working out how many different ways there were
to do them up starting with button number one. There were six
different ways. I did the same with buttons two, three and four.
Each of them had six different ways.</div>
<div>So, all together there were twenty four different ways to do
the four buttons up.</div>
<div>I tried the same method with three buttons. All together there
were six different ways to do the three buttons up.</div>
<div>I realised that all the different ways for four buttons to be
done up (twenty four) was the number of buttons (four) multiplied
by the amount of ways a jacket could be buttoned up with one fewer
button (six).</div>
<div>So, the way to work out how many ways there are to do up the
jacket is the amount of buttons it has multiplied by the number of
ways you can do it up with one less button.</div>
<div>For four buttons: </div>
<div>$4 \times 6 = 24$</div>
<div>(number of buttons) $\times$ (the number of ways to do it up
with one less button) = amount of ways to do it up.</div>
<br></br>
<br></br>
<p><span class="editorial">Very well noticed, Ella. Dan and G
from St Saviour's also suggested this was a good way to calculate
the number of ways. </span></p>
<p><span class="editorial">William from North Molton looked at it
slightly differently. He said:</span></p>
<div>I worked out that you have to times all the numbers you are
using then you will get the answers e.g.</div>
<div>$1-4 = 1 \times 2 \times 3 \times 4 =
24$.</div>
<div>$1-5 = 1 \times 2 \times 3 \times 4 \times 5 =
120$.</div>
<div>$1-6 = 1 \times 2 \times 3 \times 4 \times 5 \times 6 =
720$.</div>
<br></br>
<br></br>
<p><span class="editorial">Krystof from Uhelny Trh in Prague
used a special symbol to write this down:</span></p>
<br></br>
$2! = 2 \times1$<br></br>
$3! = 3 \times 2 \times1$<br></br>
$4! = 4 \times 3 \times 2 \times1$ etc.<br></br>
<br></br>
<p><span class="editorial">Krystof explained, therefore, that for
$n$ buttons there would be $n!$ different ways of buttoning them
up.</span></p>
<br></br>
<p class="editorial">Alex from Maidstone Grammar expressed the
total number of ways of doing up a button slightly differently
again:</p>
<br></br>
There is a pattern to do with the factor of the numbers.<br></br>
The first amount of sets is $6$, which equals $3 \times 2$.<br></br>
The second set is a larger figure, $3 \times 8$.<br></br>
Finally, $5$ buttons is a huge $3 \times 40$.<br></br>
Each time, the second factor is multiplied larger.<br></br>
It is first multiplied by $4$, to give $8$, and then $5$ to make
$40$.<br></br>
Therefore, $6$ buttons should equal $3 \times (40 \times 6)$ which
is $3 \times 240$, giving $720$ combinations.<br></br>
<br></br>
<p class="editorial">I wonder whether you can see how these
different methods of expressing the total number of ways of doing
up the buttons are connected? If I used Alex's or Krystof's
or William's or Ella's method, would I get the same answer, say for
$10$ buttons? Why? </p>
<p class="editorial">Well done to you all.</p>
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<h2>Button-up Some More</h2>
<br></br>
You might like to have a look at <a href="http://nrich.maths.org/7227&part=">Button-up</a> before trying this problem.<br></br>
<br></br>
I have a jacket which has four buttons.<br></br>
Sometimes, I do the buttons up starting with the top button. Sometimes, I start somewhere else. <br></br>
How many different ways of buttoning it up can you find?<br></br>
<br></br>
Look back at the number of different ways you found for buttoning up three buttons and four buttons.<br></br>
Can you predict the number of ways of buttoning up a coat with five buttons?<br></br>
Six buttons ...? <br></br>
<br></br>
<p><span style="font-style: italic;">This problem was inspired by an idea from Bernard Murphy.</span></p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>In <a href="http://nrich.maths.org/7227&part=">Button Up</a>, the focus was on working in a systematic way. <a href="http://nrich.maths.org/7350&part=">This follow-up problem</a> 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.</div>
<br></br>
<h3>Possible approach</h3>
<div>It would be a good idea to introduce the problem in a similar way to that suggested in the <a href="http://nrich.maths.org/7227&part=note">teachers' notes of Button Up</a>.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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? </div>
<div> </div>
<div>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.</div>
<br></br>
<h3>Key questions</h3>
<div>How do you know you have found all the ways?</div>
<div>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?</div>
<div>How will you record what you're doing?</div>
<br></br>
<h3>Possible extension</h3>
<div>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? </div>
<div>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?</div>
<br></br>
<h3>Possible support</h3>
<div>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.</div>
<br></br>
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<mdoxml version="1.0">How do you know you have found all the ways? <br></br>
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? <br></br>
How will you record what you're doing?<br></br></mdoxml>
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Button-Up Some More
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Using, Applying and Reasoning about Mathematics
Working systematically
Using, Applying and Reasoning about Mathematics
Generalising
Decision Mathematics and Combinatorics
Combinations
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