Button-Up

7227 /www/nrich/html/content/id/7227/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> My coat has three different buttons.<br></br> <br></br> Sometimes, I do them up starting with the top button. Sometimes, I start somewhere else.<br></br> <br></br> How many ways can you find to do up my coat?<br></br> <br></br> How will you remember them?<br></br> <br></br> Do you think there are any more? How do you know?<br></br> <br></br> <p style="font-style: italic;">This problem was inspired by an idea of Bernard Murphy.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">You found several different ways to help find the solution to this problem. </span></p> <p class="editorial">Lydia and Megan from Moorfield drew pictures of the buttons and numbered each button according to the order it was done up. They found six ways:</p> <br></br> <br></br> <mdo:image height="240" width="300" alt="" src="meganlydiasnapshot..jpg"></mdo:image><br></br> <br></br> <p><span class="editorial">Some of you described the buttons as 'top', 'middle' and 'bottom' then made a list of all the possible ways of doing them up. For example, Abbie from Oakthorpe Primary said:</span></p> <p> </p> <div>If we start with the top button:</div> top middle bottom<br></br> top bottom middle<br></br> <br></br> If we start with the middle button:<br></br> middle bottom top<br></br> middle top bottom<br></br> <br></br> If we start with the bottom button:<br></br> bottom top middle<br></br> bottom middle top <br></br> <br></br> <p><span class="editorial">Then there were those of you who labelled your buttons as $1$, $2$ and $3$, like Yousef at Levendale Primary who wrote:</span></p> <div> </div> <div>$132, 123, 213, 231, 312, 321 = 6$ times</div> <br></br> <br></br> <p class="editorial">Karnan from Stag Lane Junior School explained how he knew he had all the possibilities:</p> These are all the combinations for the buttons. You can be sure because all you have to do is:<br></br> 1. See how many combinations there are for buttoning the top button on first.<br></br> 2. Then, you have to multiply by three for three possible starting positions.<br></br> <br></br> <p class="editorial">Well done all of you. Kurtis from Moorfield School and Demi from Tudhoe Grange rightly pointed out that we were presuming we wanted to do up all three buttons. Kurtis asks:</p> Perhaps you could find out how many ways there are if you were allowed to do up $1$, $2$ or $3$ buttons? <br></br> <br></br> <p class="editorial">What a great question, Kurtis! </p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Button-up</h2> <br></br> My coat has three different buttons.<br></br> <br></br> Sometimes, I do them up starting with the top button. Sometimes, I start somewhere else.<br></br> <br></br> How many ways can you find to do up my coat?<br></br> <br></br> How will you remember them?<br></br> <br></br> Do you think there are any more? How do you know?<br></br> <br></br> <p style="font-style: italic;">This problem was inspired by an idea of Bernard Murphy.</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>Young children often find working in a systematic way very difficult. <a href="http://nrich.maths.org/7227&part=">This problem</a> gives a real context and an opportunity to record, leading to the very beginnings of proof.</div> <br></br> <h3>Possible approach</h3> <div>Have available a selection of objects which the children could choose to use to represent buttons.</div> <div> </div> <div>You may wish to introduce the problem using your own coat/jacket, or that of a child. Button it up in two different ways and ask the children what was different. Listen for comments to do with the order of buttoning. Ask the children how we could remember what was different. Invite them to talk in pairs about what they would do and then share the children's ideas about recording.</div> <div> </div> <div>Pose the problem as suggested and invite the children to work in pairs to solve it. As the children are working, look out for a pair that has developed a useful recording method. Ask them to copy each solution on to a separate large piece of paper, ready for use with the whole class. (Alternatively if you have an interactive whiteboard, you could ask the pair to record their solutions as separate draggable objects.)</div> <div> </div> <div>After a suitable length of time, bring the whole group together. Invite enough children to hold all the big pieces of paper at the front and ask whether we have found all the different ways of buttoning. How do we know? Is there any way of rearranging the pieces of paper to help us be sure?</div> <div> </div> <div>Give the children the chance to repeat the rearranging activity with their own way of recording.</div> <br></br> <h3>Key questions</h3> <div>What could you change about the buttoning?</div> <div>Do you always have to start in the same place?</div> <div>How will you remember the ways you've found?</div> <div>Is your recording different to other people's?</div> <br></br> <h3>Possible extension</h3> <div>You may like to look at the follow-up to this problem, <a href="http://nrich.maths.org/7350&part=">Button Up Some More</a>, which encourages children to look at having more buttons. </div> <br></br> <h3>Possible support</h3> <div>Some children may have difficulty with the recording of this activity. This is likely to stem from the fact that the most obvious way of recording implies that the buttons move. For example, buttoning green then red then blue could be recorded on the coat as:</div> <div> </div> <div> </div> <div style="text-align: left;"> <mdo:image alt="three buttons" height="226" src="buttons1.jpg" width="109"></mdo:image></div> <div> </div> <div>Or:</div> <div> </div> <div> <mdo:image alt="" height="246" src="buttons3.jpg" width="122"></mdo:image></div> <div> </div> <div>which implies that the buttons have moved.</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">What could you change about the buttoning? <br></br> Do you always have to start in the same place? <br></br> How will you remember the ways you've found? <br></br></mdoxml> 5 3 1 0 0 0 0 Button-Up My coat has three buttons. How many ways can you find to do up all the buttons? Using, Applying and Reasoning about Mathematics Working systematically Decision Mathematics and Combinatorics Combinations Admin Lower primary mapping document