Clone of Steel Cables

7797 /www/nrich/html/content/id/7797/ 1 2011-09-16T11:56:22 <?xml version="1.0" encoding="ISO-8859-1"?> <mdoxml version="1.0"><h4>If you are a teacher, click <a href="http://nrich.maths.org/7760&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...</h4> Cables can be made stronger by compacting them together in a hexagonal formation. Here is a 'size 5' cable made up of 61 strands: <mdo:image alt="cable cross section, one cable in the middle with four rings of cables around it" src="cable1.png"></mdo:image> How many strands are needed for a size 10 cable? How many for a size n cable? Can you justify your answer? Once you've had a go at the problem, click below to see how some 15 year old students worked on it. Can you explain their reasoning? Group 1 <div class="toggle"><img alt="student's picture of hexagon split into three quadrilaterals, two 5*5 rhombuses and a 4*4 rhombus" "="" data-cke-saved-src="/content/id/7760/student3a.png" src="/content/id/7760/student3a.png"> <mdo:image alt="table showing dimensions of the three quadrilaterals and total T for a size 2 up to 10 cable and a size n cable" src="student3b.png"></mdo:image> <mdo:image alt="T=n^2+n^2-n+n^2-2n+1 = 3n^2-3n+1 or 3n(n-1) +1" src="student3c.png"></mdo:image></div> Group 2 <div class="toggle"><mdo:image alt="student's picture of cable with horizontal arrows showing row lengths n, n+1, n+2 up to 2n-1 in the middle and then decreasing back down to n" src="Student1a.png" style="padding-bottom: 220px;"></mdo:image> <mdo:image alt="student's method is adding up pairs of rows to make n pairs of (3n-1), with the 2n-1 row counted twice, giving a total 3n(n-1) +1" src="student1b.png"></mdo:image></div> Group 3 <div class="toggle"><mdo:image alt="student's picture of hexagon split into six triangles" src="student2a.png" style="padding-bottom: 200px;"></mdo:image> <mdo:image alt="student's method is to use the formula for the (n-1)th triangular number n(n-1)/2, multiply by six, and then add 1 for the cable in the centre" src="student2b.png"></mdo:image></div> Group 4 <div class="toggle"><mdo:image alt="student's picture of hexagon showing four rings and one cable in the centre" src="student4a.png"></mdo:image> <mdo:image alt="1, 6*1, 6*2, 6*3, 6*4 ... 6*(n-1). We noticed that the area of each ring followed this pattern. To find the total we needed to add the area of each ring. 1 + 6*1 + 6*2 + 6*3 ... 6(n-1)=1+6(1+2+3+...+(n-1) 1+6(n(n-1)/2) 1+3n(n-1)" src="student4b.png"></mdo:image></div> Which of the four approaches makes the most sense to you? What do you like about your favourite approach? Can you think of any other approaches? Notes and Background Hexagonal packings are often used for strength or efficiency.</mdoxml> <mdoxml version="1.0"> <h3>Why do this problem?</h3> Many students are accustomed to using number patterns in order to generalise. <a href="http://nrich.maths.org/7760">This problem</a> offers an alternative approach, challenging students to consider multiple ways of looking at the structure of the problem, through making sense of other people's approaches, an important part of working mathematically. The powerful insights from these multiple approaches can help us to derive general formulae, and can lead to students' appreciation of the equivalence of different algebraic expressions. This problem and <a href="/6675">Christmas Chocolates</a> are essentially the same question but presented in a different way. <h3>Possible approach</h3> Start by showing this image: <mdo:image src="cable1.png"></mdo:image> "Cables can be made stronger by compacting them together in a hexagonal formation. Here is a 'size 5' cable. Can you work out, without counting every strand, how many strands it contains?" Give students a short time to consider this and then discuss their ideas in pairs, before bringing the class back together to share their different methods. Hand out <a href="/content/id/7760/steel%20cables%20intro.pdf">this worksheet</a>, together with <a href="/content/id/7760/steel%20cables%20templates.pdf">these templates</a>, and ask: "How many strands are needed for a size 10 cable?" "While you are working on this, keep in mind how your method could be adapted to work out the number of strands needed for any size of cable." While students are working, circulate and observe any interesting methods that students are using. When the class is ready, bring them back together and invite those students with an interesting method to explain what they did to the rest of the group. "Here is some work done by groups of 15 year old students who were asked to find a formula to work out the strands needed for a size n cable." Arrange the class in groups of four, and hand out <a href="/content/id/7760/steel%20cables-groups.pdf">this worksheet.</a> "I'd like each of you to take one of the sheets and make sense of the method used by one of the groups. Then when you are ready, you need to explain the method to the rest of your group. Finally, the group needs to decide which of the four approaches makes the most sense to you, and be prepared to justify your choice to the rest of the class." To finish off, each group can explain to the rest of the class which approach they chose, and why. Here are some key questions that could be used to help groups who are struggling to make sense of the different methods: "Is there a quick way of adding up all the numbers from 1 to n?" "Look at the picture for a size 5 cable. What might the group have drawn for a size 6 cable?" <h3> </h3> <h3>Possible extension</h3> Challenge students to come up with alternative, elegant ways of computing the number of strands in a size n cable. The problems <a href="http://nrich.maths.org/6590&part=">Summing Squares</a> and <a href="http://nrich.maths.org/325&part=">Picture Story</a> lead to formulae for some intriguing sequences through analysis of structure. <h3>Possible support</h3> <a href="http://nrich.maths.org/2290&part=">Seven Squares</a> gives lots of simple contexts where formulae emerge by looking at structure rather than number sequences. <a href="http://nrich.maths.org/2274&part=">Picturing Triangle Numbers</a>, <a href="http://nrich.maths.org/6703&part=">Mystic Rose</a>, and <a href="http://nrich.maths.org/6708&part=">Handshakes</a> introduce students to summing consecutive numbers, which is a feature of several of the solutions used by the groups in the problem. </mdoxml> <mdoxml version="1.0"> Do you know a quick way of adding up all the numbers from 1 to n? If not, take a look at the problem <a href="/2274">Picturing Triangle Numbers</a>. If you are finding it difficult to make sense of the different groups' work, try to work out what they might have drawn for a size 6 cable. </mdoxml> 2 3 0 0 0 1 0 Clone of Steel Cables Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions? Using, Applying and Reasoning about Mathematics Generalising