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Steel Cables

Stage: 4 Challenge Level: Challenge Level:1

Do you know a quick way of adding up all the numbers from 1 to n?
If not, take a look at the problem Picturing Triangle Numbers.


Below are student diagrams from the problem page, but they now also show the working that the students used to explain their diagrams. Can you explain their reasoning?

Group 1

student's picture of hexagon split into three quadrilaterals, two 5*5 rhombuses and a 4*4 rhombus table showing dimensions of the three quadrilaterals and total T for a size 2 up to 10 cable and a size n cable T=n^2+n^2-n+n^2-2n+1 = 3n^2-3n+1 or 3n(n-1) +1

Group 2

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 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

Group 3

student's picture of hexagon split into six triangles 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

Group 4

student's picture of hexagon showing four rings and one cable in the centre 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)

 

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.