Lines, Regions and Crossovers
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Coursework Title: Lines Candidate name: Shoaib Muhammad Candidate number: 5115 Centre name: Geoffrey Chaucer Technology College Centre number: 10818 Plan Firstly In this coursework I will draw six or more lines which will cross each other and while doing this I hope to get as much crossover points as I can, as well as I will try to get the maximum regions. I will try to avoid any sort of double intersecting i.e. intersecting over a ready made crossover point. I will try and keep all my lines as consistent (placement). To make sure I do this I will draw 1 line and then copy past it and then add another line this will help me keep my lines the same i.e. consistent. But I will vary the number of lines I use. I will draw six line models (each having one more line than the previous one) the first model that I will create will have only one line the second will have two lines the third will have three lines etc. Secondly I will try to figure out the formulas that will find the nth term for the crossover points, open regions, closed regions and the total regions. I also plan to find the relationships / sequence between any two of these characteristics of the line(s).
of Lines 1 2 3 4 5 6 Closed Regions 0 0 1 3 6 10 Total Regions 2 4 7 11 16 22 Difference -2 -4 -6 -8 -10 -12 The nth term The nth term is useful in order to predict the next number in any particular sequence without actually going through the trouble of working it out, which in this case means drawing the lines and counting the number of open regions for example. It can be worked out by using the formula below: A+B (n-1) +0.5(n-1) (n-2) C Where, A = 1st term in the sequences below. B = The 1st difference between the two different terms in the sequences below. C = the 2nd difference in the sequences below which is always constant. Working out the nth term for closed regions Sequence 0 0 1 3 6 10 1st Difference 0 1 2 3 4 2nd Difference 1 1 1 1 Consequently, A = 0 B = 0 C = 1 The equation for closed regions is = 0+0(n+1) +0.5(n-1) (n-2)1 Simplified = (0.5n2)-(1.5n) +1 Formula for nth term = 0.5n2-1.5n+1 Working out the nth term for crossover points Sequence 0 1 3 6 10 15 1st Difference 1 2 3 4 5 2nd Difference 1 1 1 1 Consequently, A = 0 B = 1 C = 1 The equation for crossovers is = 0+1(n+1)
I was able to prove my hypothesis correct and was able to fulfil my aim of working out the nth term for many regions. It took me a while to draw the diagrams as I wanted each crossover to give me the maximum number of open and closed regions. But once I had the diagrams, I simply put the results in a table and then looked for any obvious or underlying sequence. Problems and hurdles I faced while undergoing the investigation: * I had a bit of a problem trying to make sure that I didn't double intersect the lines because if I did that would have ruined all my investigation as it wouldn't have been a fair examination to make sure I didn't do this I double checked all my lines before I went ahead in the investigation. * I also had a bit of a problem trying to find a complex pattern but after a while I managed to find it. How would I improve my investigation next time: * I would give my self a longer time span in order to complete my investigation which will help me enhance the quality of my work. * I would try to find more complex patterns in order for me to do this I will have analyze all my work more carefully.
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