Courseworks

Maths Coursework – N Lines

Please note that this coursework depends heavily on the graphics available in the attached word document.RULESTo find out the regions it is:Lines + lines + closed regions(n+n= closed regions)To find out crossovers it is:Lines ? (lines – 1)2n? (n-1)2To find out open regions it is:Lines ? 2(n?2)To find out closed regions it is:Region – open regions3 lines12 3 45 6 77 regions6 open regions1 closed region3 crossovers5 lines123456 78910 11 12 1316 regions10 open region15 6 closed regions10 crossovers14166 lines1 2 3 4 567 8 9 1013 1411 12161718 1920 regions15 crossovers12 open regions10 closed regions202122This is what I predicted7 lines1 2 3 4 5 611 13 1410 157 8 917 18 19 202921 22 21 Crossovers29 regions16 25 14 open regions15 closed regions23 242627 28My prediction was correctIntroductionWe have been asked to draw a number of lines which cross over each other.e.g.12 394 10 11 576 8This is an example of 4 lines. Each line has to go over all the other lines. There are 6 cross over points. There are 11 regions, 1-8 are open regions and 9-11 are closed regions.We had to these trying lines of 3-6, and choose the one with the maximum cross over points. We have to put this on a table and come up with a rule to find out how many cross over points, how many regions and how many open and closed regions it has.RESULTS TABLElinesCrossoversregionsOpenregionsClosedRegions3376146118351016106615221210Prediction721291415WHAT I NOTICEDI noticed that the crossover points are triangle numbers.I noticed that the regions go up one more than it did the last time.e.g.7>411>516>622>729This also happens to the closed regions.The open regions go up by two every time.If you know the ones before you can work out what the next one will be.When you work out the crossovers, you multiply the number of lines by one less than the number of lines because each line crosses over all the other lines but it doesn’t cross over its own line that is why you have to minus one. You have to divide it by 2 because each cross over has two lines going through it and it has been counted twice, if you don’t divide it, it will have twice as much as it should have.

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