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GCSE Mathematics

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Introduction

GCSE Mathematics Coursework Number Grid By Gareth Davies Introduction For this task I was presented with a 10x10 number grid (see below). Within this grid was a box drawn around four numbers. I was asked to find the product of the top left and the bottom right numbers or corners in that box. And again with the top right and bottom left numbers. Finally I was asked to calculate the difference. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 To calculate the product difference, firstly I must multiply. ...read more.

Middle

1. 43 44 45 53 54 55 63 64 65 Product difference = (45 - 63) - (43 x 65) = 40 2. 27 28 29 37 38 39 47 48 49 Product difference = (29 x 47) - (27 x 49) = 40 3. 71 72 73 81 82 83 91 92 93 Product difference = (73 x 91) - (71 x 93) = 40 Again my hypothesis was verified (Absolute proof would involve checking every possible combination). I carried on using 4x4, 5x5, 6x6 and 7x7 boxes. And decided that the product difference were always constant. To show my results in a clearer way here is a table. Box Size Product Difference 2x2 3x3 4x4 5x5 6x6 7x7 10 40 90 160 250 360 In an effort to find a pattern I wrote the product differences in sequence form. 10 40 90 160 250 360 v v v v v First difference 30 50 70 90 110 v v v v Second difference 20 20 20 20 This illustrates that the second difference is constant. From this, I could establish a formula. ...read more.

Conclusion

- (1x33) Product difference = 112 2. 15 16 17 18 19 22 23 24 25 26 29 30 31 32 33 36 37 38 39 40 43 44 45 46 47 (19x43) - (15x47) Product difference= 112 Once again to establish a formula I wrote the product differences I sequence. 7 28 63 112 v v v First difference 21 35 49 v v Second difference 14 14 To obtain my formula I must use N as I'm using the second difference, also as explained previously half the constant must be used, therefore the formula would be 7N . Thus verifying my previous hypothesis. Conclusion I can now safely say that to predict a product difference I could use this formula: Product difference= Grid size (ie. 10x10 would be 10, 7x7 would be 7) x Nth term(Nth term being 2x2 box would be 1, 3x3 box would be 2) Eg. To predict the Product difference of a 3x3 box within a 10x10 grid: =10N = 10 x 3 = 90 Evaluation Although my formula is quite efficient, I have found that it doesn't work for irregular size boxes ie. 2x3 or 3x4 etc. If I had extra time I could consider investigating this problem further eg adapting the formula, maybe factorizing it. 1 ...read more.

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