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  • Level: GCSE
  • Subject: Maths
  • Word count: 2891

Investigating when pairs of diagonal corners are multiplied and subtracted from each other.

Extracts from this document...

Introduction

Investigating when pairs of diagonal corners are multiplied and subtracted from each other. Introduction: In this coursework I shall be investigating when pairs of diagonal corners are multiplied and subtracted from each other. To do this I shall take square grids and then select 3 square boxes from each grid and multiply the opposite corners then find the difference, then I will change the box size, to try and find a pattern or formula. Then I will change the grid size and then once again take boxes of different sizes and multiply the opposite corners and find the difference. I will try and find a formula for and square box on and size square grid. Then I shall investigate further by using rectangle boxes instead of square. 34 x 45 = 1530 35 x 44 = 1540 Difference = 10 68 x 79 = 5372 69 x 78 = 5382 Difference = 10 81 x 92 = 7452 82 x 91 = 7462 Difference = 10 In a 2 x 2 box on a 10 x 10 grid the difference is 10. Algebraic Method x x + 1 x + 10 x + 11 x(x + 11) = x� + 11x (x + 1)(x + 10) = x� + 10 + x +10x = x� + 11x + 10 x� + 11x - x� + 11x + 10 = 10 1 x 23 = 23 3 x 21 = 61 Difference = 40 36 x 58 = 2088 38 x 56 = 2128 Difference = 40 73 x 95 = 6935 75 x 93 = 6975 Difference = 40 In a 3 x 3 box on a 10 x 10 grid the difference is 40. Algebraic Method x x + 2 x + 20 x + 22 x(x + 22) = x� + 22x (x + 2)(x + 20) = x� + 20x + 2x + 40 = x� + 22x + 40 x� + 22x - x� + 22x + 40 = ...read more.

Middle

N = Any Number (b -1) = Box size - 1 Let x = (b - 1) n n + x n + 8x n + 9x n(n + 9x) = n� + 9xn (n + 1)(n + 8) = n� + 8xn + xn + 8x� = n� + 9xn + 8x� n� + 9xn - n� + 9xn + 8x� = 8x� 8x� = 8(b-1) � I will now do the same process on an 11 x 11 grid to see if there is a pattern. 14 x 26 = 364 15 x 25 = 375 Difference = 11 28 x 40 = 1120 29 x 39 = 1131 Difference = 11 76 x 88 = 6688 77 x 87 = 6699 Difference = 11 In a 2 x 2 box on an 11 x 11 grid the difference is 11. Algebraic Method x x + 1 x + 11 x + 12 x(x + 12) = x� + 12x (x + 1)(x + 11) = x� + 11x + x + 11 = x� + 12x + 11 x� + 12x - x� + 12x + 11 = 11 5 x 29 = 145 7 x 27 = 189 Difference = 44 42 x 66 = 2772 44 x 64 = 2816 Difference = 44 89 x 113 = 10057 91 x 111 = 10101 Difference = 44 In a 3 x 3 box on an 11 x 11 grid the difference is 44. Algebraic Method x x + 2 x + 22 x + 24 x(x + 24) = x� + 24x (x + 2)(x + 22) = x� + 22x + 2x + 44 = x� + 24x + 44 x� + 24x - x� + 24x + 44 = 44 4 x 40 = 160 7 x 37 = 259 Difference = 99 78 x 114 = 8892 81x 111 = 8991 Difference = 99 74 x 110 = 8140 77 x 107 = 8239 Difference = 99 In a 4 x 4 box on an 11 x 11 grid the difference is 99. ...read more.

Conclusion

= n� + 9xn + ny (n + y)(n + 9x) = n� + 9xn + ny + 9xy n� + 9xn + ny - n� + 9xn + ny + 9xy = 9xy 9xy = 9(L - 1) (W - 1) I have noticed a pattern in my formulas. The number that comes before the brackets is the same as the grid size. So I can change that number to G when G = Grid Size. So the formula for any box size on any grid size: G (L - 1)(W - 1) Proving Formula: n n+(W - 1) n + G(L - 1) n + G(L -1) + (W - 1) N = Any Number (L - 1) = Length - 1 (W - 1) = Width - 1 Let x = (L - 1) Let y = (W - 1) n n + y n + Gx n + Gx + y n(n + Gx + y) = n� + nGx + ny (n + y)(n + Gx) = n� + nGx + ny + Gxy n� + nGx + ny - n� + nGx + ny + Gxy = Gxy Gxy = G (W- 1)(L - 1) Conclusion Formula Description 10(b - 1) � This calculates the difference of any square box on a 10 x 10 grid. 9(b - 1) � This calculates the difference of any square box on a 9 x 9 grid. 8(b - 1) � This calculates the difference of any square box on an 8 x 8 grid. 11(b - 1) � This calculates the difference of any square box on an 11 x 11 grid. G(b - 1) � This calculates the difference of any square box on any grid. 10(L - 1)(W - 1) This calculates the difference of any rectangle box on a 10 x 10 grid. 9(L - 1)(W - 1) This calculates the difference of any rectangle box on a 9 x 9 grid. G(L - 1)(W - 1) This calculates the difference of any rectangle box on any grid. Limitations: ...read more.

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