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Investigating patterns in number grids. I am investigating the factors affecting the products of the diagonally opposite corners of a square or rectangle, drawn on a square number grid. I will provide evidence using algebra. 2 x 2 squares 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 70 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2 x 11 = 22 \ Difference = 10 1 x 12 = 12 / 1 +1 -1 2 + 10 +11 +9 + 10 - 10 -9 -11 - 10 11 +1 -1 12 89 +1 -1 90 + 10 +11 +9 + 10 - 10 -9 -11 - 10 99 +1 -1 100 12 +1 -1 13 + 10 +11 +9 + 10 - 10 -9 -11 - 10 22 +1 -1 23 The differences are all the same x +1 -1 x + 1 + 10 +11 +9 + 10 - 10 -9 -11 - 10 x + 10 +1 -1 x + 11 On any square grid draw a square around 4 numbers ie: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 The difference between the top left number in the small square multiplied by the bottom right number (1x6) and bottom left number multiplied by the top right number (5x2) is the size of the grid (1x6) - (5x2) = 4 4x4 grid x x+1 x+10 x+11 x(x+11) = x²+11x = \ Difference = 10 (x+10)(x+1) = x²+x+10x+10 = / ^ This square will be a 10 x 10 grid! ^ x x+1 x+4 x+5 1 2 3 4 x x+1 x+2 x+3 ^ This...

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