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  • Level: GCSE
  • Subject: Maths
  • Word count: 3576

Number Grids

Extracts from this document...

Introduction

Shaun Lalley 10 Bingham Maths Coursework: Number Grids 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 2 x 2 Grids I will start by investigating what happens if we take a 2x2 grid from a 10x10 master grid. I will multiply the top right number by the lower left. Then I will subtract the lower right number multiplied by the top left e.g. a b c d axd=ab cxb=cd ab-cd 1 2 11 12 1 x 12 =12 22 - 12=10 2 x 11 =22 83 84 93 94 84 x 93 =7812 7812 - 7802=10 83 x 94 =7802 89 90 99 100 90 x 99 =8910 8910 - 8900=10 89 x 100 =8900 9 10 19 20 10 x 19 =190 190 - 180=10 9 x 20 =180 The answer is always 10. I predict that my next 2 x 2 grid will result in the answer 10. 6 7 16 17 6 x 17 =102 102 - 92=10 7 x 16 =92 My prediction was correct. I will now prove that all 2 x 2 grids taken from a 10 x 10 master grid result in the answer 10, using algebra. X X+1 X+10 X+11 (X+1)(X+10) = X2 + 11X + 10 X(X + 11) =X2 + 11X (X2 + 11X + 10) - (X2 + 11X) = 10 This proves that all 2 x 2 grids taken from a 10 x 10 master grid result in the answer 10. ...read more.

Middle

= X2 + 55X + 250 X(X + 55) = X2 + 55X (X2 + 55X + 250) - (X2 + 55X) = 250 n x n grid Now, using algebra I will prove that an nxn grid will result in 10(n-1)2 x x+n-1 x+10(n-1) x+11(n-1) [x + (n-1)][x+ 10(n-1)]-x[x + 11(n-1)] As I do not know how to multiply out brackets which contain more than two terms. Therefore I will rename n-1, it will now be called T. So: n-1=T (x+T)(x+10T)-x(x+11T) = (x2+10T2+Tx+10Tx)-(x2+11Tx) = x2+10T2+11Tx - (x2+11x) = x2+10T2+11Tx - (x2+11Tx) =10T2 =10(n-1)2 This means that an nxn grid would result in 10(n-1)2. This means my prediction was correct. This is the result for any nxn grid taken from a 10x10 master grid. I will now experiment with different sized master grids. 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 I will start by investigating 2x2 grids. 1 2 7 8 2x7=14 1x8=8 14-8= 6 29 30 35 36 30x35=1050 29x36=1044 1050-1044= 6 25 26 31 32 26x31=806 25x32=800 806-800=6 As with the 10x10 master grid the same result is occurring. However the result is now 6 and not 10. Using algebra, I will now prove that all 2x2 grids taken from a 6x6 master grid result in an answer of 6. x x+1 x+6 x+7 (x+1)(x+6)=x2+7x+6 x(x+7)=x2+7x (x2+7x+6)-(x2+7x) =6 I will now investigate what happens if I use a 3x3 grid. 1 2 3 7 8 9 13 14 15 3x13=39 1x15=15 39-15=24 19 20 21 25 26 27 31 32 33 21x31=651 19x33=627 651-627=24 15 16 17 21 22 23 27 28 29 17x27=459 15x29=435 459-435=24 Once again, I keep getting the same result. I will now use algebra to prove that all 3x3 grids result in an answer of 24 x x+1 x+2 X+6 x+7 x+8 x+12 x+13 x+14 (x+2)(x+12)=x2+14x+24 x(x+14)=x2+14x (x2+14x+24)-(x2+14x) ...read more.

Conclusion

This is the formula for 2xn rectangles; I will now attempt to find a formula for a 3xn rectangle. I will start by investigating 3x4 rectangles. 1 2 3 4 11 12 13 14 21 22 23 24 4x21=84 1x24=24 84-24=60 6 7 8 9 16 17 18 19 26 27 28 29 9x26=234 6x29=174 234-174=60 15 16 17 18 25 26 27 28 35 36 37 38 18x35=630 15x3=570 630-570=60 I predict that my next 3x4 rectangle will result in an answer of 60. 25 26 27 28 35 36 37 38 45 46 47 48 58x45=1260 25x48=1200 1260-1200=60 Using algebra I will prove that this is the result for all 3x4 rectangles taken from a 10x10 master grid. x x+1 x+2 x+3 x+10 x+11 x+12 x+13 x+20 x+21 x+22 x+23 (x+3)(x+20)=x2+23x+60 x(x+23)=x2+23x x2+23x+60- x2-23x=60 I will now investigate 3x5 rectangles. 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 5x21=105 1x25=25 105-25=80 5 6 7 8 9 15 16 17 18 19 25 26 27 28 29 9x25=225 5x29=145 225-145=80 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30 10x26=260 6x30=180 260-180=80 Using algebra I will prove that this is the result for all 3x5 rectangles taken from a 10x10 master grid. x x+1 x+2 x+3 x+4 x+10 x+11 x+12 x+13 x+14 x+20 x+21 x+22 x+23 x+24 (x+4)(x+20)=x2+24x+80 x(x+24)=x2+24x x2+24x+80- x2-24x=80 I will now investigate 3x6 grids. 1 2 3 4 5 6 11 12 13 14 15 16 21 22 23 24 25 26 6x21=126 1x26=26 126-26=100 5 6 7 8 9 10 15 16 17 18 19 20 25 26 27 28 29 30 10x25=250 5x30=150 250-150=100 15 16 17 18 19 20 25 26 27 28 29 30 35 36 37 38 39 40 20x35=700 15x40=600 700-600=100 Using algebra, I will now prove that 100 is the result for all3x6 rectangles taken from a 10x10 master grid. x x+1 x+2 x+3 x+4 x+5 x+10 x+11 x+12 x+13 x+14 x+15 x+20 x+21 x+22 x+23 x+24 x+16 (x+5)(x+20)=x2+25x+100 x(x+25)= x2+25x x2+25x+100-x2-25x=100 1 ...read more.

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