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  • Level: GCSE
  • Subject: Maths
  • Word count: 3043

Number Grid

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Introduction

Number Grid 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 First of all we need to pick a box of numbers. Like the one shown on the diagram above. This will then be used to find the product of the top left and the bottom right number in this box. 12(n) 13(n+1) 22(n+9) 23(n+11) This shows the top left number as the lowest number which will then be multiplied against the bottom right number to find the product of the two numbers. 12 x 23 = 276 Then we need to get the product of the top right and the bottom right number to see if there is any difference in the two numbers and then see if there is any pattern or general rule to gain from this box/shape. We will be using the top right number multiplied against the bottom left number. 13 x 22 = 286 As you can see there is a difference of 10 between these two numbers, but to see if this is correct with all the other boxes in the square I will check it with another random box. 57(n) 58(n+1) 67(n+9) 68(n+11) We will follow the same method and times the top left with the bottom right first then the top right and the bottom left afterwards. We will then see if the difference is 10, which I predict it will be. ...read more.

Middle

To test this I will use examples and see whether or not this algebra is correct or not. Example C: n = 14 14 ? = 196 + 182 (13n) = 378 14 x 27 = 378 Example C (1): n = 14 n? + 13n +12 14 ? = 196 + 182 + 12 = 390 15 x 26 = 390 Difference = 12 This proves that my rule for each side of the box works. Now I need to find the grid general rule. So I will first look at every single result and see if I can find any recurring factors. In every grid the number (top left number) is always called n. In every grid the n is squared to form n? and then added by the grid scale plus one (1gs) which is also multiplying the top left number. So this gives us n? + 1gsn. Then it is added by the grid number in the second product to forum n? + 1gsn +1 + gs so therefore the general rule for the first product of the top left number multiplied by the bottom right number is: n? + 1gsn The general rule for the second product of the top right number multiplied by the bottom left number is: n? + 1gsn + gs Now I will need to test this general rule to make sure this is correct. Example D: n = 9 gs = 7 n? + 1gsn 9? + (8x9) 81 + 72 = 153 9 x 17 = 153 Example D: n = 9 gs = 7 n? + 1gsn + gs 9? + (8x9) + 7 81 + 72 +7 = 160 10 x 16 = 160 Difference = 7 This proves that my general rule for grid size works. Now I need to change a different variable to extend my investigation further. I will now extend the square's area to see if I can find a general rule for any size of square. ...read more.

Conclusion

This also means that the number difference between the top left number and the top right number makes the result double in this one because it is two, therefore in 4 x 2 it will be 3 to a result of 30 as the difference between the two products. A 5 x 2 rectangle will make the difference between the two products 50. So the general rule for the size of rectangle will be: S = rectangle size Sm = Rectangle size - 1) n(n + 10 + Sm ) = n? + 10n + Smn (n + Sm)( n + 10) = n? + 10n + Smn + 10Sm= n ? + 10Smn? +10Sm To test my general rectangle size rule I will do some examples. This will also show you how this general rule works. Example B: n = 4 S = 4 n? + 10n + Smn 4? + (10 x 4) + ( 3 x 4) 4? + 40 + 12 16 + 40 + 12 = 68 4 x 17 = 68 Example B (1): n = 4 S = 4 n? + 10Smn? +10Sm 4? + (10 x 3 ?) + (10 x 3) 4 ? + 52 + 30 16 + 52 + 30 Difference = 98 This proves that my general rule for rectangle size works. I have now finished my investigation. I have investigated into three different variables, the first was changing the grid size, the second was changing the size of the square and the third was changing the shape. I have proved my work for every variable and I have got a general rule for each of the variables which work. If I was to do this investigation again I would change the shape I used into a bigger rectangle or possibly even a rhombus. Overall my investigation was a success because I found out and proved all of the general rules and I completed the work asked of me by this investigation. ...read more.

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