# Number Grids

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Introduction

Number Grids The Problem The problem is to investigate the differences of corner numbers on a multiplication grid. Introduction To solve this problem I will have to choose several examples of squares on a grid: E.g. 2x2, 3x3, 4x4 To work this out I will need to take the opposite corners of the square and subtract them from the other sum of the two corners. This is easier seen in the diagram shown below: *The RED numbers are always multiplied then subtracted from the sum of the YELLOW numbers. 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 This is an example of a 2x2 square. I will then make an equation with this: Difference= (2x11)-(1x12) = 10 The difference for this example is ten. 2x2 Squares I will use on grid to use several example of 2x2 squares by placing them randomly on the grid. ...read more.

Middle

=90 (9x36)-(6x39) =90 (67x94)-(64x97)=90 The difference for a 4x4 grid is always going to be no matter where you place the square on a 10x10 grid. We can again prove that the difference will always be 90 using algebra. Let 'N' be the number at the top left of the square. N N+3 N+30 N+30 We then change this into the equation (N+3)(N+30)-N (N+30) Multiply out the brackets N�+30N+3N+90-N�-30N If this equation is fully worked out the only number left will be 90, which is the constant of a 4x4 grid. Quadratic Equations I can now take the results I have and put them into a table Square size Difference First Difference Second Difference 1x1 0 10 2x2 10 20 30 3x3 40 20 50 4x4 90 The constant difference is the third one when dealing with squares. Using this we can construct a quadratic formula that would be able to tell us the difference for any sort of square that would be put on a 10x10 number grid. The quadratic formula is UN= AN�+BN+C We now need to work out what A, B and C are. The formulas for working these out are: A= 2nd difference / 2 B= 1st difference-3A C= 1st term - (a+b) A=10 B= -20 C=10 So now the equation reads UN=10N�-20N+10 We can test this by using the square, for example if we use the 3x3 grid, the nth term is 3, if we insert this into the formula: UN=10x3�-20x3+10 UN=90-60+10 UN=40 This formula will find the difference of the to two opposing corners on a 10x10 grid for any square shape. ...read more.

Conclusion

4xN Grid I am doing this rectangle grid to see if my predictions are correct, and if they are I will be able to construct a formula for any grid where I know what 'M' is. 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 These rectangles and their differences are: M N Difference First Difference 4x3 60 30 4x4 90 30 4x5 120 30 4x6 150 My prediction was correct. We need to multiply the nth term by 30 this time then subtract the first difference which is 30. So we can construct the formula D=30N-30 which can be reduced to the formula D=30(N-1) Now that my predictions have been proven correct we can now work out any rectangle as long as we know the Mth term. For example if we want a rectangle that had the constant M as 7 then we could make the formula D=60(N-1). But now we want a formula so that we can be given an Nth and an Mth term and from that we can work out the difference of the opposing corners. ...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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