# Math Grid work

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Introduction

Number Grids Coursework – Maths – Mr Danes

## Maths Coursework

## Introduction

We are looking at number grids and are using the numbers in the edges of any square or rectangle. I am multiplying the opposite edged numbers and subtracting the smaller number from the bigger one.

## This is a 10 x 10 square

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 |

I pick any square from this grid and do the multiplying and subtracting and see what I get. I multiply the top right hand number to the bottom left and number and then subtract the number from the other two numbers multiplied from it.

2 x 2 square

24 | 25 |

34 | 35 |

46 | 47 |

56 | 57 |

89 | 90 |

99 | 100 |

I notice that I get the same number for any same size square so to prove this I will use X and prove why the answer is always 10. I can use a square in which the boxes are the edges of the square on the grid.

X | X + 1 |

X + 10 | X + 11 |

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+1)(X+10)…which is = X2+11X+10…and…

X(X+11)…which is = X2 +11X

[X2+11X+10]- [X2 +11X] = 10

X can be any number and will always equal 10!

3 x 3 square

36 | 37 | 38 |

46 | 47 | 48 |

56 | 57 | 58 |

X | ## X + 2 |

X + 20 | X + 22 |

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+2)(X+20)…which is = X2+22X+40…and…

X(X+22)…which is = X2 +22X

[X2+22X+40]-[X2 +22X] = 40

X can be any number and will always equal 40!

4 x 4 square

4 | 5 | 6 | 7 |

14 | 15 | 16 | 17 |

24 | 25 | 26 | 27 |

34 | 35 | 36 | 37 |

X | X + 3 |

X + 30 | X + 33 |

Middle

54

55

61

62

63

64

65

71

72

73

74

75

81

82

83

84

85

91

92

93

94

95

X | X + 4 |

X + 40 | X + 44 |

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+4)(X+40)…which is = X2+44X+160…and…

X(X+44)…which is = X2 +44X

[X2+44X+160]-[X2 +44X] = 160

X can be any number and will always equal 160!

### Working out a formula for any square size on a 10 by 10 number grid

The sequence I get is 10, 40, 90 and then 160.

So I can do the use a square to find out a formula that works out N for any square size on a 10 by 10 number grid.

I do what I do with these figures if they were numbers:

1. [X+(S-1)][X+10(S-1)

I get X2+11X(S-1)+10(S-1)2

And…

2. X[X+11(S-1)]

I get X2+ 11X(S-1)

I then subtract 2 from 1…

[X2+11X(S-1)+10(S-1) 2] - [X2+ 11X(S-1)]

And whatever that is left over must be the formula, so therefore the formula to work out N for any square size in a 10 by 10 grid square is…

N= 10(S-1)2

##### Test

When the square size is 2…

89 | 90 |

99 | 100 |

Using the formula, (2-1) 2 = 1.. x 10 = 10

###### When the square size is 5…

2 | 6 |

42 | 46 |

Using the formula, (5-1) 2 = 16… x 10 = 160

The formula works!

### What happens if I change the grid size?

I think that I have come to the stage where I can use algebra to work out what the number each time will be. So, I don’t have to show examples with numbers all the time.

## A 6 by 6 grid square on a 2x2 square

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 |

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+1)(X+6)…which is = X2+7X+6…and…

X(X+7)…which is = X2 +7X

## [X2+7X+6]-[X2 +7X]= 6

## X can be any number and will always equal 6 on a 2 by 2 square on a 6 by 6 grid.

## An 8 by 8 grid square on a 2x2 square

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 | 51 | 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 |

Conclusion

N= g(S-1)2

##### Test

When the square size is 2 and the grid size is 10

89 | 90 |

99 | 100 |

Using the formula, (2-1) 2 = 1.. x 10 = 10

###### When the square size is 3 and the grid size is 6

2 | 4 |

14 | 16 |

Using the formula, (3-1) 2 = 4… x 6 = 24

The formula works!

#### Final formula: for any size square/rectangle in any grid square

Common sense tells me that, this is basically replacing the S by the length and width of any square/rectangle.

I do what I do with these figures if they were numbers:

1. [X+(w-1)][X+g(L-1)]

I get X2+gX(L-1)+X(w-1)+g(w-1)(L-1)

And…

2. X[X+g(L-1)+(w-1)]

I get X2+ gX(L-1)+X(w-1)

I then subtract 2 from 1…

[X2+gX(L-1)+X(w-1)+g(w-1)(L-1)] - [X2+ gX(L-1)+X(w-1)]

The only thing left over is g(w-1)(L-1)

And whatever that is left over must be the formula, so therefore the formula to work out N for any square/rectangle in a g by g grid square is…

N= g(w-1)(L-1)

##### Test

When the grid size is 10 and the length and width are 2

89 | 90 |

99 | 100 |

Using the formula, 10(2-1)(2-1) = 10x1x1= 10

###### When the grid size is 5, length is 2 and width is 4

2 | 5 |

7 | 10 |

Using the formula, 5(4-1)(2-1) = 5x3x1 = 15

The formula works!

***

By: Haroon Motara

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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