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Introduction

Thomas Shaw

Introduction

In this piece of course work I will investigate different shapes in different sized number grids. The shapes I will look at will be the square, the rectangle, the rhombus and the parallelogram. I will investigate these shapes in two different number grids, one a 10x10 grid and the other a 5x5 grid. I will draw these shapes on the number grids at random points and take the corner numbers and then multiply the opposite corners. From these results I will attempt to work out a formula for how the size of shape affects the result taken from the table.

I will take the number from the bottom left corner of the shape and the top right corner of the shape and multiply these numbers together. This set of numbers will be represented by the colour RED. I will then take the top left corner and the bottom right corner and multiply these two numbers together. This set of numbers will be represented by the colour BLUE. I will then subtract the RED result from the BLUEwhich will give me the number I need to work out my formula. The number gained from the subtraction will be called the difference.

All my results will be taken from either a 5x5 grid like this:

 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

Or a 10x10 grid like this:

 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

Middle

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85 x 49 = 4165

45 x 89 = 4005

Difference = 160

 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

43 x 7 = 301

3 x 47 = 141

Difference = 160

 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

82 x 46 = 3772

42 x 86 = 3612

Difference = 160

Formula

 X X + 4 X + 40 X + 44

A = (X + 40) (X + 4)

= X² + 4X + 40X + 160

= X² + 44X + 160

B = X (X + 44)

=X² + 44X

Difference = A – B

= (X² + 44X + 160) – (X² + 44X)

=160

10x10 grid - square formula

These are the differences from my different squares on the 5x5 grid:

 Square 2x2 3x3 4x4 5x5 Difference 10 40 90 160

If we look at the difference’s we notice that they all end in zero’s so we can summarise that the formula involves a multiplication of 10. We also know that the grid is 10x10 so if G = Grid Size (Which is 10 NOT 100). Therefore if D = Difference and G = Grid Size we can make the following formula:

D = (Square size - 1)² x G

This table will show the formula in action to show how it produces the formula:

 Square Formula 2x2 10 = (2 - 1)² x 10           1 x 10           10 3x3 40 = (3 - 1)² x 10           4 x 10           40 4x4 90 = (4 - 1)² x 10           9 x 10           90 5x5 160 = (5 - 1)² x 10             16 x 10             160

If this Formula is correct it should allow us to work out what the difference of a 6x6 square is:

D = (6-1)² x 10

25 x 10

250

Therefore, if this formula is correct the difference of a 6x6 square would be 250. I will now do 2 6x6 square to show the formula is correct.

6x6

 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

62 x 17 = 1054

12 x 67 = 804

Difference = 250

 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

95 x 50   = 4750

45 x 100 = 4500

Difference = 250

Both the Differences for my 6x6 matched my prediction so we can summarise that the formula is correct.

Rectangle – 5x5 Grid

3x2

 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

6 x 13 = 78

11 x 8 = 88

Difference = 10

 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

18 x 25 = 450

23 x 20 = 460

Difference = 10

 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

8 x 15 = 120

13 x 10 = 130

Difference = 10

Formula

 X X + 2 X + 5 X + 7

A = (X +5) (X + 2)

= X² + 2X + 5X + 10

= X² + 7X + 10

B = X (X + 7)

= X² + 7X

Difference = A – B

= (X² + 7X + 10) – (X² + 7X)

= 10

4x3

 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

11 x 24 = 264

21 x 14 = 294

Difference = 30

 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

2 x 15 = 30

12 x 5 = 60

Difference = 30

 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

12 x 25 = 300

22 x 15 = 330

Difference = 30

Formula

 X X + 3 X + 10 X + 13

A = (X + 10) (X + 3)

= X² + 3X + 10X + 30

= X² + 13X + 30

B = X (X + 13)

= X² + 13X

Difference = A – B

= (X² + 13X + 30) – (X² + 13X)

= 30

5x5 Rectangle Formula

These are my differences from my different rectangles on the 5x5 grid:

Conclusion

Difference = 9

 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

87 x 79 = 6873

78 x 88= 6864

Difference = 9

 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

11 x 3 = 33

2 x 12 = 24

Difference = 9

Formula

 X X + 1 X + 9 X + 10

A = (X + 9) (X + 1)

= X² + X + 9X + 9

= X² +10X + 9

B = X (X + 10)

= X² + 10X

Difference = A – B

= (X² + 10X + 9) – (X² + 10X)

= 9

3x3

 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

24 x 8 = 192

6 x 26 = 156

Difference = 36

 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

21 x 5 = 105

3 x 23 = 69

Difference = 36

 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

82 x 66 = 5412

64 x 84 = 5376

Difference = 36

Formula

 X X + 2 X + 18 X + 20

A = (X + 18) (X + 2)

= X² + 2X + 18X + 36

= X² +20X + 36

B = X (X + 20)

= X² + 20X

Difference = A – B

= (X² + 20X + 36) – (X² + 20X)

= 36

4x4

 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

31 x 7 = 217

4 x 34 = 136

Difference = 81

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