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  • Level: GCSE
  • Subject: Maths
  • Word count: 2683

Maths Coursework - Number Grid

Extracts from this document...

Introduction

Kayleigh McCormack

Maths Coursework – Number Grid

For this task I will first be looking at a number grid from 1 to 100, like the one below :

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I will start my investigation by looking at 2 by 2 squares. I will draw a square around 4 numbers, find the product of the bottom left and top right numbers and the product of the top left and bottom right numbers, then calculate the difference between the 2 products. I will see if there are any patterns and if so I will try to work it out algebraically.

I will then look at changing the size of the squares to see if there are any patterns.  I will try looking at 3 by 3 squares, 4 by 4 squares and 5 by 5 squares; I will do the same with these squares as I have with the 2 by 2 squares, I will find the products of the top left and bottom right and the bottom left and top right numbers then calculate the difference between them.

Once I have fully investigated the patterns within the squares and found an algebraic formula for the patterns I will look at rectangles.  I will start by looking at a 3 by 2 rectangle and looking for patterns there; if I find a pattern I will try to work out a formula for this pattern.  I will then try changing the size of the rectangles and looking for patterns there.  I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles.

...read more.

Middle

Blue Square :

18

40

720

38

20

760

40

I have noticed that the difference of the products in each square is always forty.

I have worked out the formula for this pattern below:

N

N+2

N+20

N+22

(N+2)(N+20) - N(N+22) = D

N2 + 20N + 2N + 40 - N2 - 22N = D

40 = D

Difference = 10

This shows that the difference between the two products in each square is always -40; however I am only interested in the number not the sign in front of it (+/-) so I have shown the difference as 40 rather than -40.

I will next look at 4 by 4 squares within a 10 by 10 grid.  The squares I am looking at are highlighted in the grid below.  I chose these squares randomly; there is no particular reason why I have chosen them.

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I am now going to calculate the products of the diagonals and the difference of these products for each square in the grid above.  I will display my findings in a table to help me identify a pattern.

Top Left

Bottom

Product

Bottom

Top Right

Product

Difference

Right

Left

Yellow Square :

1

34

34

31

4

124

90

Green Square :

26

59

1534

56

29

1624

90

Purple Square :

62

95

5890

92

65

5980

90

I have noticed that the difference of the products in each square is always ninety.

I have worked out the formula for this pattern below:

N

N+3

N+30

N+33

N(N+33) - (N+3)(N+30) = D

N2 + 33N - N2 - 30N - 3N - 90 = D

-90 = D

Difference = 90

This shows that the difference between the two products in each square is always -90; however I am only interested in the number not the sign in front of it (+/-) so I have shown the difference as 90 rather than -90.

        Now that I have investigated squares within a 10 by 10 grid and found algebraic formulas to explain the patterns I saw, I will move on to investigate rectangles within a 10 by 10 grid. I will first be looking at 3 by 2 rectangles.  The rectangles I am looking at are highlighted in the grid below.

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I am now going to calculate the products of the diagonals and the difference of these products for each rectangle in the grid above.  I will display my findings in a table to help me identify a pattern.

Top Left

Bottom

Product

Bottom

Top Right

Product

Difference

Right

Left

Yellow Rectangle :

1

13

13

11

3

33

20

Green Rectangle :

34

46

1564

44

36

1584

20

Purple Rectangle :

58

70

4060

68

60

4080

20

Blue Rectangle :

71

83

5893

81

73

5913

20

I have noticed that the difference of the products in each square is always twenty.

I have worked out the formula for this pattern below:

N

N+2

N+10

N+12

...read more.

Conclusion

Top Left

Bottom

Product

Bottom

Top Right

Product

Difference

Right

Left

Yellow Rectangle :

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25

25

21

5

105

80

Green Rectangle :

43

67

2881

63

47

2961

80

Purple Rectangle :

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40

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36

20

720

80

Blue Rectangle :

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100

7600

96

80

7680

80

I have noticed that the difference of the products in each square is always eighty.

I have worked out the formula for this pattern below:

N

N+4

N+20

N+24

N(N+24) - (N+4)(N+20) = D

N2 + 24N - N2 - 20N - 4N - 80 = D

-80 = D

Difference = 80

This shows that the difference between the two products in each rectangle is always -80; I have shown the difference as 80 rather than -80 as I am only interested in the number and not the sign in front of it (+/-).

        I am now going to look at 4 by 5 rectangles within the same 10 by 10 grid.  The rectangles I am looking at are highlighted in the grid below.  I chose these rectangles randomly.

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I am now going to calculate the products of the diagonals and the difference of these products for each rectangle in the grid above.  I will display my findings in a table so it will be easier to identify the pattern.

Top Left

Bottom

Product

Bottom

Top Right

Product

Difference

Right

Left

Yellow Rectangle :

1

44

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41

4

164

120

Green Rectangle :

16

59

944

56

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1064

120

Purple Rectangle :

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95

4940

92

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5060

120

I have noticed that the difference of the products in each square is always one hundred twenty.

I have worked out the formula for this pattern below:

N

N+3

N+40

N+43

N(N+43) - (N+3)(N+40) = D

N2 + 43N - N2 - 40N - 3N - 120 = D

-120 = D

Difference = 120

This shows that the difference between the two products in each rectangle is always -120; I have shown the difference as 120 rather than -120 as I am only interested in the number and not the sign in front of it (+/-).

...read more.

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