# Maths-Number Grid

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Introduction

Ranvir Kandola 10.5

Maths Coursework-Algebra!

Objective: - I will carry out an investigation to reveal the relationship

between the differences of products found in different sized

square grids.

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 will be using this 10 x 10 grid above, during the time of this coursework.

I will begin this investigation by starting off with a 2 × 2 grid and extend it

gradually, until I have reached a 10 × 10 square grid. I will use a highlighted

grid in each investigation, to show an example and prediction for the other

square grids of the same size.

Method:-

1. I will multiply the bottom left number by the top right number, to give

me the product of these 2 numbers.

2. I will then multiply the bottom right number by the top left number,

to give me the next product.

3. I will get the 2 products found and subtract the smaller product away

from the larger product, to give me the product difference.

4. I will use this result found, to investigate further and find out if the same

product difference occurs in each and every grid of the same size.

5. Finally, I will use the algebraic method, to test if my predictions were

correct.

2 × 2 Grid Example:-

I have highlighted above the 2 × 2 grid I will be using for this example. I will

now use this grid to come up with a prediction for the other grids of the same

length and width. I will do this by multiplying 63 and 54 together to give me a

product of 3402.

Middle

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

As can be seen above, a 5 × 5 square grid has been highlighted. I will use

this square grid below to help me reach a prediction for the product difference

of a 5 × 5 grid. I will firstly begin, by multiplying together the bottom left

number which is 45 by the top right number which is 9, The product comes to

405. I will then work out the next product number by multiplying together the

bottom right number by the top left number, which gives me a result of 245.

Conclusively, I will minus 245 away from 405 to give me a product difference of

160.

5 | 6 | 7 | 8 | 9 |

15 | 16 | 17 | 18 | 19 |

25 | 26 | 27 | 28 | 29 |

35 | 36 | 37 | 38 | 39 |

45 | 46 | 47 | 48 | 49 |

5 × 5 Grids:-

45 | 46 | 47 | 48 | 49 |

55 | 56 | 57 | 58 | 59 |

65 | 66 | 67 | 68 | 69 |

75 | 76 | 77 | 78 | 79 |

85 | 86 | 87 | 88 | 89 |

26 | 27 | 28 | 29 | 30 |

36 | 37 | 38 | 39 | 40 |

46 | 47 | 48 | 49 | 50 |

56 | 57 | 58 | 59 | 60 |

66 | 67 | 68 | 69 | 70 |

33 | 34 | 35 | 36 | 37 |

43 | 44 | 45 | 46 | 47 |

53 | 54 | 55 | 56 | 57 |

63 | 64 | 65 | 66 | 67 |

73 | 74 | 75 | 76 | 77 |

I have realised that from the 5 × 5 grids that I have drawn, there is always

a product difference of 160, the various different numbers in the box does

not affect the product difference of 160.

6 × 6 Grid Example:-

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 |

As can be seen in the following 10 × 10 grid, I have highlighted the 6 × 6

square grid, I will be predicting and working on. Below I have drawn a grid

and I will multiply 93 by 48 to give me a product of 4464. I will then multiply

98 by 43 to give me a product of 4214. Finally, I will subtract 4214 away from

4464 to give me a product difference of 250.

6 × 6 Grid Example:-

43 | 44 | 45 | 46 | 47 | 48 |

53 | 54 | 55 | 56 | 57 | 58 |

63 | 64 | 65 | 66 | 67 | 68 |

73 | 74 | 75 | 76 | 77 | 78 |

83 | 84 | 85 | 86 | 87 | 88 |

93 | 94 | 95 | 96 | 97 | 98 |

6 × 6 Grids:-

12 | 13 | 14 | 15 | 16 | 17 |

22 | 23 | 24 | 25 | 26 | 27 |

32 | 33 | 34 | 35 | 36 | 37 |

42 | 43 | 44 | 45 | 46 | 47 |

52 | 53 | 54 | 55 | 56 | 57 |

62 | 63 | 64 | 65 | 66 | 67 |

22 | 23 | 24 | 25 | 26 | 27 |

32 | 33 | 34 | 35 | 36 | 37 |

42 | 43 | 44 | 45 | 46 | 47 |

52 | 53 | 54 | 55 | 56 | 57 |

62 | 63 | 64 | 65 | 66 | 67 |

72 | 73 | 74 | 75 | 76 | 77 |

3 | 4 | 5 | 6 | 7 | 8 |

13 | 14 | 15 | 16 | 17 | 18 |

23 | 24 | 25 | 26 | 27 | 28 |

33 | 34 | 35 | 36 | 37 | 38 |

43 | 44 | 45 | 46 | 47 | 48 |

53 | 54 | 55 | 56 | 57 | 58 |

I have calculated the difference between the products for all of the 6 × 6

grids and now know the difference between the products is exactly 250. The

size of the grid does not affect the product difference of 250.

A Table to show the product differences of different grid sizes:-

Grid Sizes | Difference | |||

2 × 2 | 1 0 | |||

3 0 | ||||

3 × 3 | 4 0 | 2 0 | ||

5 0 | ||||

4 × 4 | 9 0 | 2 0 | = 10 n² | |

7 0 | ||||

5 × 5 | 1 6 0 | 2 0 | ||

9 0 | ||||

6 × 6 | 2 5 0 |

Above, I have drawn a table to show the product differences of different grid

sizes. From these differences, I have worked out the rule is 10 n².

I worked this out by writing down the differences for each of the 5 different

grid sizes and then tried to find differences within the product differences,

from which I continued to do so until I reached 10n².

I will now use this rule to come up with a specific rule, to help me work

out if the product differences are correct.

Below I have drawn a Quadratic Sequence table to help me come up with a

rule; this rule will tell me how I will work out the algebra for each of the grid

sizes. This rule will also tell me if I have come up with the correct product

differences.

Quadratic Sequence!

1 | 2 | 3 | 4 | 5 | 6 | |

Sequence | 0 | 10 | 40 | 90 | 160 | 250 |

10n² | 10 | 40 | 90 | 160 | 250 | 360 |

Difference | -20 -10 | -40 -30 | -60 -50 | -80 -70 | -100 -90 | -120 -110 |

The Rule is:- 10n² - 20n + 10 N=Length of Square. |

I have now found out that the Nth term is 10n² - 20n +10. I will now use

this rule further to investigate and reach a prediction for the 7 × 7 grid. This

term will tell me if this is the correct rule to follow. I think that that the 7 × 7

grid will have a product difference of 360. I think this because:-

= 1 0 × 7 ² = 4 9 0

= 2 0 × 7 = 1 4 0

= 4 9 0 – 1 4 0 = 3 5 0

= 3 5 0 + 1 0

= 3 6 0

3 | 4 | 5 | 6 | 7 | 8 | 9 |

13 | 14 | 15 | 16 | 17 | 18 | 19 |

23 | 24 |
| 26 | 27 | 28 | 29 |

33 | 34 | 35 | 36 | 37 | 38 | 39 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

53 | 54 | 55 | 56 | 57 | 58 | 59 |

63 | 64 | 65 | 66 | 67 | 68 | 69 |

(A + 2) | (A + 8) |

(A + 62) | (A+68) |

Conclusion

multiply this answer by 10, to give us the product difference.

Differences between products, with Algebra:-

2 × 4 Grid:-

( L – 1 ) ( W – 1) × 10

( 2 – 1 ) ( 4 - 1 ) × 10

= 1 × 3 × 1 0 = 3 0

(A + 1) | (A + 4) |

(A + 11) | (A +14) |

I have proved that the product difference in a 2 × 4 grid is always 30. This

algebra method above shows me that the different numbers in the same

rectangular grid has no impact on the product difference, because the result

is always 30.

3 × 6 Grid:-

( L – 1 ) ( W – 1) × 1 0

( 3 – 1) ( 6 – 1 ) × 1 0

= 2 × 5 × 10 = 1 0 0

3 × 6 Grid Algebra:-

(A + 2) | (A + 7) |

(A + 22) | (A + 27) |

This 3 × 6 grid above shows the algebra for it, the product difference for this

grid always has a product difference of 100.

4 × 8 Grid:-

( L – 1 ) ( W – 1) × 1 0

( 4 – 1) ( 8 – 1 ) × 1 0

= 3 × 7 × 1 0 = 2 1 0

(A + 3) | (A + 10) |

(A + 33) | (A + 40) |

Above, I have drawn a 4 × 8 rectangular grid, which shows me the total

product difference of 210.

Conclusion/Evaluation!

I have completed my investigation on finding the difference between

products in rectangles and have come to the conclusion that the correct

rule was established, ( L – 1 ) ( W – 1) × 10. In this investigation I had various

different sized rectangles and I had to multiply the bottom left number by the

top right number and multiply the bottom right number by the top left number.

After I did this, I subtracted away the first product found from the second

product found, from this I worked out the product difference for each rectangle

gird. I have also realised that the different numbers put in the same sized

grid do not affect the product difference. The coursework I have produced,

does meet the requirements of my aim. My aim was to find the product

difference in different sized rectangle grids. During this coursework I did not

come across any anomalous results.

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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