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

# Maths Coursework - Number Grid

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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 :

 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 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.

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.

 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 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.

 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 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

Conclusion

 Top Left Bottom Product Bottom Top Right Product Difference Right Left Yellow Rectangle : 1 25 25 21 5 105 80 Green Rectangle : 43 67 2881 63 47 2961 80 Purple Rectangle : 16 40 640 36 20 720 80 Blue Rectangle : 76 100 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.

 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 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 44 41 4 164 120 Green Rectangle : 16 59 944 56 19 1064 120 Purple Rectangle : 52 95 4940 92 55 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 (+/-).

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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