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• Level: GCSE
• Subject: Maths
• Word count: 1212

# number grid

Extracts from this document...

Introduction

NUMBER GRID

COURSE WORK

 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

To investigate the patterns generated from using rules in a square grid

The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

In order to find the difference of n x n square first step is to take nxn square and then multiply the corners diagonally. For example take 2x2 square and multiply its corners diagonally.

 25 26 35 36

25 x 36= 900             26 x 35= 910                                   910 - 900= 10

 77 78 87 88

77 x 88 = 6776              78 x 87= 6786                     6786 - 6776= 10

After a few results I observed that

Middle

61

62

63

64

31 x 64 = 1984              34 x 61=2074       2074 – 1984 = 90

 n th term SIZED DIFFERENCE 1 2 x 2 10 2 3 x 3 40 3 4 x 4 90

I put my results into table and observed that the difference is square number because the box and the grid are square shaped.

I predict that for a 5 by 5 square the difference between the products will be 160.

 46 47 48 49 50 56 57 58 59 60 66 67 68 69 70 76 77 78 79 80 86 87 88 89 90

46 x 90 = 4140        50 x 86 = 4300              4300 - 4140 = 160

I will now attempt to prove my results algebraically.

After doing this I will check my answer using the nxn formula

2x2 square

 23 24 33 34

n            n+1

n+10       n+11                                             (24x33)- (23x34)

792-782=10

(n+1) (n+10)-n (n+11)

n2 +10n + n+10 - n2 +11n                                                                    n2 +11n +10 - n2 +11n =10

In both ways the difference is

5x5 square

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45

n                 n+4

n+40         n+44

n (n+44) - [(n+4) (n+40)]                                                   (5x41)-(1x45)

n2+44n - n2+40n + 4n +160                                                   205-45= 160

n2+44n +160 - n2+44n =160

## In both ways the difference is 160

After doing this I will check if the nxn formula will work in a bigger grid size like 11x11

 1 2 12 13

Conclusion

S-1 x G = D

Sized = 2 x 3   G = 10   D = 20

2-1 =1

3-1 =2

1 x 2 =2

2 x 10 =20

 37 38 39 40 47 48 49 50 57 58 59 60

37 x 60 =2220    40 x 57 = 2280 2280 - 2220 = 60

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

SIZED = 3 x 4       G = 10   D = 60

3-1 = 2

4-1 = 3

2 x 3 = 6

6 x 10 = 60

I could prove that this formula can work for both rectangle and square.

 53 54 63 64

53 x 64 = 3392       54 x 63 = 3402      3402 - 3392 = 10

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 2 x 2       G = 10    D = 10

2-1 = 1

2-1 = 1

1 x 1 = 1

1 x 10 = 10

FINALLY

In this project I found that not all formula can work for rectangle and square.

But I found a formula can work for rectangle and square it does not matter the size of the box and I will proof that it does not matter for grid sizes too.

 28 29 30 40 41 42 52 53 54

28 x 54 = 1512    30 x 52 = 1560 1560 - 1512 = 48

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 3 x 3 G = 12    D = 48

3-1 = 2

3-1 = 2

2 x 2 = 4

4 x 12 = 48

If I were to extend this project further I would try and do cube in three dimensions.

ISRAA AL-RIFAEE                                                    -  -

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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# Related GCSE Number Stairs, Grids and Sequences essays

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1. ## Maths - number grid

major trend forming, I am going to increase the size of my rectangles to and 8x5. 30x63 - 23x70 1890 - 1610 Difference = 280 I am confident that the defined difference of 280 for any 8x5 is correct. I will use algebra to ensure this is true.

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1. ## number grid investigation]

= n2+33n Stage B: Bottom left number x Top right number = (n+30)(n+3)= n2+3n+30n+90 = n2+33n+90 Stage B - Stage A: (n2+33n+90)-(n2+33n) = 90 When finding the general formula for any number (n), both answers begin with the equation n2+33n, which signifies that they can be manipulated easily.

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1. ## Number grid Investigation

x 81 = 5103 5103 - 5063 = 40 The difference is 40. 34 35 36 44 45 46 54 55 56 34 x 56 = 1904 36 x 54 = 1944 1944 - 1904 = 40 The difference is still 40.

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

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