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• Level: GCSE
• Subject: Maths
• Word count: 1769

# Investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 grid

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Introduction

Naila Parveen        Maths Coursework        year 11

Maths Coursework-Diagonal Difference

Introduction

I am given a 10 by 10 grid. I am going to find the diagonal difference of different size grids (For e.g. 3 by 3, 4 by 4) within the 10 by 10 grid, by multiplying the opposite corners which results in two answers, we then deduct these two to get a final answer for that size.

This is the grid that I will use to help me investigate.

 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

Aim

I am going to investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 grid.  I will then try to find a formula which relates to the diagonal difference of each square, I will then further this investigation by trying to find the diagonal difference of an 11 by 11 grid and a 12 by 12 grid and find the formula and see if it is the same. I will also do an extension by doing a rectangle and a square and then find the diagonal difference and the formula for this.

I am going to find the formula by finding

Middle

32

33

34

41

42

43

44

11 x 44 = 484

14 x 41 = 574

Diagonal difference: 90

Now I am going to try a 5 by 5 grid.

 51 52 53 54 55 61 62 63 64 65 71 72 73 74 75 81 82 83 84 85 91 92 93 94 95

51 x 95 = 4845

55 x 91 = 5005

Diagonal difference: 160

Now I am going to try out work out the algebraic formula for working out the diagonal differences for all squares.

So for a 6 by 6 grid I predict that the diagonal difference would be; 250.

To show this I will do a number grid and also in algebra.

The general difference formula that I predict is (n-1)2 x 10.

Now to show that it works!

 45 46 47 48 49 50 55 56 57 58 59 60 65 66 67 68 69 70 75 76 77 78 79 80 85 86 87 88 89 90 95 96 97 98 99 100

45 x 100 = 4500

50 x 95 = 4750

Diagonal difference: 250

Now to show this in an algebraic form.

 x x+1 x+2 x+3 x+4 x+5 x+10 x+11 x+12 x+13 x+14 x+15 x+20 x+21 x+22 x+23 x+24 x+25 x+30 x+31 x+32 x+33 x+34 x+35 x+40 x+41 x+42 x+43 x+44 x+45 x+50 x+51 x+52 x+53 x+54 x+55

(x+50) (x+5)-x(x+55)

= x2+50x+5x+250-(x2+55x)

= x2+55x+250-x2-55x

= 250

My formula works. So if I were to do a 7 by 7 I would do 62x10.

Now I am going to put my results in a table.

 Size of square Differences 2 x 2 10 12x10 3 x 3 40 22x10 4 x 4 90 32x10 5 x 5 160 42x10 6 x 6 250 52x10

I have shown that for a 5 by 5 square grid you will have to 42x10 to get the answer, so if I wanted to do a 7 by 7 grid I would have to do; 62x10

7 x 7 it would be (7-1)2x10

= 62 x 10

=36 x 10

=360

As I have said before that the general formula is:

(n-1)2x10

So the grid would look like this:

Now I a going to try out the rectangular grids. For this I will try to use an algebraic formula for each grid.

I am going to start of with a 2 by 3 rectangular grid.

 35 36 37 45 46 47

35 x 47 = 1645

45 x 37 = 1665

Diagonal difference: 20

Now in algebra:

 x x+1 x+2 x+10 x+11 x+12

Conclusion

Now I am going to try a 2 by 5 rectangular grid.

 32 33 34 35 36 42 43 44 45 46

32 x 46= 1472

36 x 42= 1512

Diagonal difference: 40

 x x+1 x+2 x+3 X+4 x+10 x+11 x+12 x+13 x+14

(x+4) (x+10)-x(x+14)

= x2+14x+10x+40-(x2+14x)

= x2+14x+40-x2-14x

= 40

Now I am going to put my results in a table, for the working out that is shown above.

 Sizes of rectangles Width (w) Length (L) Differences 2 2 10 1 x 10 2 3 20 2 x 10 2 4 30 3 x 10 2 5 40 4 x 10 2 x L 10 (L-1) (L-1)x10

During the investigation I have discovered that my research is correct and when I observed my results using algebra the outcomes were the same as to when I used numbers.

Now I am going to change the width to 3 and keep the length the same.

Now I am going to try a 3 by 4 rectangular grid.

 65 66 67 68 75 76 77 78 85 86 87 88

65 x 88 = 5720

85 x 68 = 5780

5780-5730 = 60

Now I am going to try a 3 by 5 rectangular grid

 63 64 65 66 67 73 74 75 76 77 83 84 85 86 87

63 x 87=5481

83 x 67=5561

Diagonal difference: 80

Now I am going to try a 3 by 6 rectangular grid.

 63 64 65 66 67 68 73 74 75 76 77 78 83 84 85 86 87 88

68 x 83 = 5644

63 x 88 = 5544

Diagonal difference: 100

Now I am going to do a 3 by 7 rectangular grid.

 22 23 24 25 26 27 28 32 33 34 35 36 37 38 42 43 44 45 46 47 48

22 x 48 = 1056

42 x 28 = 1176

Diagonal difference: 120

Now I am going to do a table to show my results.

 Sizes of rectangles Width (w) Length (L) Differences 3 4 60 6 x 10 3 5 80 8 x 10 3 6 100 10 x 10 3 7 120 12 x 10 3 x L 10 (L-1) (L)x10

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