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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

Extracts from this document...

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.

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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

...read more.

Middle

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41

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44

11 x 44 = 484

14 x 41 = 574

Diagonal difference: 90

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

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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!

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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)image01.pngimage02.pngimage00.pngimage00.png

          = 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:image03.png

image04.png

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.  

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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

...read more.

Conclusion

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

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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

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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.

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65 x 88 = 5720

85 x 68 = 5780

5780-5730 = 60

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

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63 x 87=5481

83 x 67=5561

Diagonal difference: 80

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

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68 x 83 = 5644

63 x 88 = 5544

Diagonal difference: 100

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

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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

...read more.

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