# In this piece course work I am going to investigate opposite corners in grids

Extracts from this document...

Introduction

Mathematics Course Work

## Opposite Corners

## Introduction

In this piece course work I am going to investigate opposite corners in grids. I will start by investigating a 7x7 grid. Within this grid I will use 2x2, 3x3, 4x4, 5x5, 6x6 and a 7x7 grid. I will do this to find whether I can find a pattern. I will do this by multiplying the two opposite corners together then subtracting them. I will try to find the patterns and do a formula that will work for all grid sizes and shapes. I will experiment shapes and sizes of all different grids.

### Prediction

I predict that in a 7x7 grid all the opposite corners will be a multiple of 7 and in an 8x8 grid they will be a multiple of 8 and so on. They will only do this if I multiply the two opposite corners then subtract the two from each other.

To check my hypothesis I will use 6x6, 7x7, 8x8 and maybe if I have time I will do a 9x9 and 10x10 grid. Also I will be looking at all different shapes and sizes. I hope to find a formula for all grids and all shapes and sizes.

Middle

7

9

10

11

12

13

14

16

17

18

19

20

21

23

24

25

26

27

28

30

31

32

33

34

35

37

38

39

40

41

42

Again my answer is a multiple of 7. If I divide my answer by 25 I will get 7.

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 |

252 is again a multiple of 7. 7x36=252. So far all my answers for a 7x7 main grid size is seven.

### Rectangles

I am now going to do some rectangles to see if I get the same as my last examples. The rectangles I am going to do are 2x3, 2x4 and 3x4

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 |

15 | 16 | 17 |

22 | 23 | 24 |

14 is a multiple of 7.

36 | 37 | 38 | 39 |

43 | 44 | 45 | 46 |

21 is a gain a multiple of 7.

4 | 5 | 6 | 7 |

11 | 12 | 13 | 14 |

18 | 19 | 20 | 21 |

42 is a multiple of 7 as well.

I have the same out come as my last examples.

### Analysis

After doing my main grid size of 7 x 7 I found that all my answers were multiples of 7.

Grid size | Difference | Pattern |

2x2 | 7 | 1x1x7 |

3x3 | 28 | 2x2x7 |

4x4 | 63 | 3x3x7 |

5x5 | 112 | 4x4x7 |

6x6 | 175 | 5x5x7 |

7x7 | 252 | 6x6x7 |

I can see that there are two patterns here with the grid that I have made. I can see that the pattern is between the grid size and the difference.

### Pattern 1

The first pattern that I can see is:

1 3 2

4 5 2

9 7 2

16 9 2

25 11 2

36

I can see that the difference between the differences go up in two’s. This must mean that the answer is a multiple which is true.

### Pattern 2

I have found that if you take the length and the width of the grid size for example 3x3 then subtract the width and the length each by 1, I get 2x2 then x 2x2 by 7 I get the same pattern as in my table.

Eg1,

- 6x6
- - 1 from each of the width and the length
- I get 5x5
- Then I x that by 7
- I get 5x5x7
- Which adds up to what I got in my table.

Eg2,

- 2x4
- - 1 from each of the length and the width
- I get 1x3
- Then I x that by 7
- I get 1x3x7
- Which adds up to the same as my rectangle I did in my 7x7 grid.

If I make:

- N= width
- M= length
- G= main grid size
- D= difference

I feel that I can make a formula

(Width – 1) x (length – 1) x grid size = difference

so………

(N-1) x (m-1) x G = D

I predict that if I were to use this formula I will be able to work out the differences a lot easily without having to multiply then subtract. To see if my hypothesis is correct I will do an 8x8 grid, 9x9 grid and 10x10 grid.

8x8 grid

In this section of my course work in am going to see if my formula works for an 8x8 grid. In the next section I will be doing a 9x9 grid and eventually a 10x10 grid.

2x2

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 |

28 | 29 |

36 | 37 |

29x36=1044

28x37=1036

1044-1036=8

I have the answer as 8. Which is the main grid size.

3x3

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 |

Conclusion

5x5

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 |

11 | 12 | 13 | 14 | 15 | |||

19 | 20 | 21 | 22 | 23 | |||

27 | 28 | 29 | 30 | 31 | |||

35 | 36 | 37 | 38 | 39 | |||

43 | 44 | 45 | 46 | 47 |

15x43=645

11x47=517

645-517=128

16x8 = 128. Which is a multiple of 8.

Now it is time to check my formula again.

(N-1) x (M-1) x G = D

(4-1) x (4-1) x 8 = D

D = 72

3x3x8=72

My formula has worked again.

(N-1) x (M-1) x G = D

(5-1) x (5-1) x 8 = D

D = 128

4x4x8=128

Again my formula has worked.

9x9 grid

I will now see if it works with a 9x9 grid this time only with a 2x2 and a 9x9.

2x2

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 |

20 | 21 | |||||||

29 | 30 |

21x29=609

20x30=600

609-600=9

The answer is 9, the main grid size

9x9

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 |

9x73=657

1x81=81

657-81=576

576 divided by 9 = 64 64x9= 576 which is the answer for this grid size. It is still a multiple of 9.

It is now time to test my formula with the last 9x9 grids.

(N-1) x (M-1) x G = D

(2-1) x (2-1) x 9 = D

D = 9

1x1x9=9

My formula has worked again.

(N-1) x (M-1) x G = D

(9-1) x (9-1) x 9 = D

D = 648

8x8x9=576

Again my formula has worked.

I don’t think that I have to prove my answers now with a 10x10 grid so I wont. I think you can see that I have succeeded in my course work. I am very happy in my out come and I feel that I have done very well.

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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## Here's what a teacher thought of this essay

This is a reasonably well structured investigation. It uses a wide variety of experimental examples to support a discovered pattern. To improve this more algebraic expressions to represent the differences need to be included.

Marked by teacher Cornelia Bruce 18/04/2013