# The purpose of this investigation is to look at diagonal differences on different sizes of grids

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Introduction

Number grid

The purpose of this investigation is to look at diagonal differences on different sizes of grids. I am going to do this by multiple the numbers in the corners of the grids and then subtracting there answers to get the over all difference.

To carry out this investigation I will start by extracting a 2 by 2 grid by my original 10 by 10 grid. I will then carry out by multiplying the corner numbers and subtracting the smaller number by bigger the number so I can gain an answer. To further this investigation I will extend the grid size to a 3 by 3 grid and I will be doing exactly the same method as the other grid. I will then continue this by extending the grid size to a 4 by 4, 5 by 5, 6 by 6 and so on.

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 |

2 by 2 grids

- 1x12=12

2x11=22

22-12=10

- 17x28=476

18x27=486

486-476=10

- 24x35=840

25x34=850

850-540=10

- 31x42=1302

32x41=1312

1312-1302=10

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 |

3 by 3 grids

- 7x29=203

Middle

2 by 2

10

3 by 3

40

30

4 by 4

90

50

20

5 by 5

160

70

20

The table above shows that the answer always ends in 0 and the second difference will always be 20. Also the table shows all the numbers are multiples of 10. I am now going to predict the next two grids size by using the first method by looking at patterns and rules of the above table.

One of the first patterns I can see is the second difference is always repetitive. From this I can predict the next two grid sizes from

Looking at the patterns from above.

6 by 6 | 250 | 90 | 20 |

7 by 7 | 360 | 110 | 20 |

Method two

Is the formula which is;

(N-1)² x grid size

I am now going to predict 6 by 6.

N represents the grid size e.g. 6 by 6.

Then you – 1 so you go back down two the grid size which is 5 by 5. Then you times 5 by 5 which equals 25. With this answer you then time it by the actual grid size which is 10. After that you come up with 250.

I am now going to predict 7 by 7.

N represents the grid size e.g. 7 by 7.

Then you – 1 so you go back down two the grid size which is 6 by 6. Then you times 6 by 6 which equals 36. With this answer you then time it by the actual grid size which is 10. After that you come up with 360.

I am now going to test my predation, by doing the same thing as I did for the other number grids.

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 |

6 by 6 grids

- 5x60=300

10x55=550

550-300=250

- 11x66=726

16x61=976

976-726=250

- 31x86=2666

36x81=2916

2916-2666=250

- 35x90=3150

40x85=3400

3400-3150=250

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 |

7 by 7 grids

- 1x67=67

7x61=427

427-67= 360

- 4x70=280

10x64=640

640-280= 360

- 31x97=

3007

37x91=3367

3367-3007=360

- 34x100=3400

40x94=3760

3760-3400=360

In conclusion I have found out that my predictions where right. So now I’m going to do the same thing all over again but I am going to change the grid from 10 by 10 to 11 by 11.

Extension investigation for Number grid

The purpose of this investigation is to look at diagonal differences on different sizes of grids. I am going to do this by multiple the numbers in the corners of the grids and then subtracting there answers to get the over all difference. From my previous investigation I did a 10 by 10 grid.

To carry out this investigation I will start by extracting a 2 by 2 grid by my original 9 by 9 grid. I will then carry out by multiplying the corner numbers and subtracting the smaller number by bigger the number so I can gain an answer. To further this investigation I will extend the grid size to a 3 by 3 grid and I will be doing exactly the same method as the other grid. I will then continue this by extending the grid size to a 4 by 4, 5 by 5, 6 by 6 and so on.

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 |

Conclusion

I am now going to test my predation, by doing the same thing as I did for the other number grids.

6 by 6 grids

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 |

- 1x51=51

6x46=276

276-51=225

- 4x54=216

9x49=441

441-216=225

- 28x78=2184

33x73=2409

2409-2184=225

- 31x81=2511

36x76=2736

2736-2511=225

I predict that the 6 by 6 would equal 225 and I got it right by using the formula that I had worked out to predict the answer for the 6 by 6 grids.

7 by 7 grids

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 |

- 1x61=61

7x55=385

385-61=324

- 3x63=189

9x57=513

513-189=324

- 19x79=1501

25x73=1825

1825-1501=324

- 21x81=1701

27x75=2025

2025-1701=324

I also predicted that the 7 by 7 would equal 324 and I got it right by using the formula that I had worked out to predict the answer for the 7 by 7 grids.

Conclusion

I thought that this course work went well because I wrote a lot of pages. In this course work I found out the formula to working out the number grids and that there were patterns that helped me work out the answer. I would apply the same formula to bigger grids like 11 by 11.

Betrand Dickson 11A Maths Coursework

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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