# Number Grid Coursework

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Introduction

Number Grid Coursework

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 |

- A box is drawn around 4 Numbers
- Find the product of the top left number and bottom right number in this box
- Do the same with the top right and bottom left numbers
- Calculate the difference between these products

For my piece of coursework as shown above I will need to work out the product of the numbers multiplied from the top left and bottom right numbers and the same with the top right and bottom left numbers and example is shown below:

To work out the Product I will need to multiply 12 by 23 and 13 by 22

12 x 23 = 276

To work out the difference I will

need to subtract 286 from 276 :

13 x 22 = 286276 – 286 = 10

I will now start my coursework with a 10 x 10 grid and then develop it further by doing different shaped grids and sizes also I will show that my coursework is coming along successfully by working out the algebraic equation which should work for a certain sized grid.

10 x 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 |

2 x 2

12x23 = 276

13x22 = 286 difference = 10

64x75 = 4800

65x74 = 4810 difference = 10

38x49 = 1862

39x48 = 1872 difference = 10

N | N+1 |

N+10 | N+11 |

X = N x (N + 11) = (N x N) + (N + 11) = N² + 11N

Y = (N + 1) X (N + 10) = (N x N) + (N x 10) + (N x 1) + (1 x 10) = N² + 11N + 10

Both X and Y have N² + 11N

Middle

DIFFERENCE

10

40

90

160

1st DIFFERENCE

40-10=30 90-40=50 160-90=70

2nd DIFFERENCE

50-30=20 70-50=20

11 x 9

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 |

2 x 2

24x36 = 864

25x35 = 875 difference = 11

60x72 = 4320

61x71 = 4331 difference = 11

21x33 = 693

22x32 = 704 difference = 11

N | N+1 |

N+11 | N+12 |

X = N x (N + 12) = (N x N) + (N x 12) = N² + 12N

Y = (N+1) x (N + 11) = (N x N) + (N x 11) + (N x1) + (1 x 11) = N² + 12N + 11

3 x 3

27x51 = 1377

29x49 = 1421 difference = 44

67x91 = 6097

69x89 = 6141 difference = 44

N | N+1 | N+2 |

N+11 | N+12 | N+13 |

N+22 | N+23 | N+24 |

X = N x (N + 24) = (N x N) + (N x 24) = N² + 24N

Y = (N + 2) x (N + 22) = (N x N) + (N x 22) + (N x 2) + (2 x 22) = N² + 24N + 44

4 x 4

14x50 = 700

17x47 = 799 difference = 99

51x87 = 4437

54x84 = 4536 difference = 99

N | N+1 | N+2 | N+3 |

N+11 | N+12 | N+13 | N+14 |

N+22 | N+23 | N+24 | N+25 |

N+33 | N+34 | N+35 | N+36 |

X = N x (N + 36) = (N x N) + (N x 36) = N² + 36N

Y = (N + 3) + (N + 33) = (N x N) + (N x 33) + (N x 3) + (3 x 33) = N² + 36N + 99

5 x 5

17x65 = 1105

21x61 = 1281 difference = 176

47x95 = 4465

51x91 = 4641 difference = 176

N | N+1 | N+2 | N+3 | N+4 |

N+11 | N+12 | N+13 | N+14 | N+15 |

N+22 | N+23 | N+24 | N+25 | N+26 |

N+33 | N+34 | N+35 | N+36 | N+34 |

N+44 | N+45 | N+46 | N+47 | N+48 |

X = N x ( N + 48) = (N x N) + (N x 48) = N ² + 48N

Y = (N + 4) x (N + 44) = (N x N) + (N x 44) + (N x 4) + (4 x 44) = N² + 48N + 176

-- I will now attempt to work out the difference of a grid (8 x 8) without doing the multiplication before hand to test my equation--

N | N+1 | N+2 | N+3 | N+4 | N+5 |

N+22 | N+23 | N+24 | N+25 | N+26 | N+27 |

N+33 | N+34 | N+35 | N+36 | N+37 | N+38 |

N+44 | N+45 | N+46 | N+47 | N+48 | N+49 |

N+55 | N+56 | N+57 | N+58 | N+59 | N+60 |

N+66 | N+67 | N+68 | N+69 | N+70 | N+71 |

X = N x (N + 71) = (N x N) + (N x 71) = N² + 71N

Y = (N + 5) x (N + 55) = (N x N) + (N x 55) + (N x 5) + (6 x 55) = N² + 71N +275

From the equation I predict that the difference in the 8x8 grid will be 275

25x85 = 2125

30x80 = 2400 difference = 275

21x71 = 1491

16x76 = 1216 difference = 275

Below I have worked out the Nth term for all the grid sizes (2x2 – 5x5)

GRID | 2x2 | 3x3 | 4x4 | 5x5 |

DIFFERENCE | 11 | 44 | 99 | 176 |

1st DIFFERENCE | 44-11=33 99-44=55 176-99=77 | |||

2nd DIFFERENCE | 55-33=22 77-55=22 |

8 x 9

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 |

2 x 2

19x28 = 532

20x27 = 540 difference = 8

52x61 = 3172

53x60 = 3180 difference = 8

N | N+1 |

N+8 | N+9 |

X = N x (N + 9) = (N x N) + (N x 9) = N² + 9N

Y = (N + 1) x (N + 8) = (N x N) + (N x 8) + (N x 1) + (8 x 1) = N² + 9N + 8

3 x 3

42x60 = 2520

44x58 = 2552 difference = 32

21x39 = 819

23x37 = 851 difference = 32

N | N+1 | N+2 |

N+8 | N+9 | N+10 |

N+16 | N+17 | N+18 |

Conclusion

49x63 = 3087 difference = 120

N | N+1 | N+2 | N+3 | N+4 | N+5 | N+6 |

N+10 | N+11 | N+12 | N+13 | N+14 | N+15 | N+16 |

N+20 | N+21 | N+22 | N+23 | N+24 | N+25 | N+26 |

X= N x (N + 26) = (N x N) + (N x 26) = N² + 26N

Y = (N + 6) + (N + 20) = (N x N) + (N x 20) + (N x 6) + (6 x 20) = N² + 26N + 120

-- I will now attempt to work out the difference of a grid (3 x 8) without doing the multiplication before hand to test my equation--

N | N+1 | N+2 | N+3 | N+4 | N+5 | N+6 | N+7 |

N+10 | N+11 | N+12 | N+13 | N+14 | N+15 | N+16 | N+17 |

N+20 | N+21 | N+22 | N+23 | N+24 | N+25 | N+26 | N+27 |

X = N x (N + 27) = (N x N) + (N x 27) = N² + 27N

Y = (N + 7) + (N + 20) = (N x N) + (N x 20) + (N x 7) + (7 x 20) = N² + 27N + 140

From the equation I predict that the difference in the 3x8 grid will be 140

3 x 8

12x39 = 468

19x32 = 608 difference = 140

42x69 = 2898

49x62 = 3038 difference = 140

Below I have worked out the Nth term for all the grid sizes (3x4 – 3x7)

GRID | 3x4 | 3x5 | 3x6 | 3x7 |

DIFFERENCE | 60 | 80 | 100 | 120 |

Divide by 10 | 6 8 10 12 | |||

Multiples | 2x3 2x4 2x5 2x6 |

I can now use these to work out a general rule for rectangle grids on a

10 x 10 Grid.

Grid | 3 x 4 | 3 x 5 | 3 x 6 | 3 x 7 |

Width – 1 | 3 – 1 = 2 | 3 – 1 = 2 | 3 – 1 = 2 | 3 – 1 = 2 |

Length – 1 | 4 – 1 = 3 | 5 – 1 = 4 | 6 – 1 = 5 | 7 – 1 = 6 |

Length x Width x 10 | 3 x 2 x 10 = 60 | 2 x 4 x 10 = 80 | 2 x 5 x 10 = 100 | 2 x 6 x 10 = 120 |

I have worked out a rule for the rectangular grid which seems to work

10 x (L – 1) (W - 1)

Zuhaib Ahmad

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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