# To investigate the relationship between a number in a "T" shape placed on a grid of numbers, and the total of all the numbers in that "T" shape.

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Introduction

Maths Coursework

Objective: To investigate the relationship between a number in a “T” shape placed on a grid of numbers, and the total of all the numbers in that “T” shape.

Introduction: During this investigation I will focus mainly on the relationship between the size of the “T” against the size of the grid, and I will use this as a basis to find out any link between “T” total and “T” number.

Hypothesis: I believe there will be a link between the size of the “T” and the grid size.

I think there will also be a link between the size of the T and the grid size. These will either be linear or quadratic.

Methodology:

- To begin, I will keep the “T” size constant and will change the side length of the square grid I place it on by one square each time. Then from these results I will investigate the link.
- Then I will change the “T” size and investigate how that links to the size of the grid.

Middle

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N= 20 21 22 23 24 25 26

T= 37 42 47 52 57 62 67

5 5 5 5 5 5

Link: 5n – 63

Equations:

7 by 7 = 5n - 49

8 by 8 = 5n – 56

9 by 9 = 5n – 63

10 by 10 = 5n – 70

Each of these begins with 5n and ends with a multiple of 7. The exact multiple is the same as the length times 7. This makes the link between each total 5n – 7L, where L is length of one grid side This enables us to work out the “T” total and “T” number for that specific sized “T” on any grid size. The number 5 is the amount of numbers inside the “T”.

Second “T” size (6 numbers)

The first part of the equation for this set of results will be 6n.

4 by 4 grid. 6 numbers in the “T”

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9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

N= 14 15

T= 36 42

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Link: 6n – 48

5 by 5 grid. 6 numbers in the “T”

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 |

N= 17 18 19

T= 42 48 54

- 6

Link: 6n – 60

6 by 6 =

N= 20 21 22 23

T= 48 54 60 66

6 6 6

Equations:

4 by 4 = 6n – 48

5 by 5 = 6n - 60

6 by 6 = 6n - 72

7 by 7 = 6n - 84

8 by 8 = 6n - 96

9 by 9 = 6n - 108

10 by 10 = 6n – 120

The link between each of these is the equation 6n – 12L

As before L equals the length of one of the grid sides, and the 6 is the amount of numbers in the T.

There is a pattern: an – bL

“a” is always the amount of numbers in the T.

Third “T” size (4 numbers)

4 by 4 grid. 4 numbers in the “T”

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

N= 6 7

T= 12 16

4

Link: 4n – 12

5 by 5: 4n – 15

6 by 6: 4n - 18

7 by 7: 4n - 21

8 by 8: 4n - 24

9 by 9: 4n - 27

10by 10: 4n – 30

The link between these is the equation: T = 4n – 3L

“T” number links

N-5 | N-4 | N-3 |

N | ||

Equations for the sizes of “T”s so far

4 numbers: T = 4n – 3L

4

5 numbers: T = 5n – 7L 1

5

6 numbers: T = 6n – 12L 1

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7 numbers: T = 7n – 18L 1

7

8 numbers: T = 8n – 25L 1

8

9 numbers: T = 9n – 33L 1

9

10 numbers: T = 10n – 42L

0 1 2 3 4

3 3 7 12 18

4 5 6

0.5 1 1

3 = 0.5 x 12 + b x 1 + 0

3 = 0.5 + b

3 – 0.5 = 2.5

a = 0.5

b = 2.5

c = 0

Changing the grid size:

0.5n2 + 2.5n + 0

This links with this number in the equations

T = 4n –3L

Formula for working out the “T” total on any grid size and any sized “T” (end solution)

T = 4n – (0.5n2 +2.5n) x L

1 1 2 3 4

3 4 5 6 7

1 1 1 1

n + 3

T = (X + 3) x n – (0.5n2 + 2.5n) x L

Y = x2 5x

+

2 2

T = 4n – 3L

T = ? n – ( x2 5x )

+

2 2

T = (y + X) x n – ((0.5n2 + 2.5n) x L)

Third Variable.

Width of the T.

1st T width

width of 5

grid size 6 by 6

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Conclusion

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N: 17 18 19

T: 42 49 56

7 7

The link here is therefore 7n – 77

I predict that on an 8 by 8 grid the outcome will be 7n – 88 and also that the link with this sized T will be 7n – 11g

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 |

N: 19 20 21 22

T: 45 52 59 66

7 7 7

Prediction #1 correct: 7n – 88

6 by 6: 7n - 66

7 by 7: 7n - 77

8 by 8: 7n - 88

9 by 9: 7n - 99

10 by 10: 7n - 110

Second Prediction correct: 7n – 11g

2nd T size

9 numbers in the T

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 |

N: 20 21

T: 60 69

9

Therefore link: 9n – 120

Prediction: The link for a 9 by 9 grid will be 9n –135

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 |

N: 22 23 24

T: 63 72 81

- 9

Prediction correct: 9n – 135

The link for all this sized T is 9n-15g

5n – 7g

4

7n – 11g

4

9n – 15g

An - ?g A being amount of numbers in the t. N being the t number therefore the

Unknown. G being the length of the grid.

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