# In this investigation Im going to find out relationships between the grid sizes and T shapes within the relative grids, and state an explanation to generalize the finding using the T-Number

Extracts from this document...

Introduction

Salwa Mohammed

T-Total Coursework

Introduction: In this investigation I’m going to find out relationships between the grid sizes and T shapes within the relative grids, and state an explanation to generalize the finding using the T-Number (n) (the number at the bottom of the T-Shape), the grid size to find the T-Total (t) (Total of all number added together in the T-Shape), with different grid sizes.

Aim: I will investigate the relationship between the T-total, using grids of different sizes to translate the t-shape to different positions within the grid moving it horizontally, vertically over the grid.

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 |

From this we can see that the first T shape has a T number(n) of 30, and the T-total (t) adds up to 87 (11+12+13+21+30). With the second T shape with a T number of 31, the T-total adds up to 92, by looking at the two results a trend can be seen therefore suggesting the larger the T number the larger the total.

By looking at the T-Shapes we can plot a table of results.

T-Number (n) | T-Total (t) |

30 | 87 |

31 | 92 |

32 | 97 |

33 | 102 |

34 | 107 |

By looking at my table of results a pattern can be seen between the T-Number and the T-Total, there’s also a relationship between the T-Number and the T-Total because a trend occurs as you move it over different parts of the grid and it gives a ratio of 1:5.

Middle

75

76

77

78

79

80

81

By looking at the grid I plotted a table of results for the T Shape going vertically.

T-Number (n) | T-Total (t) |

30 | 87 |

39 | 132 |

48 | 177 |

57 | 222 |

66 | 267 |

When going vertically on the 9x9 grid the table of result shows a ratio of 1:5 between the T-Number and the T-Total.

So no matter how the T-Shape is moved, either horizontally or vertically the T-number and the T-Total are in the same ratio of 1:5.

Therefore, we can conclude that: On a 9x9 grid any T-Total can be found using t = 5 n – 63 were nis the T-Number.

Finding relationships on grids using 8x8

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 |

If we take this 8x8 grid with a T-Number of 29 we get the T-Total of 89 (12+13+14+21+29), if we generalize this straight away using the same method’s used in before for a 9x9 grid we achieve the formula:

t = n – 8 + n – 17 + n – 16 + n – 15

t = 2n – 8 + 3n – 48

t = 5n – 56

Testing this out using 29 as n we get:

t = (5 × 29) – 56

t = 145 – 56

t = 89

This proves the formula because I obtain the same answer as before.

This can also be shown in this form,

n -17 | n -16 | n -15 |

n -8 | ||

n |

From the table below we can see that the T-number increases by 1 and the T-Total increases by 5 giving it a ratio of 1:5 this is the same as in the 9x9 grid.

T-Number (n) | T-Total (t) |

27 | 79 |

28 | 84 |

29 | 89 |

30 | 94 |

31 | 99 |

I plotted the same grid but going vertically to see if it will any different to what I obtained in the 9x9 grid.

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 |

I also plotted a table of results to shows the trend between the T-Number and the T-Total and this is what I obtained.

T-Number (n) | T-Total (t) |

27 | 79 |

35 | 119 |

43 | 159 |

51 | 199 |

59 | 239 |

Conclusion

t = 5n – (7 x g)

t = 5n – 7g

As we have taken 5n (the number of numbers), from that we take the grid size times 7 (as they all are multiples of 7), and therefore we can state:

Any T-Total of a T-Shape can be found if you have the 2 variables of a T-Number and a grid size for T shapes that extends upwards, using the formula t = 5n – 7g.

Another area that we can investigate is that of differing grid sizes such as 3x8 and 4x6 to see if they will make a difference to the formula.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 |

So if we work this grid out using the same method as before we get a T-Total of 47, and a formula of:

t = n + n – 4 + n – 9 + n – 8 + n – 7

t = 5n – 24

This formula is the same as the 4x4 grid as it has a number of 24, identical to a 4x4 grid, we can predict that grid width is the only important variable, but we will need to prove this.

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

10 | 11 | 12 |

13 | 14 | 15 |

16 | 17 | 18 |

19 | 20 | 21 |

22 | 23 | 24 |

On a 4x6 grid with a T-Number of 17 the T-Total is 64, again working out the formula leads to

t = n + n – 3 + n – 7 + n – 6 + n – 5

t = 5n – 21

Again this is the same number as found by the predictions for a 6x6 grid found earlier, therefore we can state that:

5n – 7g can be used to find the T-Total (t) of any Grid size, regular (e.g. 7x7) or irregular (e.g. 7x11), with the two variables of grid width (g) and the T-Number (n).

This student written piece of work is one of many that can be found in our GCSE T-Total section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month