# Investigate the relationship between the T-total and the T-number.

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Introduction

Sonny Kumar Maths Coursework

T-Totals

## Tasks

1) Investigate the relationship between the T-total and the T-number

2) Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T- number and the grid size.

3) Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.

## Investigation into T-shapes

Looking at the 10-10 grid below and the T-shape drawn on it,

The total number of the numbers on the inside of the T-shape is called the T-total

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 |

Middle

88

89

90

91

92

93

94

95

96

97

98

99

100

Using random examples I am going to show the relationship between the T-total and T-number.

T-number69-50+48+49+59+69=275(t-total)

T-number32-11+13+12+22+32=90(t-total)

## T-Number | ## T-total |

## 22 | ## 40 |

## 23 | ## 45 |

## 24 | ## 50 |

## 25 | ## 55 |

## 26 | ## 60 |

## 27 | ## 65 |

## 28 | ## 70 |

## 29 | ## 75 |

As you can see from this table every time the t-number goes up 1 the t-total goes up 5. From this I can tell that to find formula connecting these two together I will have to multiply one of them by 5. So then I decided to multiply the t-number by 5.

## T-Number | ## T-number times 5 | ## Difference between 5 times t-number and t-total | ## T-total |

## 22 | ## 110 | ## 70 | ## 40 |

## 23 | ## 115 | ## 70 | ## 45 |

## 24 | ## 120 | ## 70 | ## 50 |

## 25 | ## 125 | ## 70 | ## 55 |

## 26 | ## 130 | ## 70 | ## 60 |

## 27 | ## 135 | ## 70 | ## 65 |

## 28 | ## 140 | ## 70 | ## 70 |

## 29 | ## 145 | ## 70 | ## 75 |

## This table shows that 5 times the t-number minus 70 equals the t-total.

## So the formulae relating the t-number and the t-total is-

## T-number=N T-total=T

## T=5N-702) Different sizes and relationship between the T-total and the T- number and the grid size

I know this works for the grid 10 by 10 but I'm not exactly sure if it'll work for any other grids because there are more intervals in between the first number of each line. So I might have to change a value in my formula. I will try my formula with two different sized grids.

Here is a 9 by 9 grid for which I will test my 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 | 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

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 |

T- Number =20

Orange T-Total = 7+8+9+14+20=58

Gold T-Total = 20+21+22+28+16=107

Orange-Gold=49

This shows that I have to add 49 to the formula for my relationship to work.

So now my formula will become- T=5N-7W+49

Next I am going to use a 7 by 7 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 |

T- Number =23

Orange T-Total = 8+9+10+16+23=66

Gold T-Total = 23+24+25+32+18=122

Orange-Gold=56

This shows that I have to add 56 to the formula for my relationship to work.

So now my formula will become- T=5N-7W+56

I am going to show my results in a table.

Grid width | Orange T-Total | Gold T-Total | Difference between the two T-Total |

5 | 25 | 67 | 42 |

6 | 58 | 107 | 49 |

7 | 66 | 122 | 56 |

Conclusion

My formula’s had to be changed each time I started the next task. Which I found confusing at first but then I ‘got into it’.

For my final formula I have made it to be

Where T=T-total, N=T=number, W=Grid width

T=5N-7W

For example-

The t-number=26 in 7 by 7 grid

T=(5x26)-(7x7)

T=130-49

T=81

Then if there’s a transformation-

T= (5N-7W)+((W+1)x7)

For example-

The t-number=25 in 8 by 8 grid and then rotated 90 degree’s clockwise.

T= (5N-7W)+((W+1)x7)

T=(125-56)+(9x7)

T=169+63

T=232

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

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