# T-Totals Maths

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Introduction

T-Totals Coursework

My aim is to investigate the relationship between the T-Number and the T-shape on a varying size of grid.

1 | 2 | 3 |

4 | ||

5 |

To the left is a basic T-shape. In this investigation, the number in bold which is “5” is the T-Number.

The sum of the all the numbers in the T-shape is the T-Total.

For Example:

1+2+3+4+5 = 10.

Therefore the T-Total for this T-Shape is 10.

Using this information I can now begin my investgation.

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 |

I am going to start my investiation on a 9 by 9 grid. This shows all the numbers from 1 to 81.

Firstly in my investigation, I am going to find a formula that relates my T-Number to my T-Total, firstly with a 9x9 grid and then onto a grid of any size.

To work out my formula I have drawn two T-Shapes on my grid. The first t-shape in green has a T-Number of 20. The other T-Shape highlighted in pink has been translated one number to the right giving it a T-Number of 21.

I then worked out the T-Total for both shapes.

Green T-Shape: 1+2+3+11+20 = 37

Pink T-Shape: 2+3+4+12+21 = 42

T-Number | T-Total |

20 | 37 +5 |

21 | 42 +5 |

22 | 47 +5 |

23 | 52 +5 |

24 | 57 |

Middle

T-Total would be 57.

I then calculated this and my prediction turned out to be correct.

This is because there is 5 numbers in the T-Shape and each number in the T-Shape has to go up by one when it it translated to the right therefore adding five more onto the total. For this same reason, five has to be included as the main focus of my algebraic formula.

## The Algebraic Formula

I could then, using the process of elimination and trial and error come up with an algebraic formula. However, I concluded that there must be a more logical way.

I therefore looked at my grid again.

Starting from my T-number, which in this case is called n, I have calculated this algrbraic t-shape. in the boxes of the shape, the numbers inside are related to the T-Number (n).

n - 19 | n - 18 | n - 17 |

n – 9 | ||

n |

In this case on my 9x9 grid, the square directly above the T-Number (which is 20) is nine less. This makes that square n-9.

Conclusion

5n – 7t = T-Total.

I will now go back to the T-Totals from my 9 x 9 grid and try my new and improved formula on both of them.

9 x 9 T-Shape One.

T-Number = 20

T-Total = 37

For this Formula I am going to explain why this works.

(5 x 20) – (7 x 9) = 37.

9 x 9 T-Shape Two.

T-Number = 21

T-Total = 42

(5 x 21) – (7 x 9) = 42.

So far, with the 9 x 9 grid, the formula works. I am now going to try it out on the 8 x 8 grid T-Totals.

8 x 8 T-Shape One.

T-Number = 18

T-Total = 34

(5 x 18) – (7 x 8) = 34

8 x 8 T-Shape Two.

T-Number = 19

T-Total = 39

(5 x 19) – (7 x 8) = 39.

As the formula has worked on both grids.

This has indefinatly proved again that my formula is correct.

To prove this fianally I am going to test my formula on two T-Shapes from a 7x7 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 |

I have chosen two T-Shapes to try my formula on.

7 x 7 T-Shape One.

T-Number = 16

T-Total = 31.

(5 x 16) – (7 x 7) = 31

7 x 7 T-Shape Two

T-Number = 17

T-Total = 36

(5 x 17) – (7x7) = 36.

I now have a formula that relates the T-Number to the T-Shape regardless of the T-Shapes size as long as the formula is followed corectly.

Vikki Baker 10 Bragg.

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

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