# T-totals. For my T-totals maths coursework I will investigate the relationship between the T-total and T-number, the T-total and T-number and grid size and the T-shape in different positions.

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Introduction

Maths Coursework – T-totals

Introduction

For my T-totals maths coursework I will investigate the relationship between the T-total and T-number, the T-total and T-number and grid size and the T-shape in different positions.

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 |

Looking at this T-shape drawn on a 9x9 grid,

The total of the numbers inside the T-shape is 2+3+4+12+21=42

This is called the T-total.

The number at the bottom of the shape is the T-number. The T-number for this shape is 21.

## Part 1

For the first part of my coursework I must investigate a relationship between the T-total and the T-number. To do this I have chosen the following T-shapes:

1 | 2 | 3 |

11 | ||

20 | ||

4 | 5 | 6 |

14 | ||

23 |

&

I noticed that the difference between each number in the T-shape and the T-number was always the same no matter what T-shape you use.

(N = T-number )

23-4 | 23-5 | 23-6 |

23-14 | ||

23 |

20-1 | 20-2 | 20-3 |

20-11 | ||

20 |

&

=

With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:

N-19 | N-18 | N-17 |

N-9 | ||

N |

Middle

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

By doing the same as in the 9 x 9 grid you will get a T-shape like this:

1 | 2 | 3 |

8 | ||

14 |

=

N-13 | N-12 | N-11 |

N-6 | ||

N |

This means that the formula to find the T-number in a 6 x 6 grid is:

T = 5N – 42.

### Now we must put this formula to the test,

## For example, If I put the T-number 27 into the formula it looks like this:

5 x 27 – 42 = T

135 – 42 = T

T = 93

Therefore, the T-total of a T-shape with a T-number of 27 on a 6 x 6 grid should be 93.

14 | 15 | 16 |

21 | ||

27 |

T-total = 14+15+16+21+27=93

Therefore my formula was proven correct! √

## Part 2

In the last part I noticed that not only did the centre column in each T-shape for the 9 x 9 grid go up in 9’s but also the centre columns in each T-shape for the 6 x 6 grid went up by 6. This means that I can work out a formula for any T-total on any grid size by using the calculations below:

N-(2G-1) | N-(2G) | N-(2G+1) |

N-G | ||

N |

(G = Grid size)

This means that to find the T-number on any T-shape on any size graph you must use the formula:

### T = 5N – 7G

Now I must put this formula to the test

I will use a 5 x 5 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 |

I will use the T-number 18.

T = 5N – 7G

T = 5 x 18 – 7 x 5

T = 90 – 35

T = 55

This means that the T-total for a T-shape with T-number 18 on a 5 x 5 grid should be 55.

7 | 8 | 9 |

13 | ||

18 |

T = 7+8+9+13+18=55

## I have proven that it works on a square grid but now I will see if it will work on a rectangle grid. I will use a 4 x 6 grid and use the T-number 19.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 |

T = 5N – 7G

T = 5 x 19 – 7 x 4

T = 95 – 28

T= 67

10 | 11 | 12 |

15 | ||

19 |

T = 10+11+12+15+19=67

This means that my formula is correct and now I can make a formula that gives me the T-number if I have the T-total. This formula is:

N = T + 7G

5

## Now I will test this formula using a 4 x 4 grid and using a

T-total of 42.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

Conclusion

This relationship will enable me to find formulas for T-shapes in any positions on any grid:

N | ||

N+7 | ||

N+13 | N+14 | N+15 |

This means that to find the T-total or T-number of an “upside down” T-shape on a grid size 7 x 7 you must use this formula:

### T = 5N + 49

### Now we must check this. For example, if I use the T-number 33 into the formula it looks like this:

## T = 5N + 49

T = 5 x 33 + 49

T = 165 + 49

T = 214

33 | ||

40 | ||

46 | 47 | 48 |

T = 33+40+46+47+48=214

Therefore my formula is correct.

Now I know that to find the formula of the T-shape “upside down” or “opposite” you must turn the mathematical signs opposite I can now find the formula for any size grid without doing much working out:

N | ||

N+G | ||

N+2G+1 | N+2G | N+2G-1 |

This means that the formula to find a T-total of an upside down T-shape on any size grid is:

### T = 5N + 7G

## Now I must test this formula. I will use a 8 x 8 grid and the T-number 28:

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 |

T = 5N + 7G

T = 5 x 28 + 7 x 8

T = 140 + 56

T = 196

28 | ||

36 | ||

43 | 44 | 45 |

## T = 28+36+43+44+45=196

Therefore my formula is correct.

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

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