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

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Introduction

Mathematics GCSE Coursework

Part 1

Investigating the relationship between the T-total and the T-number.

This is the 9 by 9 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 |

Let T be the T-number and Y be the T-total.

1 | 2 | 3 |

11 | ||

20 |

We focus on the T-shape drawn above first.

If the other numbers in the T-shape other than the T-number are taken away from the T-number, a T-shape like the one drawn below is found:

T-18-1 | T-18 | T-18+1 |

T-9 | ||

T |

Because of the grid size, the centre column of the T-shape is decreasing by 9 up the column from the T-number at the bottom. Thus a formula can be worked out to find any T-total with the T-number given:

## Y = T + T – 9 + T – 18 + T – 18 + 1 + T – 18 – 1

## Y = 5T – 63

## Test

For the T-shape drawn before, the T-number is 20 and the T-total is:

20 + 11 + 2 + 1 + 3 = 37

Let us see if the formula works.

## Y = 5T – 63

Y = 5(20) – 63

## Y = 37

Two more tests on this formula are to be done for accuracy.

15 | 16 | 17 |

25 | ||

34 |

T-number = 34 and T-total = 34 + 25 + 16 + 15 + 17 = 107

## Y = 5T – 63

## Y = 5(34) – 63

Y = 107

49 | 50 | 51 |

59 | ||

68 |

T-number = 68 and T-total = 68 + 59 + 50 + 49 + 51= 277

## Y = 5T – 63

## Y = 5(68) – 63

Y = 277

### Y=5T – 63

The formula for T-total is tested for 3 different samples, which is proved to be worked for a 9 by 9 grid.

Now the T-shape is going to be translated to different positions in the 9 by 9 grid and the relationship between the T-total and the T-number is going to be further investigated.

1 | 2 | 3 |

11 | ||

20 |

Take the T-shape drawn above as the original position.

If this T-shape is translated by the vector 3 , the new T-number and the T-total will be

-1

changed to 32 and 97 respectively.

13 | 14 | 15 |

23 | ||

32 |

If the T-shape is then translated by the vector 2 , the new T-number and the T-total will

-2

be changed to 40 and 137 respectively.

21 | 22 | 23 |

31 | ||

40 |

A formula of this translation of the T-shape can be worked out by working the vectors with the original T-number.

Let a

Middle

Therefore:

The formula for T-total in an 8 by 8 grid worked out before:

Y= 5T – 56

The formula for new translated T-shape in an 8 by 8 grid:

y = 5T – 56

y = 5(T + a – 8b) – 56

## Test

28 | 29 | 30 |

37 | ||

45 |

Take this T-shape above as the original position.

If it is translated by the vector -3 , the new T-shape will be:

-2

41 | 42 | 43 |

50 | ||

58 |

The T-total of this new T-shape is 58 + 50 + 42 + 41 + 43 = 234

Let us see if the formula works.

y = 5(T + a – 8b) – 56

y = 5[45 + (–3)– 8(– 2)] – 56

y = 234

42 | 43 | 44 |

51 | ||

59 |

Take this T-shape above as the original position.

If it is translated by the vector 4 , the new T-shape will be:

4

14 | 15 | 16 |

23 | ||

31 |

The T-total of this new T-shape is 31 + 23 + 15 + 14 + 16 = 99

Let us see if the formula works.

y = 5(T + a – 8b) – 56

y = 5[59 + 4 – 8(4)] – 56

y = 99

After investigating the two formulae for new translated T-total for a 9 by 9 grid and an 8 by 8 grid, a general formula for new translated T-total for any grid of different sizes can be worked out:

Let G be the width of the grid.

y = 5(T + a – G b) – 7G

This general formula is basically based on the formula for T-total on any grid of any width worked out previously, and further developed by adding the vectors of the translation.

#### Part 3

Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the

T-total, the T-numbers, the grid size and the transformations.

(9 by 9 grid)

First of all, the T-shape is going to be translated in 90° clockwise. To start with, take the T-number as the centre of rotation.

The original position:

1 | 2 | 3 |

11 | ||

20 |

The new T-shape will be:

13 | ||

20 | 21 | 22 |

31 |

If the other numbers in the T-shape other than the T-number are taken away from the T-number, a T-shape like the one drawn below is found:

T+2-G | ||

T | T+1 | T+2 |

T+2+G |

Conclusion

Transformation of 90° clockwise:

## Y = 5T + 7

## Y = 5(T – G + 1) + 7

Transformation of 180° clockwise:

## Y = 5T + 7G

## Y = 5(T – 2G) + 7G

Transformation of 90° anti-clockwise:

## Y = 5T – 7

Y = 5(T – G + 1) – 7

Then, the general formulae for all combinations of transformations and translations with the centre box of the T-shape as the centre of rotation other than the T-number will be:

Combination of transformation of 90° clockwise:

Y = 5(T + a – G b) + 7

## Y = 5(T + a – G b – G – 1) + 7

Combination of transformation of 180° clockwise:

## Y = 5(T + a – G b) + 7G

## Y = 5(T + a – G b – 2G) + 7G

Combination of transformation of 90° anti-clockwise:

## Y = 5(T + a – G b) – 7

## Y = 5(T + a – G b – G + 1) – 7

These formulae are formed by merging the 2 parts, T + a – G b and T – G + 1 of the formulae for translation and transformation with the centre box of the T-shape being the centre of rotation respectively.

## Test

Using a 7 by 7 grid this time.

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 |

Take the T-shape below as the original position:

3 | 4 | 5 |

11 | ||

18 |

Take the centre box as the centre of rotation.

If this T-shape is translated by the vector -2 , and then followed by a transformation of

0

90° clockwise, the new T-shape will be:

3 | ||

8 | 9 | 10 |

17 |

T-number = 8 and T-total = 8 + 9 + 10 + 3 + 17 = 47

Let us see if the formula works.

## Y = 5(T + a – G b – G – 1) + 7

## Y = 5(18 + (–2) – 7(0) – 7 – 1) + 7

Y = 47

Using the same original position, if it is translated by the vector 2 , and then followed by

-1

a transformation of 180°clockwise, the new T-shape will be:

13 | ||

20 | ||

26 | 27 | 28 |

T-number = 13 and T-total = 13 + 20 + 27 + 26 + 28 = 114

Let us see if the formula works.

## Y = 5(T + a – G b – 2G) + 7G

## Y = 5[18 + 2 – 7(–1) – 2(7)] + 7(7)

## Y = 114

If it is then translated by the vector 0 , and then followed by a transformation of 90°

-3

anti-clockwise, the new T-shape will be:

24 | ||

31 | 32 | 33 |

38 |

T-number = 33 and T-total = 33 + 32 + 31 + 24 + 38 = 158

Let us see if the formula works.

## Y = 5(T + a – G b – G + 1) – 7

Y = 5(18 + (0) – 7(–3) – 7 + 1) – 7

Y = 158

These formulae are tested for 3 different samples and are proved to be worked combinations of transformations of those 3 angles.

Conclusion:

Final mark: 8 8 8 (full mark)

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

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