# T-Total Course Work

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Introduction

GCSE COURSE WORK T-TOTALS

GCSE MATHS T-Total Course Work

PART 1

I am investigating the relationship between the T-total and T-number on a 9 x 9 grid by using a variety of T-Shapes

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 | 21 |

1 | 2 | 3 | T-number = 20 Numbers: 1 2 3 11 20 20-1 20-2 20-3 20-4 20 T-19 T-18 T-17 T-9 T T-total = 37 |

11 | |||

20 |

2 | 3 | 4 | T-number = 21 Numbers: 2 3 4 12 21 21-2 21-3 21-4 21-12 21 T-19 T-18 T-17 T-9 T T-total = 42 |

12 | |||

21 |

(T-19) + (T-18) + (T-17) + (T-9) + T

= 5 x T - 63

= 5T – 63

I tested that when:

T-Number = 80

T-Total = 337

337 = 5T – 63

337 = (5 x 80) – 63

= (5 x 80 = 400) – 63

= 400 -63

= 337

337 = 337

Formula is correct.

I also tested it when:

T-Number = 20

T-Total = 27

5T-63

= (5x20)-63

= 100-63

=73

Formula Works, it is correct

This proves that the Formula = 5T– 63 works and is correct for any T-Shape on a 9 x 9 grid

This shows that the connection between the T-Total and the T-Number is 5T-63:

Let T-number = T

Let T-total = n

T | n |

20 21 22 =63 | 37 42 47 |

Formula = 5T – 63

Below is a t-shape, and in each cell how the number is connected with T

T-19 | T-18 | T-17 |

T-9 | ||

T |

PART 2

I am now going to investigate the relationship between the T-total and T-number and the grid size by

using a variety of grids and T-Shapes at different positions

8x8 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 |

1 | 2 | 3 | T-number = 18 Numbers: 1 2 3 10 18 18-1 18-2 18-3 18-10 18 T-17 T-16 T-15 T-8 T T-total = 34 |

10 | |||

18 |

Middle

To find out the number 3, we have to subtract 1 from the formula used for number 2 as it is to the right of the number 2 and needs a smaller amount to be subtracted from the T-Number in order to give 3. So, (T-(2G-1)) should equal 3… (20-(2x9-1))= 20-17 = 3. I have now found a formula for each value within the T-Shape. Now, in order to give me the value 5T-7G I added the formulae for each value in the T-Shape like so, (T) + (T-G) + (T-2G) + (T-2G+1) + (T-2G-1) = 5T-7G.

11

20

(T) = 20

(T-G) = 11

(T-2G) = 2

(T-(2G-1)) = 3

(T-(2G+1)) = 1

Add then all = 5T-7G

Here I have constructed a random GxG Grid to prove that my Formula of 5n-7G will work on any Grid size, and any position

15 x 15 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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

32 Is my T-Number

T = 32

5T = 5x32 = 160

G = 15

7G = 7x15 = 105

I now have enough data to prove that my formula works

5T - 7G

160 – 105 = 55

T-Total = 1+2+3+17+32 = 55

So my Formula works because the T-Total = 55, and my Formula gives a T-Total for the T-Shape as 55.

9 x 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 |

T =59

5T = (5x59) = 295

G = 9

7G = (7x9) = 63

T-Total = (40+41+42+50+59) = 232

I will now apply my formula onto this T-Shape on the 9x9 Grid to prove it will work on any Grid size with a T-Shape at any position

5T-7G

(5x59)-(7x9)

= 295-63

= 232

Is equal to the T-Total, so my formula 5T-7G Works.

The relationship between the T-Total, T-Number and Grid Size are all summed up into the formula 5T-7G.

This formula can be used on any Grid Size, with the T-Shape at any position.

PART 3

I am going to investigate the relationship between the T-Total, the T-Numbers, the Grid Size, and different transformations.

I am going to use the transformation of translation on the following T-Shape to the right on this 10 x 10 grid

10x10 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 | 387 | 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 |

1 | 2 | 3 | T-Number = 22 Numbers: 1 2 3 12 22 22-1 22-2 22-3 22-12 22 T-21 T-20 T-19 T-10 T T-total = 40 |

12 | |||

22 | |||

3 | 4 | 5 | T-Number = 24 Numbers: 3 4 5 14 24 24-3 24-4 24-5 24-14 24 T-21 T-20 T-19 T-10 T T-total = 50 |

14 | |||

24 |

I will now work out a general formula for a T-Shape that has been translated to the right, for any grid size because I can see the connection with a translation of 2 to the right and the grid.

(T-2G)+1 | (T-2G)+2 | (T-2G)+3 |

(T-G)+2 | ||

T+2 |

((T-2G)+1) + ((T-2G)+2) + ((T-2G)+3) + ((T-G)+2) + (T+2)

=5T-7G+10

I will now get a formula by drawing part of a GxG grid and using a T-shape that has been translated x times to the right.

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 |

1 | 2 | 3 | T-Number = 32 Numbers: 1 2 3 17 32 32-1 32-2 32-3 32-17 32 T-31 T-30 T-29 T-15 T T-total = 55 |

17 | |||

32 |

7 | 8 | 9 | T-Number = 38 Numbers: 7 8 9 23 38 38-7 38-8 38-9 38-23 38 T-31 T-30 T-29 T-15 T T-total = 85 |

23 | |||

38 |

9 | 10 | 11 | T-Number = 40 Numbers: 9 10 11 25 40 40-9 40-10 38-11 40-25 40 T-31 T-30 T-29 T-15 T T-total = 95 |

25 | |||

40 |

Let x = number of times/squares translated to the right

I will now work out a general n that has been translated x times to the right

(T-2G)+x-1 | (T-2G)+x | (T-2G)+x+1 |

(T-G)+x | ||

(T+x) |

((T-2G)+x-1) + ((T-2G)+x) + ((T-2G)+x+1) + ((T-G)+x) + (T+x)

=5T-7G+5x

This formula should work out the T-Total for the translated shape that has moved 6 times to the right, It is the Lined shape.

T = 32

G = 15

X = 6

= (5x32)-(7x15)+(5x6)

= 160 – 105 + 30

= 85

Which is the T-Total for the Translated T-Shape, this proves the formula works.

((T-2G)+x-1) + ((T-2G)+x) + ((T-2G)+x+1) + ((T-G)+x) + (T+x) =5T-7G+5x = my formula |

Conclusion

To make sure the formula works on any Grid I am going to test it on a 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 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 21 |

T= 20

G= 9

X=3

Y=5

5T-7G+5x+((5xG)xY)

= (5x20) – (7x9) + (5x3) + (45x5)

=100 – 63 + 15 + 225

=277

Which is the T-Total for the t-shape on the 9x9 Grid

49+50+51+59+68=277

This proves that the formula 5T-7G+5x+((5xG)xY)

will work on any grid with any translation of x spaces to the right and y places down.

I have also realised that this formula ONLY works on right and down, in order for it to work on left and up you will need to change the values of “x” and “y” from +ve to –ve. This will give you the ability to go left and up. I also noticed that to move x places to the left and y places down, you need to change the “x” to –ve and leave the “y” as +ve. And vice versa.

THIS IS NOT NEEDED, JUST INCASE YOU WANT TO WORK ON ROTATION INSTEAD OF TRANSLATION, JUST THE BEGINNING OF IT.

I will now try and find therelationship between the T-Total, the T-Numbers, the Grid Size, and the rotation of the T-Shape.

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 | 387 | 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 |

I have rotated the T-Shape 90* Clockwise at point 32. I will now try and find the relationship between the T-Total, T-Number, Grid size and the rotation.

T=22, T-Total = 40

T=33, T-Total = 172

Omar El-Anis

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

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