# T-total Investigation

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Introduction

Zakwan Ahmed

10 A

Maths Coursework

Mr Uddin

T-total Investigation

Aim: Find a rule connecting the T-total and the T- number.

Investigation- Part one:

(A,B,C)

Investigation- Part Two:

Investigation- Part Three:

3

1 | 2 | 3 |

12 | 2 | |

22 |

I placed the 3by2 T on the beginning of the 10by10 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 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

I then moved the T along the 10by10 grid 3 times in order to find a pattern. These are the 3 positions in which I put my 3by2 T on:

Position 1: Position 2:

1 | 2 | 3 |

12 | ||

T-no | 22 | |

2 | 3 | 4 |

13 | ||

T-no | 23 |

Position 3:

3 | 4 | 5 |

14 | ||

T-no | 24 |

The bottom number of the T is always the T-number. I added all the numbers inside the T- shape for each position and this gave me the T- total for e.g. for the first position 1 + 2+ 3+12+22 =40 and this is the T-total for the first position. I then put the T-no of each position in one column and the T-total of each position in another column to see the difference between the T- no’s and the T-totals.

T-no T-total

22 40 position 1

+1 + 5

23 45 position 2

+1 + 5

24 50 position 3

I saw that the difference between the T-no’s is 1 and the difference between the T-totals is 5. The first part of my formula is therefore 5T. It is 5T because that is how many numbers inside the T. I multiplied the difference between the T-total which is 5 by the T-no. Underneath is my working out:

- x 5 =110
- x 5=115
- x 5=120

I then took away the T-total numbers so that I could find the difference between the T-no and the numbers in each of the 4 squares inside the T.

110- 40 =70

115- 45 =70

120- 50 = 70

Middle

T

I then added up the G terms together to get 7G which is the final expression.

(2G +1) + (2G) + (2G +-1) + (G) = 7G

So therefore T = 5T – 7G

I will now test this formula on a 10by10 grid to see it works:

Checking:

(5 x 22) – (7 x 10) = 40

110 - 70 = 40

From 5t come 5 which is multiplied by the T-no from my first position which is 22.

7 comes from 7G and is multiplied by the grid size which is 10.

By checking out this formula I can see that it works on a 10by10 grid.

9 by 9

I can now explain the expression using algebra.

G = 9

T-2G+1 | T – 2G | T- 2G-1 |

T- G | ||

T |

I added up all the G terms from inside the T to get 7G the final expression.

(2G+1) + (2G) + (2g-1) + (G) = 7G

T = 5T – 7G

I will now test out this formula to see if it works on a 9by9 grid.

Checking:

(5 x 20) – (7 x 9) = 37

100 - 63 = 37

20 + 11 + 3 +2 + 1 = 37

By checking this formula on 9by9 grid I can see that it works.

8 by 8

I can now explain the expressions using algebra.

G = 8

T = T-no

T-2G-1 | T-2G | T-2G+1 |

T-G | ||

T |

I added up all the G terms from inside the T and I got 7g as the final expression.

(2G – 1) + (2G) + (2G +1) + (G) = 7G

So the formula for T is: T = 5T – 7G

I will now test this formula out to see if it works on an 8by8 grid.

(5 x 18) – (7 x 8) = 34

90 - 56 = 34

1 + 2 + 3 + 10 + 18 = 34 This formula is correct.

This formula works on an 8by8 grid.

ROTATION

First I rotated the 3by2 T by 90o on a 10by10 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 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

Position 1 Position 2

3 | |||

11 | 12 | 13 | |

23 | |||

4 | |||

12 | 13 | 14 | |

24 | |||

Position 3 Position 4

5 | |||

13 | 14 | 15 | |

25 | |||

6 | |||

14 | 15 | 16 | |

26 | |||

I drew a table to write drown my results

T-NUMBER | T-TOTAL |

11 | 62 |

12 | 67 |

13 | 72 |

14 | 77 |

N-8 | |||

N | N+1 | N+2 | |

N+12 |

Formula: 5N+7

Test The Formula:

I picked any number from the 10by10 grid e.g. 34 and used the formula

5 x 34 +7 = 177

For me to make sure it was right I counted all the numbers in the T to check if it totaled to 177.

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 |

I added: 34+35+36+26+46=177

And it was right so this meant that the formula works

I rotated the 3by2 T by 90o on a 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 |

Position 1 Position 2

4 | |||

11 | 12 | 13 | |

22 | |||

3 | |||

10 | 11 | 12 | |

21 | |||

Position 3 Position 4

6 | |||

13 | 14 | 15 | |

24 | |||

5 | |||

12 | 13 | 14 | |

23 | |||

T-NUMBER | T-TOTAL | ||

10 | 57 | ||

11 | 62 | ||

12 | 67 | ||

13 | 72 | ||

N-7 | |||

N | N+1 | N+2 | |

N+11 |

Formula: 5N+7

Test The Formula:

I picked any number from the 9 by 9 grid e.g. 30 and used the formula

5 x 30 +7 = 157

For me to make sure it was right I counted all the numbers in the T to check if it totaled to 157.

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

T-no T-total

21 48 position 1

1 7

22 55 position 2

1 7

23 62 position 3

I saw that the T-no is going up in 1 and the T-total is going up in 7. Therefore 7 is the first part of my formula. Once again I multiplied the difference of the T-total which is 7 by the T-no. I then took away the T-total from my answer.

This is my working out:

21 x 7 = 147 147 – 48 = 99

22 x 7 = 154 154 – 55 = 99

23 x 7 = 161 161- 62 = 99

My formula for a 5by2 T on a 9by9 grid is now 7T – 99.

I can put expressions inside the T to check if my formula is correct.

T - 20 | T - 19 | T - 18 | T-17 | T -16 |

T – 9 | ||||

T |

I saw that the centre column is going up in 9’s because of the grid size. I added all the expressions from inside the T. This is what I did:

T-total = T-20 + T – 19 + T – 18 + T – 18 + T – 16 + T – 9 + T

= 7T– 99

I will now test out this formula to find out if it is correct.

T = 51 T-total = 7 x 51 – 99 = 258

31 + 32 + 33 + 34 + 35 + 42 + 51 = 258 This formula is correct.

I can now use algebra to explain the expressions.

G = 9 the grid size

T-2G-2 | T-2G-1 | T – 2G | T-2G+1 | T -2G+2 |

T – G | ||||

T |

I added up all the G terms inside the T and I got 11G as the last part of my formula.

(2G -2) + (2G-1) + (2G) + (2G+1) + (2g+2) + (G) = 11G

So T = 7T – 11G

I will now check to see if this formula works on a 9by9 grid.

(7 x 21) – (11 x 9) = 48

147 - 99 = 48

1 + 2 + 3 + 4 + 5 + 12 + 21 = 48 This formula is correct.

This formula is correct and it works on a 9by9 grid.

I now know the general formula for a 3by2 T on any size grid and a 5by2 T on any size grid.

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

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