# T-Totals. Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

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Introduction

Maths Course Work

T-Totals

Introduction

Looking at a grid of 9*9, with a t-shape you can see that the totals inside the T-shape.=37 E.g.1+2+3+11+21=37.

This is called a T-total =37

And T-number is the number on the T-shape =20

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 |

## Task

I have been set a task to: -

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

- Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

- Use grids of different sizes again. Try other transformation and combinations of transformations. Investigate relationship between the T-total, the T-numbers, the grid size and the transformations.

Standard 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 | 81 |

If I input the results from several T-shape into a table it look something like this

T-number | 20 | 21 | 22 | 23 | 24 | 25 | 26 | |

T-total | 37 | 42 | 47 | 52 | 57 | 62 | 67 |

From this table I can quite clearly see that there is a difference of 5 between adjacent T-numbers.

I can now a simple process of trial and improvement to obtain what I believe the formula to be I will then test my theoretical formula to see if it is correct.

Because there is a difference of 5 I believe that is a logical place to begin the formula.

Middle

71

72

73

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77

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T-number | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |

T-total | 43 | 48 | 53 | 58 | 63 | 68 | 73 |

From this table I can see that the difference between the T-total is still 5 so the first bit of the formula is the same. Now to save time I will use a shorter method to work out the total difference between the T-totals and all the other numbers in the shape. in the last formula this number was 63.

24-1=23

24-2=22

24-3=21

24-13=11

TOTAL=77

Know I presume that this number is the variable that changes when the grid size does. I now have to change the formula to allow the grid size to be times or added to another number to reach my difference of 77. So I did this and obtained the formula, g being grid number.

5t-(7g)

I will now try this on another grid size to test my formula.

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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 |

With my formula our will work out the T-total for T-number 22 on a 6*6 grid

5*22-(7*6)

=68

My formula has been proven to work on grids of a smaller size. I have now workout how to work out any upright t on any size grid just by working out how to obtain the last number in the formula.

I will now rotate the T-shape on its sun and see what part of the formula changes. I will then try to work out a formula for all rotations.

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 have to conclude that my formula will be changed in some way. There is no numbers change so I will try and change the minus sign to a plus and I will see if that works. The formula will be

5t+63 or 5t+(7*g)

I will now test this formula on another t-shape and on a different 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 |

Conclusion

T-number=12

12+11+10+19+1=53

My formula has been proven to work.

If I now rotate the shape through 180 degrees. I predict that the formula will be the same ecept I now will change the minus in the formula to a plus.

I will now test this theory

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 |

So the formula I will use is

5*10+7=57

T-number=10

10+11+12+3+21=57

This formula has been proven to be correct

I will now try to work a formula that I can use with vectors to work out the T-number and then use the formulas that I have already obtained to work out the rest.

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 |

This shape has been translated by a vector of +4.

-3

Now if my original T-number is 20 and I have moved I across right by 4 for every one square across I have added one to my original t- number. The same is for the other vector but for every one square up or down the grid number is added or subtracted.

So in this case

X- along in 1’s

Y- up r down in g

Therefore

(t+x)+(yG)

I will now test this

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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 |

This T-shape has been moved by vector +3

-4

So if I use my formula

Red T-number=32

(32+3)+(4*9)=71

Which is the Blue T-number now t work out the T-total all I have to do I use my formula from the first section.

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

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