# To investigate the relationship between the T-total and the T-number.

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Introduction

T-total Coursework

- To investigate the relationship between the T-total and the T-number.
- Use grids of different sizes. Translate the T-shape to different positions. Investigate the relationship between the T-total and the T-number and the grid size.
- Use grids of different sizes. Try other transformations and combinations of translations. Investigate relationships between the T-total, the T-number and the translations.

Relationships between T-number (x) and T-total (t) on a 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 |

From the 9 by 9 grid we can see that the first T-shape highlighted in green has a T-number of 20 which is the number located at the bottom of the T-shape and the T-total (t) which is all the numbers in the T-shape added together equals 37 (20+11+1+2+3). With the second T-shape with a T-number of 23, the T-total adds up to 52, you can see that the larger the T-number the larger the total.

If you plot all the other T-shapes and put the information into a table about the T-total and T-number you can really see a pattern and start to work out the 1st part of the formula.

T-number (x) | T-total (t) |

20 | 37 |

21 | 42 |

22 | 47 |

23 | 52 |

24 | 57 |

25 | 62 |

26 | 67 |

29 | 82 |

30 | 87 |

31 | 92 |

32 | 97 |

33 | 102 |

The table proves that the bigger the T-number is bigger the T-total is larger; the T-numbers are arranged in order of size and the T-totals gradually get larger with the T-number.

Middle

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

Translation number | T-number | T-total | Equation used t = 5x – 7g | Difference by moving up 1 |

0 | 34 | 128 | (5 x 34) – (7 x 6) | 30 |

Up1 | 28 | 98 | (5 x 28) – (7 x 6) | 30 |

Up 2 from original | 22 | 68 | (5 x 22) – (7 x 6) | 30 |

Up 3 from original | 16 | 38 | (5 x 16) – (7 x 6) |

In this there is a common trend of 30. Both 45 and 30 are multiples of 5. I worked out that if you translate the T-shape up one square you time’s the grid size (g) by 5

For a movement upwards you use this formula (-5g) where g is the grid size

To get the number you take away, for example, on a 10 by 10 grid we can predict that if you make a translation of +1 you will have to take away 50 from the original T total.

If we take a T-shape that has a T-number of 79 we can work out the T-total by the formula

t = 5x – 7g

t = (5 x 79) – (7 x 10)

t = 395 – 70

t = 325

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 |

If we move the translation up one square you should get the result of:

t = 5x – 7g

t = (5 x 79) – (7 x 10)

t = 395 – 70.

t = 325 – 5g

t = 325 – 50

t = 275.

We can now check this by using the original formula filling it in with the data for a T-shape with a T-number of 69.

t = 5x – 7g

t = (5 x 69) – ( 7 x 10)

t = 345 – 70

t = 275

This proves that you have to use -5g for an upward translation of 1. If you wanted to do a translation of up3 you would then have a formula of p(–5g) where p= number of translations. So if you wanted to do a translation of up3 you would have to put the 3 in where the p is and then work out the rest of the formula.

The full formula looks like this.

t = (5x – 7g) – p(5g) p = number of translations

x = T-number

g = grid size

t = T-total

Translations (Horizontal)

We can then use the same method to work out the formula for a translation horizontally. If we start with a grid of 9 by 9.

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 we start with the T-number of 39 this gives us a T-total of

t = 5x – 7g

t = (5 x 39) – (7 x 9)

t = 195 – 63

t= 132

And then translate it horizontally by 1 to the right this gives us a T-number of 40 and a T-total of

t = 5x – 7g

t = (5 x 40) – (7 x 9)

t = 200 – 63

t= 137

As we can see there is a difference of +5 as the T-shape moves horizontally right 1 square. If we then do more translations of +1 right from the T-shape with a T-number of 40, and put all the results into a table you can get an idea of where to start a formula.

Translation number | T-number | T-total | Equation used t = 5x – 7g | Difference by moving Right |

0 | 39 | 132 | (5 x 39) – (7 x 9) | +5 |

Right1 | 40 | 137 | (5 x 40) – (7 x 9) | +5 |

Right 2 from original | 41 | 142 | (5 x 41) – (7 x 9) | +5 |

Right 3 from original | 42 | 147 | (5 x 42) – (7 x 9) |

So if we want to write a formula for a translation of a T-shape on a five by five grid right +1 or +p we get the formula of:

t = 5x – 7g + (p5) p = number of translations

x = T-number

g = grid size

t = T-total

We now need to look at grids of different sizes to get a universal equation for a translation on the horizontal axis.

10 by 10

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 |

Conclusion

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 |

Centre of Rotation (v) | Rotation (degrees) | Direction | T-total (t) | Difference compared to original T-total |

15 | 0 | 33 | 0 | |

15 | 90 | Clockwise | 82 | +49 |

15 | 180 | Clockwise | 117 | +35 |

15 | 270 | Clockwise | 68 | -49 |

There is no obvious equation to be seen from these two sets of results but if we break them down we can see there are 3 different equations for the 3 types of rotation. These are best shown by the explanation below. Where the T total is taken as the centre of rotation.

OLIVER, I DON’T KNOW IF THIS FOLLOWING BIT SHOULD GO NEXT – IT DOESN’T SEEM TO FOLLOW ON PROPERLY.

As we know that a 180 degree flip will be a “negative” equation of the other flip, we only need to work out one kind of shape, therefore we shall work out formulas for Tc shapes.

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

Again using the “old” method we get a T-total of 27. We have to start from scratch to make a formula so we can follow the steps used to find the original formula for Ta shapes.

t = 5 + (5 – 1) + (5 – 2) + (5 + 1) + (5 + 4)

So if we substitute 5 with v we get:

t = v + (v – 1) + (v – 2) + (v + 1) + (v + 4)

t = 2v – 1 + 3v + 3

t = 5v + 2

Therefore:

On a 3 x 3 grid the formula 5v + 2 can be used to work out the T-total (t) where v is the middle number.

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 |

Oliver Lamb

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

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