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• Level: GCSE
• Subject: Maths
• Word count: 3725

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

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Introduction

T-total Coursework

1. To investigate the relationship between the T-total and the T-number.
2. 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.
3. 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 usedt = 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

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