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  • Level: GCSE
  • Subject: Maths
  • Word count: 4517

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

Extracts from this document...

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.

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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

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42

22

47

23

52

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57

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62

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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.  From this we are able to work out some parts of the formula for a 9 by 9 grid. Taking the T-number of 20 as an example, we can say that the T-total is gained by:

t = (20 - 19) + (20 - 18) + (20 - 17) + (20 - 9) + (20 - 0) = 37

As there are 5 numbers in each T-shape, we have to use five lots of twenty.

...read more.

Middle

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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.

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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

...read more.

Conclusion

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Again using the old method we get a T-total of the rotated shape is 92.  We have to start from scratch to make a formula so we can follow the steps used to find the original formula for the T shape.

t = (17 + 0) + (17 + 1) + (17 + 2) + (17 – 3) + (17 + 7)

So if we substitute 17 with r we get:

t = r + (r + 1) + (r + 2) + (r + 1) + (r + 4)

t = 5r + 8

t = 5r + 8

On any grid the formula 5r + 8 can be used to work out the T-total (t) where r is the T number. The formula t = 5r + 8 can be used to find the T-total (t) of any T shape that has been rotated 90 degrees clockwise where r is the T number.

Following the 180 degree flip rule we can predict that for T shapes which has been rotated by 270 degrees the formula is t = 5r – 8. As it is a flip of the T shape with a rotation of 90 degrees clockwise.    

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Working this out by hand, we get a T-total of 63 (7 + 12 + 13 + 14 + 17) for this T-shape.  If we use the formula, we get:

t = (5 x 14) + 8

t = 70 + 8

t = 78

The formula t = 5x + 8 can be used to find the T-total (t) of any shape rotated 90 degrees clock wise on any sized grid, where x is the T number. The formula t = 5x – 7 can be used to find the T-total (t) of any shape rotated 270 degrees clock wise on any sized grid, where x is the T-number. Therefore we can state from a standard T-shape position the following equations can be used to generate the T-total from x.

These are all of the formulas used to find any rotation.

Rotation (degrees)

Direction

Ending (y)

0

Clockwise

-7g

90

Clockwise

+8

180

Clockwise

+7g

270

Clockwise

-8

0

Anti-clockwise

-7g

90

Anti-clockwise

-8

180

Anti-clockwise

+7g

270

Anti-clockwise

+8

I have also rotated and flipped the formulas so that we can have formulas for an anti clockwise rotation.

Oliver Lamb

...read more.

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