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

To 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)

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37

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42

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47

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52

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57

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62

26

67

29

82

30

87

31

92

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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.

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Middle

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Translation number

T-number

T-total

Equation used

t = 5x – 7g

Difference by moving up 1

0

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128

(5 x 34) – (7 x 6)

30

Up1

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98

(5 x 28) – (7 x 6)

30

Up 2 from original

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68

(5 x 22) – (7 x 6)

30

Up 3 from original

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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  

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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

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132

(5 x 39) – (7 x 9)

+5

Right1

40

137

(5 x 40) – (7 x 9)

+5

Right 2 from original

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142

(5 x 41) – (7 x 9)

+5

Right 3 from original

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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

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Conclusion

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Centre of Rotation (v)

Rotation (degrees)

Direction

T-total (t)

Difference compared to original T-total

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Clockwise

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Clockwise

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+35

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270

Clockwise

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-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.

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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.

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Oliver Lamb

...read more.

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