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  • Level: GCSE
  • Subject: Maths
  • Word count: 3569

T-totals, Relationships between the T-number and the T-total on a 9 x 9 grid.

Extracts from this document...

Introduction

Mathematics GCSE      T-totals     Alex Pavlou

1).  Relationships between the T-number and the T-total on a 9 x 9 grid.

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T-totals

T-numbers

21

42

26

67

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147

66

267

80

337

86

367

The difference in each T-shape is:  N-19  N-18  N-17

                                                                   N-9

                                                                   N

If we take the T-shape:  2  3  4

                                          12

                                          21

we can create the sum:   t=21+(21-9)+(21-19)+(21-18)+(21-17)

As there are 5 numbers in the T-shape we need 5 lots of 21, the number above 21 is 12, which is 9 less than 21, the other numbers are 2,3 and 4 which is 9 less than 21.  Therefore we arrive to the conclusion:  N-19  N-18  N-17

                                                                          N-9

                                                                           N

To prove this we use the T-shape:  61  62  63

                                                              71

                                                              80

T=80-19+80-18+80-17+80-9+80

T=337

We can do the same for the T-shape:  47  48  49

                                                                   57

                                                                   66

T=66-19+66-18+66-17+66-9+66

T=267

To find the formula of the relationship between the T-number and the T-total we use N for the T-number.

T=N+(N-9)+(N-19)+(N-18)+(N-17)

T=5N-9-54

T=5N-63

Examples of this formula are :

In the case of the T-shape:  52  53  54

                                                  64

                                                  71

T=5(71)-63

T=355-63

T=292

When we add the numbers on the calculator we get 292

In the case of the T-shape:  64  65  66

                                                   74

                                                   83

T=5(83)-63

T=425-63

T=352

When we add the numbers on the calculator we get 352

2).  Relationship between the T-number, T-total and the grid size using different size grids.

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If we use the same formula as in question 1 for the T-shape:  10  11  12

                                                                                                        19

         27

we get 79 as the T-total.  To achieve this we use the formula:

...read more.

Middle

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Centre of rotation

Rotation (degrees)

T-total

Difference compared to original (41)

11

0

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0

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90

57

+16

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180

69

+28

11

270

53

+12

To find the formula we must break the results down.

180°

As the T is upside down the equation is easy to find.

The formula is 5c+2g

We get this by:                C-7

                                          C

                               C+6  C+7  C+8

T=C+C-7+C+6+C+7+C+8

T=5N+14

T=5N+(14divided by 7) (the sum is divided by 7 because this is the grid size)

T=5N+2g

To prove this we use a grid size 4 x 4.

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3+7+10+11+12=43

Through these numbers we find the formulas

T=(5x7)+(2x4)

To double check we use the grid size 5 x 5

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2+7+11+12+13=45

T=(5x7)+(2x5)

This proves that T=5c+2g can be used to find the T-shape at 180°.

90° and 270°

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3+4+5+6+9=27

The formula can be found from:           c-2

                                                   c-1  c   c+1

                                                               c+4

Therefore if we substitute c for numbers we get:

T=(5+1)+(5+4)+(5-2)+(5-1)+5

T=27

Therefore we find the formula T=(c-1)+c+(c+1)+(c-2)+(c+4)

                                                 T=5c+2

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To prove the formula we use a 4 x4 grid:

3+7+5+6+11=32

T=5c+2

T=(5x6)+2

T=30+2

T=32

This proves that the formula T=5c+2 can be used to find the T-total of any 270° shape where c is the centre of rotation.

The relationship between the 90° and the 270°is very similar so for a 90° flip the formula must be T=5c-2.

To prove this we use the grid size 6 x 6.

3+9+15+10+11=48

When we use the formula:

T=5c-2

T=(5x10)-2

T=50-2

T=48

Therefore proving that the formula is T=5c-2 for a 90° flip and can find the T-total

Now we can form the table:

Rotation (Degrees)

Direction

Equation

0

Clockwise

T=5c-2g

90

Clockwise

T=5c+2

180

Clockwise

T=5c+2g

270

Clockwise

T=5c-2

We can also find different formulas.

180°

To find this formula we use the 3 x 3 grid.

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2+5+7+8+9=31

We can find the T-total through:  N

                                                    N+3

                                           N+5  N+6  N+7

T=2+(2+3)+(2+5)+(2+6)+(2+7)

T=31

Therefore we discover the formula.

T=N+(N+3)+(N+5)+(N+7)+(N+6)

T=5N+21

T=5N+7g

This formula is also a T-shape formula turned around.  To prove this we use the grid size 4 x 4.

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2+6+9+10+11=38

T=5N+7g

T=5(2)+7(4)

T=10+28

T=38

Therefore proving that we can use the formula T=5N+7g to find the total of any grid size.

90° and 270°

To find the formula for this we use grid size 4 x 4.

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3+5+6+7+11=32

               N-2

N  N+1  N+2

              N+6

T=N+(N+1)+(N-2)+(N+2)+(N+6)

T=5N+7

To prove this we use the grid size 7 x 7.

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5+10+11+12+19=57

By using the formula we find: T=5N+7

                                                 T=5(10)+7

                                                 T=50+7

                                                 T=57

We get the T-total.

By this we know that by using the formula T=5N+7 we can the T-total on any grid size.

To find the T-total of a 90° flip we simply turn the equation around giving us T=5N-7.

To prove this we us the grid size 5 x 5.

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2+7+12+8+9=38

Using the formula: T=5N-7

                               T=5(9)-7

                               T=45-7

                               T=38

Thus proving the formula T=5N-7 can be used on any grid size to find the T-total.  Now we can create a table:

Rotation (degrees)

Direction

Equation

0

Clockwise

T=5N-7g

90

Clockwise

T=5N-7

180

Clockwise

T=5N+7g

270

Clockwise

T=5N+7

45° and 225°

The rule for a diagonal shape is much the same as a straight shape.  First I will find the relationship on a 9 x 9 grid.

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45°

19+29+39+21+13=121

The difference in each T-shape is      N

                                            N+6    N+8

                                                  N+16

                                                        C+26

T=N+N+6+N+8+N+16+N+26

T=5N+56

On a 7 x 7 grid the formula is:

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8+16+10+4+24=62

The difference is:     N

                 N+4   N+6

                       N+12

                             N+20

T=N+N+2+N+6+N+12+N+20

T=5N+42

The overall formula of a 45° is T=5N+(7g-7)

To prove this we use grid six 10 x 10:

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11+22+33+13+4=83

T=5N+(7g-7)

T=20+(70-7)

T=20+63

T=83

This proves that the formula for a 45° angle is T=5N+(7g-7)

225°

To find the formula for this we will start with a 9 x 9 grid:

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7+17+27+25+33=109

The difference is:  N-26

                                       N-16

                                N-8          N-6

                            N      

T=N+N-8+N-16+N-26+N-6

T=5N-56

On a 7 x 7 grid the formula is:

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...read more.

Conclusion

T=25+(70+7)

T=25+77

T=102

Now we can create a table of results for rotation:

Rotation (degrees)

Direction

Equation

0

Clockwise

T=5N-7g

45

Clockwise

T=5N+(7g-7)

90

Clockwise

T=5N-7

135

Clockwise

T=5N-(7g+7)

180

Clockwise

T=5N+7g

225

Clockwise

T=5N-(7g-7)

270

Clockwise

T=5N+7

315

Clockwise

T=5N+(7g+7)

Larger T

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Now that we have worked out the formulas of normal T’s we enlarge the shape.  If we double the T-shape so the volume is 4 times bigger the grid shows the new shape.

1+2+3+4+5+6+9+10+11+12+13+14+19+20+27+28+35+36+43+44=342

The T-number is 158 as the bottom 4 numbers make up the T-number and the T-total is 342.

The difference in each number (the 4 groups of numbers) is: N-136  N-128  N-120

                                                                                                               N-64

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T=N+N-64+N-136+N-120+N-128

T=5N-448

To prove this we use:

T=5N-448

T=6x158-448

T=342

Now we must find the overall formula.  We now use a 12 x 12 grid:

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1+2+3+4+5+6+13+14+15+16+17+18+27+28+39+40+51+52+63+64=478

The T number is 230 and the T-total 478.  The difference in the T-shape is:

N-200  N-192  N-184

             N-96

               N

T=N+N-96+N-192+N-184+N-200

T=5n-672

To prove this:

T=5N-672

T=1150-672

T=478

To find the overall formula we must make a table of results

Grid size

Equation

Difference

6 x 6

T=5N-336

0

7 x 7

T=5N-392

+56

8 x 8

T=5N-448

+56

9 x 9

T=5N-524

+56

10 x 10

T=5N-560

+56

11 x 11

T=5N-616

+56

With this table we notice a pattern and that is that every the difference in each is 56.

The formula can be easily formed:

T=5N-(7gx8)

To prove this we use a 15 x 15 grid:

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1+2+3+4+5+6+16+17+18+19+20+21+33+34+48+49+63+64+78+79=580

T=5N-(7gx8)

T=1420-(105x8)

T=1420-(840)

T=580

This proves that our formula is correct.

...read more.

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