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  • Level: GCSE
  • Subject: Maths
  • Word count: 4177

I have been set the task to find formulas that I need to find t the "T-Total" which is explained below:

Extracts from this document...

Introduction

Introduction

I have been set the task to find formulas that I need to find t the “T-Total” which is explained below:

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Have a look at the T-shape highlighted on the nine by nine number grid above.

The total of the number in the T-shape is:  1+2+3+11+20=37

This is called the “T-Total” (which this time added up to 37).

The number at the bottom of the T-shape is called the “T-number”

The T-number for the shape is 20 this time.

        For the task that I have been set I have to translate the T-shape to different parts of the grid and work out the T-Total.  I have to find the relationships with the T-total and the T-number, and find out a formula.

        To take this task further I will use grids of different sizes. Translate the shapes to different positions.  Then I will work out the T-Total.  Then I will find the relationship between the T-Total, T-number and the grid size, and find out a formula.

        To take this task even further I will use grids of different sizes again.  Try other transformations and combinations of different transformations.  I will find the relationships between the T-Total, the T-numbers, the grid size and the transformations and find a formula.

Relationships between the T-Total and the T-number

9x9 Grid

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number.

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        The table below shows the results that I have gathered from the T-shapes on the 9x9 grid above.

T-Number

20

21

22

23

24

T-Total

37

42

47

52

57

        I will try and work out the formula by finding the difference of the numbers and producing a formula.

...read more.

Middle

14

The T-total for this shape is 28 and the T-number is 14.  I will use this to work out the formula for this grid.

28=5n-42

28=5x14-42

28=70-42

I will use this formula for another T-shape and see if it works for that T- number as well. I will test 15 for the T-number where the total is 33.  

33=5n-42

33=5x15-42

33=75-42

The formula also works for this.

6x6 Grid formula to find the T-Total = 5n-49

The General Formula for different Grid sizes

        I am going to find the relationship between the T-Total, the T-number and the grid size. I will try and find a formula to work out the T-Total from the T-number and the grid size.

The Formulas that I have gathered

Grid Size (G)

Formula

9x9

T=5n-63

8x8

T=5n-56

7x7

T=5n-49

6x6

T=5n-42

Each of these formulas are

T=5n-x

x= The number that is subtracted

In all of these formulas I have noticed that each of the subtractions made is the grid size multiplied by 7.  I.e. For a 6x6 grid use the number 6. E.g. 5n-(7x6)

Below is the formula that I have worked out for the general formula of different grid sizes.

T=5n-(7xG)

G= Grid size i.e. for a 6x6 grid use the number 6. E.g. 5n-(7x6)

Here are the results that I have gathered from the tests that I have done.

9x9

T-Number

20

21

22

23

24

T-Total

37

42

47

52

57

8x8

T-Number

18

19

20

21

22

T-Total

34

39

44

49

54

7x7

T-Number

16

17

18

19

20

T-Total

31

36

41

46

51

6x6

T-Number

14

15

16

17

T-Total

28

33

38

43

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9 grid

5x20-(7x9)= 37

The formula has worked for the 9x9 grid

8x8 grid

5x18-(7x8)=34

The formula has worked for the 8x8 grid

7x7 grid

5x16-(7x7)=31

The formula has worked for the 7x7 grid

6x6 grid

5x14-(7x6)=28

The formula has worked for the 6x6 grid

The general grid size formula to find the T-Total = 5n-(Gx7)

Relationships between the T-Total and the T-number using rotations of 180°

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 180°.

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The table below shows the results that I have gathered from the T-shapes on the 9x9 grid above.

T-Number

2

3

4

5

T-Total

73

78

83

88

        I will try and work out the formula by finding the difference of the numbers and producing a formula.

Here is an example:

21

19

20

11

2

Here is the example shown as a formula:

n+  19

n+ 18

n+  17

n+9

n

n= T-number

Below is my working out for the formula:

T= n+n+19+n+18+n+17+n+9

T= 5n+19+18+17+9

T= 5n+63

T= T-Total

n= T-number

The formula is T=5n + 63 for the 9x9 grid.

I will test this formula be using the T-shape below.

.

21

20

19

11

2

The t-total for this shape is 73 and the T-number is 2.  I will use this to work out the formula for this grid.

73=5n+63

73=5x2+63

73=10+63

I will use this formula for another T-shape and see if it works for that T- number as well. I will test 3 for the T-number where the total is 73.  

78=5n+63

785x3+63

78=15+63

The formula also works for this.

9x9 Grid formula to find the T-Total = 5n+56

8x8 Grid

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 180°.

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The table below shows the results that I have gathered from the T-shapes on the 8x8 grid above.

T-Number

2

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5

T-Total

66

71

76

81

        I will try and work out the formula by finding the difference of the numbers and producing a formula.

Here is an example:

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2

Here is the example shown as a formula:

n+

15

n+

16

n+  17

n+8

n

n= T-number

Below is my working out for the formula:

T= n+n+15+n+16+n+17+n+8

T= 5n+15+16+17+8

T= 5n+56

T= T-Total

n= T-number

The formula is T=5n + 56  for the 8x8 grid.

I will test this formula be using the T-shape below.

.

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10

2

The t-total for this shape is 66 and the T-number is 2.  I will use this to work out the formula for this grid.

66=5n+56

66=5x2+56

66=10+56

I will use this formula for another T-shape and see if it works for that T- number as well. I will test 3 for the T-number where the total is 71.  

71=5n+56

71=5x3+56

71=15+56

The formula also works for this.

 8x8 Grid formula to find the T-Total = 5n+56

The General Formula for different Grid sizes

        I am going to find the relationship between the T-Total, the T-number and the grid size when the t-shape is rotated 180°. I will try and find a formula to work out the T-Total from the T-number and the grid size.

The Formulas that I have gathered

Grid Size (G)

Formula

9x9

T=5n+63

8x8

T=5n+56

Each of these formulas are

T=5n+x

x= The number that is added on

In all of these formulas I have noticed that each of the additions made is the grid size multiplied by 7.  I.e. For an 8x8 grid use the number 8. E.g. 5n+(7x8)

Below is the formula that I have worked out for the general formula of different grid sizes.

T=5n+(7xG)

G= Grid size i.e. for a 9x9 grid use the number 9. E.g. 5n+(7x9)

Here are the results that I have gathered from the tests that I have done.

9x9

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9

T-Number

2

3

4

5

T-Total

73

78

83

88

8x8

T-Number

2

3

4

5

T-Total

66

71

76

81

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9 grid

5x2+(7x9)= 66

The formula has worked for the 9x9 grid

8x8 grid

5x2+(7x8)=73

The formula has worked for the 8x8 grid

The general grid size formula to find the 180°

T-shape T-Total = 5n+(Gx7)

Relationships between the T-Total and the T-number using rotations of 90° Clockwise

9x9 Grid

I am going to find the relationship between the T-Total and the T-number on a 9x9 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 180°.

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The table below shows the results that I have gathered from the T-shapes on the 9x9 grid above.

T-Number

10

11

12

13

T-Total

57

62

67

72

I have worked out the formula as shown below:

n-6

n+

2

n+  10

n+1

n

...read more.

Conclusion

 for the 9x9 grid.

8x8 Grid

I am going to find the relationship between the T-Total and the T-number on an 8x8 grid and try and find a formula to work out the T-Total from the T-number, when the t-shape is rotated 45° clockwise.

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The table below shows the results that I have gathered from the T-shapes on the 8x8 grid above.

T-Number

25

26

27

28

T-Total

76

81

86

91

I have worked out the formula as shown below:

n-23

n-

14

n-  5

n-7

n

The formula is T=5n-49 for the 8x8 grid.

The General Formula for different Grid sizes

        I am going to find the relationship between the T-Total, the T-number and the grid size when the t-shape is rotated 45° clockwise. I will try and find a formula to work out the T-Total from the T-number and the grid size.

The Formulas that I have gathered

Grid Size (G)

Formula

9x9

T=5n-56

8x8

T=5n-49

Each of these formulas are

T=5n-x

x= The number that is added on

In all of these formulas I have noticed that each of the additions made is the grid size minus 1 multiplied by 7.  I.e. For an 8x8 grid use the number 7. E.g. 5n-7(8-1)

Below is the formula that I have worked out for the general formula of different grid sizes.

T=5n-7(G-1)

G= Grid size i.e. for a 9x9 grid use the number 9. E.g. 5n-7(9-1)

I will test this formula that I have worked out on each of the first T-numbers on each of the grid sizes.

9x9 grid

5x28-7(9-1)= 84

The formula has worked for the 9x9 grid

8x8 grid

5x25-7(8-1)=49

The formula has worked for the 8x8 grid

The general grid size formula to find the clockwise 45°

T-shape T-Total = 5n-7(G-1)

A Table of Formulas that I have found

Translation

Formula

None

5n-(Gx7)

180°

5n+(Gx7)

90° Clockwise

5n+7

90° Anti-Clockwise

5n-7

45°Clockwise

5n-7(G-1)

image00.png

By Luke Scowcroft 10-0
...read more.

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