# T-Totals. To figure out an equation for different grid sizes, I have to find the relationship between grid sizes and the T total. I will now let S= Grid Size.

Extracts from this document...

Introduction

Page

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 |

T-Totals

Part 1

By using algebra I can find the equation for T Total for grid size 9

T Total= n-19+n-18+n-17+n-9+n= 5n-63

T Total= 5n-63 is the relationship between the T total and the T number. This means for example that if the T number is 20 my formula predicts a T Total of 5× 20– 63= 37 which agrees with my earlier calculations.

Part 2

Equation for different grid sizes

To figure out an equation for different grid sizes, I have to find the relationship between grid sizes and the T total. I will now let S= Grid Size.

I get this T

T Total= n+n-S+n-2S+n-2S+1+n+2S-1= 5n-7S

This means that the equation is T total= 5n-7S where S is the grid size so if for example the T number is 24 and the grid size is 11 then the T total will be (5× 24– 11× 7)= 43. I can check whether this works.

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 |

The totals are the same so my formula seems to work.

Translation

I will now try and figure out the relationship between translation and the T Total.

Middle

79

The resulting T total is 22+23+24+32+41= 142.

Using my equation for a translated t total, New T total= T+5x+5(yS), for a T total of 37, a grid size of 9 and translation of then the equation will be T=37+15+90= 142 which the same as the T total on the grid, so my formula works.

Another way of doing this is by representing the number in each square by a letter.

Again I will use to represent the vector movement, S for the grid size T for the new T Total.

a+x+yS+ b+x+yS+ c+x+yS+ d+x+yS+ e+x+yS=

a+b+c+d+e+5x+5yS= New T Total

a+b+c+d+e is equal to the old T Total which I will call T so the formula is

T+5x+5yS= New T Total

This gives the same equation as in the previous result so I know works as I showed it in the previous result.

Rotation

I will now try and work out the relationship between the T number and the T total when it is rotated about the original T number.

n= T number

S= Grid size

T= New T total

I will try a 90º clockwise rotation

n+n+1+n+2+n+2-S+n+2+s= 5n+7

T=5n+7

This is the T for a 180º clockwise rotation

n+n+S+n+2S-1+n+2S+n+2S+1=7n+7S

T= 7n+7S

This is the T for a 270º clockwise rotation

n+n-1+n-2+n-2-S+n-2+S= 7n-7

I can show that these formulas work by drawing a table.

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 |

The formula for the 90º clockwise rotation is T= 5n+7 so the T number is 20 so the T total would be (5×20) +7= 107. From the table I can see that the T total is 25+26+27+17+37= 107 so my formula works.

The formula for the 180º clockwise rotation is T= 7n+7S so the T number is 20 so the T total would be (7×20) + (7×10)= 210. From the table I can see that the T total is 25+35+45+46+44= 210 so my formula works.

The formula for the 270º rotation clockwise is T= 5n-7 so the T number is 20 so the T total would be (5×20) -7= 93. From the table I can see that the T total is 25+24+23+13+33= 107 so my formula works.

I will now try and figure out the relationship between the T number and the T total in a 90º clockwise rotation about a point on the table.

T number= n

New T total= T

Grid Size= S

Horizontal distance from the T number

To the point of rotation= x

Vertical distance from the T number

To the point of rotation = y

= Distance from the point of rotation

= Rotation Point

= The T number

To figure it out I would have to

I can prove this by drawing a 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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

Conclusion

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

I will now make a T for a rotation of 270º clockwise in the same format as the last one

n+yS+x+xS-y+n+yS+x+xS-y-1+n+yS+x+xS-y-2+n+yS+x+xS-y-2-S+ n+yS+x+xS-y-2+S=

5n+5yS+5x+5xS-5y-7= T

T= 5(n+yS+x+xS-y)-7

To show that it works, I made an example.

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 |

Limits on T

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 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 |

This student written piece of work is one of many that can be found in our GCSE T-Total section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month