# T-totals, Main objective of this project of T-totals coursework is to find an inter-relationship between the T-total and the T-number.

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Introduction

Sotirios Kopitsas

Warren Comprehensive School

Candidate number: 7037

Coursework project: Subject: T-totals and the T-number

Preface

This booklet contains information about GCSE mathematics coursework 2007-2008. Main objective of this project of T-totals coursework is to find an inter-relationship between the T-total and the T-number. The booklet uses methods and tools of analysis required in order to resolve and explore the various objectives of the coursework.

T-totals are used in GCSE mathematics coursework in order to improve the numerical quality in the student’s mind and his/her technical thinking. In this booklet, numbers including the T-number within the t-shape are used to calculate the t-total and eventually to find an overall pattern of the results of the analysis of the t-totals which involves an algebraic rule. This rule is used to prove that the T-total and the grid size follow a specific pattern as it is explained in this booklet.

Furthermore, there are other technical methods that affect the t-total structure.

Middle

Therefore, the rule is 5n-42

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 | 4 |

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 | 109 | 110 | 111 | 112 |

T-number: T-total:

- 24+17+9+10+11=71

55 55+48+40+41+42=226

65 65+58+50+51+52=276

104 104+97+89+90+91=471

Therefore, the rule is 5n-49

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 | 63 | 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 |

T-number: T-Total:

26 26+18+9+10+11=74

23 23+15+6+7+8=58

54 54+46+37+38+39=214

74 74+66+57+58+59=284

Therefore, the rule is 5n-56

Evaluation of Results

1) General rule findings

The above results are generalized in formulas (rules) for each grid size and presented in a following table:

Grid Size: | Rule |

6 | 5n-42 |

7 | 5n-49 |

8 | 5n-56 |

9 | 5n-63 |

By observation of the above table the general rule applied on the above grid sizes is then: 5n-7g where g=6 to 9.

2) Interrelation between T-totals and Translation vectors

By observation between the T-totals and translation vectors in a typical 9-size grid it appears that the T-totals are directly connected with the T-shape moving sideways and vertically in the grid as follows:

Examples in the following vectors

1 (moving T-shape by one grid horizontally to the right) the T-totals increase by 5

0 each time

-1 (moving T-shape by one grid horizontally to the left) the T-totals decrease by 5

0 each time

0 (moving T-shape by one grid vertically down) the T-totals increase by 45 each time

1

Conclusion

From the above observations a general rule may be applied as follows:

0 T-total increase by 45y

y

Similarly

x T-total increase by 5x

0

Here is an example of the method of translation vectors that I have applied:

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 |

109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 |

118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 | 126 |

T-number: T:total:

30 30+21+11+12+13=87

1 t: number=31

0 T-total=31+22+12+13+14=92

2 t-number=32

0 T-total=32+23+13+14+1=97

0 t-number=39

1 T-total=39+30+21+20+22=132

0 t-number=48

2 T-total=48+39+29+30+31=177

By using of the rule 5n-63 in the translation vectors the t-totals calculated as follows for example in the case of T-number=30:

1 5(30)-63+5(1)=87+5

0

2 5(30)-63+5(2)=87+10

0

0 5(30)-63+45(1)=87+45

1

0 5(30)-63+45(2)=87+90

2

Below we give an example of the application of the rule when T-shape moves horizontally and vertically (right across and downwards) as follows:

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 |

109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 |

If T-number=30 and assuming that the T=shape moves according to the below translation vector::

2 5(59)-63+5(2)+45(3)=232+10+135

3

The general rule for the two-dimension movement of the T-shape is the following:

5n-63+5(x)+45(y)

Page of

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

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