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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:

1.        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

Rule finding:

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)

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