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• Level: GCSE
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# My investigation is to see the relationship between the T-Total and the T-Number of T-Shapes and to find a rule for it so that the T-Total can be easily found.

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Introduction

Sabrina Ul-Hasan 10R

## Introduction

My investigation is to see the relationship between the T-Total and the T-Number of T-Shapes and to find a rule for it so that the T-Total can be easily found.

First I am going to draw a 9 by 9 grid and pick T-Shapes out for example the T-Number would be the number witch is at the bottom of the T-Shape.

e.g.

 N -191 N -182 N-173 N-911 N20

The T-Total would be the number you would get by adding all the figures inside the T-Shape together.

e.g.

 N-191 N-182 N-173 N-911 N20

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

TN=1 + 2 + 3 + 11 +20 = 37

Therefore the T-Total = 37

Here is the grid and my working out for it: -

## A 9 by 9 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
 N-192 N-183 N-174 N-912 N21

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

TN= 2 + 3 + 4 +12 + 21 =

T21 = 42

 N-193 N-184 N-175 N-913 N22

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

3 + 4 + 5 + 13 + 22 =

T22=47

 N-194 N-185 N-176 N-914 N23

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

4 + 5 + 6 + 14+ 23=

T23=52

 N-195 N-186 N-177 N-915 N24

Middle

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

 N-171 N-162 N-153 N-810 N18

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

1 + 2 +N- 3 +10 +18 =

T18=34

 N-172 N-163 N-154 N-811 N19

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

2 + 3 + 4 + 11 + 19=

#### T19=39

 N-173 N-164 N-155 N-812 N20

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

3 + 4 + 5 +12 + 20=

#### T20=44

Here is the results table for the 8 by 8 grid :-

 T-Number T-Total 18 34 19 39 20 44

The T-Total increase’s by 5 so the rule for the
8 by 8 grid would be found in the similar way to the 9 by 9 grid.

To work out the rule for this grid I multiplied 5 by the T-Number and took away the T-Total.

e.g.

 1 2 3 10 18

5 x 18 = 90

90 – 34 = 56

As a result the rule for the 8 by 8 grid is :-

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

(By adding together all the n’s you get 5n)

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

(By doing the equation you get left with –56)

5n – 56

Now I am going to do a 10 by 10 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 98 89 90 91 92 93 94 95 96 97 98 99 100
 N-211 N-202 N-193 N-1012 N22

The working out for the rule

TN= N+(N-10)+(N-20)+(N-21)+(N-19)

1 + 2 + 3 + 12 + 22 =

#### T22=40

22 x 5 = 110

110 – 40 = 70

Conclusion

Results table

 Grid size Rule 8 by 8 5n -56 9 by 9 5n – 63 10 by 10 5n – 70 11 by 11 5n – 77 g by g 5n – 7g

As you can see from the results the rule increases by 7 in each grid. Also the last figure is obtained by multiplying the grid size by 7.

For example:-

The rule for the 8 by 8 grid is 5n – 56

7 x 8 = 56

###### Therefore to show this in algebraic form it would be:

7g

 N - (g x 2) +1 N-(g x 2) N - (g x 2) -1 N-g N

Were as the algebraic equation for all of it would be:

5n – 7g

Now I am going to test this rule to see if it works on any grid size. I will do this on a 12 by 12 grid.

12 by 12 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
 N-251 N-242 N-233 N-1214 N26

5t – 7g is my rule here is the working out

5 x 26 =130

7 x 12 =84

Therefore I think the rule for the 12 by 12 grid will be :

130 – 84

 N-251 N-242 N-233 N-1214 N26

Now I am going to see if I am right

TN=N+(N-12)+(N-24)+(N-23)+(N-25)

TN= 1+ 2+ 3 + 14 + 26=46

5 x 26 = 130

130 – 46 = 84

Therefore this proves that the rule works on any size grid.

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

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