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• Level: GCSE
• Subject: Maths
• Word count: 2322

# T-Totals - All the things T said

Extracts from this document...

Introduction

Chan-woo Kim

Mathematics coursework

T-Totals - All the things T said

-Word definition

*T-Total

eg)

 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

Looking at T-shape drawn on 5 by 5 number grid.

The sum of the numbers inside the shape is 1+2+3+7+12=25

25 is called 'T-Total'.

*T-Number

The number at the bottom of the shape (the number which is shown by the shaded square) is called 'T-Number'.

The T-Number for the T-shape above is 12.

-The aim of this project is to investigate the relationship between T-Total and T-Number.

-I use 9 by 9 number grid to investigate the relationship.

 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

-Then I pick some samples to find what I am going to find.

Sample 1)

 1 2 3 11 20

T-Number = 20

T-Total = 1+2+3+11+20 = 37

Sample 2)

 2 3 4 12 21

T-Number = 21

T-Total = 2+3+4+12+21 = 42

Sample 3)

 3 4 5 13 22

T-Number = 22

T-Total = 3+4+5+13+22 = 47

-I arrange these samples in the table.

 T-Number 20 21 22 T-Total 37 42 47

-As we notice, when T-Number increases by 1, its T-Total also increases by 5. It shows that there is a certain pattern.

-If I define T-Number = N

 N-19 N-18 N-17 N-9 N

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

-To check whether it is right, I substitute N for 20.

5N-63 = 5*20-63 = 100-63 = 37.

Middle

T-Number

22

23

24

T-Total

40

45

50

-As we can see, when T-Number increases by 1, T-Total also increases by 5. It is obvious that there is a certain pattern.

-If I define T-Number = N

 N-21 N-20 N-19 N-10 N

T-Total = N+N-10+N-20+N-21+N-19 = 5N-70 = 5N-70

-To check whether it is right, it is easy and quick that I substitute N for 22.

5N-70 = 5*22-70 = 110-70 = 40

-When T-Number is 22, T-total is 40 and equation fits. Consequently the relationship between T-Number and T-Total is :

T = 5N-70

 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

-Now I use 11 by 11 number grid to find the relationship between T-Number and T-Total.

Sample 1)

 1 2 3 13 24

T-Number = 24

T-Total = 1+2+3+13+24 = 43

Sample 2)

 2 3 4 14 25

T-Number = 25

T-Total = 2+3+4+14+25 = 48

Sample 3)

 3 4 5 15 26

T-Number = 26

T-Total = 3+4+5+15+26 = 53

-I arrange the samples in the table.

 T-Number 24 25 26 T-Total 43 48 53

-Again we notice, when T-Number increases by 1, T-Total also increases by 5. Now it is certain that there is a pattern.

-If I define T-Number = N

 N-23 N-22 N-21 N-11 N

T-Total = N+N-11+N-22+N-23+N-21 = 5N-77

-Now I am going to check whether it it true. To do it, I substitute N for 24.

5N-77 = 5*24-77 = 120-77 = 43

-When T-Number is 24, T-Total is 43 and the equation does make 43 when N is 24. Therefore the relationship between T-Number and T-Total is :

T = 5N-77

-Now, I arrange these 3 equations that I have found from my the investigation.

 Grid 9*9 10*10 11*11 Equation 5N-63 5N-70 5N-77

-Look at the numbers -63, -70, and -77. They increase by -7 when the size of grid increases by 1.

-From these numbers, we can find a certain interesting point.

-63 = -7*9

-70 = -7*10

-77 = -7*11

-When we look at these results, we can notice that - 7 is constant and the number which we use to multiply it is the same as grid number. For example, -63 is from the equation of 9 by 9 grid and - 63 = -7*9. The same relationship is found in all the examples and therefore the relationship between T-Number, T-Total, and number grid is :

T-Number = N, T-Total = T, Grid =G

T = 5N-7G

-Now, I turn T-shape 90 degree clockwise and investigate the relationship between T-Number and T-Total.

 * *

Original                   Rotated one

* = T-Number

-I use 9 by 9 grid to find the relationship.

Sample 1)

 3 10 11 12 21

T-Number = 10

T-Total = 10+11+12+3+21 = 57

Sample 2)

 4 11 12 13 22

T-Number = 11

T-Total = 11+12+13+4+22 = 62

Sample 3)

 5 12 13 14 23

T-Number = 12

T-Total = 12+13+14+5+23 = 67

-I arrange these samples in the table.

 T-Number 10 11 12 T-Total 57 62 67

-It shows that there is a certain pattern as when T-Number increases by 1, T-Total increases by 5.

-If I define T-Number = N

 N-7 N N+1 N+2 N+11

T-Total = N+N+1+N+2+N+11+N-7 = 5N+7

-I check whether it is a suitable equation, by substituting N for 10

5N+7 = 5*10+7 = 57

-As I checked above, it is the right equation. Therefore the relationship between T-Number and T-Total is :

T = 5N+7

-I Change the grid to 10 by 10 and investigate the relationship.

Sample 1)

 3 11 12 13 23

Conclusion

 3 14 24 25 26

T-Number = 3

T-Total = 3+14+25+24+26 = 92

Sample 3)

 4 15 25 26 27

T-Number = 4

T-Total = 4+15+26+25+27 = 97

-I arrange these results in the table.

 T-Number 2 3 4 T-Total 87 92 97

-As we can notice, it is possible that there is a certain pattern because when T-Number increases by 1, T-Total increases by 5.

-If I define T-Number = N

 N N+11 N+21 N+22 N+23

T-Total = N+N+11+N+22+N+21+N+23 = 5N+77

-To check whether it is the right equation, I substitute N for 2.

5N+77 = 5*2+77 = 87

-The result above shows that this equation does make 87 when N is 2 so that it is the right equation. Therefore the relationshio between T-Number and T-Total is :

T = 5N+77

-Now, I arrange these 3 equations that I have found from the investigation.

 Grid 9*9 10*10 11*11 Equation 5N+63 5N+70 5N+77

-Look at the numbers 63, 70, 77. They increase by 7 when the size of grid increases by 1.

-From these numbers, we can find an interesting point.

63 = 7*9

70 = 7*10

77 = 7*11

-When we look at these results, we can notice that 7 is constant and the number which we use to multiply it is the same as the grid number. For example, 63 is from the equation of the 9 by 9 grid and 63 = 7*9. The same relationship is found in all the examples and therefore the relationship between T-Number, T-Total, and number grid is :

T-Number = N, T-Total = T, Grid = G

T = 5N+7G

-What I have found from the investigation is that T-Total increases by a certain algebraic pattern linked with T-Number and Grid number and algebraic pattern is changed when number grid is changed and/or T-shape is rotated. Therefore the relationship between T-Number and T-Total is variable.

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

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