# T-Totals - All the things T said

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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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