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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)

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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.

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

Sample 1)

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11

20

T-Number = 20

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

Sample 2)

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12

21

T-Number = 21

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

Sample 3)

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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.

...read more.

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

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

Sample 1)

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T-Number = 24

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

Sample 2)

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T-Number = 25

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

Sample 3)

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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)

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10

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21

T-Number = 10

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

Sample 2)

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T-Number = 11

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

Sample 3)

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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)

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...read more.

Conclusion

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T-Number = 3

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

Sample 3)

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T-Number = 4

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

-I arrange these results in the table.

T-Number

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T-Total

87

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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.

...read more.

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