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Maths- T-Totals

Extracts from this document...

Introduction

Contents

  • Aim and Method
  • Grid 1: 9 by 9
  • Across the Grid
  • Gown the grid Diagonally
  • Down the grid
  • Across the Grid:
  • Grid 2: 5 by 5
  • Grid 3: 6 by 6
  • Grid 4: 7 by 7
  • Grid 5: 10 by 10
  • Grid 6: 11 by 11
  • Extension: Transformations Grid 7: 9 by 9 (Going across)
  • T at 180 degrees
  • T at 90 degrees
  • T at 270 degrees
  • Conclusion

Aim:

I will investigate the relationship between the T-total (all the numbers in the T added up) and T-number (the number at the end of the T), using grids of different sizes to translate the t-shape to different positions within the grid. I will try different combinations of transformations later within the coursework.

Method:

Part 1

Investigate the relationship between the T-total and the T-number.

Part 2  

Use grids of different sizes. Translate the T-shape to different positions. Again investigate the relationship within the different sizes of the grids.

Part 3

Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-Total, and T-Numbers on the grid of different sizes.        

Grid 1: 9 by 9 – Across (      )

In the T below, I noticed that the number at the end of the T (the T-Number) was 20. I will now go across this grid writing down the T-Number on one side and the T-Total in the other aswell showing my working by putting all the numbers within that T.I will also write the trend of the T-Number and T-Total in the table and express I in the nth term after I have acquired results from various points in the 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

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

20

1+2+3+11+20= 37

+1

+5

21

2+3+4+12+21= 42

+1

+5

22

3+4+5+13+22= 47

+1

+5

23

4+5+6+14+23= 52

+1

+5

24

5+6+7+15+25= 57

+1

+5

25

6+7+8+16+25= 62

+1

+5

26

7+8+9+17+26= 67

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

47

28+29+30+38+47= 172

+1

+5

48

29+30+31+39+48= 177

+1

+5

49

30+31+32+40+49= 182

+1

+5

50

31+32+33+41+50= 187

+1

+5

51

32+33+34+42+51= 192

+1

+5

52

33+34+35+43+52= 197

+1

+5

53

34+35+36+43+53= 202

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

74

55+56+57+65+74= 307

+1

+5

75

56+57+58+66+75= 312

+1

+5

76

57+58+59+67+76= 317

+1

+5

77

58+59+60+68+77= 322

+1

+5

78

59+60+61+69+78= 327

+1

+5

79

60+61+62+70+79= 332

+1

+5

80

61+62+63+71+80= 337

+1

+5

Formula Expressed in the nth term

...read more.

Middle

16

3+4+5+10+16= 38

+1

+5

17

4+5+6+11+17= 43

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

26

13+14+15+20+26= 88

+1

+5

27

14+15+16+21+27= 93

+1

+5

28

15+16+17+22+28= 98

+1

+5

29

16+17+18+23+29= 103

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

32

19+20+21+26+32= 118

+1

+5

33

20+21+22+27+33= 123

+1

+5

34

21+22+23+28+34= 128

+1

+5

35

22+23+24+29+35= 133

+1

+5

Formula Expressed in the nth term

As the tables show every time the T-Number increases by 1 the T-Total increases by 5. I will express in the nth term by using these common differences and start on the basis of ‘if n=1’ but change accordingly if appropriate.

At the T-Number, the common difference is 1.

n=

1

2

3

4

T=

14

15

16

17

For example take 14 and 15 subtract them (15 -14) and you get 1.

Therefore the formula of this is 1n ± C = T-Number/ n ± C = 14, 14 -1=13.

Therefore nth term= n+13=14

At the T-Total, the common difference is 5.

n=

1

2

3

4

T=

28

33

38

43

For example take 28 and 33 subtract them (33-28) and you get 5. Therefore the formula of this is 5n ± C = T-Total/ 5n ± C = 28, (if n=14, the T- Number) 14 x 5= 70 – 28=42

Therefore nth term=5n-42=28.

Across- Grid 4: 7 by 7

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

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

16

1+2+3+9+16=31

+1

+5

17

2+3+4+10+17= 36

+1

+5

18

3+4+5+10+18= 41

+1

+5

19

4+5+6+11+19= 46

+1

+5

20

5+6+7+13+20= 51

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

30

15+16+17+23+30= 101

+1

+5

31

16+17+18+24+31= 106

+1

+5

32

17+18+19+25+32= 111

+1

+5

33

18+19+20+26+33= 116

+1

+5

34

19+20+21+27+34= 122

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

44

29+30+31+37+44= 32

+1

+5

45

30+31+32+38+45= 37

+1

+5

46

32+33+34+39+46= 42

+1

+5

47

34+35+36+40+47= 47

+1

+5

48

35+36+37+41+48= 52

+1

+5

Formula Expressed in the nth term

As the tables show every time the T-Number increases by 1, the T-Total increases by 5. I will express in the nth term by using these common differences and start on the basis of ‘if n=1’ but change accordingly if appropriate.

n=

1

2

3

4

T=

16

17

18

19

At the T-Number, the common difference is 1.

For example take 16 and 17 subtract them (17 -16) and you get 1.

Therefore the formula of this is 1n ± C = T-Number/ n ± C = 16, 16 -1=15.

Therefore nth term= n+15=16.

At the T-Total, the common difference is 5.

n=

1

2

3

4

T=

31

36

41

46

For example take 21 and 36 subtract them (31-36) and you get 5. Therefore the formula of this is 5n ± C = T-Total/ 5n ± C = 31, (if n=16 (T-Number) ) 16 x 5= 80,  80 -16= 64

Therefore nth term= 5n-64=22

Across - Grid 5: 10 by 10

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

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43

44

45

46

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48

49

50

51

52

53

54

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56

57

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64

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71

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79

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87

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90

91

92

93

94

95

96

97

98

99

100

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

22

1+2+3+12+22= 40

+1

+5

23

2+3+4+13+23= 45

+1

+5

24

3+4+5+14+24= 50

+1

+5

25

4+5+6+15+25= 55

+1

+5

26

5+6+7+16+26= 60

+1

+5

27

6+7+8+17+27= 65

+1

+5

28

7+8+9+18+28= 70

+1

+5

29

8+9+10+19+29= 75

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

52

31+32+33+42+52= 130

+1

+5

53

32+33+34+43+53= 135

+1

+5

54

33+34+35+44+54= 140

+1

+5

55

34+35+36+45+55= 145

+1

+5

56

35+36+37+46+56= 150

+1

+5

57

36+37+38+47+57= 155

+1

+5

58

37+38+39+48+58= 160

+1

+5

59

38+39+40+49+59= 165

+1

+5

T-Number

T-Total

Trend of         T-Number

Trend of       T-Total

92

71+72+73+82+92= 130

+1

+5

93

72+73+74+83+93= 135

+1

+5

94

73+74+75+84+94= 140

+1

+5

95

74+75+76+85+95= 145

+1

+5

96

75+76+77+86+96= 150

+1

+5

97

76+77+78+87+97= 155

+1

+5

98

77+78+79+88+98= 160

+1

+5

99

78+79+80+89+99= 165

+1

+5

Formula Expressed in the nth term

As the tables show every time the T-Number increases by 1, the T-Total increases by 5. I will express in the nth term by using these common differences and start on the basis of ‘if n=1’ but change accordingly if appropriate.

At the T-Number, the common difference is 1.

n=

1

2

3

4

T=

22

23

24

25

For example take 22 and 23 subtract them (22 -23) and you get 1.

Therefore the formula of this is 1n ± C = T-Number/ n ± C = 22, 22 - 1=21

Therefore nth term= n+21=22

At the T-Total, the common difference is 5.

n=

1

2

3

4

T=

40

45

50

55

...read more.

Conclusion

  • For a 9 by 9 grid – T-number= n+19=20, T-Total=5n-63=37
  • For a 5 by 5 grid – T-number= n+16=17, T-Total=5n-10=75
  • For a 6 by 6 grid – T-number= n+13=14, T-Total=5n-42=28
  • For a 7 by 7 grid – T-number= n+15=16, T-Total=5n-64=22
  • For a 10 by 10 grid – T-number= n+21=22, T-Total=5n-70=40
  • For a 11 by 11 grid – T-number= n+23=24, T-Total=5n-79=41
  •  For a 180 degree transformation at a 9 by 9 grid - T-number=n+9=10, T-Total=5n+7=57
  • For a 90 degree transformation at a 9 by 9 grid - T-number= n+1=2, T-Total=5n+63=73
  • For a 270 degree transformation at a 9 by 9 grid – T-number=n+11=12, T-Total= 5n-7=53

As you can see this supports my prediction showing all T-Numbers include 1n/n and T-totals include 5n.

From my results, I can also predict that T-Shapes going down a grid would have a nth term including 9n for T-Numbers and 45n for T-totals. This is shown in my results by: the nth term for T-Numbers going down a 9 by 9 grid is 9n+11=20 and the nth term for a T-Total going down a 9 by 9 grid is 45n-8=37.

Also going diagonally down a grid I can predict that T-Shapes have a nth term including 10n for T-Numbers and 50n for T-Totals. This is shown by in my results by: the nth term for T-Numbers going diagonally down a 9 by 9 grid is 10n+10=20 and the nth term for a T-Total going down a 9 by 9 grid is 50n -13=37.  

...read more.

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