# Maths- T-Totals

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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

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 | 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 |

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 |

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.

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

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