# T-shape - investigation

Extracts from this document...

Introduction

Math’s GCSE Course work

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 |

## Task

Translate the T shape to different positions on the grid.

- Investigate the relationship between the T-total and the T-number.
- Use different sized grids. Translate the T shape to different position and investigate the relationships between the T-total the T-numbers and the grid size.
- Use grids of different sizes again. Try other transformation and combinations of transformations. Investigate relationships between the T-total, the T-total, the gridsize and the transformations.

Part 1

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 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |

T-total | 37 | 87 | 137 | 187 | 237 | 287 | 337 |

From my results I would say that every time the T- number goes up by 10 the T-total goes up by 50.

I am now going to try and work out a formula for the relationships between the T-total and the T-number for anywhere on the grid. I am going to do this by working out all of the numbers in relation to N (T-number.)

1 | 2 | 3 | n-19 | n-18 | n-17 | |

11 | = | n-9 | ||||

20 | n |

Now I will add up all of the N’s and all of the numbers to hopefully get a formula.n+n-9+n-18+n-19+n-17= 5n-63.

Middle

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

1 | 2 | 3 | n-21 | n-20 | n-19 | |

12 | = | n-10 | ||||

22 | n |

62 | 63 | 64 |

73 | ||

83 |

N=83 83x5-70=345

62+63+64+73+83=345

36 | 37 | 38 |

47 | ||

57 |

N=57 57x5-70=215

57+47+37+36+38=215

My formula does work anywhere on the grid so now I am going to work out the formula for 8 by 8 and see if I can find something that relates them all together.

This is the first T-shape on an 8by 8 grid:

1 | 2 | 3 | n-17 | n-16 | n-15 | |

10 | = | n-8 | ||||

18 | n |

Adding up all the Ns and the numbers I get: 5n-56. So if N is 18 then the formula becomes 18x5-56=34 and 1+2+3+10+18=34. I shall now test the formula on other points in the grid.

41 | 42 | 43 |

50 | ||

58 |

With this shape n is 58: 58x5-56=234 and 58+50+42+41+43=234. I shall try it with one more shape just to make sure.

14 | 15 | 16 |

23 | ||

31 |

N is 31 so the formula is: 31x5-56=99 and 31+23+14+15+16=99. I have proved now that the formula works for this 8by 8 grid. I am now going to put all of my results into a table and see if there is a link for them.

8 by 8 | 9 by 9 | 10 by 10 |

5n-56 | 5n-63 | 5n-70 |

By looking at my results I can see that each time the grid size goes up by 1 the formula changes as the second number goes up by 7. And also the grid size (g) x7 =the second number. I predict that the formula will be 5n-7g. I am going to check it by working it out the same way as I did the other formulae by working out the T-shape in relation with N but this time I’m going to add G (grid size).

1 | 2 | 3 | n-2g-1 | n-2g | n-2g+1 | |

10 | = | n-g | ||||

18 | n |

Adding up all of the Ns and the Gs and the numbers I get: 5n-7g. I am now going to test this formula on the T-shapes I used before. Firstly I will test the 8 by 8 grids.

1 | 2 | 3 |

10 | ||

18 |

N=18 and G=8 so the formula is 5x22-7x8=34 and 18+10+2+1+3=34

41 | 42 | 43 |

50 | ||

58 |

N=58 G=8. 5x58-7x8=234. 58+50+42+41+43=234.

14 | 15 | 16 |

23 | ||

31 |

N=31 g=8. 31x5-7x8=99 31+23+14+15+16=99

I have now proved that the formula works on an 8 by 8 grid now ill try it on 9 by 9.

9 by 9

55 | 56 | 57 |

65 | ||

74 |

N=74 5x74-9x7=307

74+65+56+55+57=307

1 | 2 | 3 |

11 | ||

20 |

5x20-7x9=37 20+11+2+3+1=37

11 | 12 | 13 |

21 | ||

30 |

Conclusion

Normal= 5n-7g Rotated 90o= 5n+7

To see a link I need one more formula so I will work out the formula for a shape rotated by 180o.

34 | |||||

42 | |||||

49 | 50 | 51 | |||

n | |||||

n+g | |||||

n+2g-1 | n+2g | n+2g+1 |

This become equal to 5n+7g I am going to test this on 2 shapes from each grid size to check if it works.

First I will try 8 by 8.

34 | ||

42 | ||

49 | 50 | 51 |

34x5+7x8=226 and the sum of T-shape is 34+42+40+49+51=22

22 | ||

30 | ||

37 | 38 | 39 |

22x5+7x8=166 and the T-total is 22+30+38+37+39=166

Now I will try it 9 by 9.

2 | ||

11 | ||

19 | 20 | 21 |

2x5+9x7=73 and 2+11+20+21+19=73

34 | ||

43 | ||

51 | 52 | 53 |

5x34+9x7=233 and 34+43+51+52+53=233 so the formula works on 9 by 9 I will now try it on 10 by 10.

54 | ||

64 | ||

73 | 74 | 75 |

5x54+7x10=340 and 54+64+74+73+75=340

78 | ||

88 | ||

97 | 98 | 99 |

5x78+7x10=460 78+88+98+99+97=460 I have now proved that my formula works on all sized gids so now I shall compare the three formulae I have.

Normal=5n-7g rotated 90o clockwise=5n+7 rotated 180o=5n+7g

Degrees | Formulae | |

0 | 5n-7g | 5n+7(-g) |

90 | 5n+7 | 5n+7(1) |

180 | 5n+7g | 5n+7(g) |

270 | 5n-7 | 5n+7(-1) |

I predict that 270 is 5n+7. I will now check.

1 | n-g-2 | |||||

9 | 10 | 11 | = | n-2 | n-1 | n |

17 | n+g-2 |

5n-7so I was correct.

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

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month