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• Level: GCSE
• Subject: Maths
• Word count: 5822

# T-Shapes Coursework

Extracts from this document...

Introduction

T-TOTALS

An MYP Investigation

Set: 24th January

Due: 8th February

Woodside Park International School

Amrit Morokar 11k

In this investigation, I will aim to find any relationships between grid sizes and T shapes within their relative grids, I will show and explain all generalizations I can find, using the T-Number (Tn or n) (the number at the bottom of the T-Shape) and the grid width (g) to find the T-Total (Tt or t) (Total of all number added together in the T-Shape), with different grid sizes, transformations, rotations, enlargements and combinations of all three.

NB:Throughout my investigation, I will sometimes be referring to the T-total as Ttand the T-number as Tn.

We will first be conducting our investigation on a 9x9 grid, and finding our T-shapes horizontally and vertically on it. This is the blank table on which we will draw our first T-shape, and from here I will continue to show you translations horizontally until I get as many as I can, then I will go on and translate the original T-shape vertically downwards, before going further and looking at different size grids, and rotations, transformations and enlargements.

We have a 9x9 Grid, which goes from 1 – 81, on which I will be conducting part one of my investigation. I have shown this below:

 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

For the T-shapes that I will be investigating, I will find out what the T-totals and the T-numbers are, and try to find a pattern or relationship between them, recording my findings along the way, in easy-to-read data charts or tables.

This is an example T-Shape that I will be using to show you how to find the T-Total and T-Number, which I took from the above 9x9 grid.

 1 2 3 11 20

T-Total: Found, by adding up all the numbers within the T-shape. The sum of these numbers equals the T-total.

Middle

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

1st T-Shape:

T-Total

1 + 2 + 3 + 11 + 20

= 37

T-Number

= 20

2nd T-Shape

 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

2nd T-Shape:

T-Total

10 + 11 + 12 + 20 + 29

= 82

T-Number

= 29

3rd T-Shape

 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

3rd T-Shape:                                                          T-Total

19 + 20 + 21 + 29 + 38

= 127

T-Number

= 38

4th T-Shape

 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

4th T-Shape:

T-Total

28 + 29 + 30 + 38 + 47

= 172

T-Number

= 47

5th T-Shape

 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

5th T-Shape:

T-Total

37 + 38 + 39 + 47 + 56

= 217

T-Number

= 56

6th T-Shape

 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

6th T-Shape:

T-Total

46 + 47 + 48 + 56 + 65

= 262

T-Number

= 65

7th T-Shape

 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

7th T-Shape:

T-Total

55 + 56 + 57 + 65 + 74

= 307

T-Number

= 74

Recordings & Findings

I have collected data from all of the possible T-shapes vertically. I will now record my findings into a table, and see what patterns I can find, and then determine whether there is a link between the T-number and the T-total.

 T-Shape Numbers Within T-Shape T-Number T-Total 1 1, 2, 3, 11, 20 20 37 2 10, 11, 12, 20, 29 29 82 3 19, 20, 21, 29, 38 38 127 4 28, 29, 30, 38, 47 47 172 5 37, 38, 39, 47, 56 56 217 6 46, 47, 48, 56, 65 65 262 7 55, 56, 57, 65, 74 74 307

Patterns we can notice:

 T-Number T-Total 20 37 29 82 38 127 47 172 56 217 65 262 74 307

We can see clearly that there is a pattern and a relationship within these numbers. For every nine the Tn goes up, the Tt goes up by forty-five.

We can understand that the numbers go up nine times as much vertically, than they do horizontally, because there are 9 rows in the grid, and the numbers increase horizontally by one. By moving the T-Shape on the vertical, each number increases by 9 each time, because it is a 9x9 grid.

This means that in the horizontal if the T-number went up by 1 each time, in the vertical T-Shapes, the T-Number would go up by 1 x 9 = 9 each time, and we apply the same thing for the T-Total; 5 x 9 = 45 each time.

This works out correct.

Using Algebraic Numbers to Find a Formula

In the Vertical

I will express these T-Shapes in algebraic form, using the nth term, and see if I can find a pattern that applies to all of the T-shapes.

 n-19 n-18 n-17 n-9 n 19 20 21 29 38
 10 11 12 20 29 n-19 n-18 n-17 n-9 n

We can see a similar pattern that applies to both T-Shapes, using the nth term, and so we are going to use these T-shapes to help us find the formula. Remember that these are the T-shapes going vertically.

Tt =n + (n-9) + (n-18) + (n-17) + (n-19)

If we multiply out the brackets, we get:

Tt =5n – (9+18+17+19)

Or

Tt = 5n – 63

Remember: ‘n’ is the T-Number in this equation.

1

2

3

4

5

Conclusion

• Contained in the first two rows of a grid, be it any size.
• Contained in the first or the last column of a grid, be it any size.

Therefore, after coming to this conclusion, we can see that it would be impossible for any of our formulae to work, if our T-Number is situated on one of the above anomalies.

We conclude that to find the formula to any T-Shape in a grid, the T-Number of that shape must not be: a) contained in the first two rows of that grid, or b) contained in the first or last column of that grid.

Conclusion

To conclude, throughout this investigation I have analyzed T-Shapes in different grid sizes and systematically (step-by-step), using many different symbols and Geometrical language relating to the T-Shape problem, to help me find a generalised formula, for working out a relationship between T-Numbers and T-totals, taking into account grid sizes, transformations, and rotations.

Throughout the investigation, which I have now conducted, I became more and more aware of the reliability of my findings, by testing my predictions, using trial and error, and checking and re-checking, until I could confirm that a reliable pattern could be established.

At the end of the day, I discovered that no matter what, after you find one generalised formula, you can find the formula to a lot of things that are along the same lines, very easily.

This investigation was useful to me because it helped me develop my ability to find formulas using algebra, and generalise them, which personally I thought was one of my  weaknesses in Maths, but it was a skill I had to draw upon during this investigation, as it was very central to what we were looking at. It has very much improved my ability. Also, this coursework has helped my mathematical vocabulary and the use of my mathematical language overall.

Thank You.

Amrit Morokar 11K

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