# T shapes. I then looked at more of these T-Shapes from the grid in sequence and then by tabulating these results I could then work out a formula.

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Introduction

Tushyam Sonecha Maths T- Shape Coursework 10B

For this coursework I have been asked to investigate and in turn solve t e relationship between two numbers. These numbers are the T-Total and T-Number. Then further more on different sized grids and with different transformations

9 * 9 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 |

I will start by trying to find the relationship between the T-Total and T- Number in a 9 by 9 grid.

1 | 2 | 3 |

11 | ||

20 |

5 | 6 | 7 |

15 | ||

24 |

I then looked at more of these T-Shapes from the grid in sequence and then by tabulating these results I could then work out a formula.

Here is a table of my results:

T- Number | T-Total |

20 | 37 |

21 | 42 |

22 | 47 |

23 | 52 |

24 | 57 |

25 | 62 |

From this set of data it is shown that there is a change in the T-Total by 5 each times so I then times the T-Number by 5 each time and then correspond to the T-Total so here is another set of results to show this.

T- Number times 5 | T-Total |

100 | 37 |

105 | 42 |

110 | 47 |

115 | 52 |

120 | 57 |

125 | 62 |

From these results I can now predict that relationship formula is that 5 times the T-Number – 63. To help prove that this is true I will now rather than calling it the T-Number I will call it ‘n’. This then means that:

Formula= 5n – 63

n-19 | n-20 | n-21 |

n-9 | ||

n |

Middle

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1 | 2 | 3 |

12 | ||

22 |

5 | 6 | 7 |

16 | ||

26 |

As I have already shown in the previous grid through calculation, I have determined that 5n is the constant difference so that to repeat a table showing data from the T- total against the T- Number will be pointless.

5n | 110 | 115 | 120 | 125 | 130 | 135 |

T- Total | 40 | 45 | 50 | 55 | 60 | 65 |

From the above table of results I can tell that the difference is 70 therefore for a 10 y 10 grids the formula will be 5n – 70. Again to prove this I will use ‘n’.

N – 21 | N - 20 | N – 19 |

N – 10 | ||

N |

Grid Size | Formula for the T- Total |

8 * 8 | 5n - 56 |

9 * 9 | 5n – 63 |

10 * 10 | 5n - 70 |

Now for the three grids that I have already done I will tabulate the formulas.

Now, by just looking at the differences and relations in the numbers I can see a pattern and that it will always be the formula 5n – 7 * Grid Width (G).

Now I will combine both ‘n’ and ‘G’ into one formula.

N – 2G-1 | N – 2G | N – 2G-1 |

N – G | ||

N |

T-Total = 5n – 7G

Now I will look at different transformations

Translation

Now I will look at the relationship between the T- Number and T-Total as the T shape is moved onto different vectors on grid.

9 * 9 Grid Translations

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 |

N – 2G-1+4 | N – 2G+4 | N – 2G+1+4 |

N – G+4 | ||

N+4 |

I will now translate the T- shape to the vector

N – 2G-1+36 | N – 2G+36 | N – 2G+1+36 |

N – G+36 | ||

N+36 |

With the above formula I can predict that the general formula for vectors is

N + A –BG

As I did more example of vector translating I found that horizontally the number change increased only by one each time. I then found out that as the shape moved vertically, the numbers changed by 9 each time, so I called IT ‘BG’.

I then, to prove my prediction used the terms in a T-shape.

n-(2G-)+A-BG | n-(2G+9-BG) | n-(2G+1)+A-BG |

n-G+A-BG | ||

n+A-BG |

Conclusion

No I will look at the Horizontal line of reflection.

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 |

Image 1- T-Number= 29 +9 T-Total =82 +171

Reflect 1-T-Number=38 T-Total =253

Image 2 – T-Number=25 +27 T-Total=62 +261

Reflect 2 – T-Number=53 T-Total=323

As in the vertical reflection another pattern has emerged and this time for each square that the shape moves down 9 is gained onto the T-Total. With the relationships compiled from previous calculations I can also work out the T-Total.

The formula for a normal T-shape on a 9*9 grid is 5N – 63 then the formula for a reflected t shape can be worked out and adapted by just flipping it over.

n | ||

n+9 | ||

N+19 | N+18 | N+17 |

It is from this depiction that we see that the formula is 5n + 63. Now to combine both together.

5N +5(G5) +63. Like before, I will change the ‘+63’to ‘+7G’. I WILL NOW USE ‘Reflect 1 and 2@ to check my formula.

Checking:

T-Total = 52 +61 +69 + 70

= 323

Formula =5(N+GS) +7G

=5(25+ (9*3) +63

= (5*52) +63

=260+63

=323

9 * 9 Grid Rotations

Finally I am going to look at the relationship between the T-Total and T-Number as they are rotated clockwise up to 270 degrees on a 9*9 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 |

N-2G-1 | N-2G | N-2G-1 | ||||||

N-G-2 | N-9 | N-G+2 | ||||||

N-2 | N-1 | N | N1 | N+2 | ||||

N+G+2 | N+9 | N+G+2 | ||||||

N+2G-1 | N+2G | N+2G |

The formula for a 90 degrees=5N +7

The formula or 180 degrees= 5N+7G

The formula for 270 degrees= 5N-7

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

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