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  • Level: GCSE
  • Subject: Maths
  • Word count: 1852

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.

Extracts from this document...

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

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image11.pngimage01.png

image20.png

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

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image58.png

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image38.png

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6image43.png

7image49.png

image58.png

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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

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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-Totalimage56.png

100

37image57.pngimage02.png

105

42

110

47

115

52

120

57

125

62

image03.png

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

image58.png

n-9

n

...read more.

Middle

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image09.png

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image12.png

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

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T- Total

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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

image58.png

N – 10

N

image13.png

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.image14.png

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

image58.png

N – G

N

image15.png

T-Total = 5n – 7Gimage18.pngimage17.pngimage16.pngimage19.png

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

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image11.pngimage01.png

N – 2G-1+4

N – 2G+4

N – 2G+1+4

image58.png

N – G+4

N+4

image21.png

image63.png

I will now translate the T- shape to the vector image59.png

N – 2G-1+36

N – 2G+36

N – 2G+1+36

image58.png

N – G+36

N+36

image21.png

image60.png

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

N + A –BGimage23.pngimage22.pngimage25.pngimage24.pngimage27.pngimage26.png

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

...read more.

Conclusion

No I will look at the Horizontal line of reflection.

image42.pngimage41.png

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image34.pngimage45.pngimage46.pngimage44.pngimage42.pngimage42.pngimage47.pngimage48.pngimage37.pngimage36.png

image50.pngimage50.png

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    +261image50.pngimage50.png

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.

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image51.png

image52.pngimage53.pngimage54.pngimage55.png

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

Page  of

...read more.

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