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  • Level: GCSE
  • Subject: Maths
  • Word count: 2022

In this investigation, I am finding out about relationships between T-shapes and the grids they are on.

Extracts from this document...

Introduction

Rizwanali Pabani

28/04/07

T-Total

In this investigation, I am finding out about relationships between T-shapes and the grids they are on.

image00.png

Introduction

Look at the T-shape drawn on the 9 by 9 grid.

The total of the numbers inside the T-shape is

1 + 2 + 3 + 11 + 20 = 37

This is called the T-Total. (Tt)

The number at the bottom of the T-shape is called the

T-number (n).

1.Investigate the relationship between the T-total and the T-number.

Looking at the T-shape drawn on the 9 by 9 grid drawn in the introduction.

The T-number (n) of this T-shape is 20.

The T-total (Tt) of this T-shape is

1 + 2 + 3 + 11 + 20 = 37

Find the difference in the numbers of the T-shape. Take the T-number to be n. We can find that the first difference is 9 and the second difference is 18 from n. the top line, there is a consecutive difference going down from left to right. As we are working with a grid that has a width of 9, the differences of the numbers as we go up in the T-shape will obviously be 9. Therefore, any T-shape used in a grid with a width of 9 should relate to the T on the right here.

image03.pngimage01.png

The Tt of this T-shape is shown here using n as the T-number. We find the Tt of a T-shape by adding all the numbers inside the T.

...read more.

Middle

      =   (5 * 56) – (7 * 10)

      =   280  -  70

      =   210

Tt   =   35 + 36 + 37 +  46 + 56

      =   210   It works!

To show this formula works, I am going to use a T-shape in a 11 by 11 grid.image06.png

Look at this T-shape. The T-number is 63. The width of the grid is 11.

Substitute these numbers into formula 2.

image07.png

Tt   =   5n – 7x

      =   (5 * 63) - (7 * 11)

      =   315  -  77

      =   238

Tt   =   63+ 52 + 40 +  41 + 42

      =   238   It works!

This shows us that formula 2 works in finding out the T-total of a T-shape when one knows the width of the grid and the T-number of the T-shape.

Formula 2:

Tt   =   5n  -  7x  

3. Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-total, the T-numbers, the grid size and the transformations.

I am now going to try and find relationships with grid sizes and the 180 degrees rotation of the T-shape with other grid sizes and then compare them.

Starting with a Grid 5 by 5.

image08.png

n   =   12

Red Tt   =   12 + 7 + 1 + 2 + 3image09.png

              =   25

180 degree clockwise rotation:image10.png

Black Tt   =   12 + 17 + 21 + 22 + 23

                 =    95

Difference between the two Tt is: 70

Now I’m going to see what happens on a 6 by 6 width Grid

n   =   14image11.png

Red Tt   =   14 + 8 + 1 + 2 + 3

              =   28image12.png

180 degree clockwise rotation:image13.png

Black Tt   =   14 + 20 + 25 + 26 + 27

                 =   112

Difference of 84

Now I’m going to see what happens on a 7 by 7 Grid

image14.png

n   =   16

Red Tt   =   16 + 9 + 1 + 2 + 3image15.png

              =   31

180 degree clockwise rotation:image16.png

Black Tt   =   16 + 23 + 29 + 30  + 31

                 =   129

Difference of 130

I

...read more.

Conclusion

Consequently I can derive a formula from this. I can use formula 2 to find out the Tt of the first T-shape then add 15 to this to get the second  Tt after reflection. This works only using this line of symmetry. The 15 in this formula is derived because as I have shown, the numbers in the T-shape go up 15 from T-shape to the reflected T-shape:  image17.pngimage25.png

Formula 6:

Tt of reflected T  =  Formula 2 + 15

                            =  5n – 7x + 15

Now I have to show that formula 6 works on another grid.

image06.png

Black Tt   =   26 + 27 + 28 + 38 + 49image22.png

                 =   168image27.png

Reflected Tt  =  5n – 7x + 15

                      =  (5*49) – (7*11) + 15

                      =  183

Checkimage28.png

Red Tt   =   29 + 30 + 31 + 41 + 52

              =   183    It works!

In conclusion I have found that formula 6 can be used to find the Tt of a reflected T-shape when we know the width of the grid and the T-number of this T-shape.

Furthermore, I think that formula 2 is a very important formula. Tt  =  5n + 7x. This shows us that any T-shape which is upright can have its T-total worked out when we know the T-number and the width of the grid it is in. Then to work out transformations of this T-shape, other formula’s can be produced by adding or subtracting something from this formula. E.g. formula 6, where 15 is added to this formula to signify that the Tt goes up 15 in the reflection of the T-shape.

I have shown that the T-total of a T-shape is dependant on the width of the grid and it’s T-number.

...read more.

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