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T-Totals maths coursework.

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Introduction

T-Totals maths coursework

I will be investigating the relationship between the t-total and the t-number of a t-shape.

image04.png

The t – total is the sum of all the figures inside the T.

The t-number is the bottom of the T even when rotated in different positions as shown on the right.

I will begin by putting some T’s on a grid with a width of 9.

The first t-number I found was 20 with a T-Total of 37

The second t-number I found was 21 with a T-total of 42

The third t-number I found was 22 with a T-total of 47.

image05.pngimage16.pngimage25.png

I shall now try a possible structure for a T with a grid width of 9 on the right and test it with a T-number of 23 below.image26.png

image27.png

When checking the structure it works. I will now try to come up with a general formula working with the t-structure.

image28.png

This sums up the basic structure of a T-shape. I will put it into a general formula.

T-total = T-2g-1+T-2g+T-2g+1+T-g+1

        = 5T – 7g

The g is the grid width; I will try the formula on different width grids to see whether the formula still applies.

5 wide Grid.

T-total = 25

T-Number = 12

...read more.

Middle

T-Number = 10

T-Total = 57

T-Number = 11

T-Total = 62image00.png

T-Number = 12

T-Total =  67

I will look at the structure of the t-shape as shown on the right to come up with a basic formula. image31.png

Using the t-structure on the right I can work out the t-total.

T-Total = T+T+1+T+2+T+2-g+T+2+g

         = 5t+7

When this formula is tested on all the t-shapes above it works.

I will now see if my formula works on different sized grids.

8 wide grid

T-total = 52

T-number = 9

Using formula T-Total = 5x9+7

7 wide grid                         = 52

T-Total = 47

T-Number = 8

Using formula T-Total = 5x8+7

                         = 47

So the formula also holds for different sized grids.

I will now begin working out a relationship/formula for a t-shape rotated on its side as shown on the right.image06.png

I began by placing the t-shapes on a 9 wide grid and gathered some results.

T-Number = 12

T-Total = 53

T-Number = 13

T-Total = 58

T-Number = 14

T-Total =  63

Before getting to the structure of the rotated t-shape I believe the t-total will be 5t-7 as the case was before that when it was rotated 180 degrees the values became positive or negative.

image07.png

Here is the t-structure for the rotated t-shape.

So the T-Total = T-2-g+T-2+T-1+T=T-2+g

                = 5t – 7

This is as I expected.

...read more.

Conclusion

image18.png

I will test this formula with a vector b=-3.

Using formula T-Total = (5x20)-(7x9) +5(3x9)

                                      = (100-63) + (5x27)

                                      = 172

Once checked the answers correspond with the formula results meaning it works. SO the formula which works out the new t-total when b of the vector is:

New T-Total = 5n-7g-5(bxg)

The formula to work out the new t-number of the new t-shape is:

New T-Number = n-(bxg)

Both these formulas hold for negative values.

I will now try the vectorimage19.png. I will use my previous formulas to get an overall formula.

image20.png

= 5 (n+a) – 7g

 = 5n-7g-5(bxg)image21.pngimage01.png

image02.png

So           will equal image23.pngimage22.png

image22.png

So the new t-total using vector           is 5n-7g+5a-5bg. The 5bg part of the formula is from the 5(bxg) of the original ob vector shown above.

=5(n+a-bg)-7g

I will test this formula or a 9 wide grid with a vector of          .image24.png

New T-Total = 5(20+2+3x9)-7x9

                     = 245-63

                     = 182

When put on a graph the answers correspond with that from the formula.

I will now try this formula on a 7 wide grid using the same vector as before.

New T-Total = 5(16+2+3x7)-7x7

                      = 5x39-49

                      = 195-49

                      = 146

When put on a graph the answers correspond with that from the formula so the formula is correct and works for grids of any width.

image03.png

        

...read more.

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