# 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.

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.

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.

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

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

Middle

T-Number = 10 T-Total = 57 | T-Number = 11 T-Total = 62 | 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.

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.

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.

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.

Conclusion

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 vector. I will use my previous formulas to get an overall formula.

= 5 (n+a) – 7g

= 5n-7g-5(bxg)

So will equal

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 .

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.

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

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