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• Level: GCSE
• Subject: Maths
• Word count: 2868

# T-Totals Coursework.

Extracts from this document...

Introduction

Jack Palmer Maths Coursework 2003

T-Totals Coursework

Maths UF 2003

By Jack Palmer

Contents

1. INTRODUCING T’s (Including T-Numbers & T- Totals)
2. MY FIRST EQUATION (Concerning 9 x 9 grids)
3. ROTATING T’s (Upside down and back to front)
4. GRID SIZE AND THE T-TOTAL
1. Original T’s
2. Upside Down T’s
1. STRETCHING T’S
2. VECTORS

Maths Coursework

T-Totals!

1)

This is a T-shape!  It allows us to gather information into algebraic formulas to explain the relationships between numbers.

This is the T-Number.  It is the central part of our research.  If you add up all the numbers in the T, you will find the T-Total!  For the T above, the T-Total will be

1 + 2 + 3 + 9 + 16 = 31.

2)

Using algebra, we can work out a formula for this T.  On a 9x9 grid a T would look like this:

From this we can see that if:

 T number = n 1 = a 2 = b 3 = c 11 = d 20 = n

a = n-19                                From this we can see that the T-Total

b = n-18                                      will equal:

c = n-17

d = n-9                                                    1 + 2 + 3 + 11 + 20 = 37

e = n

Using the algebraic formula for each of the numbers we can see that:

T-Total = (n-19) + (n-18) + (n-17) + (n-9) + (n)

= 5n-63

We can see that if we apply this formula to a 9x9 grid we can find the T-total, and we can prove this by testing it on 1 other T:

T-Total = 4 + 5 + 6 + 14 + 23 = 52

This number should = 5n – 63

T-Total = 5n – 63

= (5 x 23) – 63

= 115 – 63

= 52

We can see that the two T-Totals (shortened to TT’s) are equal.

Middle

= 5n - 7

Using this equation on the same T, from past knowledge I predict that:

5n – 7 = 53

I will now test this to see if it correct:

TT= 5n - 7

= 5 x 12 - 7

= 60 – 7

= 53

We can see that the two T-Totals are equal and therefore I can conclude that the equation for a 900 anti-clockwise rotated T that the equation is 5n - 7

Rotation of a T by 900  Clockwise

Using my past knowledge, I can predict that this T will be 5n + 7, just as 5n – 63 was countered by an upside down T with 5n + 63.  I will now try to prove this theory.

This is what a T would look like if rotated 900 clockwise.

As you can see, the T-Total is

3 + 12 + 21 + 11 + 10 = 57

From this I will try to prove algebraically prove that this 57 can correctly prove my 5n + 7 theory.

We can see form the T above that:

 T number = n 3 = a 12 = b 21 = c 11 = d 10 = n

A= n -7                                 We can see that the T-Total should =[b]

B= n-2

C= n-11                            T-Total = 19 + 10 + 1 + 11 + 12 = 53

D= n-1

N= n                                   From this we can see that algebraically 53

should equal =

TT=  (n-7) + (n-2) + (n-11) + (n-1) + (n) =

= 5n + 7

I will now test this theory:

TT= 5n + 7

= (5 x 10) + 7

= 57

We can see that this T-Total is equal to the predicted algebraic answer.

I have also noticed that the general formulas all run in a similar vein.

Conclusion

6x2 + x

We know that the first formula (ie for a 3 x 3 T) that the number of g’s is 7.  We can collate all of this into a table and see what wee can do.

 Size of T Number in Sequence Number of G’s in formula 3 1 7 5 2 26 7 3 57 9 4 100 2y + 1 Y 6y2 + y (x-1)/2 X ……………………

We can see that because y is equal to (x-1)/2, we can find a formula with x in it.  We can then use this to find the general formula.

………………………………

= 6/4 x (x2– 2x + 1) + ((x-1)/2)

= 6x2 – 12x +6    + 2n – 2

4                    4

= 6x2  - 10x + 4

4

= 3  x2 5   + 1

1. 2

I believe this to be the general formula and will now test it.  X is the size of the T.

On a 3 x 3 T:

3 x 9  x 5 x 3  + 1

1. 2

= 27  -  15  +  2

2       2      2

= 14

2

= 7

We can see that this is the number of g’s in the equation, which shows that the formula is correct.  I will test it one more time and then will be able to draw to a conclusion.

On a 5 x 5 T:

3 x 15  x 5 x 5  + 1

1. 2

= 45  -  25 +  2

2       2      2

= 18

2

= 9

We can see that this is definitely correct and as such I can come to a conclusion that the general formula for any T shape is:

3  x2 5   + 1

2        2

I have found many equations in my coursework and all of them have been important in helping me find this equation.

[a]

[b]

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