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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)

image00.png

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

image01.pngThis 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:

image12.png

From this we can see that if:

T number = n

1 = a

2 = b

3 = c

11 = d

20 = n

image12.png

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:

image16.png

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

image17.png

                                                                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.

...read more.

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.

image02.png

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.

...read more.

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]

...read more.

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