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t totals gcse grade A

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Introduction

Tom white

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Page  1- intro

Page  2- Basic t-numbers, t-shapes and t-totals

Page  3- the basic t-formula

Page  4- Algebra in a t-formula

Page  5- -

Page  6- 90˚ rotation

Page  7- -

Page  8- 180˚ rotation

Page  9- -

Page 10- 270˚ rotations

Page 11- -

Page 12- translations

Page 13- -

Page 14- -

Page 15- transformations

Page 16- -

Page 17- -

Page 18- conclusion

Page 19- -

Intro

Over these next pages I will be investigating the formulas and patterns between all t-numbers, t-shapes and t-totals in different sized grids 9x9, 8x8, 7x7 and 6x6. I will also investigate translations and transformations in t-shapes.

                                                                                        1

Basic t-numbers, t-shapes and t-totals

A t-shape is 5 cells anywhere on grids in a t shape

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The t-number is the number at the bottom of the t-shape e.g. 21, 34 and 68

The t-total is where you add up all the numbers in the t-shape e.g. 2+3+4+12+21=42

                49+50+51+59+68=276

15+16+17+25+34=97

                                                                                        2

Basic t-formulas

This is of a 9x9 grid

T-number

t-total

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222

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232

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242

You can see the t-total in going up 5 every time as the t-number is going up 1 every time their must be a link?

57x5=285

285-63=222 the t-total

What is the link between 63 and 9?

63 is 7x9

So the formula would be

T-number x 5 – (9 x 7)

5xt-number-7x9

                                                                                         3

Algebra in a t-formula

I am trying to find the number in each cell compared to the t-number

A

B

C

D

T

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24-17=7

36-29=7

41-34=7

Every time it =7

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B

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T

T represents the t-number

D= T-grid

B= T-2grid

A= T-2grid-1

C= T-2grid+1

                                                                                         4

...read more.

Middle

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Using a basic t-shape I am trying to find a formula to get the t-total after turning it 180˚ clockwise

T-number

T-total

Before 180  turn

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97

After 180  turn

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223

I will now relate each cell to the t-number (t)

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b

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d

t

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g

f

h

A=t-2g-1                       E=t+g

B=t-2g                         F=t+2g

C=t-2g+1                        G=t+2g-1

D=t-g                         H=t+2g+1

                                                                                         8

When you put this all together into a formula you get

5N-7g for the t-shape before the rotation

5N+g+2g+2g+2g-1+1 simplified would be 5n+7g

This formula will work on any size grid and any t-number for e.g.

7x7grid

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The t-number is 17 so using the formula the t-total would be 5x17-7g=26   2+3+4+10+17=26

After the 180˚ turn clockwise the t-total would be

5x17+7g=134   17+24+30+31+32=134

So the formula works on different size grids and different t-numbers

                                                                                         9

270˚ rotation

6x6grid

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Using a basic t-shape I am trying to find a formula to get the t-total after turning it 270˚ clockwise

T-number

T-total

Before 270˚ turn

23

73

After 270˚ turn

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108

I will now relate each cell to the t-number (t)

a

b

c

g

d

f

e

t

h

A=t-2g-1

B=t-2g

C=t-2g+1

D=t-g

E=t-1

F=t-2

G=t-2-g

H=t-2+g

                                                              _                         10

When you put this all together into a formula you get

5N-7g for the t-shape before the rotation

5N-1-2-2-2-g+g simplifies would be 5n-7

This formula will work on any size grid and any t-number for e.g.

8x8grid

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The t-number is 46 so using the formula the t-total would be 5x48-7g=184   29+30+31+38+46=184

After the 270˚ turn clockwise the t-total would be

5x46-7=233   46+45+44+36+52=233

So the formula works on different size grids and different t-numbers

                                                                                        11

Translations

Translations are were we can move the t-shape around the grid but before we can do this we have to create a formula for the movements.

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To move the t-shape across to the right 1 cell you add 1 onto each square so you add 5 to the t-total

So to move the t-shape 3 cells across to the right you need to add 15 to the t-total

But when you move to the left you tack 1 from each cell so you take 5 off the t-total

So to move the t-shape 3 cells across to the left you need to take 15 off the t-total

This means that all movement on the x is equal to +or- 5 so you times the grid movement eg. +1, -5, +8 by 5

                                                                                        12

X can be a – number and a + number

- If you won’t to go left

+ If u won’t to go right

So -5 means move across 5 cells to the left

And +3 means move 3 cells to the right.

And the same for the Y but we need the formulas before we can put the numbers in the coordinates.

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...read more.

Conclusion

The formula I will use is 5n+7g+5x-5gy the 5n+7g is from the 180˚!!" rotation and the 5x is the horizontal movement and the 5gy is vertical movement.

5n+7g+5x-5gy

(5 x t-number) + (7 x grid) + (5 x -6) – ((5 x 9) x +6)

Using this formula the answer is 73

2+11+19+20+21=73

So the formula is right but now we need to change it for the 270˚ rotation

                                                                                        16

I will now test the formula for the 270˚ turn and translation

9x9

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I am going to move the t shape with 66 as its t-number +6 across and +6 up an turning it 180’ clockwise

The formula I will use is 5n-7+5x-5gy the 5n-7 is from the 270˚ rotation and the 5x is the horizontal movement and the 5gy is vertical movement.

5n-7+5x-5gy

(5 x t-number) -7 + (5 x +6) – ((5 x 9) x +6)

Using this formula the answer is 83

7+25+16+17+18= 83

So the all the formulas are right and produce a table of results for all I have found

                                                                                        17

Conclusion

This table is all the formals you need to rotate a t-shape

rotated

Basic formulas

5n-7g

90˚

5n+7

180˚

5n+7g

270˚

5n-7

This table is for the movement or translation for a t-shape

movement

Number added to total

1 cell right

+5

1 cell left

-5

1 cell down

+5g

1 cell up

-5g

5N-7g+5X+5gY

Use X+Y as coronets

9x9

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                                                                                        18

X movement is +4     Y movement is +1

5N-7g+54+5g1

start

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X

+4

+4

+4

+4

+4

Y

+9

+9

+9

+9

+9

finish

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T-total =    5N-7g+54+5g1

           5x21-7x9+5x4+5x9

             105-63+20+45

                   42+65

                    107

34+25+15+16+17=107

                                                                                        19

...read more.

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