# t totals gcse grade A

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Introduction

Tom white

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

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |

28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |

46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 |

55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

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 |

57 | 222 |

58 | 227 |

59 | 232 |

60 | 237 |

61 | 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 |

7x7grid

1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

24-17=7

36-29=7

41-34=7

Every time it =7

A | B | C |

D | ||

T |

T represents the t-number

D= T-grid

B= T-2grid

A= T-2grid-1

C= T-2grid+1

4

Middle

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

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73

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81

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 | 32 | 97 |

After 180 turn | 32 | 223 |

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

a | b | c |

d | ||

t | ||

E | ||

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

1 | 2 | 3 | 4 | 5 | 6 | 7 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

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

1 | 2 | 3 | 4 | 5 | 6 |

7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 | 24 |

25 | 26 | 27 | 28 | 29 | 30 |

31 | 32 | 33 | 34 | 35 | 36 |

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 | 23 | 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

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |

49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |

57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

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.

8x8grid

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |

49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |

57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

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.

6x6grid

1 | 2 | 3 | 4 | 5 | 6 |

7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 | 24 |

25 | 26 | 27 | 28 | 29 | 30 |

31 | 32 | 33 | 34 | 35 | 36 |

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

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |

28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |

46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 |

55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

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 |

0˚ | 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

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |

28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |

46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 |

55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

18

X movement is +4 Y movement is +1

5N-7g+54+5g1

start | 21 | 12 | 2 | 3 | 4 |

X | +4 | +4 | +4 | +4 | +4 |

Y | +9 | +9 | +9 | +9 | +9 |

finish | 34 | 25 | 15 | 16 | 17 |

T-total = 5N-7g+54+5g1

5x21-7x9+5x4+5x9

105-63+20+45

42+65

107

34+25+15+16+17=107

19

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

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