• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18
  19. 19
    19
  • Level: GCSE
  • Subject: Maths
  • Word count: 4165

T-totals. I am going to investigate the relationship between the t-total, T, and the t-number, n. The t-number is always the number at the bottom of the t-shape when it is orientated upright.

Extracts from this document...

Introduction

T-Totals

1

2

3

11

20

I am going to investigate the relationship between the t-total, T, and the t-number, n.  The t-number is always the number at the bottom of the t-shape when it is orientated upright.  Here the t-number would be 20.  The t-total is the sum of the cells inside the t-shape.  Here it would be 37 as 1+2+3+11+20 = 37

I will calculate the t-total for different t-numbers on 9×9 grid.  Working algebraically, I will find a relationship that will express the t-total in terms of the t-number and the grid size.  I will test this generalisation for t-shapes in an 8×8 and 10×10 grid.  I will then transform the t-shape and investigate it’s affect on this relationship.

T-shapes in a 9×9 grid

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

n

T

T=5n-63

20

37

5(20)-63=37

23

52

5(23)-63=52

26

67

5(26)-63=67

47

172

5(47)-63=172

50

187

5(50)-63=187

53

202

5(53)-63=202

74

307

5(74)-63=307

77

322

5(77)-63=322

80

337

5(80)-63=337

The t-total increases by 135 when the t-number is increased by 27 and by 15 when the t-number increases by 3.  It thus follows that for every unit increase in the t-number there will be an increase of 5 in the t-total.  

n-20

n-18

n-19

n-9

n

The t-shapes in a 9×9 grid can be represented algebraically.  The t-total, T, can therefore be written in terms of the t-number, n, as T= 5n - 63.  Using similar reasoning we can express T in terms of n in an 8×8 and 10×10 grid.

T-shapes in an 8×8 grid

1

2

3

4

5

6

9

10

11

12

13

14

17

18

19

20

21

22

25

26

27

28

29

30

33

34

35

36

37

38

41

42

43

44

45

46

n

T

T=5n-56

18

34

5(18)-56=37

21

49

5(21)-56=52

42

154

5(42)-56=67

45

169

5(45)-56=172

The t-total increases by 120 when the t-number is increased by 24 and by 15 when the t-number increases by 3.  It thus follows that for every unit increase in the t-number there will be an increase of 5 in the t-total.  

n-17

n-16

n-15

n-8

n

The t-shapes in an 8×8 grid can be represented algebraically.  The t-total, T, can therefore be written in terms of the t-number, n, as T= 5n - 56.  

T-shapes in a 10×10 grid

n

T

T=5n-70

22

40

5(22)-70=37

25

45

5(25)-70=52

28

50

5(28)-70=67

52

190

5(52)-70=172

55

195

5(55)-70=187

58

200

5(58)-70=202

1

2

3

4

5

6

7

8

9

11

12

13

14

15

16

17

18

19

21

22

23

24

25

26

27

28

29

31

32

33

34

35

36

37

38

39

41

42

43

...read more.

Middle

-2

-3

31

224

5 {31 - 2 + 3(9) } - 7(8) = 224

10×10

1

3

55

75

5 {55 + 1 – 3(9) } - 7(10) = 75

-3

1

66

200

5 {66 – 3 – 1(9) } - 7(10) = 200

3

-2

74

405

5 {74 + 3 + 2(9) } - 7(10) = 405

-2

-2

27

145

5 {27 - 2 + 2(9) } - 7(10) = 145

The t-total for the translated t-number agrees with the results from the formula.  We can therefore say that T = 5 (n +a - bg) -7g will work for any translation of the form image13.png in any grid size.

Rotations

Rotations about the t-number

In the same way that we justified T= 5n-7g, we can rotate the t-shape about the t-number and express the other cells in terms of n (t-number) and g (grid size).  Collecting like terms will simplify a formula for the t-total, T, of the rotated t-shape.  We can then validate the formula in different grid sizes.

Rotation of 90º clockwise/ 270º anti-clockwise

n-2g-1

n-2g

n-2g+1

n-g

n

n+2-g

n

n+1

n+2

n+2+g

T= (n) + (n +1) + (n + 2) + (n + 2 –g) + (n + 2) + (n + 2 + g)

T= 5n + 7

Rotation of 180º

n-2g-1

n-2g

n-2g+1

n-g

n

n

n+g

n+2g-1

n+2g

n+2g+1

T= (n) + (n + g) + (n + 2g) + (n + 2g –1) + (n + 2g +1)

T= 5n + 7g

Rotation of 270º clockwise/ 90º anti-clockwise

n-2g-1

n-2g

n-2g+1

n-g

n

n-2-g

n-2

n-1

n

n-2+g

T= (n) + ( n – 1) +( n - 2 )+ (n – 2 – g) + (n – 2 + g)

T= 5n + 7g

Validation

g

n

90º clockwise

180º

270º clockwise

T

T=5n+7

T

T=5n+7g

T

T=5n-7

9×9

31

157

5(31)+7=157

218

5(31)+7(9)=218

143

5(31)-7=143

40

207

5(40)+7=207

263

5(40)+7(9)=263

193

5(40)-7=193

49

252

5(49)+7=252

308

5(49)+7(9)=308

238

5(49)-7=238

8×8

20

107

5(20)+7=107

156

5(20)+7(8)=156

93

5(20)-7=93

28

147

5(28)+7=147

196

5(28)+7(8)=196

133

5(28)-7=133

36

187

5(36)+7=187

236

5(36)+7(8)=236

173

5(36)-7=173

10×10

45

232

5(45)+7=232

295

5(45)+7(10)=295

218

5(45)-7=218

55

282

5(55)+7=282

345

5(55)+7(10)=345

268

5(55)-7=268

65

332

5(65)+7=332

395

5(65)+7(10)=395

318

5(65)-7=318

As the results from the formulae agree with the t-total of the rotated t-shape, we can confirm the following formulae for a rotation about the t-number in any grid size:

  • 90º clockwise: T = 5n + 7
  • 180º: T = 5n – 7g
  • 270º clockwise: T= 5n - 7

Rotation of 90º clockwise/ 270º anticlockwise about an external point

The distance from the t-number to the centre of rotation is described by the column vector image17.png where c is the horizontal distance (positive to the right and negative to the left) and d is the vertical distance (positive being upwards and negative being downwards).

The t-total of any t-shape that has been rotated 90º clockwise or 270º anticlockwise about an external point is given by (T = t-total, n = t-number, g = grid size):

T = 5 ( n + c – dg –d –cg ) + 7

Justification

n-2g-1

n-2g

n-2g+1

  n-gimage00.png

centre of rotation

      nimage01.png

d

c

The general t-shape with t-total T= 5n - 7g is clearly shown.  The distance from the t-number, n, to the centre of rotation is described by the vector image17.png.

n-2g-1+c-dg

n-2g+c-dg

n-2g+1+c-dg

n-2g-1

n-2g

n-2g+1

n-g+c-dg

n-gimage01.png

image00.png

n+c-dg

n

d

c

Translate the t-number, n, to the centre of rotation.  By using the earlier justification, a translation by the vector image17.png will add (c-dg) to every cell.  The t-number thus becomes:  

n + c – dg

n-2g-1+c-dg-d-cg

n-2g+c-dg-d-cg

n-2g+1+c-dg-d-cg

n-g+c-dg-d-cg

n+c-dg-d-cg

-d

image03.pngimage02.png

n-2g-1+c-dg

n-2g+c-dg

n-2g+1+c-dg

c

n-g+c-dg

n+c-dg

The position of t-shape following a rotation of 90º clockwise has been outlined by a dotted line.  The vector image18.png will translate the t-number,    n+ c–dg, to the position that the t-number will take when the original t-shape has been rotated.  By using the earlier justification, a translation by the vector image18.png will add (-d-cg) to every cell.  The t-number thus becomes:  

n + c – dg –d -cg

n-2g-1+c-dg-d-cg

n-2g+c-dg-d-cg

n-2g+1+c-dg-d-cg

n-g+c-dg-d-cg

n+c-dg-d-cg+2-g

n+c-dg-d-cg

n+c-dg-d-cg+1

n+c-dg-d-cg+2

n+c-dg-d-cg+2+g

Rotate the t-shape 90º clockwise about the t-number, n+c-dg-d-cg.  By using the earlier justification that a rotation of 90º clockwise about the t-number, n, has a t-total T=5n + 7, the t-total of a t-shape that has been rotated 90º about an external point becomes:

T= 5 ( n + c – dg –d -cg ) + 7.

Rotation of 180º about an external point

The t-total of any t-shape that has been rotated 180º about an external point is given by (T = t-total, n = t-number, g = grid size):

T = 5 (n + 2c – 2dg )+ 7g

Justification

We can assume that the t-number, n, has been translated to the centre of rotation by the vector image17.png.  The t-number thus becomes n+c-dg.

n-2g-1+2c-2dg

n-2g+2c-2dg

n-2g+1+2c-2dg

n-2g-1+c-dg

n-2g+c-dg

n-2g+1+c-dg

n-g+2c-2dg

n-g+c-dg

image00.pngimage01.png

n+2c-2dg

n+c-dg

d

c

The position of the t-shape following a rotation of 180º has been outlined by a dotted line.  The vector image17.png will translate the t-number, n+c–dg, to the position that the t-number will take when the original t-shape has been rotated.  By using the earlier justification, a translation by the vector image17.png will add (c-dg) to every cell.  The t-number thus becomes:  n + 2c – 2dg

n-2g-1+2c-2dg

n-2g+2c-2dg

n-2g+1+2c-2dg

n-g+2c-2dg

n+2c-2dg

n+2c-2dg+g

n+2c-2dg+2g-1

n+2c-2dg+2g

n+2c-2dg+2g+1

Rotate the t-shape 180º about the t-number,  n+2c-2dg.  By using the earlier justification that a rotation of 180º about the t-number, n, has a t-total T=5n + 7g, the t-total of a t-shape that has been rotated 180º about an external point becomes:

T= 5 ( n + 2c – 2dg ) + 7g.

Rotation of 270º clockwise/ 90º anticlockwise about an external point

The t-total of any t-shape that has been rotated 270º clockwise or 90º anticlockwise about an external point is given by (T = t-total, n = t-number, g = grid size):

T = 5 ( n + c – dg +d +cg ) - 7

Justification

We will assume that the t-number, n, has been translated to the centre of rotation by the vectorimage17.png.  The t-number thus becomes n+c-dg.

n-2g-1+c-dg

n-2g+c-dg

n-2g+1+c-dg

n-g+c-dg

n+c-dg

d

image04.png

n-2g-1+c-dg+d+cg

image05.png

n-2g+c-dg+d+cg

n-2g+1+c-dg+d+cg

-c

n-g+c-dg+d+cg

n+c-dg+d+cg

The position of the t-shape following a rotation of 270º clockwise has been outlined by a dotted line.  The vector image19.png will translate the t-number, n + c –dg, to the position that the t-number will take when the original t-shape has been rotated.  By using the earlier justification, a translation by the vector image19.png will add   (d+ cg) to every cell.  The t-number thus becomes:

n + c –dg + d +cg

n-2g-1+c-dg+d+cg

n-2g+c-dg+d+cg

n-2g+1+c-dg+d+cg

n+c-dg+d+cg-2-g

n-g+c-dg+d+cg

n+c-dg+d+cg-2

n+c-dg+d+cg-1

N+c-dg+d+cg

n+c-dg+d+cg-2+g

...read more.

Conclusion

a

b

c

d

2

2

1

1

61

322

5 { 61 + (-1)(-10) - (-1)(12) + 2 – 22} + 7 = 322

2

2

-1

1

61

422

5 { 61 + (-3)(-10) - (-1)(12) + 2 – 22} + 7 = 422

2

2

1

-1

61

442

5 { 61 + (-1)(-10) - (-3)(12) + 2 – 22} + 7 = 442

2

2

-1

-1

61

542

5 { 61 + (-3)(-10) - (-3)(12) + 2 – 22} + 7 = 542

-2

2

1

1

61

102

5 { 61 + (3)(-10) - (-1)(12) - 2 – 22} + 7 = 102

-2

2

-1

1

61

202

5 { 61 + (1)(-10) - (-1)(12) - 2 – 22} + 7 = 202

-2

2

1

-1

61

222

5 { 61 + (3)(-10) - (-3)(12) - 2 – 22} + 7 = 222

-2

2

-1

-1

61

322

5 { 61 + (1)(-10) - (-3)(12) - 2 – 22} + 7 = 322

Translate image13.pngand rotate 180º

T = 5 {n + 2 (c - a) – 2g (d - b) + a – bg } + 7g

2

-2

1

1

61

162

5 { 61 + (2)(-1) – (22)(3) + 2 +22 } + 77 = 162

2

-2

-1

1

61

142

5 { 61 + (2)(-3) – (22)(3) + 2 +22 } + 77 = 142

2

-2

1

-1

61

382

5 { 61 + (2)(-1) – (22)(1) + 2 +22 } + 77 = 382

2

-2

-1

-1

61

362

5 { 61 + (2)(-3) – (22)(1) + 2 +22 } + 77 = 362

Translate image13.pngand rotate 270º clockwise

T = 5 { n + (c - a) (1 + g) –  (d - b) (g – 1) + a – bg } – 7

-2

-2

1

1

61

428

5 { 61 + (3)(12) - (3)(10) - 2 + 22} - 7 = 428

-2

-2

-1

1

61

308

5 { 61 + (1)(12) - (3)(10) - 2 + 22} - 7 = 308

-2

-2

1

-1

61

528

5 { 61 + (3)(12) - (1)(10) - 2 + 22} - 7 = 528

-2

-2

-1

-1

61

408

5 { 61 + (1)(12) - (1)(10) - 2 + 22} - 7 = 408

The t-total for the rotated t-shape agrees with the results from the formula.  We can therefore say that our formulae are consistent.

Summary

T-total

T = 5n – 7g

Translation by vector image13.png

T = 5 ( n + a - bg ) - 7g

Rotations about an external point that is vector image17.pngfrom t-number

  • Rotate 90º clockwise                 T = 5 ( n + c – dg –d –cg ) + 7
  • Rotate 180º                                 T = 5 (n + 2c – 2dg )+ 7g
  • Rotate 270º clockwise                T = 5 ( n + c – dg +d +cg ) – 7

* for rotations about the t-number c and d will equal zero.

Combined transformation: rotation followed by a translation

  • Rotate 90º clockwise and translate image13.png

T = 5 ( n + c – dg –d –cg + a – bg ) + 7

  • Rotate 180º and translate image13.png

T = 5 (n + 2c – 2dg + a – bg )+ 7g

  • Rotate 270º clockwise and translate image13.png

T = 5 ( n + c – dg +d +cg + a – bg ) – 7

Combined transformation: translation followed by a rotation

  • Translate image13.pngand rotate 90º clockwise

T = 5 { n + (c - a) (1-g) – (d - b) (g+1) + a – bg } + 7

  • Translate image13.pngand rotate 180º

T = 5 {n + 2 (c - a) – 2g (d - b) + a – bg } + 7g

  • Translate image13.pngand rotate 270º clockwise

T = 5 { n + (c - a) (1 + g) –  (d - b) (g – 1) + a – bg } – 7

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Related GCSE T-Total essays

  1. T-Total. I will take steps to find formulae for changing the position of the ...

    This is what I came up with for any T total in the grid size of 9 x 9: - x-19 x-18 x-17 x-9 x With these numbers, I added them altogether and I got the number 63. I divided 63 by the grid size and got the answer of 7.

  2. Objectives Investigate the relationship between ...

    that are rotated, I will find out a formula that will find the T-total of rotated T-shapes. 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

  1. T Total and T Number Coursework

    This is equal to 5n-7g. To get this I added all the negative 'g's' and that gave me a total of -7. This is how I got to my seven in the formula, this proves that there must be a seven in the formula for it to work correctly.

  2. The investiagtion betwwen the relationship of the T-number and T-total

    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

  1. T totals. In this investigation I aim to find out relationships between grid sizes ...

    34 35 36 37 38 39 40 41 42 43 44 45 As we can see we have a vertical translation of the first T-Shape (where v =23) by +3. Where v = 23, t = 105, and where v = 8, t = 30 (both found by using t

  2. T totals - translations and rotations

    57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 78 79 80 81 82 Vertical translation, rotated 180 degrees clockwise My T number is 11 as you can see on my 9by9 grid and I will also be representing this as N in my equation.

  1. T-Totals. We have a grid nine by nine with the numbers starting from 1 ...

    79 80 81 Difference 12-1 =11 12-10= 2 12-19= -7 12-11 = 1 TOTAL = 7 Formula 5tn - 7 =t-total 5*12 - 7= 53 Check to see if the formula is right T-number = 12 T-total = 1 +10 +19 +11 +12 = 53 This formula has worked.

  2. T- total T -number coursework

    in this t-shape Working Out: - 70-51=19 70-52=18 70-53=17 70-61=9 TOTAL=63 Again the number turns out to be 63. This is where the 63 came from in this equation. There is also another place this 63 comes from. This is 9*7=63.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work