• 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
  20. 20
    20
  21. 21
    21
  22. 22
    22
  • Level: GCSE
  • Subject: Maths
  • Word count: 8274

T totals. In this investigation I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalizations I can find

Extracts from this document...

Introduction

Maths GCSE Coursework 2000 – T-Total

INTRODUCTION – Tell the reader what the project is all about – get a friend/family member to read it – do they understand what it’s all about?

Introduction

In this investigation I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalizations I can find, using the T-Number (x) (the number at the bottom of the T-Shape), the grid size (g) to find the T-Total (t) (Total of all number added together in the T-Shape), with different grid sizes, translations, rotations, enlargements and combinations of all of the stated.

EXPLAIN WHAT THE LETTERS STAND FOR

Relations ships between T-number (x) and T-Total (t) on a 9x9 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

From this we can see that the first T shape has a T number of 50 (highlighted), and the T-total (t) adds up to 187 (50 + 41 + 31 + 32 + 33). With the second T shape with a T number of 80, the T-total adds up to 337, straight away a trend can be seen of the larger the T number the larger the 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

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

From these Extra T Shapes we can plot a table of results.

T-Number (x)

T-Total (t)

20

37

26

67

49

182

50

187

52

197

80

337

From this table the first major generalization can be made,

The larger the T-Number the larger the T-Total

The table proves this, as the T-Numbers are arranged in order (smallest first) and the T-Totals gradually get larger with the T-Number.

From this we are able to make a formula to relate T-Number (x) and T-Total (t) on a 9x9 grid. Taking the T-number of 20 as an example we can say that the T-Total is gained by:

t = 20 + 20 – 9 + 20 – 19 + 20 – 18 + 20 – 17 = 37

The numbers we take from 20

...read more.

Middle

31

32

33

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 = 5v – 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 23 - 3).

Middle number (v)

T-Total (t)

Equation used

Difference

23

105

t = (5 x 23) + ( 2 x 5 )

25 (105 – 80)

18

80

t = (5 x 18) + ( 2 x 5 )

25 (80 – 18)

13

55

t = (5 x 13) + ( 2 x 5 )

25 (55 - 30)

8

30

t = (5 x 8) + ( 2 x 5 )

N/a

MAKE A TABLE and LOOK FOR PATTERNS – TRY TO FIND A RULE

From this we can see that 25 is the “magic” number for vertical translations by +1 on a grid width of 5, from this I can see a link with the “magic” numbers, as for as grid width of 5 it is 25, which is 5 x 5, and for a grid width of 9 it is 45 which is 9 x 5. We can also see that translations larger than 1 can be found by a(25) (were a is the figure which you want to translate by), i.e. if you wanted to translate the T-Shape vertically by +3 , the “magic” number would be found by 5x25.

To verify this we can see what the “magic” number is on a grid width of 10, we can predict it will be 50 (5 x 10).

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

34

35

36

37

38

39

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

If we note the same form of table we have used before, we can find the “magic number”, the above graph shows a vertical translation of the T-Shape by +3, were v=46, t =210 which translates to, v=16, t=60.

Middle number (v)

T-Total (t)

Equation used

Difference

46

210

t = (5 x 46) + ( 2 x 10 )

50 (210 – 160)

36

160

t = (5 x 36) + ( 2 x 10 )

50 (160 – 110)

26

110

t = (5 x 26) + ( 2 x 10 )

50 (110 - 60)

16

60

t = (5 x 16) + ( 2 x 10 )

N/a

As I predicted the “magic” number was 50, therefore I can generalize and state that;

Any vertical translation can be found by t=(5v-2g)-a(5g), were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

In terms of the T-Number (x) instead of v,

Any vertical translation can be found by t=(5(x+g)-2g)-a(5g), were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

Horizontal

Again, we shall use our standard gird size and position to establish our basic starting point;

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

Here we can see our basic staring point (v = 41 therefore t = 187), with a vertical translation to the second shape (v = 44 therefore t = 202). Straight away, we can generalize that,

When a T-Shape is translated horizontally by a positive figure its T-Total is less than the

original T-Total

If we table these results along with all the horizontal translation results from 41 to 44 (for v), we should easily see a pattern (on a grid width of 9), also adding a column for the difference between the number in that column and the once below.

Middle number (v)

T-Total (t)

Equation used

Difference

44

202

t = (5 x 44) + ( 2 x 9 )

5 (202 – 197)

43

197

t = (5 x 43) + ( 2 x 9 )

5 (197 – 192)

42

192

t = (5 x 42) + ( 2 x 9 )

5 (192 – 187)

41

187

t = (5 x 41) + ( 2 x 9 )

N/a

We can see an obvious relationship, that as the T-Shape is translated by +1 in a vertical direction the T-Total is larger by +45 than the previous T-Total. Therefore, we can state that,

As a T-Shape is translated horizontally by +1 on a grid width (g) of 9 the T-Total (t) is

+5 larger than the previous T-Total (t) (the origin)

It is obvious we can also state that:

As a T-Shape is translated horizontally by –1 on a grid (g) width of 9 the T-Total (t) is

-5 smaller than the previous T-Total (t) (the origin)

As when v = 32, t = 142 and with a translation horizontally by –1 v = 31, t = 137, and 137 – 142 = -5, therefore the above statement it correct. We now need to use the same method with different grid size to make a universal equation. I have chosen a grid width of 11 as that has a central number as does a grid width of 9, but make it vertically shorter as we do not need the lower number as they will  not be used. We know with will not make a difference to the final answer as proved in question 2.

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

As we can see, we have a horizontal translation of the first T-Shape (where v =17) by +4. Where v = 17, t = 63, and where v = 21, t = 83 (both found by using t = 5v – 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 21 to 17).

Middle number (v)

T-Total (t)

Equation used

Difference

21

83

t = (5 x 21) + ( 2 x 11 )

5 (83 – 78)

20

78

t = (5 x 20) + ( 2 x 11 )

5 (78 – 73)

19

73

t = (5 x 19) + ( 2 x 11 )

5 (73 – 68)

18

68

t = (5 x 18) + ( 2 x 11 )

5 (68 – 63)

17

63

t = (5 x 17) + ( 2 x 11 )

N/a

From this we can see that  5 is the “magic” number again as for a grid width of 9 for horizontal translations. From this an obvious relation ship can bee seen that for all grid sizes, a horizontal translation of a T-Shape by +1, makes the T-Total +5 larger, but this is only a prediction. To verify this we can see what the “magic” number is on a grid width of 10.

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

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

If we note the same form of table we have used before, we can find the “magic number”, the above table shows a vertical translation of the T-Shape by +4, were v=45, t =205 which translates to, v=49, t=225.

Middle number (v)

T-Total (t)

Equation used

Difference

49

225

t = (5 x 49) + ( 2 x 10 )

5 (225 – 220)

48

220

t = (5 x 48) + ( 2 x 10 )

5 (220 – 215)

47

215

t = (5 x 47) + ( 2 x 10 )

5 (215 – 210)

46

210

t = (5 x 46) + ( 2 x 10 )

5 (210 – 205)

45

205

t = (5 x 45) + ( 2 x 10 )

N/a

As I predicted the “magic” number was 5, therefore I can generalize and state that;

EXTEND THE PROJECT – EXPLORE what happens when you CHANGE THE PROBLEM in a SMALL WAY

Any horizontal translation can be found by t=(5v-2g)+5a, were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

In terms of the T-Number (x) instead of v,

Any horizontal translation can be found by t=(5(x+g)-2g)+5a, were x is the T-Number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

 grid width.

Combinations (diagonal)

For diagonal translation across a grid a combination of horizontal and vertical translations are used, therefore I predict that if I combine my 2 found equations for horizontal and vertical equations, I can generate a general formula for diagonal translations, which is a prediction I need to change. One simple change needs to be made to my horizontal translation equation, that as “a” was also used for the figure by which to translate (the same as the vertical translation), we have to substitute “a” with “b” in the horizontal equation otherwise we can only move the T-Shape, is fixed diagonal positions. We need now to combine the two equations, only take one instance of 5v-2g as only one T-Shape is being translated;

WRITE YOUR RULES USING ALGEBRA

t=(5v-2g)-(a(5g))-5b

To prove this equation we need to again start with our stand grid and position and try it on a combination translation.

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 above translation a combination of a horizontal equation of +3 and a vertical translation of +3 also, the origin T-Shape has a T-Total of 187, and the translated T-Shape has a T-Total of 67, using the equation we will try and generate the Second T-Total to prove our theory correct,

CHECK YOUR RULE WITH EXAMPLES

t=((5x41)-(2x9))-(3(5x9)-5x3)

t=(205-18)-(135-15)

t=187-120

t=67

My equation has been proved correct using this translation we must now try it on another grid size with another type of a combination translation, to verify that it is correct, I have chosen a grid width of 5, extended vertically to accommodate the combination translation:

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

Here we can see a translation of +3 vertically, and +1 horizontally, with the original T-Shape having a T-Total of 105, and the translated T-Shape having a T-Total of 35, using our generated formula we can see if it correct, if it can translate with a v total of 23 to the T-Total of translated shape of 35;

T=((5x23)-(2x5))-(3(5x5)-5x1)

T=(115-10)-(75-5)

T=105-70

T=35

CHECK YOUR RULE WITH EXAMPLES

This proves my equation correct as; the correct translated T-Total is generated, thus we can state;

Any combination translation (vertically and horizontally), can be found by

using the equation of t=(5v-2g)-(a(5g))-5b were v is the middle number,

g is the grid width, a is the number by which to translate vertically and

b is the number by which to translate horizontally.

It is also apparent that this equation can be used for any s

...read more.

Conclusion

T=5(68)+2

T=347

Thus proving this formula works we now need to try it again on a different grid size with a different centre of rotation.

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

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

Here we can see a rotation of 90 Clockwise with the centre of rotation being 36, the original T-Total is 170 and the rotated T-Total is 282. Using the formula we get;

T=5(36+2x10+0)+2

T=5(56)+2

T=282

Thus proving this formula works and it is obvious that it will work in the same fashion as my static (middle number as the centre) rotations, as they both find the position as the new V number then generate the t-total based on that number, therefore I can state;

CONCLUSION – SUMMARISE WHAT YOU HAVE FOUND

The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size

can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the

centre of rotation (grid value) d is the horizontal difference of v from the relative centre of rotation , is the grid width, and b is the vertical difference of v from the relative central number.

y is to be substituted by the ending required  by the type of rotation,

these are :

Rotation (degrees)

Direction

Ending (y)

0

Clockwise

- 2g

90

Clockwise

+ 2

180

Clockwise

+ 2g

270

Clockwise

– 2

0

Anti-Clockwise

– 2g

90

Anti-Clockwise

- 2

180

Anti-Clockwise

+ 2g

270

Anti-Clockwise

+ 2

In terms of x (T-Number);

The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size

can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the

centre of rotation (grid value) d is the horizontal difference of x from the relative centre of rotation,

g is the grid width, and b is the vertical difference of x from the relative centre of rotation.

y is to be substituted by the ending required  by the type of rotation,

these are :

Rotation (degrees)

Direction

Ending (y)

0

Clockwise

- 7g

90

Clockwise

+ 7

180

Clockwise

+ 7g

270

Clockwise

– 7

0

Anti-Clockwise

– 7g

90

Anti-Clockwise

- 7

180

Anti-Clockwise

+ 7g

270

Anti-Clockwise

+ 7

Philip Price 04/03/08                

[a]i

...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

See related essaysSee related essays

Related GCSE T-Total essays

  1. Noughts and Crosses Problem Statement:Find the winning lines of 3 in grids of ...

    lines of 3 in a different n x m grid- where m = n+2. 3 x 5 4 x 6 5 x 7 6 x 8 Possible solution: To work out how many winning lines of 3 there are on each grid I will draw in the lines (see appendix).

  2. T-totals, Main objective of this project of T-totals coursework is to find an inter-relationship ...

    the T-totals decrease by 5 0 each time 0 (moving T-shape by one grid vertically down) the T-totals increase by 45 each time 1 0 (moving T-shape by one grid vertically down) the T-totals decrease by 45 -1 each time.

  1. T-total Investigation

    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

  2. How many arrangements for 2 counters in a line, in grids of various sizes ...

    x (N-1) + (N-1) x (N-1). The next diagonal section of the formula (highlighted in red on the equation) works the same with the diagonal right as the diagonal left. It works in a similar way to the vertical/horizontal equation in that the number of combinations left and right need

  1. T-Shapes Coursework

    13 65 23 88 14 70 24 94 15 75 25 100 16 80 26 106 17 85 27 112 b) Here are the results of the 5 calculations for a 5x3 "T" on Width 10 Grid: Middle Number Sum of Wing Sum of Tail Total Sum (Wing + Tail)

  2. The T-Total Mathematics Coursework Task.

    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. Rotation Of "T"- Shapes Clockwise

    "T"-total = pt + 35 + 7 =30 + 35 + 7 = 72 this was correct, so I tried to use this formula on the 6 by 6 grid. "T"-total = pt + 35 + 7 =33 + 35 + 7 =75 which is not equal to 82 the next "T"-total in the sequence.

  2. Maths Coursework - T-Total

    Translating on a vector is basically moving the shape from one position to another using a 'code' to determine how far. A vector looks like (and will be referred to as) (a/b). 'a' being the distance the shape should be moved right (if a negative number is there, the shape moves left).

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