• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Using grids of different sizes, try other transformations and combinations of transformations. Also, to investigate relationships between the T-total, the T-numbers, the grids size and the transformations.

Extracts from this document...

Introduction

MATH COURSEWORK

T- totals

Part III

Aim:

        Using grids of different sizes, try other transformations and combinations of transformations.  Also, to investigate relationships between the T-total, the T-numbers, the grids size and the transformations.

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

Rotation:

A T-shape is rotated about the p number at 90 degrees, 180 degrees and 270 degrees.

T = 12 + 13 + 14 + 22 + 31

T = 92

T = 24 + 33 + 42 +32 +31

T = 162

T = 40 + 48 + 49 + 50 + 31

T = 218

T = 20 + 29 + 38 + 30 + 31

T = 148

We already know the formula for a 0º rotation.  Since it does not move, the formula is the same as the one that I have previously been using:

5p – 7g

I found this formula by making a diagram and then, filling it in appropriately (making many small equations for each box).  I then added all the small equations together to make one big, final formula.  I can use this method again to find out the formula for a 90º, 180º and 270º rotation.

There were 5 p’s, 7 g’s, (-1) and (+1).  The (-1) and (+1) cancelled each other out, leaving 5p, and 7g.  7g was subtracted from 5p because the sum of all the g’s was – 7, therefore giving me the final equation: 5p – 7g

90º 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

Observations:

  • The left box is p, the position of the T-shape.  
  • The three boxes going across are all consecutive (they are one after another), which means if the first box was p, the next box is p + 1
  • Which also means that the box after that is p + 2
  • The top box subtractsg (the grid size) because it is decreasing.  But, subtracting the grid size means that the box is right above.  For example, in this example:

p – g  = 31 – 9

= 22

22 is directly above 31 because each level increases or decreases by the grid size number.  So, since the number I want it two spaces over, I add 2.  The equation for that box is p – g + 2

  • The bottom box adds g (the grid size) because it is increasing on the grid.  But, since every level increased or decreased means you have to add or subtract the grid size number, adding g (which is 9 in this example) would give me the box directly underneath p.  In this example:
...read more.

Middle

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

6 x 6 grid:

Using the Formula:

T = 5p + 7

T = 5(22) + 7

T = 110 + 7

T = 117

T = 22 + 23 + 14 + 18 + 30

T = 117

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

7 x 7 grid:

Using the Formula:

T = 5p + 7

T = 5(24) + 7

T = 120 + 7

T = 127

T= 24 + 25 + 26 + 19 + 33

T = 127

180º 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

Observations:

  • The top box p, is the position of the T-shape
  • The box right underneath is one level down, which means the grid size number (g) must be added from p (because this grid increases as it goes down).  Therefore, the equation for this box is p + g
  • The box directly underneath that is two levels down from p, which means you multiply the grid size (g) by 2 (because you moved 2 levels), and then add it to p.  The equation for this box is p + 2g
  • Each time you go up a level the number decreases by the gird size, so therefore, each time you go down a level, the number increases by the grid size
  • The three bottom boxes are also two levels down from p, which means you multiply the grid size number (g) by 2, and then add it to p (because it has decreased levels on the grid).  Since these three numbers are consecutive numbers, the box on the left would be one less than the middle (p + 2g – 1), and the box on the right would be one more than the middle box ( p + 2g +1)

Working out the Formula:

Adding up all these small equations, I can come up with a final formula for a 180º rotation about p, to find T if we only know p.

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

T = 5p + 7g

*the (+1) and the (-1) cancel each other out

Testing the Formula:

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

5 x 5 grid:

Using the formula:

T = 5p + 7g

T = 5(12) + 7(5)

T = 60 + 35

T = 95

T = 12 + 17 + 21 + 22 + 23

T = 95

6 x 6 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

...read more.

Conclusion

p – 1 (it is one less than p because it is 1 box over) and the next one over would be p – 2  (it is 2 less than p because it is 2 boxes over)The top box is one level up from p, which means it decreases.  Subtract the grid number (g) from p because it has increased one level, which means it has decreased (because the grid increases as it goes down).  This equation is p – g – 2 The bottom box is one level down from p, which means that it increases.  Add the grid number to p, which gives you p + g – 2

Working out the Formula:

I can use these small equations and add them together, to find the final formula for a 270º Rotation about p, to find T if we only know p.

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

T = 5p – 7

*the (+g) and (-g) cancel each other out

Testing the Formula:

5 x 5 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

Using the formula:

T = 5p – 7

T = 5(14) – 7  

T = 70 – 7

T = 63

T = 7 + 12 + 17 + 13 + 14

T = 53

6 x 6 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

Using the Formula:

T = 5p – 7

T = 5(22) – 7

T = 110 – 7

T = 103

T = 14 + 20 + 26 + 21 + 22

T = 103

7 x 7 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

Using the Formula:

T = 5p – 7

T = 5(24) – 7

T = 120 – 7

T = 113

T = 15 + 22 + 29 + 23 + 24

T = 113

Formulas for rotation about p:

Formula

0º Rotation

5p – 7g

90º Rotation

T = 5p + 7

180º Rotation

T = 5p + 7g

270º Rotation

T = 5p – 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

See related essaysSee related essays

Related GCSE T-Total essays

  1. The T-Total Mathematics Coursework Task.

    We would approach the task of getting the L-total by first drawing out the simple shape below and comparing it to the shape below that. 4 10 11 12 13 All of the numbers added together would total to the L-total.

  2. Objectives Investigate the relationship between ...

    +5 T23 45 +5 T24 50 +5 As you can see the T-totals of the 3 T-shapes are: 40, 45 and 50 As you can see they increase by the integer '+5' each time, they are translated to the right.

  1. T-Shapes Coursework

    19 209 497 706 20 220 504 724 21 231 511 742 32 352 588 940 33 363 595 958 4) Data Analysis The aim of this analyis is to find the patterns that were discovered in the other sections of the investigation.

  2. T-shapes. In this project we have found out many ways in which to ...

    To prove this I will do another. 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. Maths Coursework T-Totals

    of any Tc shape on any sized grid, were v is the Middle Number. In terms of x; The formula t = 5x + 7 can be used to find the T-Total (t) of any Tc shape on any sized grid, were x is the T-Number.

  2. Maths GCSE Investigation - T Numbers

    78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127

  1. T-Totals. Firstly I am going to do a table of 5 x 5 and ...

    if it correct for any T-number 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 If n = 17 and y = T-total

  2. Number Grids.

    This proves my formula for grid 5x5 is correct. Grid 6x6 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 I found that

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