• 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. T-Shapes Coursework

    x Number of Terms The "1/2{2n + 10(l + 1)}" part of the formula gives us the mean, and so this is multiplied by l because l is the length of the tail, and therefore the number of boxes (or terms)

  2. The T-Total Mathematics Coursework Task.

    71 385 36 210 72 390 Analysis of a L-shape rotated 270 degrees clockwise on a 9 by 9 number grid From this table and the two number grids above we are now able to work out a formula for this particular type of L-shape by only having the two variables of the L-number and the grid size.

  1. Objectives Investigate the relationship between ...

    formula is better as it doesn't require an established T-total to be worked out, all I need is the T-number of the T-shape and I will be able to work it out. * T21 10 11 12 19 20 21 28 29 30 * Tn n-11 11 12 n-2 n-1

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

    Finding relationships on grids with sizes other than 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

  1. Maths Coursework T-Totals

    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. Maths GCSE Investigation - T Numbers

    128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 In the above grid, the T-number, 16, has been translated on a vector of (2, -3), to produce a new T-shape. I will need to repeat the translation and produce a third T-shape to find a pattern.

  1. Maths GCSE Coursework – T-Total

    T=5v-18 We can even say that; T=5v-2g As 9 is the grid size, and all numbers in our relationship grid were related around 9, so we can concluded that it is related to the grid size. We can test this on different positions on the 9x9 grid.

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

    This table is proving that the T shape above with the equations in it works. This example is from the T shape in a 9 x 9 grid found on the previous pages.

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