• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 3772

T-Total Course Work

Extracts from this document...

Introduction

GCSE COURSE WORK T-TOTALS

GCSE MATHS T-Total Course Work

PART 1

I am investigating the relationship between the T-total and T-number on a 9 x 9 grid by using a variety of 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

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

21

1

2

3

T-number = 20

Numbers:

1                2                  3                11                20

20-1          20-2             20-3           20-4            20

T-19          T-18             T-17          T-9              T

T-total = 37

11

20

2

3

4

T-number = 21

Numbers:

2                3                  4                12                21

21-2          21-3             21-4           21-12          21

T-19          T-18            T-17           T-9              T  

T-total = 42

12

21

(T-19) + (T-18) + (T-17) + (T-9) + T

= 5 x T - 63

= 5T – 63

I tested that when:

T-Number = 80

T-Total = 337

337 = 5T – 63

337 = (5 x 80) – 63

= (5 x 80 = 400) – 63

= 400 -63

= 337

337 = 337

Formula is correct.

I also tested it when:

T-Number = 20

T-Total = 27

5T-63

= (5x20)-63

= 100-63

=73

Formula Works, it is correct

This proves that the Formula = 5T– 63 works and is correct for any T-Shape on a 9 x 9 grid

This shows that the connection between the T-Total and the T-Number is 5T-63:

Let T-number = T

Let T-total = n

T

n

20

21

22

               =63          

37

42

47

Formula = 5T – 63

Below is a t-shape, and in each cell how the number is connected with T

T-19

T-18

T-17

T-9

T

PART 2

I am now going to investigate the relationship between the T-total and T-number and the grid size by

 using a variety of grids and T-Shapes at different positions

8x8 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

1

2

3

T-number = 18

Numbers:

1                2                  3                10                18

18-1         18-2            18-3          18-10             18

T-17         T-16            T-15          T-8                 T

T-total = 34

10

18

...read more.

Middle

T-G ”, and for the next number, which is 2 as it is directly on top of 11, we have to times the Grid size by 2 and then minus it from the T-number. Like so            T-(2xG) “ which gives the algebraic formula of T-2G. For the number 1 in the grid, we have to add 1 to the formula as 1 is to the left of 2 making the amount needed to be subtracted more. So (T – (2G+1)) = 20-(2x9+1) = 20-19 =1

To find out the number 3, we have to subtract 1 from the formula used for number 2 as it is to the right of the number 2 and needs a smaller amount to be subtracted from the T-Number in order to give 3.  So, (T-(2G-1)) should equal 3… (20-(2x9-1))= 20-17 = 3. I have now found a formula for each value within the T-Shape. Now, in order to give me the value 5T-7G I added the formulae for each value in the T-Shape like so, (T) + (T-G) + (T-2G) + (T-2G+1) + (T-2G-1) = 5T-7G.

11

20

(T) = 20

(T-G) = 11

(T-2G) = 2

(T-(2G-1)) = 3

(T-(2G+1)) = 1

Add then all = 5T-7G

Here I have constructed a random GxG Grid to prove that my Formula of 5n-7G will work on any Grid size, and any position

15 x 15 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

82

83

84

85

86

87

88

89

90

32 Is my T-Number

T = 32

5T = 5x32 = 160

G = 15

7G = 7x15 = 105

I now have enough data to prove that my formula works

5T - 7G

160 – 105 = 55

T-Total = 1+2+3+17+32 = 55

So my Formula works because the T-Total = 55, and my Formula gives a T-Total for the T-Shape as 55.

9 x 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

T =59

5T = (5x59) = 295

G = 9

7G = (7x9) = 63

T-Total = (40+41+42+50+59) = 232

I will now apply my formula onto this T-Shape on the 9x9 Grid to prove it will work on any Grid size with a T-Shape at any position

5T-7G

(5x59)-(7x9)

= 295-63

= 232

Is equal to the T-Total, so my formula 5T-7G Works.

The relationship between the T-Total, T-Number and Grid Size are all summed up into the formula 5T-7G.

This formula can be used on any Grid Size, with the T-Shape at any position.

PART 3

I am going to investigate the relationship between the T-Total,        the T-Numbers, the Grid Size, and different transformations.

I am going to use the transformation of translation on the following T-Shape to the right on this 10 x 10 grid

10x10 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

387

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

1

2

3

T-Number = 22

Numbers:

1                2                  3                12                22

22-1         22-2            22-3          22-12              22

T-21         T-20           T-19           T-10               T

T-total = 40

12

22

3

4

5

 T-Number = 24

Numbers:

3                4                  5                14                24

24-3          24-4             24-5           24-14          24

T-21          T-20            T-19           T-10              T  

T-total = 50

14

24

I will now work out a general formula for a T-Shape that has been translated to the right, for any grid size because I can see the connection with a translation of 2 to the right and the grid.

(T-2G)+1

(T-2G)+2

(T-2G)+3

(T-G)+2

T+2

((T-2G)+1) + ((T-2G)+2) + ((T-2G)+3) + ((T-G)+2) + (T+2)

=5T-7G+10

I will now get a formula by drawing part of a GxG grid and using a T-shape that has been translated x times to the right.

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

1

2

3

T-Number = 32

Numbers:

1                2                  3                17                32

32-1         32-2            32-3          32-17              32

T-31         T-30           T-29           T-15               T

T-total = 55

17

32

7

8

9

T-Number = 38

Numbers:

7                8                  9                23                38

38-7         38-8            38-9          38-23              38

T-31         T-30           T-29           T-15               T

T-total = 85

23

38

9

10

11

T-Number = 40

Numbers:

9               10                 11               25                40

40-9         40-10         38-11          40-25             40

T-31         T-30           T-29           T-15               T

T-total = 95

25

40

Let x = number of times/squares translated to the right

I will now work out a general n that has been translated x times to the right

(T-2G)+x-1

(T-2G)+x

(T-2G)+x+1

(T-G)+x

(T+x)

((T-2G)+x-1) + ((T-2G)+x) + ((T-2G)+x+1) + ((T-G)+x) + (T+x)

=5T-7G+5x

This formula should work out the T-Total for the translated shape that has moved 6 times to the right, It is the Lined shape.

T = 32

G = 15

X = 6

= (5x32)-(7x15)+(5x6)

= 160 – 105 + 30

= 85

Which is the T-Total for the Translated T-Shape, this proves the formula works.

((T-2G)+x-1) + ((T-2G)+x) + ((T-2G)+x+1) + ((T-G)+x) + (T+x)

=5T-7G+5x

= my formula

...read more.

Conclusion

To make sure the formula works on any Grid I am going to test it 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

21

T= 20

G= 9

X=3

Y=5

5T-7G+5x+((5xG)xY)

= (5x20) – (7x9) + (5x3) + (45x5)

=100 – 63 + 15 + 225

=277

Which is the T-Total for the t-shape on the 9x9 Grid

49+50+51+59+68=277

This proves that the formula 5T-7G+5x+((5xG)xY)

will work on any grid with any translation of x spaces to the right and y places down.

I have also realised that this formula ONLY works on right and down, in order for it to work on left and up you will need to change the values of “x” and “y” from +ve to –ve. This will give you the ability to go left and up. I also noticed that to move x places to the left and y places down, you need to change the “x” to –ve and leave the “y” as +ve. And vice versa.  

THIS IS NOT NEEDED, JUST INCASE YOU WANT TO WORK ON ROTATION INSTEAD OF TRANSLATION, JUST THE BEGINNING OF IT.

I will now try and find therelationship between the T-Total, the T-Numbers, the Grid Size, and the rotation of the 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

387

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

I have rotated the T-Shape 90* Clockwise at point 32. I will now try and find the relationship between the T-Total, T-Number, Grid size and the rotation.

T=22, T-Total = 40

T=33, T-Total = 172

Omar El-Anis

...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. To prove that out of town shopping is becoming increasingly popular with shoppers, and ...

    find a free parking space on the roads leading into the own centre. But in an out of town shopping centre the same family will be able to find a parking space in minutes in the vast car parks and not have to pay a penny.

  2. Maths GCSE Coursework – T-Total

    (14), as it's center of rotation, this shape a T-Total of 72. If we rotate our T-Shape by 180 and 270 degrees clockwise, again it will be easier for us to build up a profile, and some generalizations. 1 2 3 4 5 6 7 8 9 10 11 12

  1. T-totals. I am going to investigate the relationship between the t-total, T, and ...

    - (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)

  2. 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, * T19 9 10 11 17 18 19 25 26 27 * Tn n-10 10 11 n-2 n-1

  1. T shapes. I then looked at more of these T-Shapes from the grid in ...

    To prove that this is true, I will label a T shape and inside it put the term 'S'. This term stands for the number of squares the T-shape has moved. N = 19 +5 N=18+5 N=17+5 N=9+5 N+5 With all my previous workings and examples I can strongly confirm

  2. For my investigation, I will be investigating if there is a relationship between t-total ...

    I then can simplify this to: T-total = 5N-77 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. For my investigation, I will be investigating if there is a relationship between t-total ...

    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

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

    As the T shape moves down one square, 8 is added as that is the amount of squares moved. Therefore, to move down more than one square, 5 (as there are 5 squares in the T) will have to be multiplied by the grid size and the amount of squares moving down.

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