• 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
  23. 23
    23
  24. 24
    24
  25. 25
    25
  26. 26
    26
  27. 27
    27
  • Level: GCSE
  • Subject: Maths
  • Word count: 5822

T-Shapes Coursework

Extracts from this document...

Introduction

image00.pngimage01.pngimage01.pngimage00.png

T-TOTALS

An MYP Investigation

Set: 24th January

Due: 8th February

Woodside Park International School

Amrit Morokar 11k


In this investigation, I will aim to find any relationships between grid sizes and T shapes within their relative grids, I will show and explain all generalizations I can find, using the T-Number (Tn or n) (the number at the bottom of the T-Shape) and the grid width (g) to find the T-Total (Tt or t) (Total of all number added together in the T-Shape), with different grid sizes, transformations, rotations, enlargements and combinations of all three.

NB:Throughout my investigation, I will sometimes be referring to the T-total as Ttand the T-number as Tn.

We will first be conducting our investigation on a 9x9 grid, and finding our T-shapes horizontally and vertically on it. This is the blank table on which we will draw our first T-shape, and from here I will continue to show you translations horizontally until I get as many as I can, then I will go on and translate the original T-shape vertically downwards, before going further and looking at different size grids, and rotations, transformations and enlargements.

We have a 9x9 Grid, which goes from 1 – 81, on which I will be conducting part one of my investigation. I have shown this below:

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

For the T-shapes that I will be investigating, I will find out what the T-totals and the T-numbers are, and try to find a pattern or relationship between them, recording my findings along the way, in easy-to-read data charts or tables.

This is an example T-Shape that I will be using to show you how to find the T-Total and T-Number, which I took from the above 9x9 grid.

1

2

3

11

20

T-Total: Found, by adding up all the numbers within the T-shape. The sum of these numbers equals the T-total.

...read more.

Middle

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

1st T-Shape:

T-Total

1 + 2 + 3 + 11 + 20

= 37

T-Number

= 20

2nd 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

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

2nd T-Shape:

T-Total

10 + 11 + 12 + 20 + 29

= 82

T-Number

= 29

3rd 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

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

3rd T-Shape:                                                          T-Total

19 + 20 + 21 + 29 + 38

= 127

T-Number

= 38

4th 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

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

4th T-Shape:

T-Total

28 + 29 + 30 + 38 + 47

= 172

T-Number

= 47

5th 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

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

5th T-Shape:

T-Total

37 + 38 + 39 + 47 + 56

= 217

T-Number

= 56

6th 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

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

6th T-Shape:

T-Total

46 + 47 + 48 + 56 + 65

= 262

T-Number

= 65

7th 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

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

7th T-Shape:

T-Total

55 + 56 + 57 + 65 + 74

= 307

T-Number

= 74

Recordings & Findings

I have collected data from all of the possible T-shapes vertically. I will now record my findings into a table, and see what patterns I can find, and then determine whether there is a link between the T-number and the T-total.

T-Shape

Numbers Within T-Shape

T-Number

T-Total

1

1, 2, 3, 11, 20

20

37

2

10, 11, 12, 20, 29

29

82

3

19, 20, 21, 29, 38

38

127

4

28, 29, 30, 38, 47

47

172

5

37, 38, 39, 47, 56

56

217

6

46, 47, 48, 56, 65

65

262

7

55, 56, 57, 65, 74

74

307

Patterns we can notice:

T-Number

T-Total

20

37

29

82

38

127

47

172

56

217

65

262

74

307

image08.pngimage08.pngimage08.pngimage08.pngimage02.pngimage08.pngimage08.png

image02.pngimage09.pngimage10.png

image02.pngimage02.pngimage02.pngimage02.png

We can see clearly that there is a pattern and a relationship within these numbers. For every nine the Tn goes up, the Tt goes up by forty-five.

We can understand that the numbers go up nine times as much vertically, than they do horizontally, because there are 9 rows in the grid, and the numbers increase horizontally by one. By moving the T-Shape on the vertical, each number increases by 9 each time, because it is a 9x9 grid.

This means that in the horizontal if the T-number went up by 1 each time, in the vertical T-Shapes, the T-Number would go up by 1 x 9 = 9 each time, and we apply the same thing for the T-Total; 5 x 9 = 45 each time.

This works out correct.

Using Algebraic Numbers to Find a Formula

In the Vertical

I will express these T-Shapes in algebraic form, using the nth term, and see if I can find a pattern that applies to all of the T-shapes.

n-19

n-18

n-17

n-9

n

19

20

21

29

image06.png

38

10

11

12

20

image07.png

29

n-19

n-18

n-17

n-9

n

We can see a similar pattern that applies to both T-Shapes, using the nth term, and so we are going to use these T-shapes to help us find the formula. Remember that these are the T-shapes going vertically.

Tt =n + (n-9) + (n-18) + (n-17) + (n-19)

If we multiply out the brackets, we get:

Tt =5n – (9+18+17+19)

Or

Tt = 5n – 63

Remember: ‘n’ is the T-Number in this equation.

1

2

3

4

5

...read more.

Conclusion

  • Contained in the first two rows of a grid, be it any size.
  • Contained in the first or the last column of a grid, be it any size.

Therefore, after coming to this conclusion, we can see that it would be impossible for any of our formulae to work, if our T-Number is situated on one of the above anomalies.

We conclude that to find the formula to any T-Shape in a grid, the T-Number of that shape must not be: a) contained in the first two rows of that grid, or b) contained in the first or last column of that grid.

Conclusion

To conclude, throughout this investigation I have analyzed T-Shapes in different grid sizes and systematically (step-by-step), using many different symbols and Geometrical language relating to the T-Shape problem, to help me find a generalised formula, for working out a relationship between T-Numbers and T-totals, taking into account grid sizes, transformations, and rotations.

Throughout the investigation, which I have now conducted, I became more and more aware of the reliability of my findings, by testing my predictions, using trial and error, and checking and re-checking, until I could confirm that a reliable pattern could be established.

At the end of the day, I discovered that no matter what, after you find one generalised formula, you can find the formula to a lot of things that are along the same lines, very easily.

This investigation was useful to me because it helped me develop my ability to find formulas using algebra, and generalise them, which personally I thought was one of my  weaknesses in Maths, but it was a skill I had to draw upon during this investigation, as it was very central to what we were looking at. It has very much improved my ability. Also, this coursework has helped my mathematical vocabulary and the use of my mathematical language overall.

Thank You.

Amrit Morokar 11K

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

    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-Total Maths coursework

    +7 = 60+7 = 67 T- Total (7 by 7) Look at this T-shape drawn on a 7 by 7 number 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

  1. T-Totals. Aim: To find the ...

    27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 I have translated the shape (4,0). So x=4 T-number is 16. T=(5*16)-7G+(5*4) =51 T=5+6+7+13+20 =51 My formula works, but what if the translation is not y=0.

  2. T-total Investigation

    3: 3 4 5 6 7 14 T-no 23 T-total: 3+4+5+6+7+14+23 = 62 I put the T-no in one column and the T-total in the other. T-no T-total 21 48 position 1 1 7 22 55 position 2 1 7 23 62 position 3 I saw that the T-no is

  1. Urban Settlements have much greater accessibility than rural settlements. Is this so?

    This is because one of the exits leads to the A2, one to Dartford and one to Sidcup. When there are accidents on the A2, Bexley is used as a diversion. This creates even more traffic, which leads to enormous queues.

  2. Maths GCSE Coursework – T-Total

    the values we used in are table, we get the same answers, for example taking x to be 80: t = 80 + 80 - 9 + 80 - 19 + 80 - 18 + 80 - 17 = 337 And x as 52; t = 52 + 52 -

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

    It thus follows that for every unit increase in the t-number there will be an increase of 5 in the t-total. n-21 n-20 n-19 n-10 n The t-shapes in a 10�10 grid can be represented algebraically. The t-total, T, can therefore be written in terms of the t-number, n, as T= 5 n - 70.

  2. Objectives Investigate the relationship between ...

    n+n+1+n+2+n-6+n+10 =n+n+n+n+n+1+2+10-6 (gather like terms) ='5n+7' Therefore my algebraic formula for finding the T-total of any 90� rotated t-shape is '5n+7' I will now find the T-total of a T-shape rotated from a 90� angle starting position. Rotating 90� ...2 I will rotate this T35 t-shape in another 90�'s making it a 180� rotation in total.

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