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

GCSE Mathematic Coursework T-totals Aim: to find a pattern that connects the T- number with the T- total

Extracts from this document...

Introduction

GCSE Mathematic Coursework

T-totals

Aim: to find a pattern that connects the T- number with the T- 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

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

p

41

42

43

44

45

46

47

48

p+r

50

51

52

53

54

55

56

p+2r-1

p+2r

p+2r+1

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

T- number = T 40                                                         p = T-number

T- total = 263                                                              q = T-total

Grid size = 9 x 9                                                          r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

Formula for T-shape: q = 5p + 7r

Justification: (5 x 40) + (7 x 9) = 263

image00.png

Example:

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

p

42

43

44

45

46

47

48

49

p+r

51

52

53

54

55

56

57

p+2r-1

p+2r

p+2r+1

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

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- number = T41                                                          p = T-number

T- total = 268                                                              q = T-total

Grid size = 9 x 9                                                         r = grid size

Calculations:

...read more.

Middle

41

42

43

44

45

46

47

p+r

49

50

51

52

53

54

55

p+2r-1

p+2r

p+2r+1

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

T- number = T39                                                          p = T-number

T- total = 258                                                              q = T-total

Grid size = 9 x 9                                                         r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

Formula for T-shape: q = 5p + 7r

Justification: (5 x 39) + (7 x 9) = 258

image00.png

The examples above give evidence to justify that the formula (q = 5p + 7r) works for all T- shapes that are rotated 1800.

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

I will now alter the grid sizes to try and justify as to whether my formula works on different grid sizes.

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

p

36

37

38

39

40

41

42

p+r

44

45

46

47

48

49

p+2r-1

p+2r

p+2r+2

53

54

55

56

57

58

59

60

61

62

63

64

T- number = T35                                                          p = T-number

T- total = 231                                                              q = T-total

Grid size = 8 x 8                                                         r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

Formula for T-shape: q = 5p + 7r

Justification: (5 x 35) + (7 x 8) = 231

The example above justifies clearly that the formula (q = 5p + 7r) works for all T-shapes that are rotated 1800.

1

2

3

4

5

6

7

p

9

10

11

12

p+r

14

15

16

p+2r-1

p+2r

p+2r+1

20

21

22

23

24

25

Now I am going to change the grid size to a completely contrasting size, this time the grid size is 5 x 5 and a contrasting T-number will be used.

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

T- number = T8                                                         p = T-number

T- total = 75                                                              q = T-total

Grid size = 5 x 5                                                       r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

Formula for T-shape: q = 5p + 7r

Justification: (5 x 8) + (7 x 5) = 75

Although this time the T-number I have used is different from the numbers used before the outcome is still exactly the same. This justifies that the formula (q = 5p + 7r) works for any T-shape that has a rotation of 1800 regardless of the T-number or the grid size.

image00.png

Overall after using various T-numbers and different grid sizes I have come to the conclusion that the formula for T-shapes that are rotated 1800 is (q = 5p + 7r). Using the formula I can now calculate different T-totals for different grid sizes and show a general pattern. Below are the patterns from the grid sizes I used.

9 x 9 general pattern:

T40

T41

T42

T43

263

268

273

278

As the T-numbers increase by 1, the T-total increases by 5.

8 x 8 general pattern:

T35

T36

T37

T38

231

236

241

246

As the T-numbers increase by 1, the T-total increases by 5.

5 x 5 general pattern:

T8

T9

T10

T11

75

80

85

90

As the T-numbers increase by 1, the T-total increases by 5.

My formula to calculate the T-total is the same for all of these grid sizes (when the T-shape is rotated 1800). This justifies my formula, (q = 5p + 7r).

image00.png

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

p-2r-1

p-2r

p-2r+1

33

34

35

36

37

38

39

p-r

41

42

43

44

45

46

47

48

p

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

...read more.

Conclusion

) works for all standard T-shapes. Below is another example but this time the grid size is 5 x 5.

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

p-2r-1

p-2r

p-2r+1

4

5

6

p-r

8

9

10

11

p

13

14

15

16

17

18

19

20

21

22

23

24

25

T- number = T12                                                        p = T-number

T- total = 25                                                              q = T-total

Grid size = 5 x 5                                                       r = grid size

Calculations:

p + p + p + p + p = 5p

-r + -2r + -2r + -2r = -7r

-1 + 1 = 0

Formula for T-shape: q = 5p – 7r

Justification: (5 x 12) - (7 x 5) = 25

image00.png

Overall after using various T-numbers and different grid sizes I have come to the conclusion that the formula for standard T-shapes is (q = 5p - 7r). Using the formula I can now calculate different T-totals for different grid sizes and show a general pattern. Below are the patterns from the grid sizes I used.

9 x 9 general pattern:

T49

T50

T51

T52

182

187

192

197

As the T-numbers increase by 1, the T-total increases by 5.

8 x 8 general pattern:

T38

T39

T40

T41

134

139

144

149

As the T-numbers increase by 1, the T-total increases by 5.

5 x 5 general pattern:

T12

T13

T14

T15

25

30

35

40

As the T-numbers increase by 1, the T-total increases by 5.

My formula to calculate the T-total is the same for all of these grid sizes (standard T-shape). This justifies my formula, (q = 5p - 7r).

...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. Marked by a teacher

    T-total coursework

    5 star(s)

    - 7w In the example above, when the 5 numbers of the reflected T-shape are added together, the T-total is 75. Using the formula, when (n) is 22, (w) is 10 and (h) is 2, the T-total is 75, which means the formula works.

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

    I will replace the T number with a x and adjust the rest of the numbers. Once I have done that I will try and figure out the connection and relationship. I will try this with a 9 x 9 grid and the T number will be 62.

  1. T-Shapes Coursework

    formula again with another shape on the horizontal, and then go on to the vertical, and see if a new formula is needed, or there is some kind of relationship in the formulae. 2 3 4 12 21 Our formula: Tt = 5n - 63 In this case, n = 21.

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

    If we say g is the grid size (e.g. 4 for 4x4 or 5 for 5x5). 1 2 3 4 5 6 7 8 9 If we work the T-Total on this 3x3 grid the "old" way, we get 19 (8 + 5 + 1 + 2 +3), that proves a formula of: t = x + x -

  1. T totals - translations and rotations

    The number directly above this is also 8 places back on the grid so it is N-8-8 = N-16. The remaining two numbers are N-16-1 and N-16+1. Thus the T-total is: N+ (N-8) + (N-16)

  2. T Total and T Number Coursework

    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. Objectives Investigate the relationship between ...

    To start off I will find the T-total of the following T-shapes: T22 and T32 * T22 1 2 3 11 12 13 21 22 23 1+2+3+12+22 = 40 * T32 11 12 13 21 22 23 31 32 33 11+12+13+22+32 = 90 T-shape T-total Increment T22 40 T32 90

  2. T-Shapes Coursework

    Using Pattern 1 above, we can say that the Sum of the Tail = n + g 2) Using the patterns from Section 1, we can still say that the Sum of the Wing = 3n 3)

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