Maths Coursework T-Total

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

20

21

22

23

T-total

37

42

47

52

T-number

24

25

26

27

T-total

57

62

67

72

T-number

28

29

30

31

T-total

77

82

87

92

T-number

32

33

34

35

T-total

97

02

07

12

T-number

36

37

38

39

T-total

17

22

27

32

This shows that whenever the T-number moves up by 1, the T-total number goes up by 5. This is because there are 5 numbers in each T-shape.

Next we will find a pattern that will connect the T-number to the T-total which is called the nth term.

First of all, the T-totals are going up by 5 each time. If we take the first T-shape ( red ) as an example, the T-number is 20. If you multiply this by 5 you will get 100. The T-total is 37, the difference being 63. So, the nth term will be 5n - 63 (5 x T-number - 63 = T-total).

To verify this we must make sure it follows through the grid.

Turquoise = 21 x 5 = 105, 105 - 63 = 42.

Green = 25 x 5 = 125, 125 - 63 = 62.

Pink = 32 x 5 = 160, 160 - 63 = 97.

This proves that the nth term on this grid is definitely, 5n - 63.

Would these rules follow through on a different grid, would the position of the T-shape make any difference?

Here is a 7 by 7 grid to find out.

2 3 4 5 6 7

8 9 10 11 12 13 14

5 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

T-number

6

7

8

9

T-total

31

36

41

44

T-number

20

21

22

23

T-total

51

56

61

66

T-number

24

25

26

27

T-total

13

76

81

86

This shows that the rule of going up in 5's applies in all size grids but only in T-shapes that are made in the same manner.

Because of this, we can only look at the red, green and turquoise

T-shapes to find the nth term of this grid.

Again, the T-totals are going up by 5.

6 x 5 = 80, 80 - 31 = 49.

20 x 5 = 100, 100 - 49 = 51.

25 x 5 = 125, 125 - 49 = 76.

So the nth term for the 7 x 7 grid is 5n - 49.

To prove this, we will do another grid with all T-shapes made in the same manner as each other.

2 3 4 5 6 7

8 9 10 11 12 13 14

5 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

T-number

0

1

2

3

T-total

43

48

53

58

T-number

4

5

6

7

T-total

63

68

73

78

T-number

8

9

20

21

T-total

83

88

93

98

T-number

22

23

24

25

T-total

03

08

13

18

The nth term on this grid is 5n - 7.

Here are some three examples to prove this;

0 x 5 = 50, 50 - 7 = 43.

3 x 5 = 65, 65 - 7 = 58.

25 x 5 = 125, 125 - 7 = 118.

Next we will try changing the size of the T-shape as well as the grid.

2 3 4 5

6 7 8 9 10

1 12 13 14 15

6 17 18 19 20

21 22 23 24 25

T-number

7

8

9

0

T-total

3

7

21

25

T-number

1

2

3

4

T-total

29

33

37

41

The nth term here is 4n - 15. Because;

7 x 4 = 28, 28 -15 = 13.

4 x 4 = 56, 56 - 15 = 41.

Would a different shape make a difference? This time will use an L-shape.

2 3 4 5

6 7 8 9 10

1 12 13 14 15

6 17 18 19 20

21 22 23 24 25

L-number

6

7

8

9

L-total

9

2

5

8

The nth term here is 3n - 9, as shown below;

6 x 3 = 18, 18 - 9 = 9.

9 x 3 = 27, 27 - 9 = 18.

In conclusion, you can see it is not the shape that affects the correlation or the nth term, but the number of numbers in the shape.