# This Piece of course work is to investigate the relationship between the T Total and the T Number . The T shapes will be 3 squares across and 3 squares down and they will be on a 9*9 grid.

Extracts from this document...

Introduction

This Piece of course work is to investigate the relationship between the T Total and the T Number . The T shapes will be 3 squares across and 3 squares down and they will be on a 9*9 grid.

Firstly I will draw some T shapes onto the grid and see if I can spot a relationship.

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

11

20

T Number=20 T Total=1+2+3+11+20=37

14

15

16

24

33

T Number=33 T Total=14+15+16+24+33=102

52

53

54

62

71

T Number=71 T Total=52+53+54+62+71=292

55

56

57

65

74

T Number=71 T Total+55+56+57+65+74=307

Although those results could give me enough information to find a formula to find the T Total from the T Number, Now I will draw up some more T Shapes but this time all adjacent so that when I draw a table it will be easier to spot the relationship.

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

1

2

3

11

20

T Number=20 T Total=1+2+3+11+20=37

2

3

4

12

21

T Number=21 T Total=2+3+4+12+21=42

3

4

5

13

22

T Number=22 T Total=3+4+5+13=22=47

4

5

6

14

23

T Number=23 T Total=4+5+6+14+23=52

5

6

7

15

24

T Number=24 T Total=5+6+7+15+24=57

If I now put the these results into a table I will be able to see a pattern more clearly

T Number : 20 21 22 23 24

…………..……………………………..

T Total : 37 42 47 52 57

The T Total goes up in 5’s because there are 5 squares in each T and each Square in The T goes up 1 adding up to 5.

Middle

Again I will substitute my T Number with N and then change the other numbers in relation with N so I can find the formula.

7

8

9

13

18

Change this Yellow T To This Red T

N-11

N-10

N-9

N-5

N

If I add up all the N T I end up with the formaula t=5n-35

So to check this with our original answer of just adding up the answer of the formula so

T=5n-35 5n=90

90-35=55

T=7+8=9+13+18

T=55

So that proves that 5n-35 is correct on a 5*5 grid

Now I Will test out a 4*4 grid to make sure that every grid has a different formula.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

To find the formula I will again need to substitute the T number with N and the other numbers in relation with N

6

7

8

11

15

change this blue T into this Green T

N-9

N-8

N-7

N-4

N

So if you add up all the N T Shape we get the formula T=5N-28

So Now if I add up the original T shape I get:

T=6+7+8+11+15

T=47

Now I need to test if the formula gives me the same answer

T=5n-28 5n=75

T=75-28

T=47 So that proves that 5n-28 is the correct formula for a 4*4 grid.

Conclusion

90 Clockwise

T=5n+7

180

T=5n+7g

90 Anti clockwise

T=5n-7

Note that 90 Degrees Rotations do not include ‘g’ the grid size so these formulae are independent of grid size.

I will now Rotate a T shape 180 Degrees about an external point using the vectors {2}

{-1}

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

I have noticed that if u double the vector u reach the corresponding T Number straight away and if we use x and y instead of numbers we will now have 2*{ x }

{-yg}

So if we go 2x across from The T Number 30 we get to 34 and go down 2*yg

we take away two grid sizes which is equal to 18 which is also equal to going down 2 squares so we get the formula:

T=5N+7G+5(2x-2YG)

I will now apply a combined transformation so all I need to do is find the formula for the second transformation as I have the formula for a 180 degrees rotation about an external point: T=5N+7G+5(2A-2BG)

So If I Now translate by a vector {s}

{t} the formula will also include the translation {s}

{-gt} for each T Number , therefore the total; formula for the T-Total will be:

T=5n+7g+5(2x-2YG+S-GT)

Where T=T Total

N= T Number

G=Grid Size

{x} ……… is vector for point of rotation

{y}

{s}…………..is Translation Vector

{t}

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