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  • Level: GCSE
  • Subject: Maths
  • Word count: 4430

T-Totals. Firstly I am going to do a table of 5 x 5 and look at the T-totals and T-numbers.

Extracts from this document...

Introduction

Coursework Number One

T-Totals

I am going to investigate the relationship between the T-totals and the T-number. Firstly I am going to do a table of 5 x 5 and look at the T-totals and T-numbers.

Here is a 9 by 9 table

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The total number inside the T-shape is

1 + 2 + 3 + 11 + 20 = 37

The T-Number is 20

Now I am going to put the T-shape in another location

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The T-number is 21

The T-total for this T-shape is 2 + 3 + 4 + 12 + 21 = 42

I will continue to place the T-shape in different location.

The T-number is 22

3 + 4 + 5 + 13 + 22 = 47

T-total = 47

The T-number is 23

4 + 5 + 6 + 14 + 23 = 52

T-total = 52

The T-number is 24

5 + 6 + 7 + 15 + 24 = 57

T-total = 57

The T-number is 25

6 + 7 + 8 + 16 + 25 = 62

T-total = 50

The T-number is 26

7 + 8 + 9 + 17 + 26 = 67

T-total = 67

The rest of the answers will be put into a table.

T-number

T-total

T-number

T-total

T-number

T-total

20

42

43

157

66

272

21

47

44

162

67

277

22

52

45

167

68

282

23

57

46

172

69

287

24

62

47

177

70

292

25

67

48

182

71

297

26

72

49

187

72

302

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77

50

192

73

307

28

82

51

197

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29

87

52

202

75

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30

92

53

207

76

322

31

97

54

212

77

327

32

102

55

217

78

332

33

107

56

222

79

337

34

112

57

227

80

342

35

117

58

232

81

347

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82

352

37

127

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252

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142

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257

86

372

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87

377

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152

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267

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382

Here is a 5 by 5 table

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The total number inside the T-shape is

1 + 2 + 3 + 7 + 12 = 25

The T-Number is 12

Now I am going to put the T-shape in another location

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The T-number is 13

The T-total for this T-shape is 2 + 3 + 4 + 8 + 13 = 30

I will continue to place the T-shape in different location and once I have enough data I will put it into a table.

The T-number is 14

3 + 4 + 5 + 9 + 14 = 35

T-total = 35

The T-number is 15

4 + 5 + 6 + 10 + 15 = 40

T-total = 40

The T-number is 16

5 + 6 + 7 + 11 + 16 = 45

T-total = 45

The T-number is 17

6 + 7 + 8 + 12 + 17 = 50

T-total = 50

The T-number is 18

7 + 8 + 9 + 13 + 18 = 55

T-total = 55

I am now assuming that the T-total for the T-number 18 is 55 because looking at the T-total results, it is going up in 5.

T-number

T-total

12

25

13

30

14

35

15

40

16

45

17

50

18

55

19

60

20

65

21

70

22

75

23

80

24

85

I will test two of my guesses to see whether or not my assumption is correct.

...read more.

Middle

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25

I will refer these numbers to these letters below, with n being the T-number.

a

b

c

d

n

The value of a is n – 11

The value of b is n – 10

The value of c is n – 9

The value of d is n – 5

So this is how the table now looks

n -11

n - 10

n - 9

n - 5

n

I know that that everything in this T-shape is equal to the original answer therefore I can add these up to find the T-total.

y = T-total and n = T-number

The T-total is = n – 11 + n – 10 + n – 9 + n – 5 + n

y = 5n – 35

Now I will test this formula with the T-total that I have already done.

n = 14 and the y = 35

If the total was not known then I would put n into the formula

y = 5 ( 14 ) – 35

y = 70 - 35

y = 35

The formula worked but I will do one more to make sure.

If n is 19 and my y = 60  

y = 5 ( 19 ) – 35

y = 95 – 35

y = 60

Therefore the formula for any T-number with a 5 by 5 table is

 n = T-number    y = T-total

 y = 5n - 35

Now I will investigate the T-total on a 6 by 6 grid.

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The way I have worked out the formula on a 5 by 5 grid can also be used on a 6 by 6 grid.

a

b

c

d

n

n = 14

The value of a is n – 13

The value of b is n – 12

The value of c is n – 11

The value of d is n – 6

So this is how the T-shape looks

n -13

n - 12

n – 11

n - 6

n

I will add up all the value inside the T-shape

y = n – 13 + n – 12 + n – 11 + n – 6 + n

y = 5n – 42

I will test this theory to see if it correct for any T-number

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If n = 17 and y = T-total

y = 4 + 5 + 6 + 11 + 17

y = 43

So my formula is 5n - 42

y = 5n - 42

y = 5 ( 17 ) – 42

y = 85 -42

y = 43

The formula works however I will do a few more to make sure.

n = 35

y = 5n – 42

y = 5 ( 35 ) - 42

y = 175 – 42

y = 133

Check:

y = 22 + 23 + 24 + 29 + 35

y = 133

n = 15

y = 5n – 42

y = 5 ( 15 ) - 42

y = 75 – 42

y = 33

Check:

y = 2 + 3 + 4 + 9 + 15

y = 33

n = 20 and T-total = y

y = 5n – 42

y = 5 ( 20 ) - 42

y = 100 – 42

y = 58

Check:

y = 7 + 8 + 9 + 14 + 20

y = 58

Therefore the formula for any T-number with a 6 by 6 table is

 n = T-number    y = T-total

 y = 5n - 42

I will now put all my results into a table

n

5n - 42

y

14

5 ( 14 ) - 42

28

15

5 ( 15 ) - 42

33

16

5 ( 16 ) - 42

38

17

5 ( 17 ) - 42

43

18

5 ( 18 ) - 42

48

19

5 ( 19 ) - 42

53

20

5 ( 20 ) - 42

58

21

5 ( 21 ) - 42

63

22

5 ( 22 ) - 42

68

23

5 ( 23 ) - 42

73

24

5 ( 24 ) - 42

78

25

5 ( 25 ) - 42

83

26

5 ( 26 ) - 42

88

27

5 ( 27 ) - 42

93

28

5 ( 28 ) - 42

98

29

5 ( 29 ) - 42

103

30

5 ( 30 ) - 42

108

31

5 ( 31 ) - 42

113

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5 ( 32 ) - 42

118

33

5 ( 33 ) - 42

123

34

5 ( 34 ) - 42

128

35

5 ( 35 ) - 42

133

Looking at this table I notice that the T-total has a difference of 5 which the 5 by 5 table also has a difference of 5. My guess is that the 7 by 7 table also has a difference of 5.

Here is a table 7 by 7

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49

I will find the formula for this table.

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d

n

n = 16

The value of a is n – 15

The value of b is n – 14

The value of c is n – 13

The value of d is n – 7

So this is how the T-shape looks

n -15

n - 14

n – 13

n - 7

n

I will add up all the value inside the T-shape

y = n – 15 + n – 14 + n – 13 + n – 7 + n

y = 5n – 49

So now I will test this to make sure that it is correct

If n = 16 and y = T-total

y = 1 + 2 + 3 + 16 + 9

y = 31

So my formula is 5n - 49

y = 5n - 49

y = 5 ( 16 ) – 49

y = 80 -49

y = 31

The formula works and now I will see if there is a difference of 5 between each T-total.

n

y

16

31

17

36

18

41

19

46

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51

21

56

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61

23

66

24

71

25

76

26

81

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86

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111

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126

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151

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166

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171

45

176

46

181

47

186

48

191

I now realise that because n is a multiple of 5 the difference is always going to be 5.

Therefore the formula for any T-number with a 7 by 7 table is

image00.png

 n = T-number    y = T-total

 T-total = 5n - 49

So from now I would not need to do detail working out, so I will just find out the formula for all the other grids because they all follow a same pattern in working out the formula.

Here is a table 8 by 8

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a

b

c

d

n

n = 18

The value of a is n – 17

The value of b is n – 16

The value of c is n – 15

The value of d is n – 8

So this is how the T-shape looks

n -17

n - 16

n – 15

n - 8

n

I will add up all the value inside the T-shape

y = n – 17 + n – 16 + n – 15 + n – 8 + n

y = 5n – 56

Instead of writing out the whole table to find a formula I will try to find it by using only a part of a table.

Therefore the formula for any T-number with a 8 by 8 table is

 n = T-number and y = T-total

 y = 5n - 56

I see that the way I have worked out the formula previously will be the same for the rest of the grids.

Grid

Formula

5 by 5

5n - 35

6 by 6

5n - 42

7 by 7

5n - 49

8 by 8

5n - 56

9 by 9

5n - 63

10 by 10

5n - 70

11 by 11

5n - 77

12 by 12

5n - 84

13 by 13

5n - 91

14 by 14

5n - 98

15 by 15

5n - 105

16 by 16

5n - 112

...read more.

Conclusion

 y = 5n - 7 with any x by x grid, (only a sign change). This is because when the T-shape was flip the signs also changed but the formula was the same. So for the T-shape rotated at 90°, it is also being flipped therefore causing the sign to change as well from + to -.

I will do check one with a 15 by 15 grid to see if it is correct.

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The T-total for this shaded area is 108

So now I will see if my formula is correct,

y = 5n – 7, where n = 23

y = 5 ( 23 ) – 7

y = 115 – 7

y = 108

The formula works.

T-shape rotated at 270°

Therefore the formula for any T-number with any x by x grid is

n = T-number and y = T-total and x = grid number

 y = 5n + 7

Summery

Here are all the formulas for the T-Shape rotated at different degrees.

Therefore the formula for any T-number with x by x grid is

n = T-number and y = T-total and x = grid number

 y = 5n – 7x

T-shape rotated at 180°

Therefore the formula for any T-number with x by x grid is

n = T-number and y = T-total and x = grid number

 y = 5n + 7x

T-shape rotated at 90°

Therefore the formula for any T-number with any x by x grid is

n = T-number and y = T-total and x = grid number

 y = 5n + 7

T-shape rotated at 270°

Therefore the formula for any T-number with any x by x grid is

n = T-number and y = T-total and x = grid number

 y = 5n + 7

All the formulas are similar and the differences are only a sign change ( + or – ) or the formula either includes an x or not.

Zarlasht Sardar

Mathematics GCSE

...read more.

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