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Number Stairs Coursework

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Introduction

Number Stairs Coursework

Mathematics

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Number stairs coursework

Aim:

I have been given a 10 by 10 number grid (as shown below) with a stair shape drawn on it. The stair shape is a 3–step stair and the total of the numbers inside it is 194.

91

92

93

94

95

96

97

98

99

100

81

82

83

84

85

86

87

88

89

90

71

72

73

74

75

76

77

78

79

80

61

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

59

60

41

42

43

44

45

46

47

48

49

50

31

32

33

34

35

36

37

38

39

40

21

22

23

24

25

26

27

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

I am required to carry out the followings:

  • Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid for other 3-step stairs.
  • Part 2: To investigate further the relationship between the stair totals and other step stairs on other number grids.

Plan for Part 1

I shall now investigate part 1 where I will show and explain the relationship between the stair total and the position of the stair shape on the grid for other 3- step stairs. The stair shape on the diagram is a 3 step stair because both the length and the width of the stair are 3 steps. Part 1 is basically all about drawing patterns and conclusions from the relationships, and constructing formulas based on them. To do this, I will change the position of the 3-step stair from the bottom left hand corner (see blue on diagram) from positions 1-5 (see green), vertically across the grid and calculate the total of the numbers inside it.

...read more.

Middle

Total = an + b                           Therefore a= 6

 6+b = 50

     b = 50-6

     b = 44

Therefore, 6n+44 can be used as a formula for every position possible on the number grid.

Testing my Predictions

To prove that my formula works, I can test my predictions by firstly manually calculating the total of the 3-step stair, and then algebraically calculating the total using my formula finally I will randomly select three positions to test. If my formula works I should have the same total for both of the stairs;

  • Position 58 = 58 + 59 + 60 + 68 + 69 + 78 = 392  

By using my formula (6n + 44), I should get the same total, thus if n = 58,

6 x 58 + 44 = 392

  • Position 16 = 16 + 17 + 18 + 26 + 27+ 36 = 140

By using my formula (6n + 44), I should get the same total, thus if n = 16,

6 x 16 + 44 = 140

  • Position 32 = 32+ 33 + 34 +  + 27+ 36 = 140

By using my formula (6n + 44), I should get the same total, thus if n = 16,

6 x 16 + 44 = 140

Plan for Part 2

For part 2 of this task I shall investigate further the relationship between the stair totals and other step stairs on other number grids. To do this, I will change the size of the number stair (from a 1-step stair to a 5-step stair) and calculate the total of the algebraic letters inside it. This can indicate what the relationship between the totals of the number stairs are compared to their sizes.

...read more.

Conclusion

Extending the Investigation

There are a number of possible ways to extend this investigation. For example, it is possible to determine formulas for any step stair on say an 8 by 8 grid and then comparing them with those from a 10 by 10 grid. This can be further extended to finding an overall formula for any step stair on say an 8 by 8 number grid as shown below:

I shall first investigate what the formulas are for other step stairs in order to find an overall formula for any step stair possible on an 8 by 8 grid. I then compare this to other step stairs.

1-Step stair

n

Total: n

Formula: 1n

2-Step stair

n+9

n

n +1

Total: n + n + 1 + n + 9

Formula: 3n + 10

3-Step stair

n + 17

n +9

n + 10

    n

  n +1

 n + 2

Total: n + n + 1+ n + 2 + n + 9 + n + 10

n + 17

Formula: 6n + 39

4-Step stair

n + 25

n +17

n +18

n +9

n + 10

n + 11

    n

  n +1

 n + 2

n + 3

Total: n + n + 1+ n + 2+ n + 3 + n + 9 + n + 10 + n + 11+ n + 17 + n + 18 + n + 25

Formula: 10n + 96

5-Step stair

n +33

n +25

n+ 26

n +17

n +18

n + 19

n +9

n + 10

n + 11

n + 12

    n

  n +1

 n + 2

n + 3

n + 4

Total: n + n + 1+ n + 2+ n + 3 + n + 4 + n + 9 + n + 10+ n + 11+ n + 12 + n + 17 + n + 18+ n + 19 + n + 25 + n + 26 + n + 33

Formula: 15n + 190

By looking at the number stairs I can see another pattern similar to the numbers on a 10 by 10 grid.

10 by 10 grid        8 by 8 grid

Step Stair (s)

           Formulas

1

n

2

3n  + 11

3

6n  +  44

4

10n + 110

5

15n + 220

Step Stair (s)

           Formulas

1

n

2

3n  + 10

3

6n  +  39

4

10n + 96

5

15n + 190


...read more.

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