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

number stairs

Extracts from this document...

Introduction

Nazma Khan 10A

Maths coursework image00.png

Number Stairs

In this coursework, I will be investigating the relationship between the stair totals and the position of the stair shape on the grid. I will be using the 3 step stair on a 10 by 10 grid to find out a relationship, which will help me to solve any problem regarding to solve the 3 step stair. The 10 by 10 grid is order in the reverse order from 1-100, starting from 1 situated at the bottom left hand corner and the 100 at the top right hand corner. However in order to see a sequence I would be carrying out this investigation in a systematic order in order to see a pattern. The way in which I will be systematic is by starting from the smallest number and the moving 1 box to the right each time.

This is how the grid looks like on a 10 by 10 grid and also how the 3 step stair looks like:

91

92

93

94

95

96

97

98

99

100image01.png

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

...read more.

Middle

image02.png

image02.pngimage02.pngimage02.png

        17+18+19+27+28+37= 146

As you can see I have moved the step stair systematically 1 step towards the right, and also I first started off with the smallest number first, in order to see a pattern emerging.

Therefore the difference is 6. So you multiply 6 x 14 =84

Then to get the answer 128 you + 44.              84 + 44= 128

Furthermore I will be extending my investigation further by changing the size of steps, e.g. 1 step stair, 2 step stair…, on a 10 by 10 grid. I believe that as a result to this it will allow me to see a link between the stair total, therefore I will be trying to get an overall formula which will work for the entire step stair on a 10 by 10 grid.

I will be using the same method but in algebra form.

image04.png

        χ        

image04.png

image04.pngimage04.png

        χ+χ+1+χ+10 = 3χ + 11

image04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.png

                             χ+χ+1+χ+2+χ+10+χ+11+χ+20= 6χ+44

image04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.png

                 χ+χ+1+χ+2+χ+3+χ+10+χ+11+χ+12+χ+20+χ+21+χ+30 =10χ+110

image04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.png

        15χ+1+2+3+4+10+11+12+13+20+21+22+30+31+40=

                                                        15χ+220

Now that I have found the formula for the step stairs shown above, I will be taking it further in order to get an overall formula for any size step stair which works on a 10 by 10 grid.

...read more.

Conclusion

(½ n² + ½ n) χ+ 11/3 (n-1) (1/2 n²+1/2 n)

(1/2 16 + ½ 4) χ + 11/3 (4-1) (1/2 16 + ½ 4)

10χ + 11/3 x 3 x 10

=10 χ + 110

This was to find out the 4 step stair.

Now to check if it works for 8 step stair

(1/2 8 + 4) χ+ 11/3 (8-1) (½ 8 + ½ 8)

(32 + 4) χ + 11/3 (7) (32+44)

     36X 11/3 χ 7 X 36

     36X + 924                           Therefore I have found out a formula for 8 step stair

Now I am going to find out a formula for any size step on any size grid.

image04.png

image04.pngimage04.png

        χ +χ+1+χ+2+χ+3+χ+A+χ+A+1+χ+A+2+χ+2A+χ+2A+image04.pngimage04.pngimage04.png

                                        χ+3A.image04.pngimage04.pngimage04.pngimage04.png

Step stair                               Formula                Difference

n=1                                        χ       +0                    

                                          1              

n=2                                        3χ+A+1

                                                                                    3                    

n=3                                        6χ+4A+4

                                                                                    6

n=4                                        10χ+10A+10

                                                                                    10

n=5                                        15χ+20A+20

                                                                                    15

n=6                                        21χ+35A+35

Therefore by looking at the differences I can see that it is the same as the co=efficient of χ. Therefore the formula for any size grid and step stair is:

T= (1/2 n² + ½ n) χ+ 1/3 (n-1) (1/2 n² + 1/2n) a + 1/3 (n-1) (1/n² + 1/2n)

This is the formula for the grid. However I believe that you can simplify this formula which now looks like this:

T= (1/2 n² + ½ n) χ+ 1/3 (n-1) a + 1/3 (n-1)

You can simplify this formula even further:

T= ½ (n² + n) (χ +1/3 (n-1) (a+1))

In conclusion I state that the formula that I have found works for the step stair, and grid. Therefore it is useful to have theses formulas because it makes problem solve in an efficient way, therefore it allows the problem to be solved in the quickest way.

Ms Akbar

Maths

...read more.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences 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 Number Stairs, Grids and Sequences essays

  1. Number stairs

    grid size I am going to test this general formula on a 23 by 23 and a 33 by 33 Number Grid below: From the processes I have gone through before, I have devised an algebraic for any 4-step stair on any grid size.

  2. Number Grid Investigation

    There are significant patterns throughout these results. In the 3x column across the top and on the left hand side the numbers go up in multiples of 16's, this the 2 times table multiplied by 8. For the 4x column it's the 3 times table multiplied by 8...as you can

  1. Number Grid Investigation.

    that in a 6 X 6 square, the product difference will be 200. Let's try... 1 2 3 4 5 6 9 10 11 12 13 14 17 18 19 20 21 22 25 26 27 28 29 30 33 34 35 36 37 38 41 42 43 44 45 46 (1 X 46)

  2. Number Stairs Maths Investigation

    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

  1. For other 3-step stairs, investigate the relationship between the stair total and the position ...

    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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

  2. Number Stairs

    I have done my further investigation for the 3 step stair for the 9x9 grid, now I am going to undertake further investigation for 3 step stair for the 8x8 grid. Here is an 8x8 grid showing the stair total and the stair number: Stair number (n)=1 By calculating the

  1. Mathematical Coursework: 3-step stairs

    2+3+4+12+13+22=56 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

  2. number grid investigation]

    Although it is quite certain that this trend would be observed in all number boxes of this instance, it is necessary to find an algebraic formula to prove that the difference remains invariable. Any 5x5 square box on the 10x10 grid can be expressed in this way: n n+1 n+2

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