• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17
18. 18
18
19. 19
19
20. 20
20
• Level: GCSE
• Subject: Maths
• Word count: 3422

# Staircase Coursework

Extracts from this document...

Introduction

Dominik Borgolte

Math GCSE Coursework

Dominik Borgolte

Buckswood School

GCSE Mathematic coursework

“Step stairs”

Centre number:

Candidate number: 2776

## Introduction

I am going to investigate formulas that can be used to calculate the sum of the number in a staircase on a grid paper.

In my investigation I will use:

St: stand for stair total ( S stand for stair and t for the total sum )

g: stand for the size of the grid (e.g. in 12x12 grid n=12)

n: stand for stair number, which is always the bottom left number

x. stand for the size of the stairs ( e.g. a 6 step stair would be x=6 )

## ( 1 )

I want to find a formula, which makes it easier for me to calculate the total size of a 3 step stair in a 10 x 10 grid.

10 x 10 grid

 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 31 32 33 34 35 36 41 42 43 44 45 46 47 48 49 50 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 19 20 21 22 23 24 21 22 23 24 25 26 27 28 29 30 13 14 15 16 17 18 11 12 13 14 15 16 17 18 19 20 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6

St: 1+2+3+11+12+21=50

I have added all the squares of the 3 step stairs together, which gives me a result of 50, for step1.

I will now find a formula for the total, which will make it easier to calculate the sum.

I will label a stair algebraically

This is an example of stair 1

n stands for the number in the left bottom corner of the step.

In order of that if : 1 = n          2 = n + 1     and so on ................

 n+20 n+10 n+11 n n+1 n+2
 21 11 12 1 2 3

=

Middle

177

178

179

180

151

152

153

154

155

256

257

258

259

160

161

162

163

164

165

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Formula:  6n + 4g +4

g = 15

n = 97

6x97 + 4x15 + 4 = 646

Test of the formula:

St:  97+98+99+112+113+127=646

So the formula also works for any other size grid

( 3a )

4 step stair on a 10x10 grid

## I will now try to find a formula for a 4 step stair on a 10 x 10 grid

4 step stair on a 10x10 grid

 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 use the same principle like I already did with the 3 step stair on a 10x10 grid.

So I make the bottom left number n, which gives me a huge formula.

In this case

n + (n + 1) + (n + 2) + (n + 3) + (n + 10) + (n + 11) + (n + 12) + (n + 20) + (n + 21) +(n+30)

Then I add up all normal numbers and all the n`s

So in this case it would be 10 times n and all the numbers added up would be 110.

This leaves me with the final formula

St = 10n + 110

I will now test the formula to see if it is correct by using step 46

Because n= 46St = 10x46 + 110= 570

St = 46+47+48+49+56+57+58+66+67+76 = 570

So the formula is correct

## Now I will try to find a formula for a 4 step stair on any grid.

I will use a 15x15 grid for my investigation

4-step stair on 15x15 grid

 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Conclusion

But still a few improvements could be made. I could for example increase the amount of tests on different sizes of stairs and grids

I would try out a 5 or 8 step stair, or test them at different size grids like I already used, but in the end the final result, which is the overall formula

[1/2x (x+1)] n + (g+1) [1/6(x[4]3-x)] would still be the same.

I had some help from my teacher and an older classmate. I also took one formula for the tetrahedral number out of the Internet. Other than that I did everything on my own, which is highly important in my opinion.

Therefore I think overall the coursework was a clear success, because I was able to find formulas for every different size step, and every different size grid investigation I made.

In the end I was even able to find a formula that can be used for any size grid and any size stair, so I could even achieve my final goal of the coursework.

I also think that this coursework was useful for my personal abilities, because I really improved my ability to think logically, and my ability to help my self when I am sucked at a certain point, which was the case (a few times) during this coursework.

[1]

[2]

[3]

[4]

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

# Related GCSE Number Stairs, Grids and Sequences essays

1. ## Number Grids Investigation Coursework

- (top left x bottom right) = 5 x 11 - 1 x 15 = 55 - 15 = 40 To prove that this is the same in all 2 x 5 grids, I will have to use algebra. Let the top left square of my rectangle equal a, and

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

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 x-30 x-20 x-19 x-10 x-9 x-8 X X+1 X+2 X+3 For example if x = 31

1. ## Number Stairs

Which are from 1 to 5. 14 17 20 23 26 +3 +3 +3 +3 Now that I solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 3n in the formula, the extra is calculated by working

2. ## GCSE Maths Sequences Coursework

Shape 2 in the cube sequence is made up off two "start shapes", two stage one's and stage 2 from the square sequence. I believe this pattern will continue. Using this information I will be able to get the total number of cubes for the first five stages of this sequence using the information from the squares tables.

1. ## Number Grid Investigation.

5 X 7 6 7 8 9 10 16 17 18 19 20 26 27 28 29 30 36 37 38 39 40 46 47 48 49 50 56 57 58 59 60 10 ( n - 1 )

2. ## Number stairsMy aim is to investigate the relationship between the stair total and the ...

I will now test it on a random 5 step staircase from a 6 by 6 grid. 25 19 20 13 14 15 7 8 9 10 1 2 3 4 5 6 step stairs. 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55

1. ## Maths coursework. For my extension piece I decided to investigate stairs that ascend along ...

for any given stair-number of a 3-stair shape on a 5 x 5 grid, I will draw out the 3-stair shape on a 5 x 5 grid in terms of 'n' and 'g'. Grid: 5 x 5 21 22 23 24 25 16 n + 2g 18 19 20 11

2. ## Maths Coursework: Number Stairs

6x + 48 = stair total (6 x 12) + 48 = 120 12 + 13 + 14 + 23 + 24 + 34 = 120 o9xuE This is true so I conclude that the stair total for any 3-step stair on an 11 by 11 number grid is 6x + 48= stair total.

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