• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 2015

Number Stairs Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid, for ot

Extracts from this document...

Introduction

Mathematics GCSE Coursework: Number Stairs                4/28/2007

Mathematics GCSE Coursework: Number Stairs

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. The number in the bottom left corner of the stair shape labels the position of the stair shape. The aim of this part of the investigation is to find a formula to work out the s-total (stair total) by knowing the s-number (stair number).

45

35

36

25

26

27

        25 + 26 + 27 + 35 + 36 + 45 = 194

        s-number = 25

        s-total = 194

21

11

12

1

2

3

        1 + 2 + 3 + 11 + 12 + 21 = 50

        s-number = 1

        s-total = 50

22

12

13

2

3

4

        2 + 3 + 4 + 12 + 13 + 22 = 56

        s-number = 2

        s-total = 56

23

13

14

3

4

5

        3 + 4 + 5 + 13 + 14 + 23 = 62

        s-number = 3

        s-total = 62

24

14

15

4

5

6

        4 + 5 + 6 + 14 + 15 + 24 = 68

        s-number = 4

        s-total = 68

Another observation here is that in each column of the stair shape, the difference between these rows of numbers in the same column, is 10. (Apart from the last column, which only consists of 1 number.) The difference between these numbers is 10 because in the grid, each row contains 10 numbers.

Table of Results:

s-number

s-total

1

50

+ 6

2

56

+ 6

3

2

+ 6

4

68

25

194

To go from one term to the next, simply add 6 each time. However, this does not show the relationship between the s-number and the s-total. This sequence (adding/subtracting a number each time), is known as an arithmetic sequence. A formula to find out the relationship between the s-number and the s-total is: first term + common difference x ( n – 1 ). The first term here is 50, and the common difference is 6.

        50 + 6(n – 1)

=        50 + 6n – 6

=        6n + 44

25

15

16

5

6

7

        5 + 6 + 7 + 15 + 16 + 25 = 74

        (6 x 5) + 44 = 74        image00.png

        s-number = 5

        s-total = 74

26

16

17

6

7

8

        6 + 7 + 8 + 16 + 17 + 26 = 80

        (6 x 6) + 44 = 80        image00.png

        s-number = 6

        s-total = 80

46

36

37

26

27

28

        26 + 27 + 28 + 36 + 37 + 46 = 200

        (6 x 26) + 44 = 200        image00.png

        s-number = 26

        s-total = 200

After testing the

...read more.

Middle

1

2

3

1 + 2 + 3 + 6 + 7 + 11 = 30

s-grid = 5

s-total = 30

13

7

8

1

2

3

1 + 2 + 3 + 7 + 8 + 13 = 34

s-grid = 6

s-total = 34

Table of Results:

s-grid

s-total

3

22

+ 4

4

36

+ 4

5

30

+4

6

34

10

50

In the table, it shows that the s-total increases by 4 each time as the s-grid increases by 1. (When g = grid size,) this is because:

n+2g

n + g

n+g+1

n

n + 1

n + 2

 Previously, I noticed that the difference between each row in the same column is the grid size.

        n + (n + 1) + (n + 2) + (n + g) + (n + g + 1) + (n + 2g)

=        6n + 4g + (4)**

Note: 6n + 4g  because there are 4g’s in the formula (as shown above).

Now that I have found out the relationship between the s-number, s-grid, and the s-total, I can carry this investigation further by investigating the relationship between the other step stairs, s-number, s-grid, and s-totals.

I will investigate the relationship between the s-number, s-total, s-grid and s-shape (stair shape). The s-shape will be investigated by enlargement; 1 square increase in length, and 1 square increase in width.

n

The s-number remains at the bottom left corner of the stair shape. Instead of 6 squares, there are 10. Based on my previous observations, I can predict the relationship between this s-shape, the s-number, s-grid, and the s-total as: 10n + 10g + 10.

I can check my prediction:

n+3g

n+2g

n+2g+1

n+g

n+g+1

n+g+2

n

n+1

n+2

n+3

n+(n+1)+(n+2)+(n+3)+(n+g)+(n+g+1)+(n+g+2)+(n+2g)+(n+2g+1)+(n+3g)

= 10n + 10g + 10

I shall investigate the next stair shape, which will have a height of 5 squares, and a width of 5 squares, a total of 15 squares.

n+4g

n+3g

n+3g+1

...read more.

Conclusion

= 21n + 35g + 35

image01.png

It appears that this section of the formula works. Now the new current formula is:

s-total =image02.png

Important: However, this formula has its limitations. For example, the step stair may be placed on a certain area that causes the step stair to be off the grid. This means that the formula cannot calculate the s-total as there are no numbers off the grid. Perhaps another limitation can be found if it cannot calculate negative numbers.

I shall test it out on an s-size of 7, s-grid of 8, and the s-number is 20.

68

60

61

52

53

54

44

45

46

47

36

37

38

39

40

28

29

30

31

32

33

20

21

22

23

24

25

26

20 + 21 + 22 + 23 + 24 + 25 + 26 + 28 + 29 + 30 + 31 + 32 + 33 + 36 + 37 + 38 + 39 + 40 + 44 + 45 + 46 + 47 + 52 + 53 + 54 + 60 + 61 + 68

= 1064

image03.pngimage00.png

Now to test the formula out on negative numbers:

13

14

15

16

17

18

7

8

9

10

11

12

1

2

3

4

5

6

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

-13

-14

-15

-16

-17

(-12)+(-13)+(-14)+(-6)+(-7)+(0)

= -52

image04.png

It appears that negative numbers do not work with this formula, thus the formula is limited to positive integers and that the stair shape must not have any part of it exceeding the grid. However, the formula works excluding the limitations. In conclusion, the relationship between the s-number, s-grid, s-size, and s-total is:

s-total = image02.png

(With certain restrictions as mentioned above)

However, this formula can be simplified because as previously mentioned, the ‘X’s were in 2 parts of the formula, thus it can be simplified. The first part of the equation where the formula for triangle number lies, can also be simplified to:

image05.png

An extention of this investigation, could be to investigate the relationship between negative numbers, or transformation of the stair shape.

                Page

...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. Marked by a teacher

    Maths coursework- stair totals. I shall be investigating the total and difference in sets ...

    3 star(s)

    The first grid size I will choose to use is 10 x 10 and the size stairs will be a four step size. After finding the totals for each of the stairs that I will use I shall work out the formula by using the same method as before.

  2. Number stairs

    Now the algebra formula is starting to emerge and is starting to look like this: 6x + 4n My next step is to test my general formula, I am going to use 15 by 15 Number grid as it the next in my table after 14 by 14 Number Grid

  1. Number Grid Coursework

    q, on a width z grid, the difference of the two products is always "the length of the box minus 1" times the "width of the box minus 1" times the "grid width". Therefore the following equation should be satisfied with any real value of a, any real value of

  2. Number Grids Investigation Coursework

    132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 There is a 2 x 2 square highlighted.

  1. Number Grid Investigation.

    121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 Take a 4 X 4 square from the grid above. 20 21 22 23 32 33 34 35 44 45 46 47 56 57 58 59 (20 X 59)

  2. What the 'L' - L shape investigation.

    Note that 4 is the grid square and 8 is double the grid size and two rows up. The sum of the shape above gives me the last part of the formula, which is -9: 1 + 2 + (-4)

  1. Investigation of diagonal difference.

    of the cuouts is the same, the diagonal difference stays the same. I also notice that the difference between the top left number and the top right number is 2, the difference between the top left number and the bottom right number is 12, the difference between the top left

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

    97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 Table 9: x - 24 x - 12 x - 11 x x + 1 x + 2 Using the algebra equation and adding the values

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