• 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

s-number = 5

s-total = 74

 26 16 17 6 7 8

6 + 7 + 8 + 16 + 17 + 26 = 80

(6 x 6) + 44 = 80

s-number = 6

s-total = 80

 46 36 37 26 27 28

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

(6 x 26) + 44 = 200

s-number = 26

s-total = 200

After testing the

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

Conclusion

= 21n + 35g + 35

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

s-total =

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

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

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 =

(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:

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

Page

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

3 star(s)

Here is what I found out: X+x+1+x+2+x+3+x+g+x+g+1+x+g+2+x+g+g+x+g+g+1+x+g+g+g = 10x+10g+10 Looking at the above information it is visible to know that the four step stair shape follows the same simple pattern as the three step stairs in the way that the letter g is added vertically and the numbers added horizontally.

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

I said before that the number one row up from the L-Number was the grid size and that the number two rows up was double the grid size. Now the L-Shape works in any size grid. As before I stated that the sum of the axis was equal to the last part of the formula.

1. ## Number stairs

40 & 44 and show them as shown below: I know that the above values increase by the constant number [4] and also that in any 3-step grid square if the grid size increases by 1 then the constant number is added to the value (The Highest Common Factor)

2. ## Number Grids Investigation Coursework

the algebraic expression for the difference between the products of opposite corners would be: (top right x bottom left) - (top left x bottom right) = (a + 2) (a + 20) - a (a + 22) = a2 + 2a + 20a + 40 - a2 - 22a =

1. ## Investigation of diagonal difference.

+ 4 n + G n + G + 4 40 2 6 n n + 5 n + G n + G + 5 50 Table of results for vertically aligned cutouts Height of the cutout Length of the cutout Top left corner Top right corner Bottom left corner

2. ## Number Grid Investigation.

- (6 X 41) = 200 Product difference = 200. My prediction is correct and my number pattern was successful. Finding a formula for 8 wide grid. Based on my formula from a 10 wide grid. I predict that the formula for an 8 wide grid will be: 8 (n-1)�

1. ## Number Grid Coursework

If (p - 1) is the variable representing the side of the square, then the area calculated from that is (p - 1)2, or (p - 1)(p - 1). Because each of these brackets represents the side of the square, when it becomes a rectangle, it is safe to assume that the area will be: (p - 1)(q - 1)

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

47 48 49 50 51 52 53 54 55 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

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