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).

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

        s-number = 25

        s-total = 194

        1 + 2 + 3 + 11 + 12 + 21 = 50

        s-number = 1

        s-total = 50

        2 + 3 + 4 + 12 + 13 + 22 = 56

        s-number = 2

        s-total = 56

        3 + 4 + 5 + 13 + 14 + 23 = 62

        s-number = 3

        s-total = 62

        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:

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

        5 + 6 + 7 + 15 + 16 + 25 = 74

        (6 x 5) + 44 = 74        

        s-number = 5

        s-total = 74

        6 + 7 + 8 + 16 + 17 + 26 = 80

        (6 x 6) + 44 = 80        

        s-number = 6

        s-total = 80

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

        (6 x 26) + 44 = 200        

        s-number = 26

        s-total = 200

After testing the

1

2

3

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

s-grid = 5

s-total = 30

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

s-grid = 6

s-total = 34

Table of Results:

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:

← 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.

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+(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.

= 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.

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:

(-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.