# Find relationships between the stair total and the position of the stair shape on the grid for a three step stair.

Introduction

In this investigation I am going to see if I can find relationships between the stair total and the position of the stair shape on the grid for a three step stair. I will then try to work out the relationship between the stair total and the position of the stair shape on the grid for 2 step stairs to 6 step stairs. Furthermore I will try to find a formula for working out any stair size which is placed anywhere on a 10X10 grid.

For this investigation I will need:

• Calculator
• Pencil
• Pen
• Grid Tables

To carry out the investigation I will need to use algebra and substitution. If I succeed in this part of the investigation I will then try to extend my investigation further by working out the relationships between stairs and where they are placed on different sized grids. From that I should be able to work out a formula to work out any step stairs placed anywhere on any sized grid.

3 Step Stair on A 10X10 Grid

To work out this part of the investigation I used algebra and gathered the similar terms to find a formula.

21
11 12

1    2   3                               I took this stair and added the numbers together. It totaled 50

I then made my key number 1 and let x = 1

I made an equation and then gathered up the terms.

x + (x + 1) + (x + 2) + (x + 10) + (x + 11) + (x + 20)

Total is T = (xR+c) where R is the reference number and C is the coefficient

Therefore gathering the terms I found the formula to be 6x + 44

Other Step Stairs on A 10X10 Grid

By knowing this now I then went onto work out other steps on a 10x10 grid. I started by working out a 2 step stair using the method I used to work out the 3 step stair.

11

1   2

x + (x + 1) + (x + 10)

I gathered the terms and found the formula to be 3x + 11

4 Step Stair:

31

21 22

11 12 13

1    2   3  4

x + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11) + (x + 12) + (x + 20) + (x + 21) +

(x + 30)

Therefore a formula for a 4 step stair on a 10x10 grid is 10x + 110

5 Step Stair:

41

31 32

21 22 23

11 12 13 14

1    2   3   4   5

x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 10) + (x + 11) + (x + 12) + (x + 13) +

(x + 20) + (x + 21) + (x + 22) + (x + 30) + (x + 31) + (x + 40)

Therefore the formula for a 5step stair is 15x + 220

6 Step Stair:

51

41 42

31 32 33

21 22 23 24

11 12 13 14 15

1    2   3   4   5   6

x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 10) + (x + 11) + (x + 12) +

(x + 13) + (x + 14) + (x + 20) + (x + 21) + (x + 22) + (x + 30) + (x + 31) + (x + 40)

Therefore the formula for a 6 step stair on a 10x10 grid is 21x + 385

To Work Out General Formula for a 10x10 Grid

I then had to find out a formula for working out any step anywhere on a 10x10 grid.

y = 2 >> 3x + 11
3x + 33

y = 3 >> 6x + 44                        x  + 33

4x + 66                        Diff. 11

y = 4 >> 16 x + 110                        x  + 44

5x + 110                        Diff. 11

y = 5 >> 15x + 220                        x + 55

6x + 165

y = 6 >> ...