# Investigate the relationship between the stair total and the position of the stair shape on the grid.

Shaun Meddings

Maths Coursework

Plan:

I am going to investigate the relationship between the stair total and the position of the stair shape on the grid.

By doing this I hope to find a formula by using algebra .I will then try to see if the formula works by moving the stair around the grid.

Once I have done this I will further investigate the relationship between the stair total and other step stairs on different sized grids.

Predictions:

I predict that there will be a relationship between different sized step stairs on the same grids. I also believe that there will be a formula which allows us to work out the stair total through the grid size, step size and stair number.

I will use the bottom left corner number as the step stair number. I will call this n.

I will now try to find a formula for any 3 step stair on any sized grid.

3 Step stair on a 10x10 grid.

There is a difference of 6 between the stairs; this means the first part of the formula must be 6n when n is the bottom left corner number. There are also 6 steps in the stair so every 3 step stair has 6 steps.

I know the formula must be 6n + 44 as I added all the n’s and all the numbers together which left me with 6n + 44.

So the formula must be 6n + 44.

I could instantly work out the step stair totals as I could times the step stair number by 6 then add 44.

I now need to represent the grid size in the formula I will call this g.

The formula for a 3 step stair on a 10x10 grid = g x g = 6n + 4g + 4

I worked this out as I knew the step stair was on a 10x10 grid. This meant the 44 needed to be broke up to represent the grid size in it. 4 x 10 = 40 + 4 = 44. This can now be used as the formula instead of 6n + 44.

I will now see if this formula works for other 3 step stairs on different sized grids.

3 step stair on a 9x9 grid.

The formula for a 3 step stair on a 10x10 grid was: g x g = 6n + 4g + 4

I will try out this formula by using the step stair total 46 as my example of a 3 step stair on a 9x9 grid.

Formula for a 3 step stair on a 9x9 grid = 6n + 40.

g x g = 6n + 4g + 4

= 9x9 = 6 x 1 + 4 x 9 + 4 = 46

The formula also works for a 3 step stair on a 9x9 grid.

I know these are the formulas for each grid size using a 3 step stair.

So the overall formula for a 3 step stair on any sized grid =

g x g = 6n + 4g + 4

I am now going to investigate different sized step stairs on different sized grids.

I will now try to find a formula for any 2 step stair on any sized grid.

2 step stair on a 10x10 grid

There is a difference of 3 so the formula must start 3n. I also know this as there ...