Vertical movement
As you can see they all end in a 4, similarly so does the stair shape in the example given, as it ends in 194, this is as the vertical movement was working in conjunction with the example given.
I noticed that when working out the difference between the stair shapes for the overall vertical difference it came to 60.
254-194=60
194-134=60
134-74=60
To check whether the actual movement of the stair shape in another position related correctly to the vertical movement found, we placed the same size stair shape and checked to see whether that vertical difference in another part of the grid related to the difference found. If on positioning the stair shape in another part of the grid and the overall vertical difference varied it would more difficult to work any possible relationship.
As you can see in this example they all end in 6, this is a common number for the movement.
176-116=60
116-56=60
As you can see the number 60 is common in both so this means that the variables do not change when the shape is positioned in another part of the grid. This means that for every 3-step stair on the grid whether moved 1 place to the up or down, the difference will be equal to 60.
I then progressed on to the horizontal difference; I again started firstly by using the example and then positioning it to right then the left and working out the difference.
Although there is no common number here at the end of the total for each stair shape there is a common number fond when the difference is calculated.
200-194=6
194-188=6
188-182=6
This means that for every horizontal movement either left or right on the grid the total difference between each stair shape for a 3-step stair would be 6.
Positive diagonal movement on the grid:
There is no common number found here at the end of each total. The common difference is:
260-194=66
194-128=66
128-62=66
This means that when the stair shape is moved diagonally in the “y=x” direction the difference increase is 66. The total stair shape total for the stair shape will increase by 66 for every increase or decrease by 1.
Negative diagonal movement on the grid:
302-248=54
248-194=54
194-140=54
The negative diagonal difference when translating the shape in a “y=-x” manner is 54. So for every movement up 1 on the grid in that direction you can add 54 to the total.
After working out the vertical, diagonal and horizontal difference in the entire 3-step stair shapes I found that there was a common difference with all the stair shapes and that the difference found related to the position of the stair shape on the grid.
Part 2
Investigate further the relationship between the stair totals and other step stairs on other number grids.
To further my investigation I decided to observe the relationship between larger stair shapes and their position on the grid, also their relationship with each other.
2-step stair
Using the same process as the 3-step stair I began to look at the horizontal, vertical and diagonal differences. I first started with looking at the horizontal difference
All the last numbers of the individual stair totals all have a multiple of three at the end.
89-86=3
86-83=3
The total difference overall is three and the multiple of the last number is also three. The horizontal difference is equalled to three.
Vertical difference:
There is a common number that the stair totals end in, this is 6.on looking at the overall stair difference:
116-86=60
86-56=60
As you can see we have also a common number, which also is a multiple
Positive diagonal difference:
The numbers 3, 6 and 9 can be seen; they all are multiples of three. The total overall difference for the diagonal 2-step stair:
119-86=33
86-53=33
As can be seen the number 33 is also a multiple of 3.
Negative diagonal 2-step stair:
As can be seen the numbers shown at the end are 3, 6 and 9, they are all multiples of three.
Working out the common difference:
113-86=27
86-59=27
Twenty-seven is seen to be the common difference tis however is also a multiple of three.
4 step-stair
Horizontal difference:
They all are multiples of 10, so already you can see a common difference.
Working out the overall difference:
370-360=10
360-350=10
Vertical difference:
60 is a common number in this, the total difference worked out:
360-260=100
260-160=100
There is a difference of 100 for each progressional movement vertically up or down on the 4-step star grid.
Positive diagonal movement:
360-250=110
250-140=110
