Step-stair Investigation.

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William Murray

Step-stair Investigation

Cousework Submission 8th December 2003

        For my GCSE Maths coursework I was asked to investigate the relationship between the stair total and the position of the stair shape on the grid. Secondly I was asked to investigate the relationship further between the stair totals and the other step stairs on other number grids. The number grid below has two examples of 3-step stairs. I will use Algebra as a way to find the relationship between the stair total and the position of the stair on the grid. I will use arithmetic and algebra to investigate the relationships between the grid and the stair further. The variables used will be:

Position of stair on grid = X

Sum of all the numbers within the stair = S

Step Size= n

Grid size= g

The first thing I will do is find the formula for all 3-step stairs on a size 10 grid.  

I started off by making the bottom left hand number X. X is also the position of the stair on the grid. So in the diagram coloured red above X=15. I then added up the rest of the numbers in the three-step stair in terms of X. So 16= X+1, 17=X+2, 25=X+10 etc. The 3 step-stair in terms of X looks like this:

If you simplify all the Xs and all the numbers you end up with this: 6X + 44, X+X+1+X+2+X+10+X+11+X+20=6X + 44. By investigating the formula above you will find that it is the formula for all 3-step stairs on a size 10 grid. I worked this out by adding together all the numbers in the 3-step stair and then using the formula to see if the formula comes up with the total of all the numbers in the 3-step stair. In the diagram coloured red above; 15+16+17+25+26+35=134. The formula 6X + 44 comes up with the answer 134 as well proving that this formula works. I repeated this a few times to prove that the formula really did work. In the diagram coloured blue: 62+63+64+72+73+83= 417. By using the formula 6X + 44 the answer is 417, (6*62)+ 44. This proves that the formula: 6X+44 is the formula for all 3-step stairs on a size 10 grid. S= 6X+44.

The second thing I will do is find the formula for a 3-step stair on any sized grid. I realized that the difference between X and the number in the box directly above it was always the grid size number (g). For example in the diagram at the start of this write-up the number directly above X in the red stair is 10 more than X. The grid size is 10. I investigated this further by creating a size 5 grid and seeing what the difference was between X and the number directly above it.

In the diagram it shows that the number directly above X is the grid size more than X. This means that the number above X is X+g, (g= grid size). The 3-step in terms of X and g now looks like this:

When this has been simplified to create a formula it gives the result: 6X+4g+4, X+X+1+X+2+X+g+X+g+1+X+2g= 6X + 4g+ 4. To prove that this formula works I worked out the total of all the numbers in the green stair and then saw if the formula came up with the same answer. So, 3+4+5+8+9+13=42.

6X+4g+4= (6*3)+(4*5)+4 = 42. I tried this formula with a size 10 grid. If you add up all the numbers in the red stair: 15+16+17+25+26+35=134. I saw if the formula came up with the same answer; 6X+4g+4= (6*15)+(4*10)+4= 134. This proves that 6X+4g+4 works for all 3-step stairs on all grids. S=6X+4g+4.

        After finding the two formulas listed above I decided to investigate the 6X+4g+4 style formula for 4-step, 5-step, 6-step and 7-step stairs. Ofcourse the 6X+4g+4 formula doesn’t work for 4-step, 5-step, 6-step and 7-step stairs, but by working out the formula for these step stairs in the same way that I worked out the formula for 3-step stairs, I hoped to find the formula for any step size on any size grid.

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4-step stairs:

By adding up all the Xs all the gs and all the numbers I got this:

X+X+1+X+2+X+3+X+g+X+g+1+X+g+2+X+2g+X+2g+1+X+3g=10X+10g+10 This is the formula for any four-step stairs on any size grid.

To prove that this formula worked I drew two grids of difference sizes and calculated the total of all the numbers in the 4-step stair.

By using the formula 10X+10g+10=S, I worked out the total of the numbers inside the red area of the 4-step stair.

(10*36)+(10*10)+10=260+100+10= 470

Then I added up all the numbers in the 4-step stair to see if it gave ...

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