Mathematical Coursework: 3-step stairs

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In order to understand the relationship between the stairs total and the position of the stairs shape on the grid. Foremost, I needed to distinguish if the stair total had something in common. As I believe this would enable me to find a pattern, which would eventually develop into an algebraic equation. To be able to do this I had to experiment with the grid, by shifting the 3-step stairs into different direction et cetera. In addition I would’ve seen if the pattern and algebraic equation would’ve work in different size grid such as 9cm by 9cm or in different size stair shape such as 4- step stairs.

 

In order to find the pattern I will have to draw various grids, this will enable me to eventually discover the pattern. Firstly, for my first experimental grid I decide the start in the bottom corner and randomly select a few other 3-step stairs.

Because I haven’t discovered the algebraic equation, this would allow me to find the total stair number in less than minutes. I would have to find the total by using the basic method of adding.

The Calculations

         1+2+3+11+12+21= 50                                          5+6+7+15+16+25=74       

   83+73+63+64+65+74=422                                  47+48+49+57+58+67=326

I’ve come to realise that the total numbers have something in common. Due to the fact that they are; even numbers. Thus meaning an integer can be divided exactly by the number 2. With even numbers the last digit will always b; 0,2,4,6 or 8. The total numbers of the above calculations shows that the number is even. Therefore I have decided to investigate the grid further to see if it has indeed a pattern.

  1. 1+2+3+11+12+21= 50 

     

  1. 2+3+4+12+13+22=56

  1.  3+4+5+13+14+23= 62                     

  1. 4+5+6+14+15+24=68

After analysing the graph I believe that I have discovered a pattern. I decided to use make the 3-step stairs starting from 1 (which would be the bottom-corner number shaded in example) going on the 2 et cetera. I choose this method, as I believed it would allow me to find the pattern. To prove my assumption I would now make a Table of Results which will state my findings.


 The Pattern = 6


As I now have found the pattern, this would allow me to predict the following 3-step stairs if I was still using the method. In order to check if my prediction is right I would have to again add up the numbers in the 3-step stair shape.

                                                                   

  5+6+7+15+16+25

     = 74


This proves my Prediction to be correct. Therefore I’m able to find the total number of the 3-step stair by just adding 6.  However this technique would restrict me from picking out a random 3-step stair shape out of the 10cm by 10cm grid. As I would have to follow the grid side ways from term 1 towards e.g. term 20. Thus making it’s time consuming.

After analysing the grids, the table of results and my prediction I have summed up an algebraic equation which would allow me to find out the total of any 3-step stair shape in a matter of minutes.

 

The criterion of why I haven’t explained what B stands for is because I haven’t found it yet. Nevertheless using the annotated notes on the formula my formula would look like this now:

  • 6N =6n x 1= 6

  • 6+b=50

Now I would need to find the value of b in order to use my formula in future calculations. Therefore I will take the pattern number of the total:

  • 50-6=44

  • b= 44

To conclusion my new formula would be:

  • 6n+44

However as you can see we haven’t found out what the value of N is. Which is an important factor in the equation, otherwise the algebraic equation wouldn’t allow me find the total number. Hereby I’ve come to realise that the bottom corner number of the stair shape is the value of N. To certify if my theory is correct I will pick out a random step-stair out of the 10cm by 10cm grid. As shown below:

If the shaded bottom corner is the value of N, therefore causing my formula to be as following:

  • 6n+44
  • N= 73
  • 6 x 73=  438
  • 438+44
  • 482

To ensure my answer is correct I will have to calculate the 3-step stair by using the basic method of adding: 73+74+75+83+84+93= 482

This proves my theory is correctly, however to guarantee my theory is reliable and accurate I will use a calculator and I will also calculate using another 3-step stair shape of the 10cm by 10cm grid.

  • 6n+44
  • N=31
  • 6 x 31=  186
  • 186+44
  • 230

To show the importance of the bottom corner number; - (When, trying to find the total number of the 3-step stairs using an algebraic equation.)  I will demonstrate; how any other number in the stair shape wouldn’t add to the total.

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  • 6n+44                      6n+44              
  • N=33                       N=42
  • 6 x 33                      6 x 42
  • 198+44                   252+44
  •   242                          296

  • 6n+44                         6n+44
  • N=51     ...

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