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Staircase

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October 21, 2018

Problem

       Take a look at the figure below.  The outline of this figure reminds us of a staircase, so we’ll call it that from now on.

  1. Let’s determine the area of this staircase in two different ways.
    1. First, count squares in the figure to see that its area is $$1+2+3+4+5+6$$
    2. Next use triangles in the figure to see that its area is $$\frac{6^2}{2}+\frac{6}{2}$$
    3. So, explain why you have actually just figured out that $$1+2+3+4+5+6=\frac{6^2}{2}+\frac{6}{2}$$
  2. Let’s call our picture a “6 by 6 staircase”, since there are 6 columns and 6 rows of small squares.  Now draw an 11 by 11 staircase in the same style.  Repeat your thinking from #1a, b, and c, but for this new staircase.  What equation do you get?
  3. Imagine that you had a 1000 by 1000 staircase.  It’d be a pain in the neck to draw it!  
    1. How would you describe it to someone?
    2. Once again repeat your thinking from #1, but now for this new staircase.  What equation do you get?
  4. Suppose we have a staircase, but we don’t know what its “dimensions” are: it could be 6 by 6, 11 by 11, 1000 by 1000, million by million, who knows!  Since we’re missing some information here, let’s just call it an n by n staircase, where n stands in for some number bigger than 1.
    1. How would you describe this staircase to someone?
    2. One last time: repeat your thinking from #1. What equation do you get?  What makes this equation different from the other equations you found?

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