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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.

- Let’s determine the area of this staircase in two different ways.
- First, count
*squares*in the figure to see that its area is $$1+2+3+4+5+6$$ - Next use
*triangles*in the figure to see that its area is $$\frac{6^2}{2}+\frac{6}{2}$$ - So, explain why you have actually just figured out that $$1+2+3+4+5+6=\frac{6^2}{2}+\frac{6}{2}$$

- First, count
- 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?
- Imagine that you had a 1000 by 1000 staircase. It’d be a pain in the neck to draw it!
- How would you describe it to someone?
- Once again repeat your thinking from #1, but now for this new staircase. What equation do you get?

- 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.- How would you describe this staircase to someone?
- One last time: repeat your thinking from #1. What equation do you get? What makes this equation different from the other equations you found?