How many squares do you see along the border? Don't count them one by one.
First, I told students to think about an answer and put their thumb up in front of them when they thought they had it. (A great idea I learned from the MOOC.) Students didn't shout out answers in excitement and ruin it for others! And those who normally would shoot their arm up in rapid fire speed didn't intimidate others in the classroom! Next I called on students to explain how they found the answer. The five different solutions that are shown here were obtained by by 3rd period class.
I could tell by the expressions on student faces that several were surprised that there was more than one way to think about this problem. Next, I asked some followup questions with some unanticipated results.
1) Which method would you use if you had a 20 x 20 grid? Most students thought the third way was easier to use. And I always thought this method was easier, too. After all, who would want to do 18 squared in their head?? Not me.

20^{2}  18^{2} wasn't that bad if you thought of it as (20 + 18)(20  18), a difference of squares! My surprise was genuine and the fact that I didn't notice this right away wasn't so bad. My third period students had a chance to see me think aloud through an idea, and they had a chance to see that I still get excited about math when I see something "new" that I hadn't seen before!
You might be wondering why only the first equation is graphed. That is because I kept turning the second graph off and on in order for students to see that these two equations really produced the same graph.
Could I have incorporated other ideas, such as domain or the meaning of slope, into this lesson? Yes. However, my goal was to have my students begin to see math from multiple perspectives. That goal was met.
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