Math from Multiple Perspectives (Part 2)

In my previous blog post, I talked about a problem that I showed to my students on the first day of school.  This problem was to get them to think about math from multiple perspectives and it was taken directly from the Stanford University MOOC that I took over the summer called "How to Learn Math".

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 follow-up 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.

But then it dawned on me! (It only took the third time for me to teach this lesson to see it.)
20² - 18² 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!

2)  How do we know these solutions will always give the same answer?  Students easily explained how to generalize each solution based on an n x n grid, where n was the number of sides on a square.  (This can be seen on the left.) When I asked how we might show the expressions were equivalent, I was thinking that my students might simplify each expression to show they were equivalent. However, in each of my three classes, they suggested setting 2 expressions equal and solving for n. (You can see the circled expressions that we set equal in this particular class.) In each case, an identity resulted and they could explain what that meant, that the expressions always yielded the same result for any value of n.

3)  How can graphing equations show these are equivalent expressions?  This one stumped my students and unfortunately, we ran a bit short on time for me to make sure they really got this idea.  However, in two of my classes I showed Desmos as a quick graphing tool.  [I became more acquainted with Desmos at NCTM in Denver in 2012. Thanks Desmos folks for the cool trivia prizes! I love my mint green Desmos t-shirt!]  We graphed y = 4x - 4 and y = x² - (x - 2)². A screenshot of the graph is seen here.

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.