April 5, 2013

Rational functions and limits

Way back when, over February vacation, I gave my Honors PreCalculus students a rational functions packet (thanks largely to CME). One of the tricky things about teaching this class for the first time is I'm never sure how much of any topic students studied or internalized last year. Turned out vertical asymptotes and holes came back to them easily, success! One of the things I love about teaching PreCalculus is that the end goal is "do cool math, be ready for calc." The other teacher and I were chatting and said limits were just sitting there just beyond horizontal asymptotes, how could we not study them? So we did! (She's also the AP Calc teacher so we're not messing up someone else's curriculum.)

I started with the chart below which is pretty much exactly what JackieB shared. The difference was in the follow up; I didn't want to scaffold their conclusions as much as she did. Which resulted in most students not understanding what I wanted for generalizations. Next time I need to reword that section, should I ask students what the possible values are for the limits first and then ask them to generalize how to know what the limit will be?

 One of my favorite responses to question 3- it's color coded! Albeit not accurately, but we got there eventually.

After studying limits for a bit I wanted to make sure that everyone had internalized the phrasing I was using. I asked a student to translate from symbols "the limit as x approaches 2 from the right" and in both classes they used that exact phrasing. In one class I made them all say it in chorus, just because they're all so quiet and I needed to liven things up. Some students were mixing up whether the answer was the x or y-value, so I frequently found myself saying "as x gets close to blah, what does y get close to?" And of course every time there was a hole "do we care that there's a hole? Do we care about that value at all? No! Just what it's approaching"

I did limits at infinity, one sided limits at vertical asymptotes, direct substitution, and algebraically solving removable discontinuities, in that order. It made sense to me since we were approaching limits from the context of rational functions, but I'd be interested in hearing rationales for a different sequence. (Math puns!)

The blogosphere has some cool limit activities that I set up as stations on the last day of the unit.  Bowman covered up the value in question and asked students to find the limit, lift the cover and then reassess.  Quite a few students called me over to this station to tell me that the covered up part of the chart was wrong.  Even after all those time I said holes don't matter, that we did graphical piecewise functions, that I emphasized the word approach, students were still confounded by the hidden values not matching their expectation.  However, once I asked them what a limit was, or showed them a graph from the homework that matched this situation they returned to their original answers with confidence and said "this station is easy!"  Sam is right, this really is the best question ever.

I also found a cool geometry application on Irrational Cube which had some students really digging deep into their recollection of approaches to finding area and using a lot of problem solving skills.