Class begins with these specific instructions:
"In the next 10 - 15 minutes, you need to complete the rest of Asymptotic Behavior [from the last lesson homework] and begin looking for patterns. The questions to ask yourself are: How could I predict the end behavior just by examining the equation? What simple test could I use to distinguish a vertical asymptote from a removable discontinuity?" (MP 7)
Now, I'll give the class the time to wrap things up and I'll circulate around the room directing students attention to the patterns in their solutions. Specifically, I'll point out the three different end behaviors: approaching zero, approaching infinity, approaching some other number. Also, I'll point out that when the function is discontinuous, it can be in the form 0/0 or #/0.
At the end of the 15 minutes, students should be able to make the following generalizations.
I'll draw these out by asking each group to share one observation they've made or pattern they've observed. With five or six groups in the class, we're sure to hit on the essential points.
It's a tragedy when math gets reduced to rules without reasons, so I think it's absolutely essential to verify these patterns with some clear explanations.
The fundamental principle behind the explanations is the fact that the size of a fraction depends on the ratio between the numerator and denominator, not their actual values. If the denominator becomes very large relative to the numerator, then the fraction approaches zero. If the denominator becomes very small relative to the numerator, then the fraction "approaches" infinity. (See the whiteboard.) Finally, if the denominator and numerator tend toward some specific ratio, then the fraction approaches exactly that value. I'll take a few minutes to make sure the class clearly understands this principle (see the whiteboard). I'll present the ratios on the board, but I rely on the students to use their calculators to evaluate them. This keeps them involved and saves me some time at the board. If you do this, be sure to have students check one another - even with a calculator my students frequently make mistakes!
Now, what I'd like the class to do is to apply this principle to explain why a vertical asymptote exists where the function takes on the form #/0. They can usually do this without much help from me. When a student begins to propose an explanation, I'll typically call them to the front of the class so that he can point things out on a graph or give us some examples. When he's done with his explanation, I'll ask the class if they agree or if they have questions/objections/critiques. After a bit of back-and-forth in this way, I'll wrap things up for the class being sure to point out the use and importance of the principle described above.
Next, we should try to explain why a vertical asymptote doesn't have to exit (but can) where the function takes on the form 0/0. Again, they should be able to do this on their own.
Finally, we'll tackle the horizontal asymptotes (see the whiteboard). For these, I'll provide some very strong direction by suggesting that we "multiply by a clever form of 1". After this, I'll guide the students carefully to the conclusions by asking questions and responding to their answers. In the end, we should have considered all three cases (see the whiteboard).
Bear in mind that this conversation won't be easy, but I think it's vitally important to explain asymptotes.
Since the discussion & explanation will probably take up the entire class period, I'll use an exit ticket to see get some feedback from the class. This will serve as a good formative assessment and help me determine just where I need to begin tomorrow.