I just finished conic sections with my Algebra II kids. I used the Conic Cards, created by the wonderful Cindy Johnson (@Johnsonmath), with whom I get to teach in the same building! Both Kristen Fouss and Amy Gruen have used the Conic Cards and written about the awesomeness of them here and here. They are truly amazing. I was dreading teaching conic sections, but after two weeks of card matching (and the two weeks were right after Christmas break, I might add), the kids were able to knock the socks off their first test of the semester. Which leaves us all happy.
In order to emphasize the similarities and differences of the equations for conic sections, I created this flow chart that the students filled out and used once they had learned all four conics. Before the flow chart, every time a kid would say, "Mrs. Peterson, is this a hyperbola?!" I would go through the same questions with her:
"Are both variables squared?"
"Good. So you know it's not a...?"
"Awesome. Now are the squared terms being SUBTRACTED?"
"Yes. Oh! Yeah, it IS a hyperbola."
I got tired of going through these questions over and over and over. Also, I don't think the kids realized the order in which I was asking the questions, which didn't do much for them except answer the immediate question.
So, now, instead of answering their question with a string of my own questions, all I have to say is, "Do you have your flow chart out?" Much less work for me. A little more work for them. And they're reading.
The wording isn't perfect. I don't know how to succinctly differentiate between the ellipse and the circle in standard form. This is the best I came up with. Of course, then kids think as soon as an equation has fractions in it, it can't be a circle. That's not really what the wording says, but I totally understand the confusion. I combated the confusion the lazy way: all our circles' centers were (m,n) when m and n were both integers.
Another good thing: this flow chart can be easily changed to classifying conic sections in general form. I just had the students take a few extra notes on the side (such as changing different denominators to different coefficients), and they were good to go.
All in all, conic sections went very smoothly. And now onto exponential and logarithmic functions!