Classifying conics is always a fun lesson for me to teach. For me, this is the last lesson of my conics unit. I try to keep the actual lesson as short as possible. I've found that a flow chart works very well to help students organize their information for this lesson.
If I get to teach Algebra 2 or Pre-Cal again, this is what I would use as an interactive notebook page for this lesson. I wouldn't use a foldable at all, and would just have students draw the flowchart in their notebooks.