Conic Sections Flow Chart

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?"

"No."

"Good.  So you know it's not a...?"

"Parabola."

"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.

Conic Sections Flow Chart by

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!

Conic Sections: Parabolas

Much like with circles, I need some major help in the area of teaching parabolas (through the lens of conic sections).  Maybe I'm just not cut out to teach geometry.  It's quite possible.

In any case, here are a couple of things that I found/made that did work nicely:

1. This graphing paper from MathEdPage.org is awesome.  It gives a very nice low-tech option for discovering the geometric definition of a parabola.  I split the class into groups of two or three and gave each group a sheet of this graphing paper with one of the lines darkened (which is to be the directrix).  I told the students to plot seven or so points that are equidistant from the point in the middle and the darkened line.  We found a couple together, and then they were good to go for the rest.  When I gave them the following definition, they were able to fill in the  blanks no problem:
   
   
   
2. I made a little graph with sliders that shows what happens to a parabola in the form x^2=4py when you change p.  It wasn't a ton of work, but I'm still pretty proud.  Plus, I continue to absolutely adore Demos graphing calculator at abettercalculator.com.  I also love their new "Projector Mode" under Settings.

I got to borrow one of these from my college. 
I really want one.  Unfortunately they're a little pricey.

So, those are two things that worked.  The 5-10 minute intro.  But once we got to working examples, I wasn't too pleased.  I feel like I jump all over the place when I work these problems.  "What's the vertex?!  How do we find the focus from there?  And the directrix?"  I think students get it during class, but that's with me asking all the right questions at all the right times.  Ideas for making them do more of the work?