I know I mildly hated on Desmos the other day, but I didn't mean to imply the experience was bad. In fact, it's improved immensely in the 6 or 7 months since I first found it. A native app (with iCloud document storage) would be heavenly, but the currently available product worked well. The timing is great too, because historically it's all been graphing by hand with no secondary way to show I'm not making this all up. Here's how we dove into circle and ellipses.
Step 1: An Eclipse?
We started with horizontal and vertical parabolas the previous week defining things like vertex, focus, etc. For circles and ellipses, I passed out iPads and instructed them to find actual examples. I did not define what I meant by circle or ellipse or give any real guidelines to objects. I suggested image search and maps. They took screenshots or saved images to the local iPad camera roll which eventually propogates to the class Photo Stream. They were given 15-20 minutes to find these items. We spent 5-10 minutes discussing what they found. The idea was to search for a proper definition of an ellipse. Collectively they responded "it's an oval!" But we had an interesting time discussing why this is not an ellipse:
Step 2: Ladder Climbing
Now we define an ellipse and circle mathematically. In Pre-Cal I asked for them to find the standard form equations if possible. In Algebra II I just gave it to them. We discussed what was similar about these compared to standard form parabola equations. How would we locate the center? What defines the width of an ellipse? The height? How do we classify an ellipse as vertical or horizontal? They have a focus too, really?
Next, we take the generic equations and build the model in desmos:
We do this in steps together. My iPad is projected on the screens while we do this so we can get over hurdles like creating exponents and denominators. The iPad keyboard for desmos works well though, so there were only minor problems getting them to this point. Initially my thought was to have the model saved in advance, but I'm happier I had them build it on their own.
Next some discussion questions. What does each slider do? What do you have to manipulate to get a vertical ellipse? horizontal? Is it possible to make a circle? But wait, we didn't type in the generic circle equation! So you mean a circle is just a special classification of ellipse? Ah ha.
Step 2a: Explicit Equations
In Pre Cal I wanted to emphasize the ability to convert a non-standard equation to a standard one. That involves completing the square to take something like x^2+12x+y^2-7y+48.25=29 and make it more relatable. The iPads are set aside and in their notebooks they run through a set of problems that require them to convert a bunch of circles/ellipses to standard form.
We then break out desmos again to classify these crazy equations we were converting:
They entered the given version of their problem. I had them identify the center. Does the location of the center seem to correspond with any numbers present in your standard form equation? I have them enter an ellipse and determine the width and height of the ellipse. Do either of those numbers relate to aspects of your standard form equation?
Step 3: Pencil and Paper
The lesson concludes with some more work on their respective problem sets. Algebra II set off to draw ellipses (I added things like how to calculate the focus, why circles don't have them). Pre Cal will take the standard form equations they created and draw those. It's possible I will have them add to our Parabola Wall. Later on Algebra II got to learn about completing the square and dove into hyperbolas. Pre-Cal is going to explore the polar versions of conic sections for an interesting idea about the planets I have kicking around in my head.