One of our main goals today will be to classify conic sections in general form (Ax2 + Bxy + Cy2 +Dx + Ey + F = 0) without graphing or doing any calculations. I want student to notice patterns and see trends. This is also building a foundation for tomorrow when we start looking at conic sections that have been rotated.
I give students this worksheet and have them work on #1-12 with their table group to classify each conic section. I explain that I want them to try to figure each one out without doing any calculations or graphing. As they are working I will listen to the conjectures that I am hearing and will write them on the board. Here is a list of a few that I heard in one of my classes.
I start our discussion by looking at the conjectures (shown below) and seeing what the class thinks of them.
The interesting part of the conjectures is that many of them will be true, but they may not be specific enough to be a general rule. When my students talked about A or C being negative in a hyperbola, a student asked what would happen if both were negative. I made a simple example (-2x2 – 2y2 = 100) and had them think about what would happen.
After we went through all of the conjectures and came to some definite conclusions, I ask students about the B value and what conclusions we can make about that. At this point they notice that B has always been zero so far. To consider what will happen if there is a B value that is not zero, I use the sliders on Desmos to investigate. I discuss this approach in the video below.
After our discussion about the B value, I tell students that we will investigate this more in the coming days. I tell students that the rest of the worksheet is for homework and is a good summary of all of our work with conic sections so far.