Classifying Conics

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Objective

SWBAT classify a conic section by its equation.

Big Idea

How can we classify a conic section without graphing?

Launch and Explore

10 minutes

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.

  • #3 is a circle because A and C are integers.
  • #4 is a hyperbola because A or C is has a negative coefficient.
  • #8 is a parabola because there is only one squared term.
  • #11 is an ellipse because A and C are positive. 

Share

10 minutes

I start our discussion by looking at the conjectures (shown below) and seeing what the class thinks of them.

  • #3 is a circle because A and C are integers.
  • #4 is a hyperbola because A or C is has a negative coefficient.
  • #8 is a parabola because there is only one squared term.
  • #11 is an ellipse because A and C are different values.

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.

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Summarize

30 minutes

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.