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# Classifying Conics

Lesson 5 of 10

## Objective: SWBAT classify a conic section by its equation.

*50 minutes*

#### Launch and Explore

*10 min*

One of our main goals today will be to **classify conic sections in general form** (*Ax*^{2} + *Bxy* + *Cy*^{2} +*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.

#### Resources

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#### Share

*10 min*

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 (-2x^{2} – 2y^{2 }= 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.

#### Resources

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#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Conics Sections with Princess the Dog
- LESSON 2: Exploring Ellipses
- LESSON 3: Hashing Out Hyperbolas
- LESSON 4: Pondering Parabolas
- LESSON 5: Classifying Conics
- LESSON 6: Rotated Conic Sections - Day 1 of 2
- LESSON 7: Rotated Conic Sections - Day 2 of 2
- LESSON 8: Unit Review: Conic Sections
- LESSON 9: Review Game: Lingo
- LESSON 10: Unit Assessment: Conic Sections