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# Exploring Ellipses

Lesson 2 of 10

## Objective: SWBAT solve problems involving ellipses.

*50 minutes*

#### Launch

*10 min*

One thing I want students to realize in this unit is that every **conic section can be defined in three ways** – as a locus, as a cross section, and as an algebraic equation. I start by showing students slide #2 of the PowerPoint and having them discuss what each figure would look like. They usually catch on quickly that each on is a circle.

Next, on slide #3 I ask them to define an ellipse in three different ways. Most of my students do not know what a locus is, so we define it and talk about the example with the circle. If students are having trouble, I refer them back to Princess’s leash from yesterday’s lesson. I also point out the poster they made yesterday to see if that will help to give some ideas. After a few minutes of working with their table group, we will share out answers.

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#### Explore

*20 min*

On slide #4, students are asked to **graph an ellipse and find its eccentricity**. The equation is in general form, so students will have to complete the square to write it in the standard equation of an ellipse. This is a good reminder for them on how to algebraically convert from one form to another.

While they are working, I will go around and monitor to make sure students are on the right track. Here are **some questions I will ask** to make sure they are thinking about the important aspects of the ellipse:

- How did you know if the ellipse was horizontal or vertical?
- What does the
*a*value represent? The*b*value? - How do we find the foci of the ellipse?
- What is the name of each axis of the ellipse?
- What equation relates
*a*,*b*, and*c*? - What would the graph look like if
*a*and*b*were equal?

Once it is time to share, I will have a student bring their work to the document camera and explain their process. If I feel like there was confusion about the questions shown above, we will discuss as a class.

**Eccentricity** is something that many students do not remember or were never taught, so we talk about what it measures and its formula (*e = c/a*). I try to ask questions so that we can make some generalizations (*e* is less than 1 for an ellipse, equal to zero for a circle, when it is close to 1 the ellipse is very long). I discuss more about eccentricity in the video below. Here is the link for the applet.

#### Resources

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#### Summarize

*20 min*

Finally, I give students an **assignment from our textbook** to summarize the main points about ellipses. I make sure that the assignment has students find equations and graphs using a variety of given information. For example, finding the equation when the foci and vertices are given, or graphing when the eccentricity and vertices are given. I also make sure that a few questions have some application of ellipses, such as finding the equation to model a planet’s orbit.

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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