SWBAT solve problems involving ellipses.

Define an ellipse in three different ways.

10 minutes

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.

20 minutes

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.

20 minutes

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.