Exploring Ellipses

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Objective

SWBAT solve problems involving ellipses.

Big Idea

Define an ellipse in three different ways.

Launch

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.

Explore

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.

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Summarize

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.