The Ellipse (Day 2 of 3)

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Objective

SWBAT write standard form equations for ellipses when given a graph or key features of an ellipse.

Big Idea

How can the key features of an ellipse be used to find an equation for an ellipse?

Bell Work

10 minutes

Yesterday, we learned the standard equation for an ellipse.  My students did not have time to practice writing equations using known information about an ellipse. To begin this application, I begin today's class with a Bell Work problem. This problem requires students to recall the new content introduced yesterday. If my students are not sure where to begin, I will suggest they get out their reference sheets.

I expect my students to rely on the reasoning and techniques we used in the prior lessons to determine the equation. Many will be able to find the equation. Some will be confused about where to substitute the values for a^2 and b^2.  If students ask how to do this, I plan to reply by having the student sketch a graph and determine if the major axis is horizontal or vertical. Some questions I plan to ask students are:

  • What would you need to know to determine where a^2 goes?
  • Is your major axis vertical or horizontal?
  • What parameters did we use yesterday to describe attributes of the parabola?

After about 5 minutes, I will ask my students to share their results. If needed, I will follow up with a class discussion to clarify any misunderstandings.

Writing Standard Form Equations for Ellipses

15 minutes

I want students to determine how key features of the graph of an ellipse can be used to write the standard form equation. As part of this they also need to realize what key features are used to determine the equation. For example you do not need to know the foci to write the equation if you know the length of the major and minor axis.

In the examples, I chose problems that with different key features know and the students will determine the standard form equation.

The first example the students need to find the center and then determine the orientation of the ellipse. I let students work for a few minutes. As I move around the room I choose a student who has a correct answer to share the work.

Question I will ask about the problem are:

  • What was the first step in your process for writing the equation?
  • How did you find the center?
  • How did you determine the value of "a" and where it goes in your equation?
  • How did you find the parameter b?

I find the second example interesting in that it can have many different solutions. This is one where the students will need to determine the center. The x value of the center must be 0 but the y can be any number. This is a good discussion since most students will put the center at (0,0).  I will ask some of the same questions as above but will also ask the class if the center has to be at (0,0) and how will that change the equation? Will it affect the parameters a, b and c?

The last example is a problem involving planets.  The example from the Stewart text gives the students a diagram.  We are going to use literacy strategies to draw a diagram.  I like to take problems that explain the diagram in the problem and give the students the problem to determine the diagram. This is a great way to help students read mathematically.

 

Closure

10 minutes

As we end the examples I ask the students to determine what information is needed to write an equation.

  • If you are given the vertices and the endpoints on the minor axis can you find the equation?
  • If you are given the center and the foci can you determine the equation?

 I will ask other questions similar to these and we develop a table for our reasoning on what you need to write the equation.