The Ellipse (Day 3 of 3)

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SWBAT rewrite standard form equations into general form and vice versa.

Big Idea

How can I determine the equation is an ellipse by looking at the equation.


5 minutes

Today we will be working with a lot of algebra techniques as well as discussing how the different forms for the equation of an ellipse can give us information of orientation of the graph as well as the key features of the graph.

I begin having students write a standard form equation for an ellipse. This allows me to do a quick assessment of students understanding of writing equations. It will also give the class a problem to use when for converting to general form.

Some ideas that I will be checking for understanding include:

  • how to find the center if you know the vertices.
  • by know the distance between the foci what parameter can determined.
  • how can you find the parameter "b" the distance from the center to the endpoint of the minor axis.

A student shares an answer and I ask question about the above bullets to make sure students are understanding.

General form of an Ellipse

20 minutes

Once the students have found a standard form of this example I ask "How would we find the general equation?" To convert to a general form equation the students must know the a general form to determine how to convert the equation. I will share the general equation of a conic again. For this problem the students really need to focus on the making the coefficients integers.

The bell work standard form is a very basic equation. This allows me to discuss how to use algebra to convert the fractions in an equation. Some students can do this process on their own in one step while the students with weak algebra skills will need some prompting. I use multiple steps when I convert to general form so the students do not get confused.

I now questions students about this form.

  • How does this equation compare to the general form of a circle? a parabola?
  • If you remember we had a way to predict if a general equation was a circle or a parabola, how might we determine if the equation is an ellipse?

Since we have only converted one equation I am want students to make a prediction. I give students another problem to rewrite in general form. This problem requires more algebra and give students a chance to see if their hypothesis for determining an ellipse is correct.

Once the students have found the general form we discuss what they found to by the way to determine an ellipse by looking at the equation.  Most students will say the parameters A and B are both positive but not equal.  I will ask how we could write this in a mathematical expression. I remind the students that we wrote the Parabola is that AB=0. Some students will say that AB >0. I ask so if A is negative and B is negative how can the equation be an ellipse.  Students say to either factor out a negative one or multiply both sides by negative one to make them positive.


Converting from General Form to Standard Form

15 minutes

I now give students this problem to convert to standard form.

This problem is interesting to solve. On the page 2 you can see several issues the student could struggle with as they are working the problem.

1. How to deal with the coefficient on the squared term. Students are not sure what to do when the x^2 and y^2 have coefficients on them.  I remind students that we could divide but we would need to divide all the terms which would cause us some problems. If you divide by 16 then I would still have a coefficient y^2 term. If I divided by the 25 I would still have a coefficient on the x^2 term. The best way to make the coefficients 1 is to factor the x terms and factor the y terms as shown in the second step.

2. What do you add to both sides. The students understand that they need to add the same number on both sides of the equation but many only think they should add (-1)^2 not the 16(-1)^2.  This is where I emphasize the parenthesis and the (-1)^2 is inside the parenthesis so we really added 16.

3.  How do you make the equation equal 1.  The students know that they need to divide by 25 so the equation is equal to 1. To help them see we need to have one side equal 1, I have the students look at the equation on the reference sheet to see the structure of the equation.

4. What do you do with the 16 by the (x-1)^2. This is a big issue for some I discuss how we need to have the 16 change to a 1. We do this by dividing by 16 but in this situation we divide the numerator and denominator by 16.  I have found doing division is easier for students to understand over other methods. It confused the students when I used to do 16^(-1).

After we rewrite the equation the students find the center, vertices and foci.  I also ask the students to determine if the major axis is horizontal or vertical.

I want the students to work on a problem without my assistance so I put another example on the board. Students work with each other to rewrite the equation. After working for a time the solution is shared with the class. 

Determining orientation of the graph

5 minutes
As I finish the lessons on ellipses I want the students to analyze the structure of the general and standard forms. I share a table of all the equations we have worked with today.  I have the students determine the orientation of the ellipse graph.  I then ask the students some questions:
  • How did you determine the orientation?
  • Can you use the general equation to determine the orientation?
Most students begin by using the standard form to find the orientation. The students realize that since the parameter a is half the length of the major axis that the largest denominator is a. If "a" is under the x term, then the major axis is horizontal and if it is under the y term the major axis is vertical.
Using the general equation takes some thought and analysis.  Students realize they can determine the orientation with the general equation. But this time they need to look at the smaller coefficient on the squared terms.  If the smallest coefficient is with x then the major axis is horizontal, if it is with y then the axis is vertical.
Explaining why this happens helps students understand the algebra we did when converting.
We divide by the constant so the larger denominator will always occur where the smaller coefficient appears.



5 minutes

As the class ends the students are reminded of the homework assigned in yesterday's lesson.

 p. 748 # 12, 16, 22, 27, 30, 42, 48 from Larson "Precalculus with Limits, 2nd edition"

Students now have all the information to complete the problems.

Students are asked to make a 2 columned table and label one column as standard and the other as general.

Under each column students need to write what key information can be found when given each form.  This sheet is handed in at the end of the hour. To help students focus on important ideas I had 3 questions on the board