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# Ellipses

Lesson 6 of 13

## Objective: SWBAT find the equation of an ellipse. SWBAT graph an ellipse given an equation in standard or general form SWBAT solve for y in order to use graphing technology.

## Big Idea: After learning to write the equation of an ellipse, students design an elliptical pool table that would guarantee that even the worst player can always make a shot.

*50 minutes*

Students should first copy the definition of an ellipse into their Personal Dictionaries if they haven’t done so yet. Then they should reflect whether or not the image on page 1 of the Student Handout and on page 3 of the Flipchart is actually an ellipse.

If students seem stuck on how to explain whether or not the image is an ellipse, you may want to point out that the overlapping circles on the grid help us to count the diagonal distance without having to do any calculations. I am hoping that students will be able to draw on their knowledge of how they made the ellipse out of rope and chalk in the lesson, Human Conics: Circles and Ellipses.

Mainly, I want students to be reminded (by each other) that the sum of distances from each focus point to any point on the ellipse will always have the same value, in this problem 10 cm. I expect students to relate back to how two people in their team had to be foci and one person could be the point on the ellipse and the fact that the rope always retained the same length. I predict that students will be challenged of stating their experiences on paper as lessons learned. It will be a good challenge for students to accurately use the vocabulary associated with ellipses.

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#### Explanation: Ellipses

*10 min*

Display page 4 of Flipchart - Ellipse and ask students to label the key features of an ellipse on their handout. The answers are on page 5 of the Flipchart to display when students are ready to check their work.

Use pages 6-7 to walk students through finding the equation of an ellipse using only the distance formula and the definition of an ellipse. This is quite complicated. I don't actually want my students to find the equation of an ellipse using this strategy. However, it does show them how to derive the standard form and I think it would be beneficial for them to justify each step to be sure they follow the math.

As students work to justify each step in Problem 3, I predict there will be some frustration as it will be challenging for our students. My main goal here would be for students to see how the equation of an ellipse can be derived from the Distance Formula and the definition of an ellipse. It helps us to nicely introduce the standard form of an ellipse and helps students to hopefully have a greater appreciation of why we would study the standard forms of conic sections. It really is much easier to be comfortable with the structure of the standard form and be able to immediately write the equation in that form instead of deriving every equation of an ellipse using the method demonstrated in Question 3.

After students have completed Question 3 on handout pass out the Ellipse Summary. As you provide students with this resource, emphasize the relationship between the standard form equation we found in question 3 and how it relates to the graph.

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#### Application: Group Work

*20 min*

In their teams, students work on Questions 4-6 on the Student Handout. I plan to just circle and observe how the teams make progress. I will ask students to skip around and do what they can with these problems and save questions until I get to their table. Hopefully, I can cycle through all the tables twice.

#### Resources

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Cutting Conics
- LESSON 2: Name that Conic
- LESSON 3: Human Conics: Circles and Ellipses
- LESSON 4: Circles and Completing the Square (Day 1 of 2)
- LESSON 5: Circles and Completing the Square (Day 2 of 2)
- LESSON 6: Ellipses
- LESSON 7: Human Conics: Parabolas
- LESSON 8: Parabolas
- LESSON 9: Parabola Problem Partner Critiques
- LESSON 10: Hyperbolas
- LESSON 11: Non-Linear Systems of Equations
- LESSON 12: Conic Sections Test Review
- LESSON 13: Conic Sections Unit Test