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# Unit Review: Parametric Equations and Polar Coordinates

Lesson 10 of 12

## Objective: SWBAT review the important concepts of parametric equations and polar coordinates.

*50 minutes*

#### Check Homework

*10 min*

I start class today by spending a few minutes going over the homework from yesterday's lesson on polar form of conic sections. There will inevitably be a few questions on this topic so I want to clear up any misconceptions before we start reviewing for our assessment.

Usually the questions will center around a few concepts that can be elusive to some students. Here is a list of things that usually come up after the assignment:

**The focus is always located at the pole**. Students may forget this. If they must write an equation of a conic in polar form, this can be a key piece of information that is needed.**Positives and negatives can be tricky**. One simple sign can make the equation right or wrong. It is important that students realize what the addition or subtraction sign represents for these formulas.**We can always plug in points that we know are on the graph.**This is something that is often overlooked at all levels of math! If we know one point on a conic and need to find the*p*value, for example, we can plug in the point we know and use it to solve for*p*just like we did in Algebra to find the y-intercept of a linear function.

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#### Explore

*10 min*

In an attempt to solidify the knowledge of conic sections, **I assign question #1** of the review worksheet and give student about 10 minutes to work on it. I want to do this question together before I let them loose on the review for the rest of the hour since this is a good summary problem that connects what we worked on in this polar unit with our work with conic sections in the previous unit. Students will** work on this question with their table groups**. If they finish early, I instruct them that they can continue working on the rest of the review since it will be due tomorrow.

Question #1 is difficult for students because students must **move fluidly between rectangular and polar coordinate systems**. Students can usually graph it pretty easily, but it is difficult for them to find the rectangular equation of the ellipse. I think students can get "stuck" and think that if the equation is in polar form then they cannot consider any properties of rectangular form. For example, for #1 students know that (1.5, 90°) and (3, 270°) are the endpoints of the major axis of the ellipse, but may not know how to find the center of the ellipse. It is a conceptual jump to switch these points to (0, 1.5) and (0, -3) on the rectangular coordinate plane and to just find the midpoint in order to know the center.

#### Resources

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#### Share and Summarize

*30 min*

There are** two main methods that students may use** to come up with the rectangular equation of the ellipse. I will try to choose students to show both ways. If one method does not come up, you can introduce it to students and finish it up as a class.

**Method 1**

**Method 2**

Another way to find the rectangular equation is to just start with the polar form and use the conversions in order to get an equation with* x* and *y*. Some clever algebra may be needed but one way is to cross multiply so you have 6*r* + 2*r*sin*θ* = 12. Now *r* can be written as sqrt(*x*^2 + *y*^2) and *r*sin*θ* = *y*. The form that it is written in may not be recognizable as an ellipse, so method 1 may be more useful.

After going through this problem, it is time to keep working on the rest of the unit review and finish it for homework. This will be a good time for students to recap all of the important concepts that we worked on in this unit.

#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Stolen Car and Keys: An Introduction to Parametric Equations
- LESSON 2: Converting Parametric Equations
- LESSON 3: A New Way to Locate Points
- LESSON 4: Polar Distance Formula
- LESSON 5: Graphing Polar Equations
- LESSON 6: Limaçons and Roses - Day 1 of 2
- LESSON 7: Limaçons and Roses - Day 2 of 2
- LESSON 8: Polar Equations of Conics - Day 1 of 2
- LESSON 9: Polar Equations of Conics - Day 2 of 2
- LESSON 10: Unit Review: Parametric Equations and Polar Coordinates
- LESSON 11: Unit Review Game: Trashball
- LESSON 12: Unit Assessment: Parametric Equations and Polar Coordinates