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# A New Way to Locate Points

Lesson 3 of 12

## Objective: SWBAT convert from polar coordinates to rectangular or vice versa.

#### Launch

*5 min*

When we looked at rotated conic sections in an earlier lesson, my students realized that the point (1,1) could also be thought of as a rotation of 45 degrees and a distance of sqrt(2) from the origin. I remind students of this on this worksheet and use this as an** introduction to polar coordinates**. I outline my teaching strategies in the video below.

#### Resources

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#### Explore

*20 min*

Once students understand the premise behind polar coordinates, I will have them complete the table on the worksheet and** brainstorm generalizations about the conversions** from polar coordinates to rectangular and vice versa.

I encourage students to **sketch out graphs for each one and to make right triangles**. That will establish the strong connection to trigonometry. As students are working I will circulate and see how they are doing. Here is a list of things I will look for:

- Are students using Pythagorean Theorem to find the
*r*value? - If students are using inverse tangent to find the angle measure, is it in the right quadrant?
- Are students making an overgeneralization that inverse tangent will always give the correct angle when it really only works in quadrants I and IV?
- Are students sketching the locations of the points correctly?

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#### Share and Summarize

*20 min*

Once students have had time to brainstorm, I will have students share their thoughts about these conversions with the class. I find that they usually have an easier time** converting from polar to rectangular**, than from rectangular to polar. Here are the conversions we are looking for:

It is **important that students realize** that the tan(θ) = *y/x* conversion is not perfect; they will have to think about the quadrant that the point is in and adjust the point accordingly.

The remainder of the worksheet provides an opportunity for students to graph a few points on the polar axis and to predict what some simple polar equations will look like. I encourage students to list points that satisfy each equation if they do not know what it will look like. That is usually** a helpful strategy for them to envision the graph**. We will share out our thoughts after they have a chance to think about it.

Finally, I will assign some problems from our textbook with polar coordinates. I will stick to problems that** focus on single points or very simple graphs**. We will work on more complicated equations as we progress 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