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## Unit 5 - Day 8

##### All Units
###### ​Learning Objectives​
• Understand the polar system as an alternate way of describing locations by using a radius and an angle

• Use coterminal angles and reflected radii to name polar points in multiple ways

• Convert between polar and Cartesian coordinates

###### Experience First

We absolutely loved the Springboard curriculum’s approach to introducing students to polar coordinates! Keeping track of all the planes in an airspace is no small feat and polar coordinates help air traffic controllers do it! Most of the ideas for the context of this lesson come from that Precalculus textbook but we saved some of the more formal ideas for the debrief.

Students begin by locating an airplane on the polar screen and recognize that there are infinitely many places an airplane could be if it is 15 miles from the control tower. They recognize that an angle is needed to pinpoint a location.

In question 3, students explore how to convert points from Polar to Cartesian. Many students estimated these rectangular coordinates based on the graph. Push them for precise answers.

In question 4 and 5, students convert a Cartesian point into Polar coordinates. This should feel very familiar to them as they just found magnitude and direction for vectors. Students will notice that the hot air balloon in its new location does violate the rule and is just inside the 20 mile radius. Many students were able to use inverse tangent to find the reference angle for the location. Ask students if that’s how the air traffic controllers would measure the angle. They should be able to come to the conclusion that the controllers measure angles CCW from due east, they will need to subtract their found angle form 360˚.

###### Formalize Later

The idea of renaming a polar point is an important one for the debrief. While there is only one way to refer to a location in Cartesian coordinates, there are infinitely many ways to do this using polar coordinates. Students might suggest measuring clockwise with a negative angle, or using a coterminal angle. Explain what a negative radius is and have students discuss how they might need to adjust the angle.

The general formulas for conversions become the margin notes for the activity. Emphasize that when finding the sine and cosine of the angles, they can use the acute angle in the triangle, or the actual angle (210˚ vs 30˚ in question 3) and why this is.

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