Polar Coordinates (Lesson 5.5)
Unit 5 - Day 8
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
Quick Lesson Plan
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.
I begin this lesson by asking students “Where is EKHS?” (our school). I give them 15 seconds to think then I have each group share out. Some students say that EKHS is across from Starbucks, off of M-6, some give the nearest intersection, some talk about the geographical region. I have some pictures from Google maps on the projector that say we are 8.18 miles SE of Grand Rapids, and another that says we are 1855 miles Northeast of Los Angeles. I then show a slide that gives our exact longitude and latitude, and ask them why none of the groups gave the location of our school in that way. We talk about how latitude and longitude are like the x and y coordinates they are so used to from school, but there are other ways to describe a location, such as how far away and in what direction the school is from other places. I have them talk about advantages and disadvantages of referring to location in each of these ways. Then I say that today we’re going to learn about some other ways to talk about location using the polar system and that this is really helpful for aviation. I then have them start the activity portion of the lesson.
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˚.
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.