Experience First:

In this final unit of the course, students will be introduced to a new coordinate system and explore polar functions. Many of the function analysis tools students have used throughout the course will now be applied to polar functions.

 

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.

 

In the opening lesson, students will explore how air traffic controllers monitor the location of airplanes using a radar screen set up like a polar coordinate plane! Students learn to pinpoint locations using a distance from the air traffic control tower (the radius) and an angle measured counterclockwise from due east (theta, in standard position). Just like in the Cartesian coordinate system, two values are needed, for neither the radius nor the angle is sufficient to determine a plane’s unique location.

 

In questions 3-5 students use the context of monitoring an air space to convert between Cartesian and polar points. Remember that students have not been pre-taught any conversion formulas! However, they have plenty of experience identifying coordinates of points on circles (see Lesson 6.3 especially!). This prior knowledge is sufficient for students to answer questions 3-5. 

 

In question 5, students will likely be quick to recognize the need for the tangent function but may be perplexed by the negative angle in radians that their calculator produces. Use monitoring questions to help students make sense of this value. It is generally difficult for students to get a feel for the size of an angle given in radians that is not in terms of π. Being able to interpret a radian value and place it roughly between the key angles on the unit circle is a helpful skill. 

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Monitoring Questions:

• Why is having a distance and an angle sufficient for determining the plane’s location?

• How is determining the plane’s north/south and east/west position based on a distance from the origin and an angle similar to work we have already done this year? What does it remind you of?

• What would you need to know to determine if this plane is in violation of the 20-mile rule?

• Why did you choose tangent to find the angle? 

• Why did your calculator produce a negative angle when you evaluated inverse tangent?

• What quadrant is this angle in? What quadrant is the balloon’s new location in?

• We read that angle measures are measured counterclockwise from due east. How could you rewrite this negative angle so it follows these rules?

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Formalize Later:

The ability to rename a polar point is an important concept to bring up in the debrief. While there is only one way to refer to a location in Cartesian coordinates, there are infinitely many ways to refer to the same location 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.

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There are (at least) two approaches to teaching students how to convert between polar and rectangular forms. The first approach uses the reference triangle created by the point on the terminal ray. The reference triangle is created by drawing a perpendicular segment from the point to the x-axis and then drawing a horizontal segment along the x-axis to the origin. The angle that is referenced here is the acute angle that has the origin as its vertex. To find this angle, students can use tangent inverse of the absolute value of the y-coordinate over the x-coordinate (this ensures the angle is always positive). They will then need to adjust the angle for the quadrant so that the angle is measured in the conventional way, counterclockwise from the positive x-axis. In the first quadrant, no adjustment is needed. In the second quadrant, the acute angle is subtracted from π. In the third quadrant, the acute angle is added to π. In the 4th quadrant, the acute angle is subtracted from 2π. These “adjustment rules” don’t necessarily have to be memorized. A quick sketch of a unit circle will illuminate what adjustments may need to be made.

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When converting from polar to rectangular, this approach would find the lengths of the legs of the right triangle using that reference angle, and then adjust which measurements need to be negative based on the quadrant. We could summarize this approach as the right triangle trig approach. The idea is that all measurements are found as positive values, then adjusted for the quadrant as needed.

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I used the above approach for many years because I found that students were less likely to make errors when they used one consistent approach to determining the angle and the mind seems to jump almost immediately to drawing a right triangles on the plane.

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However, the AP Precalculus sequence may lend itself better to a different approach. The fact that we have defined the sine, cosine, and tangent ratios based on a point’s horizontal and vertical displacement and distance from the origin, has made finding coordinates in the second, third, and fourth quadrant much more intuitive. I don’t think students will struggle as much with making sense of the angle measure given from standard position, instead of its reference angle. Converting to rectangular coordinates is now the exact same skill as finding the coordinates of a point on a circle of any size radius (Lessons 6.3 and 6.4).

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This approach flows more naturally into the x=rcos(theta) and y=rsin(theta) formulas, because the value of theta is actually a coordinate of the polar point, not a reference angle. When students use right triangles, they are actually using some different angle alpha and then later adjusting the sign of x and y. When I would present the conversion formulas, students would take me at my word, but it was dissatisfying that the formula didn’t actually match their right triangle approach. 

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When converting from rectangular to polar coordinates, the inverse tangent will give values in the 1st quadrant or in the 4th quadrant (but using a negative angle measure). Students now understand why this is the case, so the need and strategy for finding the corresponding angle in the 2nd or 3rd quadrant feels intuitive. This is the exact same thought process they used when they had to find all solutions to an equation like tan(x)=3 on a given interval. The approach becomes much less about adjusting for each quadrant, and more about understanding why the calculator produces only one specific type of angle measure and then considering at what other angle the tangent function would have the same value.

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When using this approach, we suggest drawing a perpendicular line from the point down to the x-axis and then drawing a segment perpendicular to the y-axis through the point itself, rather than having this horizontal segment on the x-axis.