This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."
Students realize that the hypotenuse of a right triangle, the magnitude of a vector, and the r-value of a polar coordinate are three different names for the same quantity. Students realize that the direction of a vector and the theta-value of a polar coordinate are the same quantity. Students understand how to use polar equations to find r-values and how those r-values are plotted. Students understand how parameters in a polar equation affect the properties of its graph.
A couple years ago I moved straight from vectors to the polar coordinate system because the math is identical. With such recent exposure to the mechanics it just becomes a vocabulary lesson.
We start simple, taking rectangular coordinates and finding their polar equivalent. We draw some right triangle systems and define the hypotenuse as the length of the radius. The theta-value of the coordinate is just the angle relative to zero. It's the same math from vectors.
Next, what makes these things unique. Graph of some high-res polar grids. Have students make a table in 15 degree increments, 0 to 360. Give them some simple polar equations like r = 3 + 3 cos theta and r = 3 + 3 sin theta. Have them complete the table for all the values of thetas. Ignore the groans. Once finished, show them how to plot a point along the circles and then connect the dots.
To better understand the weird shapes, they do an exploration. Given a set of polar graphs and some equation frames (a+/-b cos/sin theta; m sin/cos k theta) students determine the equation associated with a graph and we have a discussion about properties. They have iPads to do the graphing.
Questions to ask: why does 5 + 3 sin theta appear the same as -5 + 3 sin theta? why do graphs have a favored orientation? what determines petal count in a rose curve? What induces a loop? is there anything odd about petal counts? how would you make a rose curve with 10 petals?
Next we get a little more technical. Students are given a second set of polar graphs. This time, no iPads to help. Come up with the equation. Using the equation you came up with, find r-values at 0º, 90º, 180º, and 270º. Use this information to make a connection between negative r-values and the appearance of loops. Discuss the angle separation of rose curve petals. Discuss how far off-axis a sin based rose curve is rotated from its cos counterpart.
One last set of polar equations. Having their fill of cardiods, limaçons, and roses, we discuss ellipses and circles. Pass out the iPads again and send them to a polar ellipse generator. Give them a set of ellipses and have them determine the appropriate parameters. Have a brief discussion about any effects they notice and give them some definitions: semi-major axis, eccentricity, and focus. The key part of this study is to get an idea for eccentricity and how it relates to the dimensions of a shape.
If you really want to keep going (and at this point it's mid-April and probably have the itch to start Calculus), you can explore parabolas and hyperbolas.
Great stuff here. After the initial discussion of equations that match graphs, the students get a chance to design their own. Then we draw them outside on the sidewalk.
When discussing polar conics, you can have some really fun discussion about the solar system. Polar ellipses are plotted using Kepler's equations, which nicely map the orbits of any celestial body you've heard about. After the exploration of making equations of arbitrary ellipses, send them to Wikipedia and lookup the semi-major axis and eccentricity for Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, Eris, and Halley's Comet. Sticking those values into a Kepler equation gets you a lovely solar system, complete with Pluto's encroachment of Neptune.
There's a great video to show as a followup that discusses how bodies interact with one another in space.