By performing three tests, we will see how to apply the properties of symmetry to polar equations. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. Symmetry is a property that helps us recognize and plot the graph of any equation. All points that satisfy the polar equation are on the graph. Recall that the coordinate pair ( r, θ ) ( r, θ ) indicates that we move counterclockwise from the polar axis (positive x-axis) by an angle of θ, θ, and extend a ray from the pole (origin) r r units in the direction of θ. Just as a rectangular equation such as y = x 2 y = x 2 describes the relationship between x x and y y on a Cartesian grid, a polar equation describes a relationship between r r and θ θ on a polar grid. (credit: modification of work by NASA/JPL-Caltech) Testing Polar Equations for Symmetry While this is of course true, we find that it obscures the idea of trying to represent the same graph in two different coordinate systems, and students tend to focus only on the procedure instead of the larger goal.Figure 1 Planets follow elliptical paths as they orbit around the Sun. Other students use the conversion formula tan θ=y/x and then plug in the angle of π/3 and solve for y. We talk about the fact that rise/run is simply the ratio of the opposite over the adjacent side, which can be found by taking the tangent of the angle. We talk about how we are used to writing lines in y=mx+b form, and then we go about finding the y-intercept (0) and the slope. They recognize that the set of these points represents the graph of the equation and it is a line passing through the origin that makes a 60˚ angle with the x-axis/polar axis. I like to have students graph the set of points where the angle is π/3, regardless of what the radius is. Many just gave the ordered pair on the unit circle at that angle. We found that our students struggled most when converting polar equations like θ=π/3 into rectangular. As time permits, allow students time to work on homework or other practice problems in their small groups, applying their new strategies. An important take-away is that equations can be manipulated so that they match the conversion formula, and then can be substituted for the variable in the other form. On the back of their recording sheets or in a notebook have students identify and write down the major strategies they used to match the cards. It may be helpful to have the conversion formulas up on the board or in a place that’s easily visible for students.Ī Desmos version of this card sort can be found here. They will then work on the final column which requires showing algebraically why the polar and Cartesian equations are equivalent. Additionally, give each student a recording sheet where they will keep track of their matches. Each group will need one set of cards and they will work together to find matching trios. We printed each set on a different color. We then had each group share one response back to the whole group.įor the next activity, you will need to print the card sort that contains 6 graphs, 6 equations in rectangular coordinates and 6 equations in polar coordinates. Then we had them share their response to their group members and make any edits they wanted to make. How would you describe to them what polar points are?” First, we gave students 5 minutes of individual writing time. We gave students the prompt: “Suppose a classmate was absent yesterday. We spent the first ten minutes of class reviewing the big ideas from yesterday’s lesson, since it was the first day they had ever seen polar coordinates.
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