Experience First

In this activity students will use Desmos to begin to explore polar graphs, specifically small circles and roses. We suggest having students work on page 1 of the activity, then debriefing page 1 before having students continue on to page 2. As students are working, maintain a stance of inquiry and curiosity. The questions in the activity are more open-ended and allow students to notice patterns and make generalizations. Our students enjoyed seeing the patterns, especially about how to determine the number of petals. Continue to challenge students to connect their noticings to the parts of the equation. A key idea over the next two lessons will be that sine and cosine both have a range of -1 to 1 and they act as modifiers of the parameters in the equation. 

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Formalize Later

While students noticed many patterns, they seemed to struggle with reasoning about the equation analytically and connecting the graphs to their specific ordered pairs. Knowledge of the unit circle, specifically at the four quadrants is critical for understanding the outputs of these polar graphs.

Graphing polar roses can be done in multiple ways. One method is to have students fill out a table, then plot the points, and use their general observations about shape to finish the graph. Students were also able to use the symmetry of the graph to graph some additional petals “for free” simply by reflecting ones they had already found. At the beginning, we recommend giving students the values of theta, specifically those that will correspond to the tips of the petals. Over time, students should be able to reason about this on their own.

Another method is to identify the angle at which the first petal occurs and use the number of petals to determine the angle between petals. By knowing the lengths of the petals and their location, students were able to draw a fairly accurate representation of the polar equation.

One common misunderstanding: while roses with cosine always have a petal on the polar axis (either positive or negative axis, depending on the sign of a), roses with sine do not always have a petal on the y-axis (only if there are an odd number of petals). Additionally, roses with an even number of petals will have both polar axis and θ=π/2 symmetry.