So I kept bumping into obstacles that I hadn’t imagined would be there. It was the first time I had taught polar coordinates in any serious way. I’m interested in exploring this territory a bit.įor the record, I was flying blind through this material last semester. My students saw some relationships that they couldn’t quite articulate. The precise conceptual nature of the relationship between polar coordinates and cartesian parametric equations is unclear to me. This seemed to be true even for students who could talk dynamically about cartesian graphs ( increasing, decreasing, approaching an asymptote-this was terminology my students could apply to cartesian graphs, but not to polar ones). But they couldn’t think about the process of tracing out a polar graph. They could make their graphing calculators display polar graphs. They could identify points one at a time. They couldn’t view a function defined in polar coordinates as a dynamic relationship. My students struggled to think about an angle as an independent variable that could change (and correspondingly a radius as a dependent variable that could change). Polar coordinates were always (in their minds) in relation to cartesian coordinates. They knew how to convert between polar and cartesian coordinates, but they didn’t seem to know why one would do that, nor did they seem to see polar coordinates as a self-contained system. When I taught Calculus 2 this past semester, it was clear that my students were struggling to sort out differences between cartesian coordinates and polar coordinates. What do we gain from graphing this in PreCalc?
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