Pre-calculus students this semester dove deep into some of the most advanced concepts in their curriculum: polar coordinates. This challenging yet visually engaging topic is offering pre-calculus students a unique way to think about mathematical relationships.

Unlike the familiar Cartesian (x,y) coordinate system where movement is described as left-right and up-down, polar coordinates utilize a "pole" as a central reference point. Students learned to describe locations based on their distance from a pole and their angle of rotation from a fixed direction.

"This is some of the most advanced work that we do in Pre-Cal here," explained teacher Kit Wallach. The transition from the intuitive x-y plane to the "loopy" nature of polar coordinates can be initially challenging. "They're so used to an x, y coordinate system where they're going left, right, up, down. And so instead, we have to think about going our distance from a central pole – a sort of in-out – and then a measurement of how far around, how much rotation we've done," Kit said.

To help students grasp this new way of thinking, the unit incorporates various physical class projects as visual representations of the concepts. As an introduction to this topic, teachers Kit and Catie Marchessault brought their students outside into the front circle ("A perfect shape for polar coordinates!" said Kit), to dance and move around a maypole,  constructed by the teachers from a large central pole and many colorful ribbons. Each student held a ribbon, weaving towards and away from the pole, while also moving around it.

"It turns out that it actually has been really helpful in helping them make those connections, so that they're thinking, you know, towards the pole, away from the pole, in or out, as opposed to up or down or left, right," Kit notes. "All of those lines that used to be parallel are now like rays out from a central point."

Another unique, hands-on demonstration used a paper fan. By pulling a pin and stretching out a gathered fan, students can visualize how a rectangular piece of paper, with its parallel lines, transforms into a circular shape where those same lines radiate out from a central point.

This connection between rectangular and polar spaces allows for the creation of stunning visual graphs. Students enjoyed graphing sessions, where they will learn to plot intricate shapes that resemble "flowers or snails or spirals." These graphs were drawn by hand and explored using Desmos, a popular online graphing calculator.

All of these creative class projects helped further solidify their understanding of this advanced mathematical concept.