It’s a gripe I’ve shared before, but I’ll repeat it—the typical high school geometry approach to introducing transformations is boring. Fresh from learning the definition of a translation, reflection, rotation, or translation, students are whisked off to the safety of the coordinate plane and asked to explore the numerical effect of reflecting a point over the x- or y-axis or rotating a point by 90° about the origin. As with so much of high school geometry, there’s an unspoken discomfort with staying in a strictly geometric realm for too long. Quick, bring back some numbers and algebra where the “real” math resides!

In our prior blog posts, we’ve described geometric transformation activities from our NSF Forging Connections project that offer students rich interactions with reflections, rotations, translations, dilations, and glide reflections. For example, our transformation dances provide students with embodied experiences of transformations, with the coordinated movement between preimage and image points being dance-like in nature.  And in our transformation games, students are given transformations with one critical element omitted (e.g., the location of a mirror line, the value of an angle of a rotation) and must estimate these locations and values.

In this post, I’d like to share two new activities where students construct rotations to trace attractive designs like the ones below. Before jumping in, you can watch this introductory video that demonstrates how they work. Aside from the aesthetic appeal of using rotations to create art, these activities also benefit from the multiple approaches students can use to draw the star patterns.

In this first websketch (below and here), the penguin travels back and forth along the arm of a star. The frog is busy as well, moving along a rotated path. Your goal is to keep adjusting the center and angle of the rotation so that the frog, one arm at a time, traces the remainder of the star. There are three pages of challenges.

In the second websketch (below and here) the penguin and frog are gone, but the goal is the same: Use the provided tools to trace the arms of the stars. Whereas the preimage point for the penguin-frog challenges was always the penguin, now, here you can choose different preimage points for each arm of the star you trace (This introductory video demonstrates some possibilities.) There are seven pages to explore.