Summary: Middle school mathematics generally explores applications of the Pythagorean theorem and lays the foundation for working with linear equations. The Grade 8 Curriculum Focal Points recommend that students "apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra." Although developing a generalized equation for a circle is commonly considered a high school topic, initial explorations tied to the Pythagorean theorem are within the grasp of middle-grades students and lay the foundation for further work at the high school level. Teachers need to help students visualize, instead of just memorize, so they can develop mathematical thinking. In so doing, students will look for different representations of known ideas, and will seek out and make "connections" that build the useful structures that lead to understanding and that provide a means for further learning and application. Different representations for relationships help students visualize mathematical concepts rather than just memorize formulas. In this article, the authors share activities that they use with their preservice elementary and middle-grades teachers to explore these relationships. (Contains 6 figures.) (ERIC)