Exploration, innovation, proof: For students, teachers, and others who are curious, keeping your mind open and ready to investigate unusual or unexpected properties will always lead to learning something new. Technology can further this process, allowing various behaviors to be analyzed that were previously memorized or poorly understood. This article shares the adventure of one such discovery of exploration, innovation, and proof that was uncovered when a teacher tried to find a smoother way to model conic sections using dynamic technology. When an unexpected pattern regarding the locus of an ellipse's or hyperbola's foci emerged, he pitched the problem to a ninth grader as a challenge, resulting in a marvelous adventure for both teacher and student. Beginning with the evolution of the ideas that led to the discovery of the focal locus and ending with the significant student-written proof and conclusion, we hope to inspire further classroom use of technology to enhance student learning and discovery.
Chris Harrow, email@example.com, a National Board Certified Teacher, is the chair of the mathematics department at the Hawken School in Cleveland, Ohio. He blogs and presents nationally on the educational uses of technology and on Computer Algebra Systems (CAS) in precollegiate mathematics. He is also the recipient of a Presidential Award for Excellence in Mathematics and Science Teaching.
Lillian Chin, firstname.lastname@example.org, attended The Westminster Schools in Atlanta, Georgia, where Harrow previously taught; she is now a student at MIT. Chin was an Intel Science Talent Search Finalist for 2013 for her development of a computer model of cell behavior in healing wounds.