Using technology to solve triangle construction problems, students apply their knowledge of points of concurrency, coordinate geometry, and transformational geometry.
Chris Harrow and Lillian Chin
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.
Kevin C. Moore and Kevin R. LaForest
A connected introduction of angle measure and the sine function entails quantitative reasoning.
Lorraine A. Jacques
Teachers have what they need-students, a data projector or an interactive whiteboard, and connection to the Internet. Teachers know what they want-students observing mathematics in action, making conjectures, and supporting their conjectures with solid reasoning. However, when using applets, teachers quickly encounter two difficulties: how to choose them and how to use them.
Exploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions.
Christopher J. Bucher and Michael Todd Edwards
In the introductory geometry courses that we teach, students spend significant time proving geometric results. Students who conclude that angles are congruent because “they look that way” are reminded that visual information fails to provide conclusive mathematical evidence. Likewise, numerous examples suggesting a particular result should be viewed with skepticism. After all, unfore–seen counterexamples render seemingly valid conclusions false. Inductive reasoning, although useful for generating conjectures, does not replace proof as a means of verification.