Using technology to solve triangle construction problems, students apply their knowledge of points of concurrency, coordinate geometry, and transformational geometry.

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### 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.

### Harold B. Reiter, John Thornton, and G. Patrick Vennebush

Through KenKen puzzles, students can explore parity, counting, subsets, and various problem-solving strategies.

### Bobson Wong and Larisa Bukalov

Parallel geometry tasks with four levels of complexity involve students in writing and understanding proof.

### Gloriana González and Anna F. DeJarnette

An open-ended problem about a circle illustrates how problem-based instruction can enable students to develop reasoning and sense-making skills.

### Michael K. Weiss and Deborah Moore-Russo

The moves that mathematicians use to generate new questions can also be used by teachers and students to tie content together and spur exploration.

### R. Alan Russell

In trying to find the ideal dimensions of rectangular paper for folding origami, students explore various paper sizes, encountering basic number theory, geometry, and algebra along the way.

### Ayana Touval

Through movement-a welcome change of pace-students explore the properties of the perpendicular bisector.

### Colin Foster

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.