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.

### Michael Tamblyn

A wonderful experience occurred in a class that I was teaching recently. It was a precalculus class, the last period of the day. The local university had brought over its cadre of preservice secondary school mathematics teachers to observe my class, so there were twenty-four additional eyes on me that day.

### Martin Griffiths

I always seek activities that might stretch my students yet would be accessible to them; that might require logical thought yet would contain counterintuitive elements; that might provide the opportunity to venture into new mathematical realms yet would have a simple starting point. This article and the activity that inspired it did indeed arise by way of a relatively straightforward problem that I proposed to one of my classes.

### Craig Barton

Students often have difficulty with the topic of straight-line graphs. Perhaps they cannot relate to the abstractness of the concepts involved. Perhaps the sheer number and complexity of the skills required—reading algebra, substituting values, rearranging formulas, dealing with negative numbers, understanding coordinates and fractions—magnifies any misconceptions or weaknesses that students may have in other areas of mathematics, rendering them unable to come to grips with the topic as a whole.

### John H. Lamb

Vector properties and the birds' frictionless environment help students understand the mathematics behind the game.

### Alison L. Mall and Mike Risinger

Our favorite lesson, an interactive experiment that models exponential decay, launches with a loud dice roll. This exploration engages students in lively data collection that motivates interest in key components of the Common Core State Standards for Mathematics: functions, modeling, and statistics and probability (CCSSI 2010).

### Douglas A. Lapp, Marie Ermete, Natasha Brackett, and Karli Powell

Algebra involves negotiating meaning between the worlds of mathematical ideas and the symbols that represent them. Here we examine classroom interactions and explorations as they relate to the connection of these worlds through the use of dynamically connected representations in a technology-rich environment.

### S. Asli Özgün-Koca, Michael Todd Edwards, and Michael Meagher

The Spaghetti Sine Curves activity, which uses GeoGebra applets to enhance student learning, illustrates how technology supports effective use of physical materials.

### Jon D. Davis

Using technology to explore the coefficients of a quadratic equation leads to an unexpected result.