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Eric L. McDowell

Enhance students' number sense and illustrate some surprising properties of this alternative operation.

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Roger Turton

Reinforce the difference between inductive and deductive reasoning using a small number of points around a circle.

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Eric Weber, Amy Ellis, Torrey Kulow, and Zekiye Ozgur

Modeling the motion of a speeding car or the growth of a Jactus plant, teachers can use six practical tips to help students develop quantitative reasoning.

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Wayne Nirode

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.

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Kevin C. Moore and Kevin R. LaForest

A connected introduction of angle measure and the sine function entails quantitative reasoning.

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Darla R. Berks and Amber N. Vlasnik

Two teachers discuss the planning and observed results of an introductory problem to help students nail a conceptual approach to solving systems of equations.

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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.

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Heather Lynn Johnson

This article explores quantitative reasoning used by students working on a bottle- filling task. Two forms of reasoning are highlighted: simultaneous-independent reasoning and change-dependent reasoning.

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Helen M. Doerr, Donna J. Meehan, and AnnMarie H. O'Neil

Building on prior knowledge of slope, this approach helps students develop the ability to approximate and interpret rates of change and lays a conceptual foundation for calculus.