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• Twelfth
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## Mediants Make (Number) Sense of Fraction Foibles

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

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## The Spot Problem Revisited

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

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## Six Principles for Quantitative Reasoning and Modeling

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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## Triangles from Three Points

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

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## Technology-Enhanced Discovery

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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## The Circle Approach to Trigonometry

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

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## Working the System

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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## Technology Tips: Using Applets for Inquiry

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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## Connecting Research to Teaching: Reasoning about Quantities That Change Together

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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## A Natural Approach to the Number e

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.