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Kent Thele

Encourage investigation of the conic-section attributes of focus, eccentricity, directrix, and semi-latus rectum using polar coordinates and projective 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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Marlena Herman

An introduction to definitions and equations of conic sections can be extended to explain the significance of the determinant.

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Wendy B. Sanchez and David M. Glassmeyer

In this 3-part activity, students use paper-folding and an interactive computer sketch to develop the equation of a parabola given the focus and directrix.

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Megan Snow

Understanding what students are thinking sometimes requires immediate action. Here are some quick, easy strategies.

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Elaine M. Purvinis and Joshua B. Fagan

In first- and second-year algebra classrooms, the all-too-familiar whine of “when are we ever going to use this in real life?” challenges mathematics teachers to find new, engaging ways to present mathematical concepts. The introduction of quadratic equations is typically modeled by describing the motion of a moving object with respect to time, and typical lessons include uninspiring textbook practice problems that portray dropping or shooting objects from given distances or at particular time intervals. For a novel approach to exploring quadratics, we chose to step outside the classroom to look at some phenomena in the field of acoustics. Our activity incorporates mathematical modeling to provide a multirepresentational view of the math behind the physics and to provide a conceptual basis for analyzing and understanding a real-world quadratic situation.

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Joanne C. Caniglia

The stunning natural beauty of Arizona, New Mexico, southern Colorado, and Utah is indicative of the American Southwest and is reflected in Southwestern baskets. Many Southwestern basket weavers use coiling as their method of construction (see fig. 1). The following problems relate mathematics to the art of basket weaving, with an emphasis on coiling.

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A set of problems of many types.

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A set of problems of many types.

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A set of problems of many types.