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November 2011, Volume 105, Issue 4

 FEATURES Is There a “Best” Rectangle?R. Alan RussellIn 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. Sketching Up the Digital DuckKathryn G. Shafer, Gina Severt, and Zachary A. OlsonTwo preservice teachers describe how using Google SketchUp, Terrapin Logo, and The Geometer’s Sketchpad fosters a deeper understanding of measurement concepts. Teaching the Perpendicular Bisector: A Kinesthetic ApproachAyana TouvalThrough movement—a welcome change of pace—students explore the properties of the perpendicular bisector. The Shape of an EllipseGregory D. FoleyEllipses vary in shape from circular to nearly parabolic. An ellipse’s eccentricity indicates the location of its foci, but its aspect ratio is a direct measure of its shape. More on Ellipses and Polar Coordinates Navigating between the DimensionsJulian F. Fleron and Volker EckeTwo classroom activities—the Flatland game and sliceforms—are useful vehicles for student exploration of the geometric interplay between the dimensions. Stop Teaching and Let Students Learn GeometryMichael J. Bossé and Kwaku Adu-GyamfiA geometry course for teachers—easily adaptable to a high school geometry class—integrates technology, reasoning, communication, collaboration, reading, writing, and multiple representations. Instructions for the EGASPExamples of the EGASP Second Look:Geometry Recreating History with Archimedes and PiLora C. SantucciStudents use modern technology to investigate historical perspectives and calculate an approximation of pi. Thinking Deeply about Area and PerimeterWayne NirodeUsing a 6-inch-square sheet of paper and a simple rule for creating a polygon, students can explore interesting area and perimeter problems.

 Departments Editorial/From the EditorsFrom the Editorial Panel: Teaching Geometry Reader ReflectionsReader Reflections - November 2011 Media ClipsMedia Clips: Traffic Accidents//Game Shows Mathematical LensMathematical Lens: Georgia on My Mind Calendar ProblemsCalendar and Solutions - December 2012 Delving DeeperDelving Deeper: Pythagorean Triples: Avenues for Exploration For Your Information/Products/PublicationsFor Your Information - November 2011 The Backpage: My Favorite LessonThe Back Page: My Favorite Lesson: Playing the River Crossing Game

 Calls for Manuscripts Call for Manuscripts: 2013 Focus Issue: Beginning Algebra: Teaching Key Concepts - November 2011 See all current Calls for Manuscripts

 Second Look - Geometry Key Ideas and Insights in the Context of Three High School Geometry ProofsThe study of proofs is one topic that that most students encounter in Geometry with varying degrees of success. Teachers often search for new and exciting ways to help students generate informal and formal proofs. This article may be used with pre-service teachers or by in-service teachers interested in exploring ways to connect students’ intuitive sense making with formal proofs. Activities for Students: Moving a Wall: Using Geometry to Measure an Imperceptible DistanceStudents conduct a mathematical experiment to determine how far they can move a solid wall when they push against it. Intended for use in any geometry or trigonometry classroom, this lesson capitalizes on students’ prior knowledge of basic circle properties, right-triangle trigonometry, and unit conversions. Translations toward Connected MathematicsThe translation principle allows students to solve problems in different branches of mathematics and thus to develop connectedness in their mathematical knowledge. Hands-On Fractals and the Unexpected in MathematicsA hands-on project in which fractal images are produced using a photocopy machine and office supplies. Article discusses how the resulting images are an example of the contraction mapping principle. Includes questions for extension investigations. Illuminations Lesson: Reflect On ThisThis lesson, adapted from an activity in Navigating through Geometry in Grades 9-12, requires students to investigate reflections using hinged mirrors. With a kaleidoscope, students will examine the interior angles of regular polygons. Geometry Standard for Grades 9-12Instructional programs from prekindergarten through grade 12 should enable all students to— Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; Specify locations and describe spatial relationships using coordinate geometry and other representational systems; Apply transformations and use symmetry to analyze mathematical situations; Use visualization, spatial reasoning, and geometric modeling to solve problems.