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April 2012, Volume 105, Issue 8

 FEATURES The Devil and Daniel’s SpreadsheetMaurice J. BurkeA tool that combines the power of computer algebra and traditional spreadsheets can greatly enhance the study of recursive processes. Second Look:Recursive Reasoning Multiple Proof Approaches and Mathematical ConnectionsZhonghong Jiang and George E. O’BrienUsing technology to explore the Three Altitudes of a Triangle problem, students devise many proofs for their conjectures. Don’t Fence Me In!Harold Reiter, Arthur Holshouser, and Patrick VennebushThis method for counting lattice octagons strengthens students’ counting skills and geometrical thinking. Practicing Algebraic Skills: A Conceptual ApproachAlex Friedlander and Abraham ArcaviIntegrating procedures and thinking processes makes learning more meaningful. Dragging and Making Sense of Invariants in Dynamic GeometryAnna E. Baccaglini-FrankIn this activity, students learn to make conjectures about properties that do not change. Enhancing Understanding of Transformation MatricesJonathan Dick and Maria ChildreyWith the Common Core State Standards’ emphasis on transformations, teachers need a variety of approaches to increase student understanding.

 Departments Reader ReflectionsReader Reflections - April 2012 Sound Off!Sound Off!: The Mathematics Teacher's Bookshelf Media ClipsMedia Clips: Midnight Rider // Snake Wedding Mathematical LensMathematical Lens: Weight, Weight, Don’t Tell Me! Calendar ProblemsCalendar and Solutions - May 2012 Technology/Technology TipsTechnology Tips: Exploring the Golden Ratio with The Geometer’s Sketchpad Delving DeeperDelving Deeper: Approximating Pi Using Similar Triangles For Your Information/Products/PublicationsFor Your Information - April 2012 The Backpage: My Favorite LessonThe Back Page: My Favorite Lesson: Crazy Algebra Objectives

 Second Look - Recursive Reasoning Developing Mathematical Power by Using Explicit and Recursive ReasoningThe use of a variety of tasks that encourage students to examine the advantages and limitations of recursive and explicit reasoning. The authors discuss the reasoning for employing different types of reasoning and how to encourage students to justify their thinking. Just Say Charge It!The use of a recursive sequence with a graphing calculator to involve students in analyzing credit card debt. Technology Tips: Using the Iterate Command to Construct Recursive Geometric SketchesThis month's article introduces readers to an advanced feature of the Geometer's Sketchpad: how to iterate geometric shapes to construct Baravelle spirals and Pythagorean trees. The "Surfing Note" sends readers to a site with applets that will generate fractals such as the Sierpinski gasket or the Koch snowflake. Illuminations Lesson: Trout PondThis investigation illustrates the use of iteration, recursion and algebra to model and analyze a changing fish population. Graphs, equations, tables, and technological tools are used to investigate the effect of varying parameters on the long-term population. Algebra Standard for Grades 9-12Instructional programs from prekindergarten through grade 12 should enable all students to— Understand patterns , relations, and functions  Represent and analyze  mathematical situations and structures using algebraic symbols Use mathematical models to represent and understand quantitative relationships  Analyze change in various contexts