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Teaching Mathematics with

The Geometers Sketchpad

Limited Reproduction Permission 2002 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the teacher who purchases Teaching Mathematics with The Geometers Sketchpad the right to reproduce activities and example sketches for use with his or her own students. Unauthorized copying of Teaching Mathematics with The Geometers Sketchpad is a violation of federal law.

The Geometers Sketchpad, Dynamic Geometry, and Key Curriculum Press are registered trademarks of Key Curriculum Press. Sketchpad is a trademark of Key Curriculum Press. All other brand names and product names are trademarks or registered trademarks of their respective holders. Key Curriculum Press 1150 65th Street Emeryville, California 94608 USA 510-595-7000 http://www.keypress.com/

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2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad ii

Contents Teaching Notes ............................................................................................................1

The Geometers Sketchpad and Changes in Mathematics Teaching.................................................. 1 Where Sketchpad Came From.................................................................................................................. 2 Using Sketchpad in the Classroom.......................................................................................................... 2 Using Sketchpad in Different Classroom Settings ................................................................................ 7 Using Sketchpad as a Presentation Tool................................................................................................. 8 Using Sketchpad as a Productivity Tool................................................................................................. 9 The Geometers Sketchpad and Your Textbook .................................................................................... 9

Sample Activities ......................................................................................................11 Angles ........................................................................................................................................................ 12 Constructing a Sketchpad Kaleidoscope .............................................................................................. 13 Properties of Reflection ........................................................................................................................... 16 Tessellations Using Only Translations.................................................................................................. 18 The Euler Segment ................................................................................................................................... 20 Napoleons Theorem ............................................................................................................................... 22 Constructing Rhombuses........................................................................................................................ 23 Midpoint Quadrilaterals ......................................................................................................................... 24 A Rectangle with Maximum Area ......................................................................................................... 25 Visual Demonstration of the Pythagorean Theorem .......................................................................... 27 The Golden Rectangle ............................................................................................................................. 28 A Sine Wave Tracer ................................................................................................................................. 30 Adding Integers........................................................................................................................................ 32 Points Lining Up in the Plane............................................................................................................. 35 Parabolas in Vertex Form........................................................................................................................ 38 Reflection in Geometry and Algebra..................................................................................................... 41 Walking Rex: An Introduction to Vectors ............................................................................................ 44 Leonardo da Vincis Proof ...................................................................................................................... 46 The Folded Circle Construction ............................................................................................................. 48 The Expanding Circle Construction ...................................................................................................... 52 Distances in an Equilateral Triangle...................................................................................................... 55 Varignon Area .......................................................................................................................................... 59 Visualizing Change: Velocity ................................................................................................................. 63 Going Off on a Tangent........................................................................................................................... 67 Accumulating Area.................................................................................................................................. 70

Also Available from Spectrum...............................................................................74

2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 1

Teaching Notes If youve read the Learning Guide, youve learned how to use The Geometers Sketchpad and youve probably discovered that the range of things you can do with the software is greater than you first imagined. For all its potential uses though, Sketchpad was designed primarily as a teaching and learning tool. In this section, we establish a context for Sketchpad in geometry teaching and offer suggestions for using Sketchpad in different ways in different classroom settings. More than 20 sample activitiestouching on a range of school mathematics topicsfollow these teaching notes. By exploring the sample documents that are installed with the software, youll find even more ideas. Try them with your students for a sense of how Sketchpad can serve your classroom best.

The Geometers Sketchpad and Changes in Mathematics Teaching The way we teach mathematicsgeometry in particularhas changed, thanks to a few important developments in recent years. Alternatives to a strictly deductive approach are available after more than a century of failing to reach a majority of students. (The National Assessment of Educational Progress found in 1982 that doing proofs was the least liked mathematics topic of 17-year-olds, and less then 50% of them rated the topic as important.) First, in 1985, Judah Schwartz and Michal Yerushalmy of the Education Development Center developed a landmark piece of instructional software that enabled teachers and students to use computers as teaching and learning tools rather than just as drillmasters. The Geometric Supposers, for Apple II computers, encouraged students to invent their own mathematics by making it easy to create simple geometric figures and make conjectures about their properties. Learning geometry could become a series of open-ended explorations of relationships in geometric figuresa process of discovery that motivates proof, rather than a rehashing of proofs of theorems that students take for granted or dont understand. In 1989, the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics (the Standards) which called for significant changes in the way mathematics is taught. In the teaching of geometry, the Standards called for decreased emphasis on the presentation of geometry as a complete deductive system and a decreased emphasis on two-column proofs. Across the curriculum, the Standards called for an increase in open exploration and conjecturing and increased attention to topics in transformational geometry. In their call for change, the Standards recognized the impact that technology tools have on the way mathematics is taught, by freeing students from time-consuming, mundane tasks and giving them the means to see and explore interesting relationships. By publishing the first edition of Michael Serras Discovering Geometry: An Inductive Approach in 1989, Key Curriculum Press joined the forces of change. Discovering Geometry, a high school geometry textbook, takes much the same approach that the creators of The Geometric Supposers espoused: Students should create their own geometric constructions and themselves formulate the mathematics to describe relationships they discover. With Discovering Geometry, students working in cooperative groups do investigations using tools of geometry to discover properties. Students look for patterns and use inductive reasoning to make conjectures. They arent expected to prove their discoveries until after theyve mastered geometry concepts and can appreciate the significance of proof. Now in its second edition, Discovering Geometry lets students take advantage of a broader range of tools, including patty papers and The Geometers Sketchpad. This approach is consistent with r