1. Geometry Theme MAA PREP WORKSHOP Laurie Burton and Maria Fung Western Oregon University July 8 th 2003

2. Overview Overview of WOU Geometry Courses  Foundations  Geometry for Elementary Teachers  Geometry for Middle School Teachers Geometry Activities  Explorations on symmetry and reflections  Visual approach to the Pythagorean Theorem  Geometer’s Sketchpad Lab Explorations Assessment

3. Geometry NCTM Standards Highlights Analyze properties and characteristics of two- and three-dimensional geometric shapes Describe spatial relationships using coordinate geometry and other representational systems Apply transformations and use symmetry to analyze mathematical situations

4. Geometry for K-8 Teachers Foundations of Elementary Mathematics sequence includes a one-quarter course on geometry Main topics covered:  Geometric figures Geometric Figures; Plane Figures Polygons and Tessellation; Space Figures; Symmetric Figures  Measurement Systems of Measurement Area and Perimeter; Volume and Surface Area  Motions in geometry Congruence and Constructions Congruence Mappings; Similarity Mappings Introduction to the Geometer’s Sketchpad Package

5. Geometry for K-8 Teachers Geometry For Elementary School Teachers is a one-quarter course for mathematics focus students Main topics covered:  Geometric Shapes and Measurement  Polygons & Circles  Perimeter & Circumference, Area, Angles  The Pythagorean Theorem  Polyhedra  Surface Area & Volume  Triangle Congruence & Similarity  Basic Geometric Constructions

6. Geometry for K-8 Teachers Geometry for Middle School Teachers is a mixed advance undergraduate and graduate course Topics covered:  Reflections and Symmetry  Tessellations  Isometries and their Relations  Perimeter, Area and the Pythagorean Theorem  Similarity  Measurement and Similarity Mappings

7. Comments on Mirrors and Symmetry Activity Introductory activity for the Geometry for Middle School Teachers Course Students use trial and error at first Hands-on experience typically leading to informal proofs Excellent conceptual understanding of reflections and lines of symmetry Good review of quadrilaterals and their properties

8. More Comments on Mirrors and Symmetry Activity This activity encourages students to “reflect on reflections” Students have fun exploring and improving their geometric intuition Increases awareness of geometric concepts (e.g. parallel, angle) Follow up: construct and critique minimal “message” for obtaining a specific shape

9. Examples of Follow-up Activities Construct a minimal message for seeing a rhombus that is not a square. Critique the following message: 1.Make a right angle. 2.Bisect the right angle. 3.Place the mirror perpendicular to the bisector in Step 2 (with the reflecting side facing the vertex of the right angle) 4.Be sure that the mirror cuts off equal lengths on the sides of the right angle.

10. Paper Folding Activity Another hands-on method of discovering lines of symmetry for various quadrilaterals Opportunity for sharing different ways of producing required shapes

11. Pythagorean Theorem Activity Find the areas of the squares by using visual methods with the distance between two dots as a unit (and not by measuring the sides and applying the formula for the area of a square) Address the case of an obtuse triangle for part 2 of this activity Show the construction for the proof to students

12. Design Your Own Work on creating an activity or lesson plans centered around the topic of tessellations Use the provided materials in your Resource Notebooks Make a poster or a presentation of your work to be shared with the group

13. Activities with the Geometer’s Sketchpad Work on the following lab activities: Glide Reflections (first part to start) One tessellation activity to start (and finish them all, time permitting)  A Tumbling Block Design  Tessellations Using Only Translations  Tessellations That Use Rotations Dissection Demonstration of the Pythagorean Theorem The Golden Rectangle One activity of More Ideas for Transformation Projects

14. Geometry Assessment I Foundational Geometry:  Concept Pages  Conceptually Centered Exams  Problems of the Week (Explorations of Geometric Relationships)  Article Reports  Textbook Homework

15. Geometry Assessment II Advanced Geometry:  Geometry Lab Reports (√-, √, √+)  Readings Reports and Reflections  Final Projects  Geo-Jeopardy Final  Vocabulary Lists  Quizzes & Exams  Extensive Homework

16. Assessment Continued Example of a grading rubric from Geometry for Middle School Teachers Course: Interpretation (0-2 pts.) Execution (0-2 pts.) Completeness (0-2 pts.) Reflection (0-2 pts.) 0: sketchy, incomplete, incorrect, or missing 1: adequate 2: thoroughly developed, excellent