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Transformational Geometry in CCSS Mathematics: Building Understanding and Engagement

This workshop aims to deepen educators' understanding of transformational geometry within the Common Core State Standards in Mathematics (CCSS-M). Participants will engage in hands-on activities that explore the geometry sequence, the significance of transformations, and their applications in teaching. We’ll discuss affines and projective transformations, and how to develop students' geometrical reasoning through technology and constructions. Join us to share experiences and strategies, and discover new ways to enhance your mathematics instruction.

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Transformational Geometry in CCSS Mathematics: Building Understanding and Engagement

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  1. Embracing transformational geometry in CCSS-Mathematics Jim Short jshort@vcoe.org Presentation at Palm Springs 11/1/13

  2. Introductions Take a minute to think about, and then be ready to share: • Name • School • District • Something you are doing to implement CCSS-M • One thing you hope to learn today

  3. Workshop Goals • Briefly explore the Geometry sequence in CCSS-M • Deepen understanding of transformational geometry and its role in mathematics • In the CCSS-M • In mathematics in general • Engage in hands-on classroom activities relating to transformational geometry • Special thanks to Sherry Fraser and IMP • Special thanks also to CMP and the CaCCSS-M Resources

  4. 1. Bring and assume best intentions. 2. Step up, step back. 3. Be respectful, and solutions oriented. 4. Turn off (or mute) electronic devices. Workshop Norms ATP Administrator Training - Module 1 – MS/HS Math

  5. Transformation Geometry • What is a transformation? • In Geometry: An action on a geometric figure that results in a change of position and/or size and or shape • Two major types • Affine – straight lines are preserved (e.g. Reflection) • Projective – straight lines are not preserved (e.g. map of the world) • School mathematics focuses on a sub-group of affine transformations: the Euclidean transformations

  6. Flow of Transformational Geometry • Ideas of transformational geometry are developed over time, infused in multiple ways • Transformations are a big mathematical idea, importance enhanced by technology Develop Understanding of Attributes of Shapes Develop Understanding of Effect of Transformations on Figures Develop Understanding of Transformations as Functions on the Plane/Space Develop Understanding of Coordinate Plane Develop Understanding of Functions

  7. Geometry Standards Progression • Share the standards with your group. Take turns reading the content standards given • Analyze the depth and complexity of the standards • Order the standards across the Progression from K – High School

  8. Geometric Transformations In CCSS-Mathematics • Begins with moving shapes around • Builds on developing properties of shapes • Develops an understanding of dynamic geometry • Provides a connection between Geometry and Algebra through the co-ordinate plane • Provides a more intuitive and mathematically precise definition of congruence and similarity • Lays the foundation for projections and transformations in space – video animation • Lays the foundation for Linear Algebra in college – a central topic in both pure and applied mathematics

  9. Golden Oldies: Constructions • “Drawing Triangles with a Ruler and Protractor” (p. 125-126) • Which of the math practice standards are being developed? • How can this activity be used to prepare students for transformations?

  10. More With Constructions • Please read through “What Makes a Triangle?” on p. 134-135 • Please do p. 136, “Tricky Triangles” • How can we use constructions to prepare students for a definition of congruence that uses transformations as the underlying notion? • What, if any, is the benefit of using constructions to motivate the development of geometric reasoning?

  11. Physical Movement in Geometry • Each person needs to complete #1 on p. 148 • Each group will then complete #2 for one of the 5 parts of #1. • What are the related constructions, and how do we ensure that students see the connections?

  12. Transformations • In any transformation, some things change, some things stay constant • What changes? • What stays constant? • What are the defining characteristics of each type of transformation? • Reflection • Rotation • Translation • Dilation

  13. Reflection Is This A Reflection? Is This A Reflection?

  14. Reflection • Do “Reflection Challenges” on p. 168 either using paper and pencil, or using Geometer’s Sketchpad (or Geogebra or other dynamic geometry system) • What is changed, what is left constant, by a reflection? • What is gained by having students use technology? What is lost by having students use technology? • ..\..\..\Desktop\Algebra in Motion\Geometric Transformations (reflect, translate, rotate, dilate objects).gsp

  15. Rotations • Do activity “Rotations” • Patty paper might be helpful for this activity • Do “Rotation with Coordinates” p. 177 • What are students connecting in this activity? • Look at “Sloping Sides” on p. 178. • What are students investigating and discovering? • ..\..\..\Desktop\Algebra in Motion\Geometric Transformations (reflect, translate, rotate, dilate objects).gsp

  16. Translations • Look at “Isometric Transformation 3: Translation” (p. 180) • Do “Translation Investigations” p. 183 • ..\..\..\Desktop\Algebra in Motion\Geometric Transformations (reflect, translate, rotate, dilate objects).gsp

  17. Dilations • Do “Introduction to Dilations” • Look at p. 189, “Dilation with Rubber Bands” • Now do “Enlarging on a Copy Machine” (p. 191-192) • “Dilation Investigations” – read over and think about p. 193 • ..\..\..\Desktop\Algebra in Motion\Geometric Transformations (reflect, translate, rotate, dilate objects).gsp

  18. Euclidean Transformations • What changed and what remained the same in the four Euclidean transformations? • Complete “Properties of Euclidean Transformations” • How do we now define congruent figures? • How do we now define similar figures?

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