EMBRACING TRANSFORMATIONAL GEOMETRY IN CCSS-MATHEMATICS Jim Short [email protected] Presentation at Palm Springs 11/1/13 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 Workshop Norms 1. Bring and assume best intentions. 2. Step up, step back. 3. Be respectful, and solutions oriented. 4. Turn off (or mute) electronic devices. ATP Administrator Training Module 1 – MS/HS Math 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 Flow of Transformational Geometry Ideas of transformational geometry are developed over time, infused in multiple ways Develop Understanding of Attributes of Shapes Develop Understanding of Coordinate Plane Develop Understanding of Effect of Transformations on Figures Develop Understanding of Transformations as Functions on the Plane/Space Develop Understanding of Functions Transformations are a big mathematical idea, importance enhanced by technology 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 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 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? 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? 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? 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 Reflection Is This A Reflection? Is This A Reflection? 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 Rotations Do activity “Rotations” Patty Do “Rotation with Coordinates” p. 177 What are students connecting in this activity? Look at “Sloping Sides” on p. 178. What paper might be helpful for this activity are students investigating and discovering? ..\..\..\Desktop\Algebra in Motion\Geometric Transformations (reflect, translate, rotate, dilate objects).gsp 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 Dilations Do “Introduction to Dilations” Look at p. 189, “Dilation with Rubber Bands” Now do “Enlarging on a Copy Machine” (p. 191192) “Dilation Investigations” – read over and think about p. 193 ..\..\..\Desktop\Algebra in Motion\Geometric Transformations (reflect, translate, rotate, dilate objects).gsp 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?