Embracing transformational geometry in ccss mathematics
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Embracing transformational geometry in CCSS-Mathematics. Jim Shortjshort@vcoe.org. 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

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Embracing transformational geometry in CCSS-Mathematics

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Embracing transformational geometry in CCSS-Mathematics

Jim Shortjshort@vcoe.org

Presentation at Palm Springs 11/1/13


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

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

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

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

  • 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

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?


  • 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


Is This A Reflection?

Is This A 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


  • 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


  • 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


  • 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

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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