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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

- 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.

ATP Administrator Training - Module 1 – MS/HS Math

- 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

- 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

- 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

- 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

- “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?

- 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?

- 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

- 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?