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EMBRACING TRANSFORMATIONAL GEOMETRY IN CCSS-
MATHEMATICS
Presentation at Palm Springs 11/1/13
Jim Shortjshort@vcoe.org
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
Introductions
3
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
Workshop Goals
4
ATP Administrator Training - Module 1 – MS/HS Math
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.
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 Coordinate Plane
Develop Understanding of Effect of
Transformations on Figures
Develop Understanding of Functions
Develop Understanding of
Transformations as Functions on the
Plane/Space
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 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
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.
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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