Congruence and Similarity through Transformations

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Congruence and Similaritythrough Transformations

• Jenny Ray, Mathematics Specialist
• Kentucky Dept. of Education
• Northern Ky Cooperative for Educational Services
• www.JennyRay.net

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The National Council of Supervisors of Mathematics

• The Common Core State Standards
• Illustrating the Standards for Mathematical
  Practice
• Congruence Similarity Through Transformations
• www.mathedleadership.org

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Defining Congruence Similarity through
Transformations
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Reflective Writing Assignment

• How would you define congruence?
• How would you define similarity?

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Definition of Congruence Similarity Used in
the CCSS
A two-dimensional figure is congruent to another
if the second can be obtained from the first by a
sequence of rotations, reflections, and
translations.
A two-dimensional figure is similar to another
if the second can be obtained from the first by a
sequence of rotations, reflections, translations
and dilations.
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Static Conceptions of Similarity Comparing two
Discrete Figures
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A Transformation-based Conception of Similarity
What do you notice about the geometric structure
of the triangles?
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Static and Transformation-basedConceptions of
Similarity
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Your Definitions of Congruence Similarity
Share, Categorize Provide a Rationale

• Static (discrete)

• Transformation-based

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Standards for Mathematical Content

• Here is an excerpt from the 8th Grade Standards
• Verify experimentally the properties of
  rotations, reflections, and translations
• Understand that a two-dimensional figure is
  congruent to another if the second can be
  obtained from the first by a sequence of
  rotations, reflections, and translations given
  two congruent figures, describe a sequence that
  exhibits the congruence between them.
• Describe the effect of dilations, translations,
  rotations, and reflections on two-dimensional
  figures using coordinates.
• Understand that a two-dimensional figure is
  similar to another if the second can be obtained
  from the first by a sequence of rotations,
  reflections, translations, and dilations given
  two similar two-dimensional figures, describe a
  sequence that exhibits the similarity between
  them.

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Standards for Mathematical Practice

1. Make sense of problems and persevere in solving
   them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
   reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
   reasoning.

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Hannahs Rectangle Problem

• Which rectangles are similar to rectangle a?

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Hannahs Rectangle Problem Discussion

• Construct a viable argument to explain why those
  rectangles are similar.
• Which definition of similarity guided your
  strategy, and how did it do so?
• What tools did you choose to use? How did they
  help you?

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Norms for Watching Video

• Video clips are examples, not exemplars.
• To spur discussion not criticism
• Video clips are for investigation of teaching and
  learning, not evaluation of the teacher.
• To spur inquiry not judgment
• Video clips are snapshots of teaching, not an
  entire lesson.
• To focus attention on a particular moment not
  what came before or after
• Video clips are for examination of a particular
  interaction.
• Cite specific examples (evidence) from the video
  clip, transcript and/or lesson graph.

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Introduction to the Lesson Graph

• One page overview of each lesson
• Provides a sense of what came before and after
  the video clip
• Take a few minutes to examine where the video
  clip is situated in the entire lesson.

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Video Clip Randy

• Context
• 8th grade
• Fall
• View Video Clip.
• Use the transcript as a reference when discussing
  the clip.

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Unpacking Randys Method

• What did Randy do? (What was his method?)
• Why might we argue that Randys concept of
  similarity is more transformation-based than
  static?
• What mathematical practices does he employ?
• What mathematical argument is he using?
• What tools does he use? How does he use them
  strategically?
• How precise is he in communicating his reasoning?

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Representing Similar Rectangles as Dilation Images
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Summary Reconsidering Definitions of Similarity
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A Resource for Your Practice
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End of Day Reflections

• Are there any aspects of your own thinking and/or
  practice that our work today has caused you to
  consider or reconsider? Explain.
• 2. Are there any aspects of your students
  mathematical learning that our work today has
  caused you to consider or reconsider? Explain.

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www.wested.org

• Laminated Field Guides Available in class sets

• Video Clips from Learning and Teaching Geometry
  Foundation Module

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Join us in thanking theNoyce Foundationfor
their generous grant to NCSM that made this
series possible!
http//www.noycefdn.org/
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NCSM Series Contributors

• Geraldine Devine, Oakland Schools, Waterford, MI
• Aimee L. Evans, Arch Ford ESC, Plumerville, AR
• David Foster, Silicon Valley Mathematics
  Initiative, San José State University, San José,
  California
• Dana L. Gosen, Ph.D., Oakland Schools, Waterford,
  MI
• Linda K. Griffith, Ph.D., University of Central
  Arkansas
• Cynthia A. Miller, Ph.D., Arkansas State
  University
• Valerie L. Mills, Oakland Schools, Waterford, MI
• Susan Jo Russell, Ed.D., TERC, Cambridge, MA
• Deborah Schifter, Ph.D., Education Development
  Center, Waltham, MA
• Nanette Seago, WestEd, San Francisco, California
• Hope Bjerke, Editing Consultant, Redding, CA

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Help Us Grow!
The link below will connect you to a anonymous
brief e-survey that will help us understand how
the module is being used and how well it worked
in your setting.
Please help us improve the module by completing a
short ten question survey at http//tinyurl.com/s
amplesurvey1