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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
The Common Core State Standards Illustrating the Standards for
Mathematical Practice:Congruence & Similarity Through
Transformations
www.mathedleadership.org
The National Council of Supervisors of Mathematics
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Common Core State Standards
Mathematics
• Standards for Content
• Standards for Practice
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Today’s Goals
• Explore the Standards for Content and Practice through video of classroom practice.
• Consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and to plan next steps.
In particular participants will:• Examine congruence and similarity defined
through transformations• Examine the use of precise language, viable
arguments, appropriate tools, and geometric structure.
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Standards for Mathematical Practice
“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.” (CCSS, 2010)
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Defining Congruence & Similarity through Transformations
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Reflective Writing Assignment
• How would you define congruence?
• How would you define similarity?
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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
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.
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Static Conceptions of Similarity: Comparing two Discrete Figures
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Corresponding side lengths of similar figures are in proportion (height 1st triangle:height 2nd triangle is equal to base 1st triangle:base 2nd triangle)
Between Figures
13 6
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Ratios of lengths within a figure are equal to ratios of corresponding lengths in a similar figure (height :base1st triangle is equal to height :base 2nd
triangle)
Within Figures
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3 6
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A Transformation-based Conception of Similarity
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What do you notice about the geometric
structure of the triangles?
National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Static and Transformation-BasedConceptions of Similarity
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Handout
National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Your Definitions of Congruence & Similarity:
Share, Categorize & Provide a Rationale
Static (discrete) Transformation-based
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Standards for Mathematical ContentHere is an excerpt from the 8th Grade Standards: 8.G.1-4
1. Verify experimentally the properties of rotations, reflections, and translations: …abc….
2. 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.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
4. 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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Hannah’s Rectangle Problem
Which rectangles are similar to rectangle a?
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Handout
National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Hannah’s Rectangle Problem Discussion
• Construct a viable argument for 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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Video Clip: Randy
• Context:
– 8th grade
– Fall
• View Video Clip
• Use the transcript as a reference when discussing the clip
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
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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Handout
National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Unpacking Randy’s Method• What did Randy do? (What was his method?)
• Why might we argue that Randy’s conception 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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Representing Similar Rectangles as Dilation Images
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Summary: Reconsidering Definitions of Similarity
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Handout
National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
A Resource for your Practice
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Handout
National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
End of Session Reflections
1. 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
Video Clips from Learning and Teaching Geometry
Foundation Module
Laminated Field Guides Available in class sets
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Join us in thanking the
Noyce Foundation
for their generous grant to NCSM that made this series possible!
http://www.noycefdn.org/
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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations
Project 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
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