Congruence and Similarity through Transformations

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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Jenny Ray, Mathematics Specialist Kentucky Dept. of Education Northern Ky Cooperative for Educational Services www.JennyRay.net 1

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Congruence and Similarity through Transformations. Jenny Ray, Mathematics Specialist Kentucky Dept. of Education Northern Ky Cooperative for Educational Services www.JennyRay.net. The National Council of Supervisors of Mathematics. The Common Core State Standards - PowerPoint PPT Presentation

Transcript of Congruence and Similarity through Transformations

Page 1: Congruence and Similarity through Transformations

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations

Congruence and Similaritythrough Transformations

Jenny Ray, Mathematics Specialist

Kentucky Dept. of EducationNorthern Ky Cooperative for Educational Services

www.JennyRay.net

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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

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

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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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A Transformation-based Conception of Similarity

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What do you notice about the geometric structure of the triangles?

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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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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:

1. Verify experimentally the properties of rotations, reflections, and translations:

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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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 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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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

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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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

Unpacking Randy’s Method• What did Randy do? (What was his method?)

• Why might we argue that Randy’s 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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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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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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations

End of Day 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

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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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations

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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:

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