# Illustrating the Standards for Mathematical

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Illustrating the Standards for Mathematical PracticeThrough Rich Tasks

Congruence and SimilarityPresented by:Jenny Ray, Mathematics SpecialistKentucky Dept. of Education/NKCESwww.JennyRay.net

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Slide2The Common Core State Standards Illustrating the Standards for Mathematical Practice:Congruence & Similarity Through Transformationswww.mathedleadership.org

The National Council of Supervisors of Mathematics

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Slide3Common Core State Standards

MathematicsStandards for ContentStandards for Practice

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Slide4Today’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 transformationsExamine the use of precise language, viable arguments, appropriate tools, and geometric structure.

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Slide5Standards 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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Slide6Standards for Mathematical Practice

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

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Slide7Defining Congruence & Similarity through Transformations

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Slide8Reflective Writing Assignment

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

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Slide9A 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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Slide10Static 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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3

6

2

Ratios of lengths within a figure are equal to ratios of corresponding lengths in a similar figure

(

height

:

base

1

st

triangle is

equal to

height :base 2nd triangle)

Within Figures

1

3

6

2

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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?

Slide12Static and Transformation-BasedConceptions of Similarity

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Slide13Your Definitions of Congruence & Similarity: Share, Categorize & Provide a Rationale

Static (discrete)

Transformation-based(continuous)

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Slide14Standards 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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Slide15Standards for Mathematical Practice

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

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Slide16Hannah’s Rectangle Problem

Which rectangles are similar to rectangle a?

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Slide17Hannah’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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Slide18Norms for Watching Video

Video clips are examples, not exemplars. To spur discussion not criticismVideo clips are for investigation of teaching and learning, not evaluation of the teacher. To spur inquiry not judgmentVideo clips are snapshots of teaching, not an entire lesson. To focus attention on a particular moment not what came before or afterVideo 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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Slide19Introduction to the Lesson Graph

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

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Slide20Video Clip: Randy

Context: 8th gradeFallView Video ClipUse the transcript as a reference when discussing the clip

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Slide21Unpacking 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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Slide22Representing Similar Rectangles as Dilation Images

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Slide23Summary: Reconsidering Definitions of Similarity

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Slide24A Resource for your Practice

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Slide25End 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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Slide26www.wested.org

Video Clips from Learning and Teaching Geometry Foundation Module

Laminated Field Guides Available in class sets

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Slide27Join us in thanking theNoyce Foundationfor their generous grant to NCSM that made this series possible!

http://www.noycefdn.org/

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Slide28Project Contributors

Geraldine Devine, Oakland Schools, Waterford, MIAimee L. Evans, Arch Ford ESC, Plumerville, ARDavid Foster, Silicon Valley Mathematics Initiative, San José State University, San José, CaliforniaDana L. Gosen, Ph.D., Oakland Schools, Waterford, MILinda K. Griffith, Ph.D., University of Central ArkansasCynthia A. Miller, Ph.D., Arkansas State UniversityValerie L. Mills, Oakland Schools, Waterford, MISusan Jo Russell, Ed.D., TERC, Cambridge, MADeborah Schifter, Ph.D., Education Development Center, Waltham, MANanette Seago, WestEd, San Francisco, California

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## Illustrating the Standards for Mathematical

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