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National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Illustrating the Standards for Mathematical Practice Through Rich Tasks Congruence and Similarity Presented by: Jenny Ray, Mathematics Specialist Kentucky Dept. of Education/NKCES www.JennyRay.net 1

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Illustrating the Standards for Mathematical Practice Through Rich Tasks. Congruence and Similarity Presented by: Jenny Ray, Mathematics Specialist Kentucky Dept. of Education/NKCES www.JennyRay.net. The National Council of Supervisors of Mathematics. The Common Core State Standards - PowerPoint PPT Presentation

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Page 1: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical PracticeCongruence and Similarity through Transformations 1

Illustrating the Standards for Mathematical PracticeThrough Rich Tasks

Congruence and Similarity

Presented by:Jenny Ray, Mathematics Specialist

Kentucky Dept. of Education/NKCESwww.JennyRay.net

Page 2: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 3: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 4: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 5: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 6: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 7: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 8: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 9: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 10: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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

2

 

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

1

3 6

2

   

Page 11: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

A Transformation-based Conception of Similarity

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

Page 12: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 13: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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

(continuous)

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Page 14: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 15: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 16: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 17: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 18: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 19: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 20: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 21: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 22: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 23: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 24: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 25: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 26: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

www.wested.org

Video Clips from Learning and Teaching Geometry

Foundation Module

Laminated Field Guides Available in class sets

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Page 27: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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Page 28: Illustrating the Standards for Mathematical  Practice Through Rich Tasks

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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