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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

The National Council of Supervisors of MathematicsNational Council
of Supervisors of Mathematics Illustrating the Standards for
Mathematical Practice Congruence and Similarity through
Transformations

Congruence and Similarity

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

The Common Core State Standards

Illustrating the Standards for Mathematical Practice:

Congruence & Similarity Through Transformations

*

*

IMPORTANT: To offer notes that include descriptive graphical representations pertaining to this module, facilitator notes are provided outside of the PowerPoint in the Facilitator’s Agenda for use with Slides 6 – 23.

Preparation for facilitating the session:

Make copies for each participant:

A copy of the PPT (formatted with two slides per page)

A copy of the participant handouts:

Video transcript (Randy)

Common Core Content Standards for Mathematics

Bulleted version of the Common Core State Standards for Mathematical Practice (If you can, it is nice to have these Standards printed on card stock and laminated!)

Download the video file (1 clip—Randy); view prior to session.

Materials needed: Using the Sorting Rectangles template, print on cardstock rectangle cutouts (sets of four rectangles in baggies; one set for each group). Facilitators should also have a straight edge, colored pencils, patty paper, and graph paper available for participants to use when they work on the math tasks.

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Defining Congruence & Similarity through Transformations

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Reflective Writing Assignment

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

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

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Static Conceptions of Similarity: Comparing two Discrete Figures

*

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

1

3

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

*

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

A Transformation-based

structure of the triangles?

Nice animation here!

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

Static and Transformation-based

Conceptions of Similarity

*

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

Your Definitions of Congruence & Similarity:

Share, Categorize & Provide a Rationale

Static (discrete)

*

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

Standards for Mathematical Content

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.

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Standards for Mathematical Practice

Reason abstractly and quantitatively.

Model with mathematics.

Look for and express regularity in repeated reasoning.

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Hannah’s Rectangle Problem

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence 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?

*

*

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

Norms for Watching Video

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.

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence 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

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Video Clip: Randy

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence 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?

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Representing Similar Rectangles as Dilation Images

*

Nice animation here, too!

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

Summary: Reconsidering Definitions of Similarity

*

*

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

A Resource for Your Practice

*

*

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

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.

*

*

Ask participants to take the final 10 minutes to respond to these questions in writing. This activity is intended to offer participants a moment to synthesize the thinking they did during this session and to leave facilitators with some evidence of where participants’ current thinking is following the session.

Before participants leave, please use the last few slides to acknowledge the generous support from WestEd, the Noyce Foundation, and the members of NCSM who made these resource materials possible.

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

www.wested.org

Laminated Field Guides Available in class sets

*

*

Source of the video is the from the Learning and Teaching Geometry Project, funded by a National Science Foundation Grant and soon to be published by WestEd. WestEd has graciously granted NCSM permission to utilize the video clips for this resource.

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Join us in thanking the

Noyce Foundation

for their generous grant to NCSM that made this series possible!

*

This series was made possible by a generous grant from the Noyce Foundation to NCSM.

*

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

NCSM Series Contributors

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

Thank you to the project contributors.

*

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

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:

Congruence and Similarity

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

The Common Core State Standards

Illustrating the Standards for Mathematical Practice:

Congruence & Similarity Through Transformations

*

*

IMPORTANT: To offer notes that include descriptive graphical representations pertaining to this module, facilitator notes are provided outside of the PowerPoint in the Facilitator’s Agenda for use with Slides 6 – 23.

Preparation for facilitating the session:

Make copies for each participant:

A copy of the PPT (formatted with two slides per page)

A copy of the participant handouts:

Video transcript (Randy)

Common Core Content Standards for Mathematics

Bulleted version of the Common Core State Standards for Mathematical Practice (If you can, it is nice to have these Standards printed on card stock and laminated!)

Download the video file (1 clip—Randy); view prior to session.

Materials needed: Using the Sorting Rectangles template, print on cardstock rectangle cutouts (sets of four rectangles in baggies; one set for each group). Facilitators should also have a straight edge, colored pencils, patty paper, and graph paper available for participants to use when they work on the math tasks.

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Defining Congruence & Similarity through Transformations

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Reflective Writing Assignment

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

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

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Static Conceptions of Similarity: Comparing two Discrete Figures

*

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

1

3

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

*

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

A Transformation-based

structure of the triangles?

Nice animation here!

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

Static and Transformation-based

Conceptions of Similarity

*

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

Your Definitions of Congruence & Similarity:

Share, Categorize & Provide a Rationale

Static (discrete)

*

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

Standards for Mathematical Content

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.

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Standards for Mathematical Practice

Reason abstractly and quantitatively.

Model with mathematics.

Look for and express regularity in repeated reasoning.

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Hannah’s Rectangle Problem

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence 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?

*

*

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

Norms for Watching Video

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.

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence 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

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Video Clip: Randy

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence 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?

*

*

*

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Representing Similar Rectangles as Dilation Images

*

Nice animation here, too!

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

Summary: Reconsidering Definitions of Similarity

*

*

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

A Resource for Your Practice

*

*

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

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.

*

*

Ask participants to take the final 10 minutes to respond to these questions in writing. This activity is intended to offer participants a moment to synthesize the thinking they did during this session and to leave facilitators with some evidence of where participants’ current thinking is following the session.

Before participants leave, please use the last few slides to acknowledge the generous support from WestEd, the Noyce Foundation, and the members of NCSM who made these resource materials possible.

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

www.wested.org

Laminated Field Guides Available in class sets

*

*

Source of the video is the from the Learning and Teaching Geometry Project, funded by a National Science Foundation Grant and soon to be published by WestEd. WestEd has graciously granted NCSM permission to utilize the video clips for this resource.

National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics

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

Join us in thanking the

Noyce Foundation

for their generous grant to NCSM that made this series possible!

*

This series was made possible by a generous grant from the Noyce Foundation to NCSM.

*

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

NCSM Series Contributors

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

Thank you to the project contributors.

*

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

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: