The National Council of Supervisors of Mathematics

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations The Common Core State Standards Illu

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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

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Module Evaluation

Facilitator: At the end of this

Powerpoint, you will find a

link to an anonymous brief

e-survey that will help us

understand how the module

is being used and how well it

worked in your setting.

We hope you will help us

grow and improve our NCSM

resources!

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Common Core State Standards

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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 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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Standards for Mathematical Practice

“The Standards for

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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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Defining Congruence &

Similarity through Transformations

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Reflective Writing Assignment

• How would you define

congruence?

• How would you define

similarity?

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A two-dimensional figure is

second can be obtained

from the first by a

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)

in a similar figure ( height : base 1st

triangle is equal to height : base 2nd

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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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Static and Transformation-based

Conceptions of Similarity

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Your Definitions of Congruence &

Similarity:

Share, Categorize & Provide a Rationale

Static (discrete) Transformation-based

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Standards for Mathematical Content

Here 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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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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Hannah’s Rectangle Problem

Which rectangles are similar to rectangle a?

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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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• 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

– Cite specific examples (evidence) from the video clip, transcript

and/or lesson graph

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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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Video Clip: Randy

• Context:

– 8th grade

– Fall

• View Video Clip.

• Use the transcript as a

reference when

discussing the clip

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Unpacking Randy’s Method

• What did Randy do? (What was his

method?)

• Why might we argue that Randy’s concept

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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Representing Similar Rectangles

as Dilation Images

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Summary: Reconsidering

Definitions of Similarity

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A Resource for Your Practice

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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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Video Clips from Learning and

Teaching Geometry Foundation Module

Laminated Field Guides Available

in class sets

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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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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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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:

http://tinyurl.com/samplesurvey1

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