The National Council of
Supervisors of Mathematics
The Common Core State Standards
Illustrating the Standards for
Mathematical Practice:
Congruence & Similarity Through
Transformations
www.mathedleadership.org
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
1
Common Core State Standards
Mathematics
• Standards for
Content
• Standards for
Practice
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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 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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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)
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Defining Congruence &
Similarity through
Transformations
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Reflective Writing Assignment
• How would you define
congruence?
• How would you define
similarity?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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.
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
8
Static Conceptions of Similarity:
Comparing two Discrete Figures
Within Figures
Between Figures
2
1
3
6
Corresponding side lengths of similar
figures are in proportion (height 1st
triangle:height 2nd triangle is equal to
base 1st triangle:base 2nd triangle)
2
1
3
6
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)
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A Transformation-based
Conception of Similarity
What do you notice about the
geometric
structure of the triangles?
10
Static and Transformation-Based
Conceptions of Similarity
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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
Transformation-based
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Standards for Mathematical Content
Here is an excerpt from the 8th Grade Standards: 8.G.1-4
1.
Verify experimentally the properties of rotations, reflections,
and translations: …abc….
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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
13
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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Hannah’s Rectangle Problem
Which rectangles are similar to rectangle a?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Video Clip: Randy
• Context:
– 8th grade
– Fall
• View Video Clip
• Use the transcript as a
reference when discussing
the clip
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Introduction to the Lesson Graph
Handout
• 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
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Representing Similar Rectangles
as Dilation Images
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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Summary: Reconsidering
Definitions of Similarity
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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A Resource for your Practice
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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End of Session 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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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www.wested.org
Video Clips from Learning
and Teaching Geometry
Foundation Module
Laminated Field Guides
Available in class sets
25
Join us in thanking the
Noyce Foundation
for their generous grant to NCSM that
made this series possible!
http://www.noycefdn.org/
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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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
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations
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