Practice Through Rich Tasks Congruence and Similarity Presented by Jenny Ray Mathematics Specialist Kentucky Dept of EducationNKCES wwwJennyRaynet 1 The Common Core State Standards ID: 312233
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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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The 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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Common Core State Standards
MathematicsStandards for Content
Standards for
Practice
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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 transformationsExamine the use of precise language, viable arguments, appropriate tools, and geometric structure.
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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)
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Standards 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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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
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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Static Conceptions of Similarity: Comparing two Discrete Figures
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Corresponding side lengths of similar figures are in proportion (
height
1
st triangle:height
2
nd
triangle is
equal to
base
1
st
triangle:
ba
se
2
nd
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
:
base
1
st
triangle is
equal to
height
:
base
2
nd
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?Slide12
Static and Transformation-BasedConceptions of Similarity
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Your Definitions of Congruence & Similarity: Share, Categorize & Provide a Rationale
Static (discrete)
Transformation-based
(continuous)
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Standards 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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Standards 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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Hannah
’s Rectangle Problem
Which rectangles are similar to rectangle a?
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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?
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Norms 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 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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Introduction 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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Video Clip: Randy
Context: 8th gradeFall
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
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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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
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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www.wested.org
Video Clips from Learning and Teaching Geometry Foundation ModuleLaminated Field Guides Available in class sets
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Join us in thanking the
Noyce Foundationfor their generous grant to NCSM that made this series possible!
http://www.noycefdn.org/
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Project 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 Arkansas
Cynthia A. Miller, Ph.D., Arkansas State University
Valerie L. Mills, Oakland Schools, Waterford, MI
Susan Jo Russell,
Ed.D., TERC, Cambridge, MADeborah Schifter, Ph.D., Education Development Center, Waltham, MANanette Seago, WestEd, San Francisco, California
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