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

Facilitator: At the end of this
Powerpoint, you will find a link to an anonymous brief esurvey 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!
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations

Mathematics
• Standards for
Content
• Standards for
Practice
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 3

• 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
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 4

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 5

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 6

National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 7

• 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 8

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.
9

Between Figures Within Figures
2
2
1
1
6 3 6 3
Corresponding side lengths of similar figures are in proportion ( height 1 st triangle: height 2 nd triangle is equal to base 1 st triangle: base 2 nd triangle)
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)
10

A Transformation-based
Conception of Similarity
What do you notice about the geometric structure of the triangles?
11

Static and Transformation-based
Conceptions of Similarity
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 12

Your Definitions of Congruence &
Similarity:
Share, Categorize & Provide a Rationale
Static (discrete) Transformation-based
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 13

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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 14

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 15

Which rectangles are similar to rectangle a?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 16

• 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?
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 17

• 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 18

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.
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 19

• 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 20

• 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
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 21

National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 22

National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 23

National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations 24

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 25

Video Clips from Learning and Teaching Geometry
Foundation Module
Laminated Field Guides
Available in class sets
26

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 27

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

The link below will connect you to a anonymous brief esurvey 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
National Council of Supervisors of Mathematics
Illustrating the Standards for Mathematical Practice
Congruence and Similarity through Transformations