1. The National Council of Supervisors of Mathematics
The Common Core State Standards
Illustrating the Standards for Mathematical Practice:
Congruence & Similarity Through Transformations
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
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)
Hannah lesson graph
Hannah’s rectangle problem
Static/transformation handout
Definitions handout
Similarity/congruence field guide (if possible, laminated copies)
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

2. Common Core State Standards
Mathematics
Standards for Content
Standards for Practice
National Council of Supervisors of Mathematics

3. I can deepen my understanding of Kentucky Core Academic Standards and mathematics pedagogy.

4. Today’s Goals
Explore the Standards for Content and Practice through video of classroom practice.
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

5. 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

6. 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.
National Council of Supervisors of Mathematics

7. Defining Congruence & Similarity through Transformations
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8. Reflective Writing Assignment
How would you define congruence?
How would you define similarity?
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9. 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
National Council of Supervisors of Mathematics

10. 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 1 3 6 2
National Council of Supervisors of Mathematics

11. A Transformation-based Conception of Similarity
What do you notice about the geometric
structure of the triangles?
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12. Static and Transformation-Based Conceptions of Similarity
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13. Your Definitions of Congruence & Similarity: Share, Categorize & Provide a Rationale
Static (discrete)
Transformation-based
National Council of Supervisors of Mathematics

14. 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.
National Council of Supervisors of Mathematics

15. 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.
National Council of Supervisors of Mathematics

16. Hannah’s Rectangle Problem
Which rectangles are similar to rectangle a?
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17. 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

18. 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.
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19. 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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20. Video Clip: Randy Context: 8th grade Fall View Video Clip
Use the transcript as a reference when discussing the clip
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21. 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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22. Representing Similar Rectangles as Dilation Images
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23. Summary: Reconsidering Definitions of Similarity
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24. A Resource for your Practice
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25. Reflections
How has your own thinking and/or practice from our work today changed?
2. What aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain.
National Council of Supervisors of Mathematics