The National Council of Supervisors of Mathematics

1. 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

2. Defining Congruence & Similarity through Transformations

3. Reflective Writing Assignment How would you define congruence? How would you define similarity?

4. 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

5. Static Conceptions of Similarity: Comparing two Discrete Figures Between Figures Within Figures 2 2 1 1 3 6 6 3 Corresponding side lengths of similar figures are in proportion (height1sttriangle:height2nd triangle is equal to base 1sttriangle:base 2nd triangle) Ratios of lengths within a figure are equal to ratios of corresponding lengths in a similar figure (height :base1sttriangle is equal to height :base2nd triangle)

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

7. Static and Transformation-BasedConceptions of Similarity

8. Your Definitions of Congruence & Similarity: Share, Categorize & Provide a Rationale Static (discrete) Transformation-based

9. 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. Standards for Mathematical Content

10. 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. Standards for Mathematical Practice

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

12. 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?

13. 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.

14. 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

15. Video Clip: Randy Context: 8th grade Fall View Video Clip Use the transcript as a reference when discussing the clip

16. 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?

17. Representing Similar Rectangles as Dilation Images

18. Summary: Reconsidering Definitions of Similarity

19. A Resource for your Practice

20. 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.

21. www.wested.org Video Clips from Learning and Teaching Geometry Foundation Module Laminated Field Guides Available in class sets

22. Join us in thanking theNoyce Foundationfor their generous grant to NCSM that made this series possible! http://www.noycefdn.org/

23. 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