National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations The Common Core State Standards Illustrating the Standards for Mathematical Practice: Congruence & Similarity Through Transformations The National Council of Supervisors of Mathematics 1

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Defining Congruence & Similarity through Transformations 2

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Reflective Writing Assignment How would you define congruence? How would you define similarity? 3

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. 4

Static Conceptions of Similarity: Comparing two Discrete Figures 5 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) Between Figures Ratios of lengths within a figure are equal to ratios of corresponding lengths in a similar figure (height :base1 st triangle is equal to height :base 2 nd triangle) Within Figures

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

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Static and Transformation-Based Conceptions of Similarity 7

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Your Definitions of Congruence & Similarity: Share, Categorize & Provide a Rationale Static (discrete)Transformation-based 8

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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. 9

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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. 10

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Hannah’s Rectangle Problem Which rectangles are similar to rectangle a? 11

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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? 12

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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. 13

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

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Video Clip: Randy Context: –8th grade –Fall View Video Clip Use the transcript as a reference when discussing the clip 15

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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? 16

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Representing Similar Rectangles as Dilation Images 17

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Summary: Reconsidering Definitions of Similarity 18

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

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations End of Day 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. 20

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

National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Join us in thanking the Noyce Foundation for their generous grant to NCSM that made this series possible! 22

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