The National Council of Supervisors of Mathematics

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 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Module Evaluation 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 Common Core State Standards Mathematics • Standards for Content • Standards for Practice National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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 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 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 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 Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Defining Congruence & Similarity through Transformations 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? National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 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 Static Conceptions of Similarity: Comparing two Discrete Figures Within Figures Between 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) 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) 10 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 Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 15 Hannah’s Rectangle Problem 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 Hannah’s Rectangle Problem Discussion • Construct a viable argument to explain why those rectangles are similar • Which definition of similarity guided your strategy, and how did it 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 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 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 Video Clip: Randy • 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 Unpacking Randy’s Method • 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 Representing Similar Rectangles as Dilation Images National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 22 Summary: Reconsidering Definitions of Similarity National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 23 A Resource for Your Practice National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations 24 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 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 www.wested.org 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 NCSM Series 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 • Hope Bjerke, Editing Consultant, Redding, CA National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Help Us Grow! 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 ... ‘processes and proficiencies’ with longstanding importance in mathematics education.” (CCSS, 2010) National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice... Conception of Similarity What you notice about the geometric structure of the triangles? 11 Static and Transformation-based Conceptions of Similarity National Council of Supervisors of Mathematics. .. resources! National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice Congruence and Similarity through Transformations Common Core State Standards Mathematics