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The National Council of Supervisors of Mathematics

Explore the connections between similarity, slope, and graphs of linear functions through video examples of classroom practice. Unpack the concept of similarity, examine viable arguments, and consider the impact of the Common Core State Standards on your mathematics program.

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The National Council of Supervisors of Mathematics

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  1. The National Council of Supervisors of Mathematics The Common Core State Standards Illustrating the Standards for Mathematical Content & Practice: Similarity, Slope and Graphs of Linear Functions www.mathedleadership.org

  2. Module Evaluation Facilitator: At the end of this Powerpoint, you will find a link to an anonymous brief e-survey 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!

  3. Mathematics Standards for Content Standards for Practice Common Core State Standards

  4. Explore the Standards for Contentand Practicethrough 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 Unpack the connection between similarity, slope and the graphs of linear functions Examine the use of viable arguments, precise language and geometric structure Today’s Goals

  5. 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

  6. Here are excerpts from the 8th Grade Standards: 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. Understand the connections between proportional relationships, lines, and linear equations. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. Standards for Mathematical Content

  7. Similarity, Slope and Graphs of Linear Functions

  8. Graphing Similar Rectangles • Create four similar rectangles. List the dimensions for each rectangle. • Plot the four rectangles on a coordinate graph by aligning the bottom left vertex of each rectangle with the origin (0,0), and orienting the rectangles in the same direction, thus nesting the rectangles. • Write the equation for the line that passes through the origin and the upper right vertices of each rectangle. Draw the line.

  9. Graphing Similar Rectangles: Discussion • Share multiple methods for creating similar rectangles, along with the definitions of similarity that guided each of these methods. • Describe how your set of similar rectangles is a representation of dilation. • What is your definition of dilation? • What is the center of dilation? On your graph, what point acts as the center of dilation? • What is a dilation line? On your graph, where are some lines of dilation? • What are some properties of the line you drew through the upper right vertices in relation to your set of similar rectangles?

  10. 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.

  11. 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.

  12. Video Clip: Brian & Macy Context: 8th grade Fall View Video Clip. Use the transcript as a reference when discussing the clip.

  13. Unpacking the Video Clip How did Brian’s group generate their set of rectangles? Are the rectangles for Brian’s group dilations of each other? How do you know? Are the rectangles for Mary’s group dilations of each other? How do you know? Why must the line that connects the upper right vertices pass through the origin?

  14. Similar Triangles Task Graph the linear function Plot the following points on a coordinate graph: (-11,-4), (-5,-1), (3,3) & (-3,0). Draw a line segment from each of the plotted points perpendicular to the x-axis. You should have created three right triangles. Make two different viable arguments that the three right triangles are similar: One viable argument using ratios One viable argument using rotation and dilation

  15. Similar Triangles: Discussion • Thinking about your ratio arguments: • Are you looking at ratios within or between figures? • What connections can you make between the ratios and slopes of the hypotenuses of the right triangles? • Thinking about your rotation & dilation arguments: • Where are the lines of dilation? • What point acts as the center of dilation in the diagram? • What connections can you make between similarity and slope?

  16. Summary:What do you notice about the geometric structure of the set of similar right triangles?

  17. 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.

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

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

  20. 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

  21. Help Us Grow! The link below will connect you to a anonymous brief e-survey 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

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