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1 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments The National Council of Supervisors of Mathematics The Common Core State Standards Illustrating the Standards for Mathematical Practice: Mathematical Modeling and Constructing Viable Arguments

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2 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments 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!

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3 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Common Core State Standards Mathematics Standards for Content Standards for Practice

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4 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Todays Goals To explore the mathematical standards for Content and Practice To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps In particular, participants will Examine opportunities to develop skill in explaining and mathematical modeling

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5 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments 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)

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6 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments 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.

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7 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Structuring the Practices

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8 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Standards for Mathematical Practice What implications might the Standards for Mathematical Practice have on your classroom? 1. Individually review the Standards for Mathematical Practice. 2.Choose a partner at your table and discuss a new insight you had into the Standards for Mathematical Practice. 3.Then discuss the following question.

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9 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Common Core State Standards for Mathematical Practice In our work today, we will examine the Standards for Mathematical Practice through a classroom vignette. What is the nature of mathematical tasks in these classrooms? What do you hear or see in a mathematics classroom that is building the mathematical practices?

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10 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Standards of Mathematical Practice through Reengagement Teacher Goals: –Students will make mathematical sense of tables and discover mathematical reasoning. –Students will go beyond the notion that the table is either right or wrong to evaluate both mathematical sense and correct modeling. Outline: 1.Gym Task (Formative Assessment) 2.DVD Rental Task (Reengagement Vignette) 3.Carnival Task (Culminating Activity)

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11 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Formative Assessment to Plan Instruction Carlo thinks he will go to the gym about 20 times a month. Calculate how much each of these options would cost Carlo for one month. Which of these options is the least expensive for Carlo? Explain. Pay as you go Pay only $6 each time you work out Regular deal Pay $50 a month and $2 each time you work out All-in-one price! Pay just $100 per month for unlimited use of our great facilities

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12 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Formative Assessment to Plan Instruction Consider the summary of 6 th grade results of the Gym task. What are some possible next steps to deal with misconceptions or misunderstandings documented in the table?

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13 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Opportunities for Content and Practice in Tasks Create verbal and tabular representations of these 3 DVD rental plans. Do the three plans ever cost the same? Explain. Movie Buster $3 per movie rented Online Flix $12 per month plus $1 per movie rented Mail Flix $18 per month regardless of the number of movies rented

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14 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Opportunities for Content and Practice in Tasks 1. Individually complete the task. 2.Then work with a partner to compare your work and discuss the following questions: a)What mathematics content is needed to compete the task? b)When using tables to model and compare the DVD plans, what information and processes will students need to use strategically? c)What aspects of the explaining and modeling Standards of Practice might students need to use to complete the task?

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15 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments The Nature of Tasks Used in the Classroom … Tasks as they appear in curricular materials Student learning Will Impact Student Learning!

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16 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments But, WHAT TEACHERS DO with the tasks matters too! Stein, Grover & Henningsen (1996) Smith & Stein (1998) Stein, Smith, Henningsen & Silver (2000) The Mathematical Tasks Framework Tasks as they appear in curricular materials Student learning Tasks as they appear in curricular materials Student learning Tasks as set up by teachers Tasks as enacted by teachers and students

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17 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Standards for Mathematical Practice: Opportunities in Task Enactment In what ways did the teachers launch increase students opportunities to begin explaining and modeling? What evidence do you see that students are building these standards of practice?

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18 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Student Work to Develop Standards for Mathematical Practice Use the questions posed by Mr. Dimas to analyze the tables generated by Student H on the next slide. Do the tables make mathematical sense, and do they match the plans? Would these tables help us tell if the three plans will ever cost the same?

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19 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Student Work to Develop Standards for Mathematical Practice Movies $3$6$9$12$15$18$21 Money $13$14$15$16$17$18$ $18$36$54$72$90$108$126 Movie Buster Online Flix Mail Flix Student H

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20 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Student Work to Develop Standards for Mathematical Practice As you watch the video, consider these questions. What evidence do you see that suggests students are developing competency with explaining and modeling? In what ways did interactions between students support their ability to develop competency with explaining and modeling?

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21 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Student Work to Develop Standards for Mathematical Practice # of Movies MBOFMF Use the questions posed by Mr. Dimas to analyze the table generated by Student A. Does the table make mathematical sense, and does it match the plans? Would this table help us tell if the three plans will ever cost the same?

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22 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Using Student Work to Develop Standards for Mathematical Practice As you watch the video, consider these questions. What evidence do you see that suggests students are developing competency with explaining and modeling? In what ways did interactions between students support their ability to develop competency with explaining and modeling?

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23 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Considering Next Steps Create verbal, graphical, and tabular representations of these carnival ticket plans. Will any of the three plans ever cost the same? Explain. Dollar Deal No Entrance Fee $1.00 per ticket Bracelet Unlimited tickets with a $12.00 bracelet Discounted Plan $4.00 Entrance Fee, with discounted tickets ($0.50/ticket)

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24 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Planning and Teaching to Develop Standards for Mathematical Practice How does the process of reengagement seen in this vignette support students in developing Standards for Mathematical Practice? What instructional decisions did Mr. Dimas make that seemed to support the development of Standards for Mathematical Practice for students?

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25 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Adapted from Lesh, R., Post, T., & Behr, M. (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier, (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp ). Hillsdale, NJ: Lawrence Erlbaum. Geometric/ Graphical Verbal - Written and Oral Tabular Contextual Symbolic Pictures Oral Language Manipulative Models Real-World Situations Written Symbols Representation Stars Elementary Secondary

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26 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Next Steps and Resources Review the implications you listed earlier and discuss with your table group one or two next steps you might take as a district, school, and classroom teacher.

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27 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Todays Goals To explore the mathematical standards for Content and Practice To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps In particular, participants will Examine opportunities to develop skill in explaining and mathematical modeling

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28 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments

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29 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments Join us in thanking the Noyce Foundation for their generous grant to NCSM that made this series possible!

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30 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments 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.

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31 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments 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 Hope Bjerke, Editing Consultant, Redding, CA

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32 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling and Constructing Viable Arguments 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:

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