1. © 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Students’ Solution Paths to Maximize Student Learning Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education High School Mathematics Algebra 2

2. © 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: learn what to monitor in student work when circulating during the Explore Phase of the lesson; learn about guidelines or “rules of thumb” for selecting and sequencing student work that target essential understandings of the lesson; and learn about focus questions that target the essential understandings.

3. © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: discuss the content standards and identify the related essential understandings of a lesson; analyze samples of student work; select and sequence student work for the Share, Discuss, and Analyze Phase of the lesson; identify “rules of thumb” for selecting and sequencing student work; and write focus questions that target essential understandings.

4. Rationale Orchestrating discussions that build on students’ thinking places significant pedagogical demands on teachers and requires an extensive and interwoven network of knowledge. Teachers often feel that they should avoid telling students anything, but are not sure what they can do to encourage rigorous mathematical thinking and reasoning. (Stein, M.K., Engle, R., Smith, M., Hughes, E. 2008. Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell) In this session, we will focus on monitoring, selecting, and sequencing student work so you can assess and advance student learning during the Share, Discuss, and Analyze Phase of the lesson.

5. © 2013 UNIVERSITY OF PITTSBURGH Triple Trouble

6. © 2013 UNIVERSITY OF PITTSBURGH The Task: Discussing Solution Paths Solve the task in as many ways as you can. Discuss the solution paths with colleagues at your table. If only one solution path has been used, work together to create others. Consider possible misconceptions or errors that we might see from students.

7. © 2013 UNIVERSITY OF PITTSBURGH Linking the Standards to Student Solution Paths The task has been selected with specific Standards for Mathematical Content and Practice in mind. Where do you see the potential to work on these standards in the written task or the solution paths?

8. The CCSS for Mathematical Content CCSS Conceptual Category – Algebra 2 Common Core State Standards, 2010 Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.1Write a function that describes a relationship between two quantities. ★ F-BF.A.1bCombine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Arithmetic with Polynomials and Rational Expressions (A-APR) Understand the relationship between zeros and factors of polynomials A-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★ ). Where an entire domain is marked with a star, each standard in that domain is a modeling standard.

9. Common Core Standards for Mathematical Practice What must happen in order for students to have opportunities to make use of the 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. Common Core State Standards, 2010

10. Five Representations of Mathematical Ideas Pictures Written Symbols Manipulative Models Real-world Situations Oral & Written Language Adapted from Van De Walle, 2004, p. 30

11. Five Different Representations of a Function Language TableContext GraphEquation Van De Walle, 2004, p. 440

12. © 2013 UNIVERSITY OF PITTSBURGH The Structure and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving 1.Generate and Compare Solutions 2.Assess and advance Student Learning MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: Different solution paths to the same task Different representations Errors Misconceptions SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT by engaging students in a quick write or a discussion of the process. Set Up of the Task Share Discuss and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

13. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Student Work

14. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Student Work Use the student work to further your understanding of the task. Consider: What do the students know? How did the students solve the task? How do their solution paths differ from each other?

15. © 2013 UNIVERSITY OF PITTSBURGH Group A

16. © 2013 UNIVERSITY OF PITTSBURGH Group B

17. © 2013 UNIVERSITY OF PITTSBURGH Group C

18. © 2013 UNIVERSITY OF PITTSBURGH Group D

19. © 2013 UNIVERSITY OF PITTSBURGH Group E 19

20. © 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Student Work

21. © 2013 UNIVERSITY OF PITTSBURGH Monitoring Sheet StrategyWho and WhatOrder

22. © 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Student Work (Small Group Discussion) Examine the students’ solution paths. Determine which solution paths you want to share during the class discussion; keep track of your rationale for selecting the pieces of student work. Determine the order in which work will be shared; keep track of your rationale for choosing a particular order for the sharing the work. Record the group’s decision on the chart in your participant handouts.

23. © 2013 UNIVERSITY OF PITTSBURGH Standards and Essential Understandings The product of two or more linear functions is a polynomial function. The resulting function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial. Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms. Two or more polynomial functions can be multiplied using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, f(x 1 )∙g(x 1 )) will be on the graph of the product f(x)*g(x). Making meaningful use of algebraic symbols means one can choose variables and construct expressions and equations from a context, table, or graph.

24. © 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Student Work (Small Group Discussion) Each team should record their group’s sequence of solution paths on the chart. Identify the student’s solution path that would be shared and discussed first, second, third, and so on. Be prepared to justify your response.

25. © 2013 UNIVERSITY OF PITTSBURGH Selecting and Sequencing Student Work (Group Discussion) Listen to each group’s rationale for selecting and sequencing student work. As you listen to the rationale, come up with a general “rule of thumb” that can be used to guide you when selecting and sequencing work for the Share, Discuss, and Analyze Phase of the lesson.

26. © 2013 UNIVERSITY OF PITTSBURGH Reflecting On Essential Understandings Which of the sequences of student work were driven by the standards and essential understandings?

27. © 2013 UNIVERSITY OF PITTSBURGH Reflecting on the Standards and the Essential Understandings The product of two or more linear functions is a polynomial function. The resulting function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial. Two or more polynomial functions can be multiplied using the algebraic representations by applying the distributive property and combining like terms. Two or more polynomial functions can be multiplied using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, f(x 1 )∙g(x 1 )) will be on the graph of the product f(x)*g(x). Making meaningful use of algebraic symbols means one can choose variables and construct expressions and equations from a context, table, or graph.

28. Common Core 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. Common Core State Standards, 2010

29. © 2013 UNIVERSITY OF PITTSBURGH “Rules of Thumb” for Selecting and Sequencing Student Work What are the benefits of using the “rules of thumb” as a guide when selecting and sequencing student work for the Share, Discuss, and Analyze Phase of the lesson?

30. © 2013 UNIVERSITY OF PITTSBURGH Pressing for Mathematical Understanding

31. © 2013 UNIVERSITY OF PITTSBURGH Pressing for Mathematical Understanding Let’s focus on one piece of student work for the Share, Discuss, and Analyze Phase of the lesson. Assume that a student has explained the work and others in the class have repeated the ideas and asked questions. Now it is time to “FOCUS” the discussion on an important mathematical idea. What questions might you ask the class as a whole to focus the discussion? Write your questions on chart paper to be posted for a gallery walk.

32. © 2013 UNIVERSITY OF PITTSBURGH Pressing for Mathematical Understanding  EU: Two or more polynomial functions can be multiplied using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, f(x 1 )∙g(x 1 )) will be on the graph of the product f(x)*g(x).

33. © 2013 UNIVERSITY OF PITTSBURGH Pressing for Mathematical Understanding Do a gallery walk. Review other groups’ questions. What are some similarities among the questions? What are some differences between the questions?

34. © 2013 UNIVERSITY OF PITTSBURGH Reflecting on Our Learning What have you learned today that you will think about and make use of next school year? Take a few minutes and jot your thoughts down.