Study Group 6 - High School Math (Algebra 1 & 2, Geometry)

1. Study Group 6 - High School Math (Algebra 1 & 2, Geometry) Welcome Back! Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises.

2. Accountable Talk Features and Indicators Accountability to the Learning Community • Actively participate in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge • Appropriate evidence for claims and arguments • Commitment to getting it right Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

3. Review of the essential understandings for the tasks: For Algebra 1: • The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values For Algebra 2: • 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. For Geometry: • Using coordinates of a midsegment of a triangle justifies that the midsegment is parallel to the side that it does not intersect because the slope of the segment containing the midpoints is the same as the slope of the segment connecting the endpoints of the third side of the triangle.

4. Let’s Go Over Bridge to Practice #5: Time to Reflect on Our Learning Plan a discussion focusing on one or more pieces of your student work that focus on the underlying mathematics from the BTP #4 task/lesson you did. For Algebra 1, Use No Place Like Home Task For Algebra 2, Use Triple Trouble Task For Geometry, Use MidsegmentTask On a piece of paper, scribe several Accountable Talk moves or questions you will use to facilitate the discussion and anticipate student responses (Remember that telling is not allowed. Plan questions that will move students toward the essential understanding) Classify the moves according to the Accountable Talk feature they supportand why (Community, Knowledge, and Rigorous Thinking) 4

5. Discuss What did you notice about planning questions and anticipating student responses? What are some things you said and did to hold students accountable to the learning community, knowledge, and rigorous thinking?

6. Step Back: Reflecting on the Benefits What are the benefits of using Accountable Talk features and indicators as a tool for reflecting on the classroom discussion? For planning?

7. Supporting Rigorous Mathematics Teaching and Learning Strategies for Scaffolding Student Understanding: Academically Productive Talk and the Use of Representations Tennessee Department of Education High School Mathematics

8. Rationale Teachers provoke students’ reasoning about mathematics through the tasks they provide and the questions they ask. (NCTM, 1991) Asking questions that reveal students’ knowledge about mathematics allows teachers to design instruction that responds to and builds on this knowledge. (NCTM, 2000) Questions are one of the only tools teachers have for finding out what students are thinking. (Michaels, 2005) Today, by analyzing a classroom discussion, teachers will study and reflect on ways in which Accountable Talk® (AT) moves and the use of representations support student learning and help teachers to maintain the cognitive demand of a task. Accountable talk® is a registered trademark of the University of Pittsburgh.

9. Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning Stein, Smith, Henningsen, & Silver, 2000

10. Session Goals Participants will learn about: • Accountable Talk moves to support the development of community, knowledge, and rigorous thinking; • Accountable Talk moves that ensure a productive and coherent discussion and consider why moves in this category are critical; and • representations as a means of scaffolding student learning.

11. Overview of Activities Participants will: • analyze and discuss Accountable Talk moves; • engage in and reflect on a lesson in relationship to the CCSS; • analyze classroom discourse to determine the Accountable Talk moves used by the teacher and the benefit to student learning; • design and enact a lesson, making use of the Accountable Talk moves; and • learn and apply a set of scaffolding strategies that make use of the representations.

12. Review theAccountable Talk Features and IndicatorsLearn Moves Associated With the Accountable Talk Features

13. The Structure and Routines of a Lesson • MONITOR: Teacher selects • examples for the Share, Discuss, • and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task Set Up the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving Generate and Compare Solutions Assess and Advance Student Learning 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: Engage students in a Quick Write or a discussion of the process. Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

14. Accountable Talk Discussion • Review the Accountable Talk features and indicators. • Turn and Talk with your partner about what you recall about each of the Accountable Talk features. • Accountability to the learningcommunity • Accountability to accurate, relevant knowledge • Accountability to discipline-specific standards of rigorous thinking

15. Accountable Talk Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

16. Accountable Talk Moves Consider: • In what ways are the Accountable Talk moves different in each of the categories? • Support Accountability to Community • Support Accountability to Knowledge • Support Accountability to Rigorous Thinking • There is a fourth category called, “To Ensure Purposeful, Coherent, and Productive Group Discussion.” Why do you think we need the set of moves in this category?

17. Accountable Talk Moves (continued)

18. Accountable Talk Moves (continued)

19. Accountable Talk Moves

20. Engage In and Reflect On a LessonBike and Truck TaskAlgebra I

21. Bike and Truck Task - Algebra I Distance from start of road (in feet) Time (in seconds) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

22. Bike and Truck Task - Algebra I • Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. • Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. • Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. • Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack and why.

23. The Cognitive Demand of the TaskAlgebra I Why is this considered to be a cognitively demanding task?

24. The Mathematical Task Analysis Guide- Algebra I Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

25. Analyzing a Lesson: Lesson ContextAlgebra I The students and the teacher in this school have been working to make sense of the Common Core State Standards for the past two years. The teacher is working on using the Accountable Talk moves and making sure she targets the mathematics standards in very deliberate ways during the lesson.

26. Instructional Goals - Algebra I Shalunda’s instructional goals for the lesson are: • students will use the language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) to describe how two quantities vary together over a range of possible values; and • students will describe how one quantity changes with respect to another.

27. Engage In and Reflect On a LessonBuilding a New Playground Task- Geometry

28. Building a New Playground TaskGeometry The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown.

29. Building a New Playground- Geometry PART A • Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. • Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B • The City Planning Commission is planning to build a third elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park. • Describe a strategy for determining a point equidistant from any three points.

30. The Cognitive Demand of the TaskGeometry Why is this considered to be a cognitively demanding task?

31. The Mathematical Task Analysis Guide- Geometry Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

32. Analyzing a Lesson: Lesson ContextGeometry The students and the teacher in this school have been working to make sense of the Common Core State Standards for the past two years. The teacher is working on using the Accountable Talk moves and making sure she targets the mathematics standards in very deliberate ways during the lesson.

33. Instructional Goals - Geometry Debbee’s instructional goals for the lesson are: • students will determine a set of points that are equidistant from two points, A and B; • students will recognize and conjecture that all such points fall on the perpendicular bisector of ; and • students will prove their conjecture.

34. Engage In and Reflect On a LessonMissing Function TaskAlgebra II

35. Missing Function Task – Algebra II If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning.

36. The Cognitive Demand of the TaskAlgebra II Why is this considered to be a cognitively demanding task?

37. The Mathematical Task Analysis Guide – Algebra II Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000) Implementing standards-based mathematics instruction: A casebook for professional development, p. 16. New York: Teachers College Press.

38. Analyzing a Lesson: Lesson ContextAlgebra II The students and the teacher in this school have been working to make sense of the Common Core State Standards for the past two years. The teacher is working on using the Accountable Talk moves and making sure she targets the mathematics standards in very deliberate ways during the lesson.

39. Instructional Goals – Algebra II Jamie’s instructional goals for the lesson are: • students will multiply two linear functions using their graphs or tables of values and recognize that, given two functions f(x) and g(x) and a specific x-value, x1, the point (x1, f(x1) ∙ g(x1)) will be on the graph of the product f(x) ∙ g(x); and • students will recognize that the product of two or more linear functions is a polynomial function having the same x-intercepts as the original functions because the original functions are factors of the polynomial.

40. Reflection Question(Small Group Discussion) As you watch the video segment, consider what students are learning about mathematics. Name the moves used by the teacher and the purpose that the moves served.

41. Reflecting on the Accountable Talk Discussion(Whole Group Discussion) Step back from the discussion. What are some patterns that you notice? What mathematical ideas does the teacher want students to discover and discuss? How does talk scaffold student learning?

42. Reflection: The Use of Accountable Talk Moves and Tools to Scaffold Student Learning What have you learned?

43. Bridge to Practice 6: Accountable Talk: Putting it all together! Choose an Instructional Task with your PLC that you will all teach. Plan the entirelesson together. Anticipate student responses. Write your assessing and advancing questions Facilitate the entire lesson with your classes. Put special attention to your (and the student’s) use of the Accountable Talk questions/moves. Bring examples of student work and some of the whole class discussion script.