1. Principles to Actions Effective Mathematics Teaching Practices The Case of Wobberson Torchon and the Calling Plans 1 Task Algebra I Principles to Actions Effective Mathematics Teaching Practices The Case of Wobberson Torchon and the Calling Plans 1 Task Algebra I This module was developed by Frederick Dillon, Math Instructional Coach, Ideastream. Video courtesy of Providemce Public School District and the Institute for Learning. These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the project team that includes: Margaret Smith (chair), Victoria Bill (co- chair), Melissa Boston, Frederick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Lynn Raith, and Michael Steele.

2. Overview of the Session Solve and Discuss the Calling Plans Task Watch the video clip and discuss what the teacher does to support her students engagement in and understanding of mathematics Discuss the effective mathematics teaching practices of pose purposeful questions and using and connecting mathematical representations

3. The Calling Plans 1 Task Long-distance company A charges a base rate of $5.00 per month plus 4 cents for each minute that you are on the phone. Long- distance company B charges a base rate of only $2.00 per month but charges you 10 cents for every minute used. How much time per month would you have to talk on the phone before subscribing to company A would save you money?

4. The Calling Plans 1 Video Context School: Feinstein High School, Providence, RI Principal: Nancy Owen Teacher: Mr. Wobberson Torchon Class: Algebra I At the time the video was filmed, Wobberson Torchon was a teacher at Feinstein High School in the Providence Public School District. The students are mainstream Algebra students. (Wobberson Torchon is currently a Principal at Providence Career and Technical Academy, in Providence, RI.)

5. Learning Goals What learning goals can you create for the lesson?

6. Mr. Torchon’s Mathematics Learning Goals Students will understand that: 1.The point of intersection is a solution to each equation (Plans A and B); 2.A system of equations may be solved with a table or a graph; 3.When the graph of one function is below the graph of another function, the function with the lower graph has y-values that are less than the other function.

7. Connections to the CCSS Content Standards National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Reasoning with Equations and Inequalities A-REI Solve Systems of Equations 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically 11.Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately. ★

8. Connections to the CCSS Content Standards National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. ★ 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. ★

9. Connections to the CCSS 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.

10. The Calling Plans 1 Task The Context of Video Clip Students are working on the Calling Plans Task. Mr. Torchon is initially working with groups of students, asking questions to help them focus on their thinking and to progress toward a solution. The tables, graphs (and, in at least one case an equation) students produce in response to the task are then posted in the classroom and a whole class discuss ensues.

11. Lens for Watching the Video First Viewing As you watch the video, make note of what the teacher does to support student learning and engagement as they work on the task. In particular, identify any of the Effective Mathematics Teaching Practices that you notice Mr. Torchon using. Be prepared to give examples and to cite line numbers from the transcript to support your claims.

12. Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking.

13. Pose Purposeful Questions Effective Questions should: Reveal students’ current understandings; Encourage students to explain, elaborate, or clarify their thinking; and Make the mathematics more visible and accessible for student examination and discussion. Teachers’ questions are crucial in helping students make connections and learn important mathematics and science concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding. (Weiss and Pasley, 2004)

14. Use and Connect Mathematical Representations Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematical concepts and procedures and as tools for problem solving (NCTM, 2014, p.24). The depth of [student] understanding is related to the strength of connections among mathematical representations that students have internalized (Pape &Tchoshanov, 2001, Webb, Boswinkel & Dekker, 2008). (NCTM, 2014, p.25)

15. Lens for Watching the Video Second Viewing As you watch the video the second time, pay attention how the questions that the teacher asks and the extent to which representations are used and connected. The viewing prompts on your handout provide specific questions to consider.

16. Lens for Watching the Video Second Viewing As you watch the video this time, pay attention to the questions the teacher asks. Specifically, which questions focus on: Gathering information? Probing thinking? What do these questions reveal about students’ current understandings? Making mathematics more visible and accessible for student examination and discussion? Encouraging reflection and justification? How do students answer these questions?

17. Lens for Watching the Video Second Viewing As you watch the video this time, pay attention to the ways in which representations are used and connected. Specifically: Does the task allow students to use multiple representations? What representations are students using? In what ways are students making connections among the mathematical representations? In what ways are students contextualizing mathematical ideas in real-world situations?

18. What have you learned and how do these ideas apply to your classroom work?