Problem Solving Via Functions Tasks

1. Problem Solving Via Functions Tasks Day 2

2. Reflecting on Functional Thinking Go to mscsummercourses2013.wikispaces.com and then to the Problem Solving via Functions Tasks page. Respond to the discussion prompt “Reflections on Day 1” on the bottom of the page. Math & Science Collaborative at the Allegheny Intermediate Unit

3. Connecting to the Standards • How do you see the Learning Progressions helping you or applying in your classroom? • Use an example to illustrate this

4. Functional thinking Based on your reading assignment: How has your thinking about functions and functional thinking changed? Why? What new insights have you gained?

5. Cubes in a Line Task How many faces (face units) are there when 2 cubes are put together sharing a face? 10 cubes? 100 cubes? t cubes?

6. Mathematical Task Questions • Predict how a student might solve this problem using: • a visual, closed method • a recursive method • a table

7. Mathematical Task Questions • Look over how you solved this problem. Why did it make sense to you to solve it this way? How is this similar/different than how you solved the Tiling the Patio task on Day 1? • What are some of the ways students might solve it? How might they use the cubes to generate the number of faces for any number of cubes? What misconceptions might they bring? • What might a teacher need to do to prepare to use this task with students?

8. Video Segment Focus Questions • What moments or interchanges appear to be interesting/important mathematically? • What about them makes this so? • For additional resources for this task, see the article Developing Algebraic Reasoning Through Generalization in your binder

9. Language Context Table Graph Equation Van de Walle, 2004 Five Different Representations of a Function

10. Representations and SMP • Where do you see evidence for each of the 5 representations of function? • Which SMPs are best illuminated by the Cubes in a Line Task? Provide evidence.

11. Linear vs. Exponential Functions

12. Linear and Exponential Models Construct and compare linear and exponential models to solve problems • Distinguish between situations that can be modeled with linear functions and with exponential functions (F-LE1a) • Linear and exponential functions should receive the bulk of attention

13. PA Model Curriculum- Algebra

14. Linear and exponential relationships • By the end of eighth grade students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation. • Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. • They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. • All of this work is grounded on understanding quantities and on relationships between them.

15. Growth Task(Mathematics in Context, Roodhardt, et al. 1998) The plants’ heights (in millimeters) over several weeks (t) appear in the tables in the figure below. h(t) and g(t) are functions that represent the height of the plants’ over time, respectively.

16. Growth Task(Mathematics in Context, Roodhardt, et al. 1998)

17. Linear vs. Exponential Functions

18. Qualitative Graphing: A Context for Exploring Functional Relationships

19. Importance of Qualitative Graphs Rather than beginning with tasks that require students to plot points on a scaled Cartesian coordinate system, students should first be introduced to qualitative graphs and asked to view them globally. This approach utilizes students’ everyday knowledge of real-world events and provides a basis for interpreting graphs of functions. Leinhardt, Zaslavsky, and Stein , 1990, p.28

20. Importance of Qualitative Graphs Introducing the function concept by using graphs or pictorial representations highlights the importance of graphs in linking representations of functions. There is a natural progression from qualitative graphs to quantitative graphs to tables to equations and students are more comfortable working with the function concept when it is introduced in this progression. Van Dyke, 2003, p. 126

21. Keisha’s Bicycle Ride Task • Solve the task • Share your story with a partner • Identify the key elements that you would be looking for in a student-generated story • Identify the misconceptions you might expect to surface as students work on this task

22. Reflection • What is functional thinking? What new insights have you gained? How has your thinking changed?

23. Functional Thinking in Your Classroom • Plan task/activity to develop/extend students’ functional thinking. This will be submitted (electronically) for inclusion on the course wikipage • Focus on one content standard and one standard for math practice • Resources are found at www.mscsummercourses2013.wikispaces.com Math & Science Collaborative at the Allegheny Intermediate Unit

24. Homework • Complete the task/activity you started working on this afternoon. • Read the Case of Robert Carter found in your binder, day 2. Complete the form Supporting Students’ Capacity to Engage in the Standards for Mathematical Practice -The Case of Robert Carter to guide your reading. Math & Science Collaborative at the Allegheny Intermediate Unit