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. REGINA’S LOGO LESSON TASK Assume the pattern continues to grow in the same manner. Find a rule or formula to determine the number of tiles in a figure of any size. size 1size 2size 3size 4

10. STRATEGY

13. VIDEO SEGMENT FOCUS QUESTIONS  What moments or interchanges appear to be interesting/important mathematically?  Pedagogically?

14. REGINA’S LOGO 2 AND 3 LESSON TASKS  Regina’s Logo 2 size 0size 1 size 2size 3 Regina’s Logo 3 size 1size 2 size 3size 4size 5

15. MATHEMATICAL TASK QUESTIONS In each of Logos 2 and 3, assume the pattern continues to grow in the same manner. Create a table and a graph of each pattern. Find a rule or formula to determine the number of tiles in any size figure. How are Regina’s Logo, Regina’s Logo 2, and Regina’s Logo 3 similar? Different?

16. THINKING ABOUT THE CHANGES With your small group, create a poster depicting the changes/ translations/ transformations among the three versions of Regina’s Logo Problem with at least one representation (arithmetic sequence, equations, tables, graphs, etc.)

17. Language TableContext GraphEquation Five Different Representations of a Function Van de Walle, 2004

18. 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.

19. REGINA’S LOGO VS. GRAPHING PROBLEM Consider: Graphing Problem Simplify and graph the following equations: 1. Y 1 = 2x + (x+2) 2. Y 2 = 2(x+1) +x 3. Y 3 = 3(x-1) + 5 How these two tasks similar? How are they different? Do the differences matter?

20. LINEAR VS. EXPONENTIAL FUNCTIONS LinearExponential Rate of change is constant Recursive: next= now + rate of change f(t+1)= f(t)+m; m is the rate of change Closed form: f(x)=mx + b, m = rate of change (slope), b= y-intercept (constant) Arithmetic sequences can be thought of linear functions whose domains are pos. integers 1, 4, 7, 10, 13,…. Cubefaces 16 210 314

21. 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

22. PA MODEL CURRICULUM- ALGEBRA The PK-12 PA Common Core Standards for Mathematics stress both procedural skills and conceptual understanding to ensure students are learning and applying the critical information they need to succeed at higher levels. The introduction at each grade level articulates a small number of critical mathematical areas that should be the focus for that grade. The Standards emphasize applying mathematical ways of thinking to real world issues and challenges. Students will be able to independently use their learning to: 1.Make sense of and persevere in solving complex and novel mathematical problems. 2.Use effective mathematical reasoning to construct viable arguments and critique the reasoning of others. 3.Communicate precisely when making mathematical statements and express answers with a degree of precision appropriate for the context of the problem/situation. 4.Apply mathematical knowledge to analyze and model situations/relationships using multiple representations and appropriate tools in order to make decisions, solve problems, and draw conclusions. 5.Make use of structure and repeated reasoning to gain a mathematical perspective and formulate generalized problem solving strategies. Algebra 1 At this level it is expected that students will formalize and expand on Algebraic concepts established in previous coursework. Students will deepen and extend their understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend. Students will engage in methods for analyzing and using functions. Students will fluently move between multiple representations of functions including but not limited to linear, exponential, and quadratics. Algebra 1 Modules The modules should equate to a full year of instruction: Module 1: Relationships Between Quantities and Reasoning with Equations Module 2: Linear and Exponential Relationships Module 3: Descriptive Statistics Module 4: Equations and Expressions Module 5: Quadratic Functions and Modeling

23. 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.

24. 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.

25. GROWTH TASK (MATHEMATICS IN CONTEXT, ROODHARDT, ET AL. 1998)

26. LINEAR VS. EXPONENTIAL FUNCTIONS LinearExponential Rate of change is constantRate of change is not constant, but it changes Rate of change Increases/decreases over time When the input increases by 1unit, the output is multiplied by a constant factor Recursive: next= now + rate of change f(t+1)= f(t)+m; m is the rate of change Recursive: next =now X positive constant g(t+1)= g(t) X b; b is not the rate of change of the sequence but it does tell us how the sequence is changing Closed form: f(x)=mx + b, m & b are constants Closed form: g(t)=aXb t, where b is the “base” that determines the rate of change; a is the “starting point” or y-intercept of the graph Arithmetic sequences can be thought of linear functions whose domains are pos. integers 1, 4, 7, 10, 13,…. Geometric sequences can be thought of as exponential functions whose domains are pos. integers 10, 20, 40, 80, ….

27. QUALITATIVE GRAPHING: A Context for Exploring Functional Relationships

28. 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 IMPORTANCE OF QUALITATIVE GRAPHS

29. 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

30. 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

31. REFLECTION  What is functional thinking? What new insights have you gained? How has your thinking changed?

32.  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 FUNCTIONAL THINKING IN YOUR CLASSROOM MATH & SCIENCE COLLABORATIVE AT THE ALLEGHENY INTERMEDIATE UNIT

33.  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. HOMEWORK MATH & SCIENCE COLLABORATIVE AT THE ALLEGHENY INTERMEDIATE UNIT