CCSS: Focus on Functions

Problem Solving Via Functions Tasks

Day 2Need graphing calculators and graph paper for today.1Reflecting on Functional ThinkingGo 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 Unit2Connecting to the StandardsHow do you see the Learning Progressions helping you or applying in your classroom? Use an example to illustrate this

Learning Progressions are documents from the authors of the CCSSM and a larger working committee that give more detail around what the standards look like in action. What students might be doing while engaging in this standard? How the standards might be related to one another? Not every standard is necessarily mentioned in the Learning progression, but by and large, most of them are.From the PARCC framework, pg 10: One of the primary uses of the progressions is to give educators and curriculum developers information that can help them develop materials for instruction aligned to the standards.3Functional thinkingBased on your reading assignment:How has your thinking about functions and functional thinking changed? Why? What new insights have you gained?

Individual then have them talk to a shoulder partner4Cubes in a Line TaskHow many faces (face units) are there when 2 cubes are put together sharing a face? 10 cubes? 100 cubes? t cubes?SAS Secondary Mathematics Teacher Leadership Academy, Year 1 (Adapted from Video Cases for Mathematics Professional Development PowerPoints)5Mathematical Task QuestionsPredict how a student might solve this problem using: a visual, closed method a recursive method a tableThere are a variety of different visual methods to use but all will simplify to Tiles = 3n + 2

Recursive will see the addition of 3 each time. NEXT = NOW + 3

Table can be viewed in a number of different ways to see any variation of 3n + 26Mathematical 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?Video Segment Focus QuestionsWhat 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

Reginas Logo Lesson TaskAssume 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 4Lets go ahead and engage in a mathematical task that may help us connect to the way the groups sorted the cards

Mike, My DVD does not include Reginas Logo 9Strategy

Does this strategy draw on a correspondence perspective or covariation perspective of functions?10Strategy

Does this strategy draw on a correspondence perspective or covariation perspective of functions?

11Strategy

Does this strategy draw on a correspondence perspective or covariation perspective of functions?

12Video Segment Focus QuestionsWhat moments or interchanges appear to be interesting/important mathematically? Pedagogically?

Mike,I do not have the video tape of this in my VCMPD materials.

From the transcript, here is my thinking.The exchange with Kiril around time 40:08 is mathematically interestingThe initial exchange with Reymond at 41:22 to about 42:20 is mathematically interestingWhen Gisele steps in around 42:35 that is pedagogically importantReymonds talk around 45 minutes is mathematically importantGisele at 45:23Rey around 45:47Gisele at 46:09Gisele at 47:1913Reginas Logo 2 and 3Lesson TasksReginas Logo 2size 0size 1size 2size 3Reginas Logo 3size 1size 2size 3size 4size 5Can think about Logo 2 and Logo 3 as horizontal shifts of original. Equation for 2 can be: t = 3(n+1) + 2 and Equation for 3 can be: t = 3(n-1) +214Mathematical Task QuestionsIn 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 Reginas Logo, Reginas Logo 2, and Reginas Logo 3 similar? Different?Can think about Logo 2 and Logo 3 as horizontal shifts of original. Equation for 2 can be: t = 3(n+1) + 2 and Equation for 3 can be: t = 3(n-1) +2 Similarities might include that they have same recursive pattern of adding 3, which is shown in the graph as the slope of the line always being 3.

Since the starting point is different in each case, the are parallel lines but are translations of one another. I thought of them as shifts along the x-axis, but if you think numerically, you might think about the shifting as along the y axis (y-intercept changes by 3 up or down).15Thinking about the changesWith your small group, create a poster depicting the changes/ translations/ transformations among the three versions of Reginas Logo Problem with at least one representation (arithmetic sequence, equations, tables, graphs, etc.)16LanguageTableContextGraphEquationFive Different Representations of a FunctionVan de Walle, 200417One thing that we worked on is making connections between different representations.In general, using different representations of a concept can help students better understand the concept. Here is a diagram that shows 5 different representations of mathematical ideas. It is the translations between and within each that can help students develop new concepts. Go through; emphasize the double-headed arrows.

These questions might not be the direction we want to go: What connections did we make when we worked on the card sort task? What additional connections could we have made?

Representations and SMPWhere 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.

Reginas Logo vs. graphing ProblemConsider: Graphing ProblemSimplify and graph the following equations:Y1 = 2x + (x+2)Y2 = 2(x+1) +xY3 = 3(x-1) + 5

How these two tasks similar? How are they different?

Do the differences matter?Do we want to ask if the same standards are illuminated by this task as marbles? Might be risky.

Perhaps we want to include a very straightforward graphing of a linear equation or evaluating a functionsomething usually associated with the textbook learning of linear functionsto illustrate a contrasting low-level task to Reginas Logo instead.

Also, where does the Tiling Patio Task come in? We have one form of it in the binder and that plus another version on share drive. Are we using it in the context of high level tasks?19Linear vs. Exponential FunctionsLinearExponentialRate of change is constant

Recursive: next= now + rate of changef(t+1)= f(t)+m; m is the rate of changeClosed 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. integers1, 4, 7, 10, 13,.

Cubefaces16210314Linear and Exponential ModelsConstruct and compare linear and exponential models to solve problemsDistinguish 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 PA Model Curriculum- AlgebraThe 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:Make sense of and persevere in solving complex and novel mathematical problems. Use effective mathematical reasoning to construct viable arguments and critique the reasoning of others.Communicate precisely when making mathematical statements and express answers with a degree of precision appropriate for the context of the problem/situation. 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.Make use of structure and repeated reasoning to gain a mathematical perspective and formulate generalized problem solving strategies.Algebra 1At 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 ModulesThe modules should equate to a full year of instruction:Module 1: Relationships Between Quantities and Reasoning with Equations Module 2: Linear and Exponential RelationshipsModule 3: Descriptive StatisticsModule 4: Equations and ExpressionsModule 5: Quadratic Functions and ModelingLinear and exponential relationshipsBy 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.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.

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

Linear vs. Exponential FunctionsLinearExponentialRate of change is constantRate of change is not constant, but it changes Rate of change Increases/decreases over timeWhen the input increases by 1unit, the output is multiplied by a constant factorRecursive: next= now + rate of changef(t+1)= f(t)+m; m is the rate of changeRecursive: next =now X positive constantg(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 changingClosed form: f(x)=mx + b, m & b are constantsClosed form: g(t)=aXbt, where b is the base that determines the rate of change; a is the starting point or y-intercept of the graphArithmetic sequences can be thought of linear functions whose domains are pos. integers1, 4, 7, 10, 13,.Geometric sequences can be thought of as exponential functions whose domains are pos. integers10, 20, 40, 80, .

Qualitative Graphing:A Context for Exploring Functional RelationshipsSMTLA, Year 2, Summer 200527Rather 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 GraphsSMTLA, Year 2, Summer 200528Importance 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. 126SMTLA, Year 2, Summer 200529Keishas Bicycle Ride TaskSolve 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

SMTLA, Year 2, Summer 200530ReflectionWhat is functional thinking? What new insights have you gained? How has your thinking changed?Plan task/activity to develop/extend students functional thinking. This will be submitted (electronically) for inclusion on the course wikipageFocus on one content standard and one standard for math practiceResources are found at www.mscsummercourses2013.wikispaces.com

Functional Thinking in Your ClassroomMath & Science Collaborative at the Allegheny Intermediate UnitComplete 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.

HomeworkMath & Science Collaborative at the Allegheny Intermediate Unit33