The Functions

Conceptual Category Progression, Model Content Frameworks,

Illustrations, and Prototypes

“Counting on the Core” A Common Core Workshop

Saturday, March 16, 2013 Kenston High School

9500 Bainbridge Road Chagrin Falls, 44023

Lynn Aring President-Elect, GCCTM

Vice-President for Secondary, OCTM Bay High School, Retired Lynn.Aring@gmail.com

The Progression The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. These documents were spliced together and then sliced into grade level standards. From that point on the work focused on refining and revising the grade level standards. The early drafts of the progressions documents no longer correspond to the current state of the standards. It is important to produce up-to-date versions of the progressions documents. They can explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. This would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks. Progressions documents also provide a transmission mechanism between mathematics education research and standards. Research about learning progressions produces knowledge which can be transmitted through the progressions document to the standards revision process questions and demands on standards writing can be transmitted back the other way into research questions.

As you read through the “PROGRESSION”, consider the following key ideas: Overview • Defining characteristic of a function: input value determines the output value. • Mathematical definition is sometimes very different and sometimes similar to common uses. • In some situations it may not be clear which variable is the input and which is the output, in other situations,

the context may determine the input variable. • Most functions in HS involve only one input variable but geometric transformations can be considered

functions of both “x” and “y”. • Some pattern standards (4.OA.5 and 5.OA.3) are preparation for the study of functions. • Sequences and Functions: patterns are sequences, sequences are functions with a domain of whole

numbers, HS students use an index to indicate which term is being discussed. • Functions and Modeling: Sometimes context and statistics are used to find algebraic expressions that

model a situation. In some situations, a function can be written using an algebraic expression. In other situations the function is best represented by a graph or table.

• Functions and Algebra: Biggest connection: solving equations can be seen as finding the intersection of graphs of functions.

Grade 8: Define, evaluate , and compare functions • Concept of function: a rule that assigns to each input exactly one output. • Students explore, interpret, and translate among functional relationships described algebraically,

graphically, numerically in tables, and verbally. • The main focus in Grade 8 is linear functions. • Students recognize linearity in a table(constant differences in input values produce constant differences

between output values) • Students use the constant rate of change appropriately in a verbal description of a context. Grade 8: Use functions to model relationships between quantities • Modeling a linear relationship involves determining the rate of change which is then related to slope • Within a class of linear functions, some are proportional and some are not. • Describing relationship qualitatively by stating what happens to the graph as the input value increases from

left to right.

HS Domain: Interpreting Functions Cluster: Understanding the concept of a function and use function notation • Formalize function notation and language for functions-domain and range. • Distinguishing functions from relations is no longer in the standards. • Key question: “Does each element of the domain correspond to exactly one element in the range?” • De-emphasize vertical line test, searching tables. • Promote fluency with function notation by interpreting function notation in context. • Promote use of function notation not only for closed formulas but also with graphs and tables. • Sequences are functions whose domain is a subset of the integers, need an index to act as an input to the

term. • Possible use of subscript notation(has been seen on PARCC examples) Cluster: Interpret functions that arise in applications in terms of the context • Functions are often described and understood in terms of their behavior—increasing, decreasing, constant,

positive, negative, zero, end behavior, etc. • Average rates of change using function notation Cluster: Analyze functions using different representations • Understand families of functions and how changing the parameters affect the graph of the function and its

key features. • Fluency with graphing linear, quadratic, and exponential functions by hand • For other families—graph simple cases by hand and the rest with technology • Re-writing expression for functions in different forms can reveal key features of the graph of the function. HS Domain: Building Functions • This domain focuses on building functions to model relationships and building new functions from existing

functions • Composition and composition of a function and its inverse are among the plus standards. Cluster: Build a function that models a relationship between two quantities • Very closely related to the algebra standard on writing equations in two variables. • Recursively defined functions are useful • Building functions from simpler functions based on context—exponential decay and a constant function for a

hot object cooling to room temperature • Arithmetic sequences are linear functions, geometric sequences are exponential functions Cluster: Build new functions from existing functions • Quadratic and absolute value functions are good contexts to understand various transformations, but the

effects of transformations should be applied abstractly to all functions. • Confusion about effect on the input variable versus effect on the graph. • Concepts of even and odd functions are related to symmetry. • Modest expectations for inverse functions—students must solve equations of the form f(x)=c, using numbers

at first and then for a general value of x. • Advanced students will need a more formal sense of families of inverse functions, compositions of a function

and its inverse function. • Advanced students will understand logarithmic functions as the inverse of exponential functions and use this

understanding to develop the properties of logarithms.

HS Domain: Linear, Quadratic, and Exponential Models Cluster: Construct and compare linear and exponential models and solve problems • Distinguishing between situations that can be modeled by linear functions and exponential functions. • Linear functions grow by equal differences over equal intervals--proof • Exponential functions grow by equal factors over equal intervals—proof • Using two points to determine both a linear function and an exponential function. Cluster: Interpret expressions for functions in terms of the situation they model • This may be harder than when the students build a function HS Domain: Trigonometric Functions • Trigonometric functions can be defined in terms of the trig ratios but only for angles between 0 degrees and

90 degrees. • Circular trigonometry extends the domains of the trig functions • Only sine, cosine, and tangent need to be discussed. Cluster: Extend the domain of trigonometric functions using the unit circle • Use of radian measure which is dimensionless. • Development of circular trigonometry using the unit circle, reference triangles, and unwrapping the unit

circle”. • Only advanced students need to develop fluency with the trig functions at “special” angles. • Symmetry of the unit circle relates to the symmetry of the graphs of trig functions. Cluster: Model periodic phenomena with trigonometric functions • All students will model periodic phenomena—most likely with sine and cosine. • Advanced students will model more complex phenomena including those with phase shifts. • Advanced students will solve equations involving trig functions and understand that there are an infinite

number of solutions. Cluster: Prove and apply trigonometric identities • All students will prove 𝑠𝑖𝑛! 𝜃 + 𝑐𝑜𝑠! 𝜃 = 1. • Advanced students will prove other trigonometric identities. ---------------------------------------------------------------------------------------------------------------------------- Information from the Model Content Frameworks

Course Overviews and the Function Conceptual Category Numerals in parentheses designate individual content standards that are eligible for assessment in whole or part. Underlined numerals (e.g., 1) indicate standards eligible for assessment on two or more end-of-course assessments. Not all CCSS-M content standards in a listed domain or cluster are assessed.

Algebra I Domain Course

Emphases Cluster

Interpreting Functions (F-IF)

Major Understand the concept of a function and use function notation (1, 2, 3) Major Interpret functions that arise in applications in terms of the context (4, 5, 6) Supporting Analyze functions using different representations (7, 8, 9)

Building Functions (F-BF)

Supporting Build a function that models a relationship between two quantities (1) Additional Build new functions from existing functions (3)

Linear, Quadratic, and Exponential Models(F-LE)

Supporting Construct and compare linear, quadratic, and exponential models and solve problems (1, 2, 3)

Supporting Interpret expressions for functions in terms of the situation they model (5)

Algebra II Domain Course

Emphases Cluster

Interpreting Functions(F-IF)

Supporting Understand the concept of a function and use function notation ( 3) Major Interpret functions that arise in applications in terms of the context (4, 6) Supporting Analyze functions using different representations (7, 8, 9)

Building Functions(F-BF)

Major Build a function that models a relationship between two quantities (1, 2) Additional Build new functions from existing functions (3,4a)

Linear, Quadratic, and Exponential Models(F-LE)

Supporting Construct and compare linear, quadratic, and exponential models and solve problems ( 2, 4)

Additional Interpret expressions for functions in terms of the situation they model (5)

Trigonometric Functions (F-TF)

Additional Extend the domain of trigonometric functions using the unit circle (1,2) Additional Model periodic phenomena with trigonometric functions (5) Additional Prove and apply trigonometric identities (8)

Mathematics I

Domain Course Emphases

Cluster

Interpreting Functions (F-IF)

Major Understand the concept of a function and use function notation (1, 2, 3) Major Interpret functions that arise in applications in terms of the context (4, 5, 6) Supporting Analyze functions using different representations (7, 9)

Building Functions (F-BF)

Major Build a function that models a relationship between two quantities (1, 2)

Linear, Quadratic, and Exponential Models(F-LE)

Supporting Construct and compare linear, quadratic, and exponential models and solve problems (1, 2, 3)

Supporting Interpret expressions for functions in terms of the situation they model (5)

Mathematics II

Domain Course Emphases

Cluster

Interpreting Functions (F-IF)

Major Interpret functions that arise in applications in terms of the context (4, 5, 6) Supporting Analyze functions using different representations (7, 8, 9)

Building Functions (F-BF)

Supporting Build a function that models a relationship between two quantities (1) Additional Build new functions from existing functions (3)

Mathematics III

Domain Course Emphases

Cluster

Interpreting Functions (F-IF)

Major Interpret functions that arise in applications in terms of the context (4, 6) Supporting Analyze functions using different representations (7, 9)

Linear, Quadratic, and Exponential Models(F-LE)

Supporting Construct and compare linear, quadratic, and exponential models and solve problems (4)

Trigonometric Functions (F-TF)

Additional Extend the domain of trigonometric functions using the unit circle (1,2) Additional Model periodic phenomena with trigonometric functions (5) Additional Prove and apply trigonometric identities (8)

Note: The Functions conceptual category contains the following “plus” standards: F-IF 7d, F-BF, 1c, 4c, 4d, and 5, F-TF 3,4,6,7, and 9

Examples of Key Advances from previous grades or courses involving Functions Course Statement Algebra I

Students in grade 8 extended their prior understanding of proportional relationships to begin working with functions, with an emphasis on linear functions. In Algebra I, students will master linear and quadratic functions. Students encounter other kinds of functions to ensure that general principles are perceived in generality, as well as to enrich the range of quantitative relationships considered in problems.

Algebra II

In Geometry, students began trigonometry through a study of right triangles. In Algebra II, they extend the three basic functions to the entire unit circle.

Algebra II

As students acquire mathematical tools from their study of algebra and functions, they apply these tools in statistical contexts (e.g., S-ID.6). In a modeling context, they might informally fit an exponential function to a set of data, graphing the data and the model function on the same coordinate axes.

Math I Students formalize their understanding of the definition of a function, particularly their understanding of linear functions, emphasizing the structure of linear expressions. Students also begin to work on exponential functions, comparing them to linear functions.

Math II A parallel extension occurs from linear and exponential functions to quadratic functions, where students also begin to analyze functions in terms of transformations.  

Math III The understandings that students have developed with linear, exponential and quadratic functions are extended to considering a much broader range of classes of functions.  

Discussion of Mathematical Practices in Relation to the Function Conceptual Category Course Statement Algebra I Modeling with mathematics (MP.4) continues to be a particular focus as students see a broader

range of functions, including using appropriate tools strategically (MP.5). Algebra I Use appropriate tools strategically (MP.5). Spreadsheets, a function modeling language,

graphing tools, and many other technologies can be used strategically to gain understanding of the ideas expressed by individual content standards and to model with mathematics.

Algebra I Look for and express regularity in repeated reasoning (MP.8). Creating equations or functions to model situations is harder for many students than working with the resulting expressions. An effective way to help students develop the skill of describing general relationships is to work through several specific examples and then express what they are doing with algebraic symbolism (A-CED). For example, when comparing two different text messaging plans, many students who can compute the cost for a given number of minutes have a hard time writing general formulas that express the cost of each plan for any number of minutes. Constructing these formulas can be facilitated by methodically calculating the cost for several different input values and then expressing the steps in the calculation, first in words and then in algebraic symbols. Once such expressions are obtained, students can find the break-even point for the two plans, graph the total cost against the number of messages sent, and make a complete analysis of the two plans.

Algebra II Construct viable arguments and critique the reasoning of others (MP.3). As in geometry, there are central questions in advanced algebra that cannot be answered definitively by checking evidence. There are important results about all functions of a certain type — the factor theorem for polynomial functions, for example — and these require general arguments (A-APR.2). Deciding whether two functions are equal on an infinite set cannot be settled by looking at tables or graphs; it requires arguments of a different sort (F-IF.8).

Algebra II Look for and express regularity in repeated reasoning (MP.8). Algebra II is where students can do a more complete analysis of sequences (F-IF.3), especially arithmetic and geometric sequences, and their associated series. Developing recursive formulas for sequences is facilitated by the practice of abstracting regularity for how you get from one term to the next and then giving a precise description of this process in algebraic symbols (F-BF.2). Technology can be a useful tool here: Most Computer Algebra Systems allow one to model recursive function definitions in notation that is close to standard mathematical notation. And spreadsheets make natural the process of taking successive differences and running totals (MP.5).

Math I Modeling with mathematics (MP.4) should be a particular focus as students see the purpose and meaning for working with linear and exponential equations and functions.

Math II Modeling with mathematics (MP.4) should be a particular focus as students see the purpose and meaning for working with quadratic equations and functions, including using appropriate tools strategically (MP.5).

Math III Modeling  with  mathematics  (MP.4)  continues  to  be  a  particular  focus  as  students  see  a  broader    range  of  functions,  including  using  appropriate  tools  strategically  (MP.5).    

Fluency Recommendations involving Functions Course Standard Statement Algebra II F-IF.3 Fluency in translating between recursive definitions and closed forms is helpful when

dealing with many problems involving sequences and series, with applications ranging from fitting functions to tables to problems in finance.

Math II F/S Fluency in graphing functions (including linear, quadratic, and exponential) and interpreting key features of the graphs in terms of their function rules and a table of value, as well as recognizing a relationship (including a relationship within a data set), fits one of those classes. This forms a critical base for seeing the value and purpose of mathematics, as well as for further study in mathematics.

Math III A/F Students should look at algebraic manipulation as a meaningful enterprise, in which they seek to understand the structure of an expression or equation and use properties to transform it into forms that provide useful information (e.g., features of a function or solutions to an equation). This perspective will help students continue to usefully apply their mathematical knowledge in a range of situations, whether their continued study leads them toward college or career readiness.

Math III F-BF.3 Students should understand the effects of parameter changes and be able to apply them to create a rule modeling the function.

Pathway Summary Table: Traditional Pathway: Algebra I-Geometry Algebra II

Integrated Pathway Math I-Math II-Math III

Assessment Limits for Standards Assessed on More Than One End-of-Course Test Two tables in the PARCC Model Content Frameworks show assessment limits for standards assessed on more than one end-of-course test. Table 2 found on pages 56-59 relates to the Traditional Pathway and Table 4 found on pages 70-79 relates to the Integrated Pathway -----------------------------------------------------------------------------------------------------------------------------------------------------

Illustrations Sampler Domain: Interpreting Functions (F-IF) Cluster: Understand the concept of a function and use function notation, Standards 1-3

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Domain: Interpreting Functions (F-IF) Cluster: Interpret functions that arise in applications in terms of the context, Standards 4-6

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Domain: Interpreting Functions (F-IF) Cluster:Analyze functions using different representations, Standards 7-9

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Domain: Building Functions (F-BF) Cluster: Build a function that models a relationship between two quantities, Standards 1-2

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Domain: Building Functions (F-BF) Cluster: Build new functions from existing functions, Standards 3-5

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Domain: Linear, Quadratic, and Exponential Models(F-LE) Cluster: Construct and compare linear, quadratic, and exponential models and solve problems, Standards 1-4

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Domain: Linear, Quadratic, and Exponential Models(F-LE) Cluster: Interpret expressions for functions in terms of the situation they model, Standard 5

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Domain: Trigonometric Functions F-TF Cluster: Extend the domain of trigonometric functions using the unit circle, Standards 1-4 No Illustrations at this time!

------------------------------------------------- Domain: Trigonometric Functions F-TF Cluster: Model  periodic  phenomena  with  trigonometric  functions,    Standards  5-­‐7

Domain: Trigonometric Functions F-TF Cluster: Prove and apply trigonometric identities, Standards 8-9 No Illustrations at this time!

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The 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. There are now some Illustrations for the Standards for Mathematical Practice. Some “Illustrations” are in video form, other “Illustrations” are PDFs that also illustrate content. Here are the “illustrations” from the Functions Conceptual Category that are also cited as good examples of a mathematical standard:

• F-­‐LE  Equal  Differences  over  Equal  Intervals  1 -----------------------------------------------------------------------------------------------------------------------------------------------------

Released High School Prototype Questions

Course? Task Title Standards Assessed Algebra I, Math II

Functions—See below F-IF.9, MP.6

Algebra I, Math II

Seeing Structure in a Quadratic Equation A-REI.4, MP.7

Algebra I, Math II

Seeing Structure in an Equation A-SSE.3, MP.7

Algebra II, Math I

Cellular Growth F-LE.2, F-BF.2, MP.2, MP.4

Algebra I, Math I

Golf Balls in Water F-BF.1, F-LE.2, F-LE.5, MP.1, MP.2, MP.4

Algebra I, Math I

Isabella’s Credit Card—See below F-IF.4, F-BF.1, MP.2, MP.4, MP.5

Algebra I, Math I

Rabbit Population F-BF.1, F-LE.2, F-LE.5, MP.4

Algebra 1, Math II

Transforming Graphs of Quadratic Functions F-BF.3, MP.3, MP.7

Isabella’s Credit Card

Resources PARCC Resources As part of item development, PARCC will be releasing sample items and assessment blueprints. These will provide greater specificity around task types on the assessments and highlight the innovations built into the PARCC Assessment System. As the materials become available, they will be published for voluntary use at http://www.parcconline.org.

Progressions The progressions are being developed by members of the Common Core State Standards working group and writing team through the University of Arizona’s Institute for Mathematics and Education. Progressions are narratives of the standards that describe how student skill and understanding in a particular domain develop from grade to grade. 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. http://ime.math.arizona.edu/progressions/  

Illustrative Mathematics Under the guidance of members of the working group as well as other national experts in mathematics and mathematics education, The Illustrative Mathematics Project will illustrate the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards and by publishing other tools that support implementation of the standards. http://illustrativemathematics.org  Common Core Tools Additional tools that continue to be developed are posted from time to time on http://commoncoretools.wordpress.com, a blog moderated by Dr. William McCallum, distinguished professor and head of mathematics at the University of Arizona and mathematics lead for the Common Core State Standards for Mathematics.  Quality Review Rubrics As part of its efforts to assist states with implementation of the Common Core State Standards, Achieve has developed a number of resources. In particular, Achieve worked with Massachusetts, New York, and Rhode Island to develop Quality Review Rubrics and review processes for educators to use in evaluating the quality of lessons and units intended to address the Common Core State Standards in mathematics and English Language Arts/Literacy. Achieve is now working with additional states through the Educators Evaluating Quality Instructional Products (EQuIP) project to use the rubrics to review instructional materials. http://www.achieve.org/EQuIP Achieve the Core Achieve the Core is a website developed by members of the Common Core State Standards writing team that provides instructional resources, including professional development modules, publishers criteria, and overviews of the key shifts in mathematics. http://www.achievethecore.org/