 Common Core High School Functions Teaching and Learning Resources

Domain: BF: Building Functions • BF-A: Build a function that models a relationship between two quantities

• BF-B: Build new functions from existing functions BF-A: Build a function that models a relationship between two quantities

Standards:

• BF-A.1. Write a function that describes a relationship between two quantities.
• BF-A.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
• BF-A.1.b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
• BF-A.1.c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
• BF-A.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with BF-A, BF-B:

 arithmetic sequence explicit function geometric sequence model composition exponent rules inverse function recursive formula for a sequence explicit formula for a sequence function operations logarithm rules transformations

Study.com: Algebra: Functions: These are video lessons on functions, each of which includes sample problems and a short quiz.  Use the following with BF-A.1:

Study.com: Algebra: Sequences and Series: This chapter contains seven video lessons, each of which includes sample problems and a short quiz.  Videos define sequences and series, how to find and classify arithmetic and geometric sequences, summation notation and mathematical series, and how to calculate arithmetic and geometric series with practice problems on these latter calculations provided.  There is a chapter multiple choice test following the last video.  Use all to address BF-A.2.

LearnZillion:

• Lesson set: Write a function that describes a linear relationship between two quantities.  Five video lessons: Identify linear relationships by exploring rectangle perimeter, define a recursive process by exploring patterns, define an explicit expression by exploring patterns, compare explicit expressions and recursive processes, connect linear relationships, explicit expressions, and recursive processes.  Aligns with BF-A.1 and BF-A.1.a.
• Lesson set: Write a function that describes an exponential relationship between two quantities.  Five video lessons: Determine if a relationship between two quantities is exponential, writing an explicit formula for an exponential relationship, writing an explicit formula for the graph of an exponential relationship, writing an exponential formula for real-life problems, writing a recursive formula for an exponential relationship.  Aligns with BF-A.1 and BF-A.1.a.
• Lesson set: Combine standard function types using arithmetic operations.  Five video lessons: Combining functions by adding and subtracting, multiplying, dividing; creating a new function from the graphs of two functions, applications of combining functions.  Aligns with BF-A.1 and BF-A.1.b.

Southeast Comprehensive Center of SEDL: Video: Build a function that models a relationship between two quantities: This video aligns with standard BF-A.1.a.  View online or download.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary. Amazing Problems: Arithmetic and Geometric Sequences: "This lesson aims to introduce arithmetic sequences and geometric sequences to secondary school students. It uses two wonderful problems to do that and it explains the laws of those two kinds of sequences."  Aligns with IF-A.3, BF-A.1.a, and BF-A.2.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 12 Sequences and Series for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook.  Use with BF-A.2.

NCTM Illuminations:

• Function Matching: Students see the graph and determine parameters of the function.  The graph might be linear, a polynomial (up to degree 4), exponential, logarithmic, or periodic (sine, cosine, or tangent).  You can choose the type of graph to generate, or display graphs randomly.  Use with BF-A.1.
• Trout Pond: This investigation illustrates the use of iteration, recursion and algebra to model and analyze a changing fish population. Graphs, equations, tables, and technological tools are used to investigate the effect of varying parameters on the long-term population.  NCTM's tool: Affine Recurrence Plotter can also be used with this activity.

Purple Math:

• Composition of Functions: Composing functions that are sets of points, composing functions with other functions, word problems using composition, inverse functions and composition.  Use with BF-A.1.c.
• Number Patterns finding the next number in a sequence, including common differences and recursions.  Use with BF-A.2.
• Sequences and Series: Terminology and notation, basic examples, arithmetic and geometric sequences, arithmetic series, finite and infinite geometric series.  Use with BF-A.2.

Shodor Interactivate: Introduction to Arithmetic and Geometric Sequences: Lesson introduces learners to these concepts.  Sequencer is the virtual manipulative for students to explore further through producing sequences by varying the starting number, multiplier, and add-on.  The numbers in the sequence are displayed on a graph, and they are also listed below the graph.  Use with BF-A.2.

Teaching Channel Video: Ferris Wheel: Trigonometric Functions.  Lesson objectives: Model how a trigonometric function describes the relationship of a Ferris wheel rider as the wheel spins at a constant rate with relationship to the height of the rider from the ground.  Discern the relationship between the given measure and the period, phase, offset and amplitude of a cosine function.  Questions for learners to consider are included.  Use with BF-A.1.b, BF-B.3, and TF-A.1.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

• US Common Core State Standard Math HSF-BF.A.1: 29 manipulatives.  Among those: Combining functions, composition of functions, exponential decay, continuous exponential growth, composition and transformations, iterating linear functions, iterating rational functions, Fibonacci numbers and the Golden Ratio, and more.
• US Common Core State Standard Math HSF-BF.A.2: 9 manipulatives.  Among those: Plot of a geometric sequence and its partial sums, summing an arithmetic progression, sum of a geometric series, arithmetic sequences, arithmetic series, and more.

Multiple Choice:

Khan Academy: Practice questions with videos.

McDougal Littell ClassZone, Algebra 2, 2011.  Use with BF-A.2:

Study.com: Algebra: Sequences and Series chapter multiple choice test.  Learners should have completed the chapter  Sequences and Series with its seven video lessons prior to completing this test.  Use to address BF-A.2.

Constructed-response:

Algebra Lab:

• Arithmetic Sequences: Lesson and practice problems with writing recursive and explicit formulas.  Aligns with BF-A.2.
• Geometric Sequences: Lesson and practice problems with writing recursive and explicit formulas.  Aligns with BF-A.2.

Khan Academy: Practice questions with videos.

MATHguide.com:

• Arithmetic Sequences: Students learn how to identify arithmetic sequences, calculate the nth term in an arithmetic sequence, find the number of terms in an arithmetic sequence and find the sum of an arithmetic sequence.  Each lesson has a quizmaster.  Fill in and check answers.  Aligns with BF-A.2.
• Geometric Sequences: Students learn how to identify geometric sequences, calculate the nth term in a geometric sequence, find the number of terms in a geometric sequence and find the sum of a geometric sequence.  Each lesson has a quizmaster.  Fill in and check answers.  Aligns with BF-A.2.

OpenEd:

Illustrative Mathematics: Functions:

• Coffee: Students use a chart to solve simultaneous equations.  This task aligns with function standard BF-A.1 and algebra standard REI-C.6.
• How old are they?: Students form expressions and form and solve an equation to solve an age problem.  This task aligns with function standards BF-A.1 and BF-B.4.
• Conference tables: Students find and extend a number pattern in a practical context and find and use a rule or formula.  This task aligns with function standards BF-A.1 and BF-B.4.

Mathematics Assessment Project: Standards: High School: Functions: https://www.map.mathshell.org/stds.php?standardid=1448 Choose from the following:

Mathematics Common Core Toolbox: PARCC Prototype Project:  Under High School Tasks, select:

• Cellular growth: "Students write geometric sequences both recursively and with an explicit formula and use the two representations to model situations. In this task, students consider the use of one or more representations to solve a real-world problem and choose the type of sequence to represent a situation."  This four-part task aligns with function standards BF-A.2 and LE-A.2 and mathematical practice standards MP.2 and MP.4.
• Golf balls in water: "Students model approximately linear data with a linear function. To do so, they must use contextual information and construct representations of linear relationships as well as interpret parameters of the linear models in the context. It makes use of common sense understandings of volume as it relates to changing the dimensions of a glass jar; thus, the mathematics is presented in a context that is novel, yet has multiple entry points for students." This three-part task aligns with function standards BF-A.1, LE-A.2, LE-B.5, and mathematical practice standards MP.1, MP.2 and MP.4.
• Isabella's credit card: Students see a page resembling a spreadsheet.  They "consider a financial context requiring multiple calculations for each month ... Students create a recursive process for calculating values within the context." They "calculate values and complete the table for the second and third months. ... the task allows for the use of either the recursive process or a description of steps that lead to a solution." Finally, they "extend their use of the recursive process or sequence of steps to find additional values. They identify the important values and interpret those values within the context to answer the questions." This three-part task aligns with function standards BF-A.1, IF-B.4, and mathematical practice standards MP.2, MP.4 and MP.5.
• Rabbit populations:  "Students analyze the situation (rabbit population growth) and create a mathematical model that represents it. Students are then presented with a second rabbit population that is slightly different and asked to compare the models for the two populations. This requires students to identify which function parameters and key features of the graph will change based on the adjustment in context. Students then use the two models together to answer mathematical questions." This three-part task aligns with function standards BF-A.1, LE-A.2, LE-B.5, and mathematical practice standard MP.4.

Mathematics Vision Project, Secondary 1 Student Edition:

• Module 3: Arithmetic and Geometric Sequences: This module contains 10 classroom tasks.  Function standards addressed in Module 2 include LE-A.1, LE-A.2, LE-B.5, and BF-A.1.  Task 1: Growing Dots; Task 2: Growing, Growing Dots; Task 3: Scott’s Workout; Task 4: Don’t Break the Chain; Task 5: Something to Chew On; Task 6: Chew On This; and Task 7: What Comes Next? What Comes Later?  align to function standard BF-A.1.
• Module 4: Linear and Exponential Functions: This module contains 10 classroom tasks.  Function standards addressed in Module 3 include IF-A.3, IF-C.7, LE-A.1, LE-A.2, LE-A.3, LE-B.5, and BF-A.1, BF-A.2.  Task 6: Growing, Growing, Gone aligns to function standards BF-A.1, BF-A.2.
• Module 5: Features of Functions: This module contains nine classroom tasks.  Function standards addressed in Module 4 include IF-A.1, IF-A.2, IF-A.3, and IF-B.4, IF-B.5, IF-C.7, and BF-A.1.  Task 5: Pooling It Together; and Task 6: Interpreting Functions align to standard BF-A.1.
• Module 7: Connecting Algebra and Geometry: This module contains six classroom tasks.  Task 4: Training Day; Task 5: Training Day Part II; and Task 6: Shifting Functions each address the function standards BF-A.1, BF-B.3, and IF-C.9.  The other three tasks address geometry standards GPE-B.4, GPE-B.5, and GPE-B.7.

Mathematics Vision Project, Secondary 2 Student Edition:

• Module 1: Quadratic Functions: This module contains 7 classroom tasks.  Function standards addressed in Module 1 include those in BF and LE. Tasks 1-6 align to function standard BF-A.1.

Mathematics Vision Project, Secondary 3 Student Edition:

NCTM's Reasoning and Sense Making Task Library: Over the Hill includes the task overview, teacher notes for its use in the classroom, and student activity sheet.  Aligns with function standard BF-A.1b, algebra standards CED-A.2, REI-C.6, and mathematical practice standards 1, 4, 5, and 7.

WeTeachNYC, developed by the New York City Department of Education: High School Algebra: Aussie Fir Tree: Unit Overview: “The Aussie Fir Tree task is a culminating task for a 2-3 week unit on algebra that uses the investigation of growing patterns as a vehicle to teach students to visualize, identify and describe real world mathematical relationships. Students who demonstrate mastery of the unit are able to solve the Aussie Fir Tree task in one class period.” Function standards addressed include IF-A.3, BF-A.1 and BF-A.1.a.  Algebra standards include CED-A.1, REI-B.4 and REI-B.4.a.  Mathematical Practices standards include MP.1, MP.3, MP.4, MP.6, and MP.7.

BF-B: Build new functions from existing functions

Standards:

• BF-B.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
• BF-B.4. Find inverse functions.
• BF-B.4.a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
• BF-B.4.b. (+) Verify by composition that one function is the inverse of another.
• BF-B.4.c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
• BF-B.4.d. (+) Produce an invertible function from a non-invertible function by restricting the domain.
• BF-B.5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Technology-enhanced investigations:

Concord Consortium STEM resources: Mathematics:  Each of the following includes an applet, student activity, teacher lesson plan, and student assessment.

Study.com: Algebra: Functions: These are video lessons, each of which includes sample problems and a short quiz.  Use the following with BF-B.3 and BF-B.4:

Teaching Channel Video: Ferris Wheel: Trigonometric Functions.  Lesson objectives: Model how a trigonometric function describes the relationship of a Ferris wheel rider as the wheel spins at a constant rate with relationship to the height of the rider from the ground.  Discern the relationship between the given measure and the period, phase, offset and amplitude of a cosine function.  Questions for learners to consider are included.  Use with BF-A.1.b, BF-B.3, and TF-A.1.

LearnZillion:

• Lesson set: Manipulate linear functions.  Eight video lessons: Vertically, horizontally translate linear equations; reflect linear equations; apply horizontal, vertical scaling transformations to linear equations; identify the value of k in translations, scaling transformation of linear equations; apply knowledge of transformations to linear equations.  Aligns with BF-B.3.
• Lesson set: Manipulate quadratic functions.  Six video lessons: Vertically, horizontally translate quadratic equations; understand vertical, horizontal scaling of quadratic equations; even and odd functions, application of quadratic function transformations.  Aligns with BF-B.3.

Saltire Software: Common Core Nuggets: There are four applets addressing vertical and horizontal translations and dilations in graphing, which align with BF-B.3.

Study.com: Algebra: Exponentials and Logarithms: There are eight video lessons in this chapter, each of which includes sample problems and a short quiz.  Videos address the basic definitions of exponentials and logarithms, exponential growth and decay, how to evaluate logarithms, logarithm properties and practice problems for those properties, and how to solve exponential and logarithmic equations.  There is a chapter multiple choice test following the last video  Use all the videos with BF-B.5.

A. Dendane: Analyze Math:  Select the following:

• Even and Odd Functions: Applet and practice problems for investigating eight given functions designed to help learners understand even and odd functions graphically and analytically.  Aligns with BF-B.3.
• Find Inverse of Rational Functions: A video illustrating how to find the inverse of rational functions.  Aligns with BF-B.4.
• Logarithm Functions: Tutorial with definition, examples, and two applets.  The first applet helps learners to explore properties of logarithms of the form  f(x)= a logB( b( x + c) ) + d by changing parameters a, b, c, d, and base B.  The second applet can be used to explore the logarithm function and its inverse, the exponential function, by changing values of parameters c and d and base B in f(x)= logB( x + c) + d. Aligns with BF-B.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 6 Rational Exponents and Radical Equations for interactive practice: Animations (Virtual manipulatives, including inverse functions), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook (Do section 6.4 on inverses).  Use with BF-B.4.

MIT BLOSSOMS: Video lessons with additional teacher and learner resources.  Descriptions are from the video summaries.

• The Quadratic Equation: It’s Hip to Be Squared: "The lesson begins with three quadratics and their graphs (three parabolas):  y = x2 - 2x + (0 or 1 or 2). The prerequisite or co-requisite is some working experience with algebra, like factoring x2 -2x into x(x-2). The objective is to connect four things: the formula for y, the graph of y (a parabola), the roots of y and the minimum or maximum of y."  Aligns with Algebra standards SSE-B.3.a, CED-A.2, and REI-B.4.b and Function standards IF-B.4, IF-C.7.a, IF-C.8.a, BF-B.3.
• Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled."  Additional simulations are included.  Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.

Thinking Mathematics:

Math Open Reference: Graphical Function Explorer: Graph up to three functions on the same set of axes using this free online graphing tool.

National Library of Virtual Manipulatives: Scroll down to Algebra (9-12) and select Function Transformations: Explore how simple transformations affect the graph of a function.

Purple Math: Use with BF-B.

• Function Transformations: Translations
• Inverse Functions: Definition and Drawing an Inverse from a graph; Is the inverse a function?; Finding inverses; Proving inverses

Shodor Interactivate: Function Flyer: This activity allows the manipulation of the constants and coefficients in any function thereby encouraging the user to explore the effects on the graph of the function by changing those numbers.  Good for studying about translations and dilations.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

• US Common Core State Standard Math HSF-BF.B.3: 24 manipulatives.  Among those: Transforming parent functions, a library of functions with transformations, compositions and transformations, graphing functions with four parameters, and more, including the following:
• Quadratic in Vertex Form: Learners manipulate the parameters of the quadratic in symmetry-vertex form to see the affect on the graph (dilation, translation horizontally and/or vertically).  Aligns with BF-B.3.
• Even and Odd Functions: This demonstration allows learners to explore the definitions of even and odd functions by selecting graphs of each type to view.  Aligns with BF-B.3.
• Function Transformations and Inverses: This is a comprehensive manipulative in which learners can choose f(x) (e.g., linear, quadratic, cubic, absolute value, square root, piecewise, exponential, logarithmic, sine, cosine, tangent, inverse sine, inverse cosine, inverse tangent) and set parameters a, b, c, and d to create g(x)=a f(b(x + c))+d.  The inverse graph can be shown, which may or may not be a function.  Aligns with BF-B.3, BF-B.4, and BF-B.5.
• US Common Core State Standard Math HSF-BF.B.4: 6 manipulatives.  Among those: Function identification game, domain and range of inverse trig functions, inverse composition rule, and the next two.
• US Common Core State Standard Math HSF-BF.B.5: 5 manipulatives.  Among those: Swing the logarithm curve around (1,0), calculating integer logarithms in different bases, rules for logarithms, elementary transcendental functions and their inverses, multiplication by adding logs.

Multiple Choice:

Khan Academy: Practice questions with videos.

MathsIsFun.com: Tutorial on the Inverse Functions.  Examples are shown.  Learners then complete 10 multiple choice problems.  Aligns with BF-B.4.

McDougal Littell ClassZone, Algebra 2, 2011: Ch. 6.4 Quiz: Use Inverse Functions

OpenEd: Find inverse functions: 5 questions that combine multiple choice and free response.  Includes related review resources.  Aligns with BF-B.4.

Wolfram Demonstrations Project:

Study.com: Algebra: Exponentials and Logarithms chapter multiple choice test.  Learners should have completed the eight video lessons in this chapter on Exponentials and Logarithms prior to taking this test.  Use with BF-B.5.

Constructed-response:

Algebra Lab: Inverse Functions: Lesson and practice problems finding inverse functions and verifying that two functions are inverses of each other.

Khan Academy: Practice questions with videos.

Georgia Standards of Excellence Framework: Pre-Calculus Unit 2: Trigonometric Functions.  This unit contains two tasks: Right Triangles and the Unit Circle, and Inverse Trigonometric Functions.  Tasks align to BF-B.4, BF-B.4.d, TF-A.3, TF-A.4, TF-B.6, TF-B.7 and are also related to TF-A.2, TF-B.5, and IF-C.7.e.

Illustrative Mathematics: Functions:

• How old are they?: Students form expressions and form and solve an equation to solve an age problem.  Aligns with BF-A.1 and BF-B.4.
• Conference tables: Students find and extend a number pattern in a practical context and find and use a rule or formula. Aligns with BF-A.1 and BF-B.4.

Mathematics Common Core Toolbox: PARCC Prototype Project:  Under High School Tasks, select:

• Transforming graphs of quadratic functions: "Students use the innovative graphical interaction to experiment and create the image of the transformed function, providing a dynamic technological replacement for paper-and-pencil sketching. While the technology is not essential to solving the problem, it provides a support for students to interact with function transformations and has significant implications for future item development—a next iteration of this task could ask students to manipulate the graph using the sliders and submit their answers through the interactive tool instead of filling in the blanks." This two-part task aligns with function standard BF-B.3 and mathematical practice standards MP.3 and MP.7.

Mathematics Assessment Project: Standards: High School: Functions: https://www.map.mathshell.org/stds.php?standardid=1448

Mathematics Vision Project, Secondary 1 Student Edition:

• Module 7: Connecting Algebra and Geometry: This module contains six classroom tasks.  Task 4: Training Day; Task 5: Training Day Part II; and Task 6: Shifting Functions each address the function standards BF-A.1, BF-B.3, and IF-C.9.  The other three tasks address geometry standards GPE-B.4, GPE-B.5, and GPE-B.7.

Mathematics Vision Project, Secondary 2 Student Edition:

• Module 4: More Functions, More Features: This module contains 7 classroom tasks.  Function standards addressed in Module 4 include IF-C.7b within tasks 1-4; BF-B.4 in tasks 5-6; and IF-B.4 in task 7.

Mathematics Vision Project, Secondary 3 Student Edition:

NCTM's Reasoning and Sense Making Task Library: Tidal Waves includes the task overview, teacher notes for its use in the classroom, and student activity sheet.  Aligns with BF-B.3, TF-B.5 and mathematical practice standards 1 and 7. Back to top Common Core Math: Intro | HS Functions Domain: IF  |  BF  |  LE  |  TF  |