Computing Technology for Math Excellence Logo

Black line

Common Core High School Functions Teaching and Learning Resources

Domain: IF: Interpreting Functions

Common Core Math

Select the cluster for resources on this page:

  • IF-A: Understand the concept of a function and use function notation

  • IF-B: Interpret functions that arise in applications in terms of the context

  • IF-C: Analyze functions using different representations

Black line

IF-A: Interpret the structure of expressions.

Standards:

 

Technology-enhanced investigations:

Mathwords.com: Math Dictionary, and words at * are from Mathsisfun.com:  Key vocabulary for this domain.  Use with IF-A, IF-B, IF-C:

amplitude exponential function maximum of a function recursive formula for a sequence
asymptote exponential growth minimum of a function step function
average rate of change extremum period of a periodic function symmetric
domain function piecewise function x-intercept
end behavior function operations range y-intercept
exponential decay increasing function and decreasing function rational function zero of a function
logarithmic
function
*
inverse
function
trig functions Fibonacci
sequence*

 

CK-12: Algebra: Concepts, videos, and practice problems for learners to complete.

Interactive Mathematics: Functions and Graphs includes an introduction to functions, functions from verbal statements, rectangular coordinates, graphs of functions including domain and range, graphs defined by tables and data, continuous and discontinuous functions, split functions, and even/odd functions.  Use with IF-A.1 and IF-A.2.

Study.com: Algebra: Functions: These are video lessons, each of which includes sample problems and a short quiz.  Use the following with IF-A.1 and IF-A.2:

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

  • Lesson set: Identify functions and find domain and range.  This is a series five video lessons: Identify functions given sets, mappings, and real world scenarios, identify functions using tables and graphs, identify functions using algebra, find the domain and range of a function, understand graphs of f(x).  Aligns with IF-A.1.
  • Lesson set: Use and interpret function notation; evaluate functions.  This is a series  of four video lessons: Understand function notation, evaluate functions written in function notation, understand restrictions on functions in terms of domain and range, evaluate mathematical situations by analyzing the domain and range of functions.  Aligns with IF-A.2.
  • Lesson set: Evaluate functions, use function notation, and interpret statements that use function notation.  This is a series of five video lessons: Understand functions in terms of inputs and outputs, understand f(x), evaluate a function given an input value, put restrictions on a domain, interpret functions in context.  Aligns with IF-A.2.
  • Lesson set: Understand sequences.  This is a series of seven video lessons: Represent an arithmetic sequence as an explicit rule, represent explicit rules as linear functions, use explicit rules to write sequences recursively, explore the Fibonacci sequence as an example of a recursive sequence, represent a geometric sequence as an explicit rule, represent an explicit rule as an exponential function, write a geometric sequence recursively.  Aligns with IF-A.3.
  • Lesson set: Recognize that sequences are functions and identify their features.  This is a series of four video lessons: Understand sequences as functions, create a recursive formula, create an explicit formula, apply recursive and explicit formulas.  Aligns with IF-A.3.

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

Purplemath: Functions versus Relations, including domain and range.  Use with IF-A.1.

Shodor Interactivate:

  • Vertical Line Test: This test helps to determine if the graph of an equation is that of a function.
  • Graphit: Create graphs of functions and sets of ordered pairs on the same coordinate plane. This is like a graphing calculator with advanced viewing options.

National Library of Virtual Manipulatives: Virtual manipulatives for grades 9-12 (Java required). Scroll down to Algebra (9-12) and select the following:

  • Function Machine: Explore the concept of functions.  Input a value into the machine and observe the output.
  • Grapher: A tool for graphing and exploring functions.

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

  • 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.
  • The Power of Exponentials, Big and Small: "Exponential growth is keenly applicable to a variety of different fields ranging from cell growth in biology, nuclear chain reactions in physics to computational complexity in computer science. In this lesson, through various examples and activities, [the presenters] have tried to compare exponential growth to polynomial growth and to develop an insight about how quickly the number can grow or decay in exponentials."  Aligns with Algebra standards SSE-B.3.c and CED-A.2 and Function standards IF-A.3, IF-C.7.(a, c, e), IF-C.8.b, and LE-A.1.(a, c).

Wolfram Alpha: Domain and Range Calculator Online, powered by Wolfram Alpha.  Students benefit from seeing the use of function notation in the answer and seeing the graph of the function.  Use with IF-A.1.

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

Multiple Choice:

Khan Academy:

Constructed-response:

Khan Academy: See Algebra I: Functions, Algebra 2 (includes several sections on functions): Practice questions with videos.

Math Warehouse: Algebra: Relations, Functions, Domain, Range etc.  The section includes lessons, interactive demonstrations, and practice problems and also includes evaluating functions, inverse of a function (including an applet), vertical line test, one-to-one functions, composition of functions, operations (add, subtract, multiply) functions, and recursive sequences/functions.  Aligns with IF-A.1, IF-A.2, and IF-A.3.

Zweigmedia.com:  Tutorials and examples using fill-in or multiple choice to test understanding.  Learners can also choose to do game versions of some topics.  Use with IF-A.1 and IF-A.2: Functions from the Numerical, Algebraic, and Geometric Viewpoints

Performance tasks:

Illustrative Mathematics: Functions:

Inside Mathematics: MARS task: Printing Tickets: This task aligns with standards IF-A.2, IF-B.4, and IF-C.9

Mathematics Assessment Project: Standards: High School: Functions:

Mathematics Vision Project, Secondary 1 Student Edition:

  • 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 1: Connecting the Dots aligns with function standard IF-A.3.
  • 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.  Complete the following:
  • Task 7: A Water Function aligns with IF-A.1
  • Task 8: To Function or Not to Function aligns with standards IF-A.1 and IF-A.3.
  • Task 4: Water Park; Task 5: Pooling it Together; Task 6: Interpreting Functions; and Task 9: Match That Function align with IF-A.2.

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.

 

Back to top

 

IF-B: Interpret functions that arise in applications in terms of the context

Standards:

Technology-enhanced investigations:

Graphing Calculator is free from Desmos.com and compatible with any computer or tablet.  See Classroom Activities for functions.

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

  • Graphing Quadratic Equations: Aligns with IF-B.4, IF-C.7.a, and IF-C.8.a.
  • Quadratic Equations in Vertex Form: Aligns with IF-B.4, IF-C.7.a, and IF-C.8.a.
  • Quadratic Word Problems Part 1: Students solve two problems involving the motion of projectile objects, modeling the motion using quadratic equations.  Students graph parabolas and use the graphs to answer questions about projectile objects. Students identify the maximum heights of the moving objects and discover how long each object is in the air before hitting the ground.  Aligns with IF-B.4, and IF-C.7.a.
  • Quadratic Word Problems Part 2: Students solve two problems involving the motion of projectile objects, modeling the motion using quadratic equations.  One problem involves profits from a business and the other, water draining from a bathtub.  Aligns with IF-B.4, and IF-C.7.a.

A. Dendane: Analyze Math: Periodic Functions: Applet to explore and understand what it means to be a periodic function.  Aligns with IF-B.4.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

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

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.
  • Tragedy of the Commons: "This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem."  Aligns with Algebra standard SSE-B.3.c and Function standards IF-B.4, IF-C.8.b, LE-A.1.a, LE-A.3, and LE-B.5.
  • Free Fall: "This video lesson is an example of "teaching for understanding" in lieu of providing students with formulas for determining the height of a dropped (or projected) object at any time during its fall. The concept presented here of creating a chart to organize and analyze data collected in a simple experiment is broadly useful."  Aligns with Algebra standard CED-A.2 and Function standards IF-B.6, IF-C.7.a, and LE-A.1.a.

Shodor Interactivate:

  • Graphit: Create graphs of functions and sets of ordered pairs on the same coordinate plane. This is like a graphing calculator with advanced viewing options.
  • Derivate: View the graph and the equation of the line tangent to any function at any point on the function.  This manipulative can be used with IF-B.6.
  • Incline: "This activity demonstrates the relationship between dependent and independent variables by allowing students to explore a real life example. After adjusting the biker's path and the initial velocity, the user can analyze the resulting speed vs. position graph."  Use with IF-B, and Algebra strand CED-A, SSE-A.

NCTM Illuminations: Drug Filtering: In this lesson, students observe a model of exponential decay on how kidneys filter blood and after performing calculations of the amount of a drug left in the body over time, they make and analyze a graphical representation of the exponential function.

Purplemath: Meaning of Slope and Y-Intercept

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

Multiple Choice:

Khan Academy: See Algebra I: Functions: Graphing Functions, Domain and Range, and Analyzing Linear Functions, and Algebra 2 (includes several sections on functions): Practice questions with videos.

Constructed-response:

Khan Academy: See Algebra I: Functions: Practice questions with videos. Range of quadratic functions: Aligns with IF-B.5.

Performance tasks:

Illustrative Mathematics: Functions:

Inside Mathematics: MARS Tasks:

  • Quadratic (2009): Students work with a quadratic function in various forms. This task aligns with function standards IF-B.4, IF-C.7, IF-C.8 and algebra standard REI-B.4.
  • Graphs (2007): Students work with linear and quadratic functions, their graphs and equations.  This task aligns with function standards IF-B.4, IF-B.5, IF-C.7 and IF-C.9.

Mathematics Assessment Project: Standards: High School: Functions: Task 430: Functions and Everyday Situations

Mathematics Vision Project, Secondary 1 Student Edition:

  • Module 4: Linear and Exponential Functions: This module contains 10 classroom tasks.  Task 2: Sorting Out the Change; and Task 3: Where’s My Change? align with function standard IF-B.6.  Other 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.
  • 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 1: Getting Ready for a Pool Party; Task 2: Floating Down the River; Task 3: Features of Functions; Task 4: Water Park; Task 5: Pooling it Together; Task 6: Interpreting Functions; and Task 9: Match That Function align with both IF-B.4 and IF-B.5.

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:

  • Module 3: Polynomial Functions: This module contains 8 classroom tasks.  Task 2 and task 6 align to function standard IF-B.4; task 3 includes standard IF-B.6.
  • Module 4: Rational Functions: This module contains 7 classroom tasks.  Multiple standards are addressed. Standards IF-B.4, IF-B.5, IF-B.6 are among those found in tasks 1-2, 4-7.

 

Back to top

 

IF-C: Analyze functions using different representations

Standards:

 

Technology-enhanced investigations:

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

  • Graphing Quadratic Equations: Aligns with IF-B.4, IF-C.7.a, and IF-C.8.a.
  • Quadratic Functions in Vertex Form: Aligns with IF-B.4, IF-C.7.a, and IF-C.8.a.
  • Quadratic Word Problems Part 1: Students solve two problems involving the motion of projectile objects, modeling the motion using quadratic equations.  Students graph parabolas and use the graphs to answer questions about projectile objects. Students identify the maximum heights of the moving objects and discover how long each object is in the air before hitting the ground.  Aligns with IF-B.4, and IF-C.7.a.
  • Quadratic Word Problems Part 2: Students solve two problems involving the motion of projectile objects, modeling the motion using quadratic equations.  One problem involves profits from a business and the other, water draining from a bathtub.  Aligns with IF-B.4, and IF-C.7.a.
  • Graphing Exponential Equations: Learn about exponential functions by graphing various equations representing exponential growth and decay. Graph these functions by connecting ordered pairs on x-y axes.  Aligns with IF-C.7.e, IF-C.8.b and BF-B.3.
  • Exponential Growth: Learn about exponential growth in real-world situations. Problems involve the application of compound interest and exponential population growth.  Aligns with IF-C.7.e, IF-C.8.b and BF-B.3.
  • Exponential Decay: Learn to use exponential decay equations and interpret graphs of real-world situations.  Problems involve the application of depreciation of an asset and radioactive decay.  Aligns with IF-C.7.e, IF-C.8.b and BF-B.3.

Ohio Resource Center on YouTube: Tutorials for High School Mathematics.  Select the following, which align with IF-C.7:

PhET Interactive Simulations:

  • Graphing Lines.  Explore slope, slope-intercept and point-slope.  Includes a game to test understanding.  Aligns with IF-C.7, IF-C.7.a and LE-A.1.a.
  • Graphing Quadratics. Explore including standard form, vertex form, and focus/directrix.  Aligns with IF-C.7, IF-C.7.a, IF-C.8, IF-C.8.a, and GPE-A.2.

Study.com: Algebra: Matrices and Absolute Value:  This is a series of eight video lessons, six of which are on absolute value.  Each video also includes sample problems and a short quiz.  Absolute value videos begin with the question, What is Absolute Value? and then include how to evaluate, solve, and graph absolute value equations.  Graphing also includes transformations, dilations, and reflections of absolute value equations.  Aligns with IF-C.7.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

Thinking Mathematics:

A. Dendane: Analyze Math:  Select the following, which align with IF-C.7:

  • Exponential Functions: HTML5 applet to explore the graphs of exponential functions.  Learners change parameters a, b, c, d, and B in equations of the form f(x)= a(Bb(x+c))+ d.
  • Logarithm Functions: HTML5 applet to explore graphs of logarithms of the form f(x)= a logB( b( x + c) ) + d by changing parameters a, b, c, d, and base B.
  • Square Root Functions: Interactive applet to explore graphs of square root functions of the form f(x) = a SQRT(x-c) + d.  Learners change parameters a, c, and d and can also determine characteristics of graphs such as domain, range, x intercept, y intercept.  There is access to a tutorial with examples of square root graphs, including square roots of quadratics.
  • Cube Root Functions: Interactive applet to explore graphs of square root functions of the form f(x) = a (x-c)1/3 + d.  Learners change parameters a, c, and d and can also determine characteristics of graphs such as domain, range, x intercept, y intercept.

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

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.
  • Free Fall: "This video lesson is an example of "teaching for understanding" in lieu of providing students with formulas for determining the height of a dropped (or projected) object at any time during its fall. The concept presented here of creating a chart to organize and analyze data collected in a simple experiment is broadly useful."  Aligns with Function standards IF-B.6, IF-C.7.a, LE-A.1.a, and Algebra standard CED-A.2.
  • The Power of Exponentials, Big and Small: "Exponential growth is keenly applicable to a variety of different fields ranging from cell growth in biology, nuclear chain reactions in physics to computational complexity in computer science. In this lesson, through various examples and activities, [the presenters] have tried to compare exponential growth to polynomial growth and to develop an insight about how quickly the number can grow or decay in exponentials."  Aligns with Algebra standards SSE-B.3.c and CED-A.2 and Function standards IF-A.3, IF-C.7.(a, c, e), IF-C.8.b, and LE-A.1.(a, c).
  • 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.
  • The Towers of Hanoi: Experiential Recursive Thinking: "This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones.  The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers."  Aligns with Algebra standard SSE-B.3.c, and Function standards IF-C.8.b and LE-A.1.a.
  • Tragedy of the Commons: "This lesson introduces students to the “Tragedy of the Commons,” an extended metaphor for problems of shared environmental or man-made resources that are overused and eventually depleted. In this metaphor, shared resources are compared to a common grazing pasture, or “commons,” on which any dairy farmer can graze as many cows as he/she wishes. If too many cows are added to the commons, they will overeat the grass in the pasture and the shared resource will become depleted – a disadvantage to everyone. In this lesson, students will be inspired to think about possible solutions to this problem."  Aligns with Algebra standard SSE-B.3.c and Function standards IF-B.4, IF-C.8.b, LE-A.1.a, LE-A.3, and LE-B.5.

Purple Math:

Teacher Tube videos:

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 94 manipulatives addressing IF-C.7, 26 manipulatives for IF-C.8, and 2 manipulatives for IF-C.9.  Among those:

Multiple Choice:

A. Dendane: Analyze Math: Graphs of Rational Functions: Self-check quiz.  Learners are given an equation of a function and select the correct graph of it.  Learners need to write down their answers and then check from answers provided.

CK-12: Analysis of Rational Functions: Finding restrictions, vertical asymptotes, breaks, end behavior, and intercepts: Practice questions with review content and video.  Aligns with IF-C.7.d.

Khan Academy: See Algebra 1: Functions; Algebra 2 (includes several sections on functions): Practice questions with videos.

Constructed-response:

Khan Academy: Practice questions with videos.

Math Warehouse: Converting the Equation of a Parabola: Vertex and Standard Forms: Tutorial and Practice Problems converting vertex form to standard form, and standard form to vertex form using completing the square method.  Aligns with IF-C.8.

Performance tasks:

Illustrative Mathematics: Functions:

Inside Mathematics: MARS Tasks:

  • Functions (pdf) or see all MARS Tasks: Functions & Relations: Course 1 Algebra: Scroll down the page to select Functions: Learners work with graphs of linear and non-linear functions.  This task aligns with function standards IF-C.7, IF-C.9 and LE-A.3.
  • Sorting Functions (pdf) or see all MARS Tasks: Functions & Relations: Course 1 Algebra: Scroll down the page to select Sorting Functions: Learners match graphs of four functions with their equations, tables, and rules.  This task aligns with function standard IF-C.7 and algebra standard CED-A.2.
  • Quadratic (2009): Students work with a quadratic function in various forms.  This task aligns with function standards IF-B.4, IF-C.7, IF-C.8 and algebra standard REI-B.4.
  • Graphs (2007): Students work with linear and quadratic functions, their graphs and equations.  This task aligns with function standards IF-B.4, IF-B.5, IF-C.7 and IF-C.9.

Mathematics Assessment Project: Standards: High School: Functions:

Mathematics Vision Project, Secondary 1 Student Edition:

  • Module 4: Linear and Exponential Functions: This module contains 10 classroom tasks.  Task 5: Getting Down to Business; and Task 9:  Up a Little, Down a Little align with function standard IF-C.7.  Other 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.
  • 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 4: Water Park; Task 5: Pooling it Together; Task 6: Interpreting Functions; and Task 9: Match that Function align with IF-C.7.
  • 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:

Mathematics Vision Project, Secondary 3 Student Edition:

 

Black line

Back to top

Common Core Math: Intro | HS Functions Domain: IF  |  BF  |  LE  |  TF  |