Common Core High School Functions Teaching and Learning Resources
Domain: BF: Building Functions
Select the cluster for resources on this page:
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:
PhET Interactive Simulations:
Function Builder. Aligns with BF-A.1 and BF-B.4.
Study.com: Algebra:
Functions: 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.
Imagine Learning Classroom (formerly LearnZillion) subscription
needed:
- 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.
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.
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:
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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.
YouTube.com: Recognizing
Patterns: Video to review arithmetic, geometric, powers, inductive reasoning
patterns. Use with BF-A.2.
Multiple Choice:
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:
HSF-F-BF
includes HSF-BF.A.1 and HSF-BF.A.2 Common Core aligned problems.
Also see the following practice questions with videos:
MATHguide.com:
-
Arithmetic Sequences and Series: 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.
Instructional videos are included. 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.
Performance tasks:
Illustrative Mathematics:
Functions: Following align with BF-A.1:
Inside Mathematics: MARS Tasks:
-
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:
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, 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.
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.
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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.
PhET Interactive Simulations:
Function Builder. Aligns with BF-A.1 and BF-B.4.
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.
Imagine Learning Classroom (formerly LearnZillion) subscription
needed:
- 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.
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:
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:
HSF-F-BF
includes HSF-BF.B.3 through HSF-BF.B.5 Common Core aligned problems.
Also see the following practice questions with videos.
Zweigmedia.com: Algebra Review Tutorials and examples using
multiple choice and fill-in to test understanding. Learners can
also choose to do game versions of some topics. Use with BF-B.5:
Logarithms.
Performance tasks:
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:
Inside Mathematics MARS Tasks:
-
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 Assessment Project:
Standards: High School: Functions:
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, and student activity sheet. Aligns with BF-B.3,
TF-B.5 and mathematical practice standards 1 and 7.
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Common Core Math:
Intro | HS Functions Domain:
IF | BF | LE | TF |
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