Common Core High School Functions Teaching and Learning Resources
Domain: IF: Interpreting Functions
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
IF-A: Interpret the structure of expressions.
Standards:
- IF-A.1. Understand that a function from one set
(called the domain) to another set (called the range) assigns to each
element of the domain exactly one element of the range. If f is a
function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The
graph of f is the graph of the equation y = f(x).
- IF-A.2. Use function notation, evaluate
functions for inputs in their domains, and interpret statements that use
function notation in terms of a context.
- IF-A.3. Recognize that sequences are functions,
sometimes defined recursively, whose domain is a subset of the integers.
For example, the Fibonacci sequence is defined recursively by f(0) =
f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
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:
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:
-
US Common Core State Standard Math HSF-IF.A.1:
24 manipulatives. Among those: Expression machine, finding
the linear function rule, iterating linear functions, rational
function graphs, polynomial graph generator, composing functions
using dynagraphs, and more.
-
US Common Core State Standard Math HSF-IF.A.2:
2 manipulatives: Expression machine, composing functions using
dynagraphs, both of which also align with IF-A.1.
-
US Common Core State Standard Math HSF-IF.A.3:
16 manipulatives. Among those: Fibonacci numbers and
the Golden Ratio, pine cone, and more.
Multiple Choice:
Constructed-response:
Khan Academy: HSF-F-IF
includes HSF-IF.A.1 through HSF-IF.A.3 Common Core aligned problems.
Also 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 tasks 4-7 per the following:
- Task 4: Water Park; Task 5: Pooling it Together; Task 6:
Interpreting Functions; and Task 9: Match That Function align with IF-A.2.
- 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.
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:
- IF-B.4. For a function that models a
relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship.
Key features include: intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and
minimums; symmetries; end behavior; and periodicity.
- IF-B.5. Relate the domain of a function to
its graph and, where applicable, to the quantitative relationship it
describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the
positive integers would be an appropriate domain for the function.
- IF-B.6. Calculate and interpret the average
rate of change of a function (presented symbolically or as a table)
over a specified interval. Estimate the rate of change from a graph.
Technology-enhanced investigations:
Graphing Calculator is free
from Desmos.com and compatible with any computer or tablet. See
Classroom Activities for
functions.
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 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:
- Lesson set:
Graph linear
functions and show intercepts, maxima, and minima.
This series has five video lessons: Find the slope of a linear
function, graph a line given its slope and a point, graph a line
in slope-intercept form align with IF-B.4. Graph a function
not in slope-intercept form, graph a line using x and y
intercepts align with IF-C.7.
- Lesson set:
Graph
quadratic functions and show intercepts, maxima, and minima.
This series of four video lessons continues this concept: Find
axis of symmetry and vertex aligns with IF-B.4. Graph
quadratic equations, graphically solve quadratic functions using
the intercepts of the graph, graph quadratic equations using a
graphing calculator align with IF-C.7.
- Lesson set:
Relate the
domain of a function to its graph. This series has
three video lessons: Understand domain restriction in linear
functions, exponential functions, and quadratic functions in
terms of real-world context. Aligns with IF-B.5.
- Lesson set:
Relate the
domain of a function to its graph, accounting for asymptotes and
restricted domains. This series has five video
lessons: Write a rational function, determine the domain of a
function, identify a vertical asymptote, identify holes in a
function, identify a discrete domain. Aligns with IF-B.5.
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:
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:
- IF-C.7. Graph functions expressed
symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.
- IF-C.7.a. Graph linear and quadratic
functions and show intercepts, maxima, and minima.
- IF-C.7.b. Graph square root, cube root,
and piecewise-defined functions, including step functions and
absolute value functions.
- IF-C.7.c. Graph polynomial functions,
identifying zeros when suitable factorizations are available,
and showing end behavior.
- (+) IF-C.7.d. Graph rational functions,
identifying zeros and asymptotes when suitable factorizations
are available, and showing end behavior.
- IF-C.7.e. Graph exponential and
logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.
- IF-C.8. Write a function defined by an
expression in different but equivalent forms to reveal and explain
different properties of the function.
- IF-C.8.a. Use the process of factoring
and completing the square in a quadratic function to show zeros,
extreme values, and symmetry of the graph, and interpret these
in terms of a context.
- IF-C.8.b. Use the properties of
exponents to interpret expressions for exponential functions.
For example, identify percent rate of change in functions such
as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and
classify them as representing exponential growth or decay.
- IF-C.9. Compare properties of two functions
each represented in a different way (algebraically, graphically,
numerically in tables, or by verbal descriptions). For example,
given a graph of one quadratic function and an algebraic expression
for another, say which has the larger maximum.
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:
- Lesson set:
Graph linear
functions and show intercepts, maxima, and minima.
This series has five video lessons: Find the slope of a linear
function, graph a line given its slope and a point, graph a line
in slope-intercept form align with IF-B.4. Graph a function
not in slope-intercept form, graph a line using x and y
intercepts align with IF-C.7.
- Lesson set:
Graph
quadratic functions and show intercepts, maxima, and minima.
This series of four video lessons continues this concept: Find
axis of symmetry and vertex aligns with IF-B.4. Graph
quadratic equations, graphically solve quadratic functions using
the intercepts of the graph, graph quadratic equations using a
graphing calculator align with IF-C.7.
- Lesson set:
Graph square
root, cube root, and piecewise-defined functions, including step
functions and absolute value functions. This series
has five video lessons, one per each function noted.
Aligns with IF-C.7.b.
- Lesson set (additional):
Graph square
root, cube root, and piecewise-defined functions, including step
functions and absolute value functions. This series
has five additional video lessons to this concept: Find domain
and range of a function using a graph, graph square root
functions using transformations, and graph cube, piece-wise, and
absolute value functions. Aligns with IF-C.7.b.
- Lesson set:
Graph polynomial functions, identifying
zeros and showing end behavior. Five video lessons:
Determine end behavior, write equation of a polynomial using its
x-intercepts, graph a polynomial by finding its x-intercepts,
graph polynomials by finding zeroes through factoring, find the
roots of a polynomial using a graphing calculator. Aligns
with IF-C.7.c.
- Lesson set:
Graph polynomial functions, identifying
zeros when suitable factorizations are available, and showing end behavior. Five additional video
lessons: Write a polynomial function, identify the impact of
context on functions, identify polynomial extrema, analyze the
physical context of a polynomial function, determine end
behavior. Aligns with IF-C.7.c.
- Lesson set:
Graph
exponential and logarithmic functions. This series has
five video lessons: Graph exponential growth, exponential decay,
and logarithmic functions; shift exponential and logarithmic
functions; stretch and reflect exponential and logarithmic
functions. Aligns with IF-C.7.e.
- Lesson set:
Factor and
complete the square in order to show zeros, extreme values, and
symmetry in quadratic functions. Five video lessons on
these concepts. Aligns with IF-C.8.a.
- Lesson set:
Use the
properties of exponents to interpret expressions and functions.
Five video lessons: Determine population using an exponential
model, determine the age of an object by modeling exponential
decay, find compound interest using an investment model, compute
depreciation, determine percent rate of change. Aligns
with IF-C.8.b.
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:
- Graphing
Overview: straight lines; absolute values & quadratics;
polynomials, radicals, rationals, & piecewise.
-
Graphing Linear Equations
-
Graphing Quadratic Functions
- Completing the Square: Finding the vertex and deriving
the vertex formula.
-
Polynomial Graphs: end behavior; zeroes & their
multiplicities; "flexing"; turning & "bumps"; and graphing
-
Asymptotes: vertical, horizontal, slant with examples
-
Graphing Rational Functions: introduction, examples, and
special cases with "holes"
-
Graphing Exponential Functions: introductory concepts,
step by step instruction, and worked examples
-
Graphing Logarithmic Functions: concepts and worked
examples
-
Trigonometric functions and their graphs: basic graphs
-
Graphing trigonometric functions: introduction, examples
with amplitude and vertical shift, and example with phase shift.
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:
-
Quadratic Equation: This Demonstration allows you to move the sliders or select
coefficients in the quadratic (standard form) to plot the quadratic equation
together with its zeros and critical points. Aligns with
IF-C.7.
-
Quadratic in Vertex Form: Learners manipulate the parameters
in a quadratic in symmetry-vertex form to see the affect on the
graph (dilation, translation horizontally and/or vertically).
Aligns with IF-C.7, IF-C.8.
-
Cubic
Polynomial: Change the coefficients in the cubic equation
and see the result on the graph, including zeroes, critical points, and points of
inflection (i.e., where the graph changes from concave up to
concave down and vice versa). Aligns with IF-C.7.
-
Piecewise Functions: Aligns with IF-C.7.
-
Visualizing square root and absolute value: Aligns with
IF-C.7.
-
Simple Rational Functions: This manipulative shows a
quotient of two quadratic polynomials. By manipulating
numeric values, learners see changes in the graph, including
asymptotes and possible "holes." Aligns with
IF-C.7.
-
Rational Function Graphs: The demonstration shows the graph
of a rational function. Choose the numerator and
denominator to create a function p(x)/q(x). You
can also manipulate the domain and range to see key features.
Aligns with IF-C.7.
-
Graphing functions with four parameters: Experiment with
linear, quadratic, cubic, square root, absolute value,
exponential, logarithm, sine, and cosine graphs. Aligns
with IF-C.7.
-
Quadratic Equation with Factored Form: Aligns with IF-C.8.
-
Completing the square of a quadratic: Aligns with IF-C.8.
-
The greatest fenced area along a barn: Aligns with
IF-C.9.
-
Global extrema on an interval: Aligns with IF-C.9.
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: HSF-F-IF
includes HSF-IF.C.7 through HSF-IF.C.9 Common Core aligned problems.
Also see 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:
-
Module 2: Logarithmic Functions: This module contains 5
classroom tasks. Tasks 3-4 align with IF-C.8.
-
Module 3: Polynomial Functions: This module contains 8
classroom tasks. Task 3 aligns to function standard
IF-B.6, IF-C.7, IF-C.9; task 6 aligns with IF-B.4, IF-C.7; tasks
7-8 include standard IF-C.7.
-
Module 4: Rational Functions: This module contains 7
classroom tasks. Multiple standards are addressed.
Standards IF-C.7 is among those found in
tasks 1-2, 4-7.
- Module
5: Modeling with Geometry: This module contains 8 classroom tasks.
Multiple standards are addressed. Standard IF-C.8 is found in task
8.
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Common Core Math:
Intro | HS Functions Domain: IF | BF | LE | TF |
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