Common Core High School Functions Teaching and Learning Resources
Domain: LE: Linear, Quadratic, and Exponential Models
Select the cluster for resources on this page:
LE-A: Construct and compare linear
and exponential models and solve problems
Standards:
- LE-A.1. Distinguish between situations that
can be modeled with linear functions and with exponential functions.
- LE-A.1.a. Prove that linear functions
grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal
intervals.
- LE-A.1.b. Recognize situations in which
one quantity changes at a constant rate per unit interval
relative to another.
- LE-A.1.c. Recognize situations in which
a quantity grows or decays by a constant percent rate per unit
interval relative to another.
- LE-A.2. Construct linear and exponential
functions, including arithmetic and geometric sequences, given a
graph, a description of a relationship, or two input-output pairs
(include reading these from a table).
- LE-A.3. Observe using graphs and tables
that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or (more generally) as
a polynomial function.
- LE-A.4. For exponential models, express as
a logarithm the solution to abct = d where
a, c, and d are numbers and the base b
is 2, 10, or e; evaluate the logarithm using technology.
Technology-enhanced investigations:
Mathwords.com: Math Dictionary:
Key vocabulary for this domain. Use
with LE-A, LE-B:
CK-12: Algebra: Concepts, videos, and practice problems for learners
to complete:
Interactive Mathematics:
Exponential and Logarithmic Functions delves into definitions,
graphs, logarithm laws, logs in base 10, natural logs, exponential and
log equations, also includes applications and examples. Use with LE-A.
Ohio Resource Center on YouTube: Tutorials for High School
Mathematics: Logarithms.
Definition of and computation with both common logarithms and base e
natural logarithms.
Imagine Learning Classroom (formerly LearnZillion) subscription
needed:
- Lesson set:
Distinguish
between linear functions and exponential functions: This is a
series of seven video lessons that also include guided practice on this
topic. Videos demonstrate how to model exponential growth
situations with 2 variables; how to determine the end behavior of a
polynomial or exponential function, and the greater of two polynomial
functions; how to compare polynomial and exponential growth; name the
3rd ordered pair belonging to a linear or exponential function. Aligns
with LE-A.1.
- Lesson set:
Prove how linear
functions grow (and exponential functions): This is a series
of two video lessons that also include guided practice on this topic of
how linear and exponential functions grow. Aligns with LE-A.1.a.
- Lesson set:
Prove that,
over equal intervals, linear functions grow by equal differences
and exponential functions grow by equal factors. Five
video lessons: Identify linear functions by linking equal
differences to equal intervals, graph an exponential function,
understand the rate of change in exponential functions,
distinguish between linear and exponential functions by
examining intervals and using tables. Aligns with
LE-A.1.a.
- Lesson set:
Recognize when
one quantity changes at a constant rate per unit interval of
another. Four video lessons: Compare relationships
between quantities using linear models, interpret d=rt by
graphing its linear function, find average rates of change using
data in tables, calculate average rate of change using d=rt.
Aligns with LE-A.1.b.
- Lesson set:
Understand exponential growth and decay: This is a series
of four video lessons that also include guided practice on this topic.
Videos demonstrate writing exponential growth and decay functions,
determining which of two growing quantities will exceed the other at a
given point in time. Aligns with LE-A.1.c.
- Lesson set:
Recognize when
one quantity grows or decays by a constant percent rate per unit
interval of another. Four video lessons: Graph
depreciation and half-life of an element using an exponential
decay model, graph the spread of behaviors using an exponential
growth model, compare exponential growth models using graphs.
Aligns with LE-A.1.c.
- Lesson set:
Express
solutions to exponential models as logarithms; evaluate using
technology. Five video lessons: Evaluate logarithms
using exponentials, solve for time in a continuously compounding
function, solve real-world problems using properties of
logarithms, use an exponential formula to solve a population
problem, solve a half-life model. Aligns with LE-A.4.
MIT BLOSSOMS: Video lessons with additional teacher
and learner resources. Descriptions are from the video summaries.
- 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.
- 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).
- 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.
PhET Interactive Simulations:
Graphing Lines. Aligns with LE-A.1.a.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives. Note: Within the
Wolfram Demonstration Project are
9 manipulatives addressing LE-A.1,
19 manipulatives for LE-A.2,
1 manipulative for LE-A.3, and
3 manipulatives for LE-A.4. Among those:
YouTube.com: Introduction to Linear Functions video
illustrates the concept and how linear functions represent a
constant rate of change.
Multiple Choice:
Khan Academy: Practice questions with videos.
Constructed-response:
Khan Academy:
HSF-F-LE
includes HSF-LE.A.1 through HSF-LE.A.4 Common Core aligned problems.
Also see the following practice questions with videos.
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 LE-A.1
and LE-A.2:
-
Functions and Models: Part A: Cost, revenue, and profit models;
Part B: Demand, supply, and time-change models
-
Linear functions and models: In three parts, learners get the
basics of slope and intercepts; finding the equation of a line; then
applications of linear models.
Performance tasks:
Illustrative Mathematics:
Functions:
Inside Mathematics: MARS Task:
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.
Mathematics Assessment Project:
Standards: High School: Functions:
Mathematics Vision Project,
Secondary
Mathematics 1 Student Edition:
Task 2: Growing Dots;
Task 3: Growing, Growing Dots;
Task 4: Scott’s
Workout;
Task 5: Don’t Break the Chain;
Task 6: Something to Chew
On;
Task 7: Chew On This; and
Task 8: What Comes Next? What Comes
Later? align to both standards LE-A.1 and LE-A.2.
Task 9: What Does It Mean?; and Task 10: Geometric
Meanies; Task 11: I Know … What Do You Know? align to standard LE-A.2.
Mathematics Vision Project,
Secondary
Mathematics 1 Student Edition:
Task 2: Shh! Please be discreet aligns to standards LE-A.1 and LE-A.2.
Task 3: Linear, Exponential or Neither aligns to standard LE-A.3.
Task 4: Getting Down to Business aligns to standards LE-A.2 and LE-A.3.
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LE-B: Interpret expressions for
functions in terms of the situation they model
Standards:
- LE-B.5. Interpret the parameters in a
linear or exponential function in terms of a context.
Technology-enhanced investigations:
A. Dendane: Analyze Math:
Exponential Functions: HTML5 applet to
explore the properties of 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.
MIT BLOSSOMS: Video lesson with additional teacher and learner
resources. Description is from the video summary.
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.
PhET Interactive Simulations:
Projectile Motion. Aligns with LE-B.5.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulative:
Final number of bacteria cells. Aligns with LE-B.5.
Multiple Choice:
Constructed-response:
Khan Academy:
HSF-F-LE
includes HSF-LE.B.5 Common Core aligned problems.
Also see the practice questions with videos in
Construct exponential models: Aligns with LE-A.1 and LE-B.5.
Performance tasks:
Back to top
Common Core Math:
Intro | HS Functions Domain:
IF | BF | LE | TF |
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