Wentzville School District
Algebra II
Unit 6 - Rational Functions
Unit Title: Rational Functions
Course: Algebra II
Brief Summary of Unit: In this unit, students will graph and analyze the properties of rational (reciprocal) functions.
Students will simplify rational expressions and solve rational equations.
Textbook Correlation: Glencoe Algebra II Chapter 8 Sections 1-4 and 6, however, section 6 will need to be supplemented
with more problems similar to page 576: 1-4 and some rational proportions.
Time Frame: 3-4 weeks
WSD Overarching Essential Question
Students will consider…
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How do I use the language of math (i.e. symbols,
words) to make sense of/solve a problem?
How does the math I am learning in the classroom
relate to the real-world?
What does a good problem solver do?
What should I do if I get stuck solving a problem?
How do I effectively communicate about math
with others in verbal form? In written form?
How do I explain my thinking to others, in written
form? In verbal form?
How do I construct an effective (mathematical)
argument?
How reliable are predictions?
Why are patterns important to discover, use, and
generalize in math?
How do I create a mathematical model?
How do I decide which is the best mathematical
tool to use to solve a problem?
How do I effectively represent quantities and
relationships through mathematical notation?
How accurate do I need to be?
WSD Overarching Enduring Understandings
Students will understand that…
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Mathematical skills and understandings are used
to solve real-world problems.
Problem solvers examine and critique arguments
of others to determine validity.
Mathematical models can be used to interpret and
predict the behavior of real world phenomena.
Recognizing the predictable patterns in
mathematics allows the creation of functional
relationships.
Varieties of mathematical tools are used to
analyze and solve problems and explore concepts.
Estimating the answer to a problem helps predict
and evaluate the reasonableness of a solution.
Clear and precise notation and mathematical
vocabulary enables effective communication and
comprehension.
Level of accuracy is determined based on the
context/situation.
Using prior knowledge of mathematical ideas can
help discover more efficient problem solving
strategies.
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When is estimating the best solution to a
problem?
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Concrete understandings in math lead to more
abstract understanding of math.
Transfer
Students will be able to independently use their learning to…
know that operations and understandings developed in arithmetic can be extended to algebraic systems.
know that rational functions and their graphs can be used to model real world phenomena.
Meaning
Essential Questions
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How do I know when a rational expression is
simplified?
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What strategies would you use to solve a rational
equation?
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Why is a solution extraneous?
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Under what circumstances would you obtain an
extraneous solution?
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How does the placement of the vertical asymptote
correspond to the equation and the domain?
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How is a transformed function related to its parent
function (reciprocal function)?
Understandings
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Rational equations have solutions, some of which
may be extraneous.
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Rational expressions can be rewritten in different
forms to analyze their properties and make them
more useful to perform operations upon.
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Working with algebraic rational expressions and
equations follows the same properties/rules as
numerical rational expressions and equations.
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There are real world situations that can be
modeled using rational functions.
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Horizontal asymptotes help to describe end
behavior of a function.
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Rational expressions can be simplified in a more
meaningful context.
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Transformations of functions can be an effective
tool to graph and describe functions efficiently.
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Each component of a rational function provides
information about the corresponding graph, and
vice versa.
Acquisition
Key Knowledge
Key Skills
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Domain
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Simplify rational expressions
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extraneous solutions
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Rational expressions and equations
Rewrite rational expressions (factoring, multiplying
by a common denominator, etc.) so that two or
more rational expressions can be added,
subtracted, multiplied, divided, etc.
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reciprocal function
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horizontal asymptote
Graph reciprocal functions using transformations
with and without technology.
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vertical asymptote
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Interpret the meaning of the key features of the
rational function from a graph or table in relation
to the problem. (Key features may include
intercepts, asymptotes (horizontal and vertical)
and end behavior.)
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Use technology to discover the effects of
parameter changes to reciprocal functions.
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Given a rational function, determine the practical
domain of the function as it relates to the
numerical relationship it describes as well as how
it relates to its graph.
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Solve simple rational equations.
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Identify extraneous solutions.
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Solve real-world problems involving rational
functions using multiple strategies.
Standards Alignment
MISSOURI LEARNING STANDARDS
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions
may arise.
A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x),
b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or,
for the more complicated examples, a computer algebra system.
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are
the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions,
make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.★
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
F.IF.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.★
F.BF.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.
A.SSE.1 Interpret expressions that represent a quantity in terms of its
context.★
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret
P(1+r)n as the product of P and a factor not depending on P.
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Show Me-Standards
Goal 1: 1, 4, 5, 6, 7, 8
Goal 2: 2, 3, 7
Goal 3: 1, 2, 3, 4, 5, 6, 7, 8
Goal 4: 1, 4, 5, 6
Mathematics: 1, 4, 5