Download 2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence

2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence
Unifying Concept: Polynomial, Rational, and Radical Functions
Mathematics Content Focus:
Students will determine the best method to simplify and
New book Chapters 3, 5
solve rational equations, and will justify each step used
Old book Chapters 6, 7, 9
in their solution(s).
Students will also apply the meaning of the zeroes, yintercept and asymptotes of a rational function to the
context a real world problem. Students will sketch a
graph of the radical function using the key features of
the graph to include: zeroes, end behavior, and
asymptotes, and be able to justify each step used to
solve an equation.
Rational Functions
Mathematical Practice Focus:
-Operations
Mathematically Proficient Students…
-Solving Rational Equations
1. Make sense of problems and persevere in solving
-Graphing
them.
2. Reason abstractly and quantitatively.
Radical Functions
3. Construct viable arguments and critique the
-Properties of Exponents
reasoning of others.
-Solving Radicals Equations
4. Model with mathematics.
-Graphing
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
Target Standards
A-SSE.A.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2
(polynomial
– (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
only)
A-SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the
quantity represented by the expression. ★
A-REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
A-REI.D.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y =
(polynomial
g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g.,
and rational
using technology to graph the functions, make tables of values, or find successive
only)
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions. ★
F-IF.B.4
For a function that models a relationship between two quantities, interpret key features of graphs
(polynomial
and tables in terms of the quantities, and sketch graphs showing key features given a verbal
only)
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-IF.C.7c
Graph polynomial functions, identifying zeros when suitable factorizations are available, and
showing end behavior.
N-RN.A.1
Explain how the definition of the meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for a notation for radicals in terms of
rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 =
5(1/3)3 to hold, so (51/3)3 must equal 5.
N-RN.A.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
A-APR.B.2
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder
Polynomials
-Operations
-Factoring
-Graphing/Transformations
-Remainder Theorem
-Solving equations
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2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence
A-APR.B.3
on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
A-CED.A.1
(rational
only)
Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential
functions.
Quarter Major Clusters
Arizona considers Major Clusters as groups of related standards that require greater emphasis than some of the
others due to the depth of the ideas and the time it takes to master these groups of related standards.
A-SSE.A Interpret the structure of expressions
Essential Concepts
Essential Questions
 Complicated expressions can be interpreted by viewing
 How does using the structure of an expression
parts of the expression as single entities.
help to simplify the expression?
 Structure within expressions can be identified and used
 Why would you want to simplify an expression?
to factor or simplify the expression.
A-SSE.B Write expressions in equivalent forms to solve problems
Essential Concepts
Essential Questions
 The solutions of quadratic equations are the x-intercepts of
the parabola or zeros of quadratic functions.
 Factoring methods and the method of completing the square
reveal attributes of the graphs of quadratic functions.
 Factoring a quadratic reveals the zeros of the function.
 Completing the square in a quadratic equation reveals the
maximum or minimum value of the function.
 Properties of exponents are used to transform expressions
for exponential functions.




What are the solutions to a quadratic equation and
how do they relate to the graph?
What attributes of the graph will factoring and
completing the square reveal about a quadratic
function?
How are properties of exponents used to transform
expressions for exponential functions?
Why would you want to transform an expression for
an exponential function?
A-REI.A Understand solving equations as a process of reasoning and explain the reasoning
Essential Concepts
Essential Questions
 Simple rational and radical equations can have
 Give an example of a simple rational or radical
extraneous solutions.
equation that has an extraneous solution and
explain why it is an extraneous solution.
A-REI.D Represent and solve equations and inequalities graphically
Essential Concepts
Essential Questions
 Solving a system of equations algebraically yields an
 Why are the x-coordinates of the points where
exact solution; solving by graphing or by comparing
the graphs of the equations y = f(x) and y = g(x)
tables of values yields an approximate solution.
intersect equal to the solutions of the equation
f(x) = g(x)?
 The x-coordinates of the points where the graphs of the
 Why does graphing or using a table give
equations y = f(x) and y = g(x) intersect are the
solutions of the equation f(x) = g(x).
approximate solutions?
 In what situations would you want an exact
solution rather than an approximate solution or
vice versa?
F-IF.B Interpret functions that arise in applications in terms of the context
Essential Concepts:
Essential Questions:
 Key features of a graph or table may include intercepts;
 How can you describe the shape of a graph?
intervals in which the function is increasing, decreasing
 How can you relate the shape of a graph to the
or constant; intervals in which the function is positive,
meaning of the relationship it represents?
negative or zero; symmetry; maxima; minima; and end
 How would you determine the appropriate
behavior.
domain for a function describing a real-world
 Given a verbal description of a relationship that can be
situation?
modeled by a function, a table or graph can be
constructed and used to interpret key features of that
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2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence
function.
 The intervals over which a function is increasing,
decreasing or constant, positive, negative or zero are
subsets of the function’s domain.
 Graphs can be described in terms of their relative
maxima and minima; symmetries; end behavior; and
periodicity.
F-IF.C Analyze functions using different representations
Essential Concepts:
Essential Questions:
 Key features of a graph or table may include intercepts;  How can you compare properties of two
intervals in which the function is increasing, decreasing
functions if they are represented in different
or constant; intervals in which the function is positive,
ways?
negative or zero; symmetry; maxima; minima; end
 How do different forms of a function help you to
behavior; asymptotes; domain; range and periodicity.
identify key features?
 The graph of a trigonometric function shows period,
 How do you determine which type of function
amplitude, midline and asymptotes.
best models a given situation?
 The graph of a polynomial function shows zeros and
end behavior.
 A function can be represented algebraically, graphically,
numerically in tables, or by verbal descriptions.
N-RN.A Extend the properties of exponents to rational exponents
Essential Concepts:
Essential Questions:
 Rational exponents are exponents that are fractions.
 How do you use properties of rational
exponents to simplify and create equivalent
 Properties of integer exponents extend to properties of
forms of numerical expressions?
rational exponents.

Why are rational exponents and radicals
 Properties of rational exponents are used to simplify
related to each other?
and create equivalent forms of numerical expressions.
 Given an expression with a rational exponent,
 Rational exponents can be written as radicals, and
how do you write the equivalent radical
radicals can be written as rational exponents.
expression?
A-APR.B Understand the relationship between zeros and factors of polynomials
Essential Concepts:
Essential Questions:
 The Remainder theorem says that if a polynomial p(x) is  How can you determine whether x – a is a
divided by x  a , then the remainder is the value of the
factor of a polynomial p(x)? Why does this
work?
polynomial evaluated at a.
 How do you determine how many zeros a
 Saying that x – a is a factor of a polynomial p(x) is
polynomial function will have?
equivalent to saying that p(a) = 0, by the zero property
of multiplication.
 What information do you need to sketch a
rough graph of a polynomial function?
 Any polynomial of degree n can be factored into n
binomials of the form x – c, with possibly complex
 How are the zeros of a polynomial related to its
values for c.
graph?
 If p(a) = 0, then a is a zero of p.
 Extension: Why is it true that p(x)/(x – a) has a
remainder of p(a)?
 If a is a zero of p, then a is an x-intercept of the graph of
y = p(x).
 The values and multiplicity of the zeros of a polynomial,
along with the end behavior, can be used to sketch a
graph of the function defined by the polynomial.
A-CED.A Create equations that describe numbers or relationships
Essential Concepts:
Essential Questions:
 Equations and inequalities can be created to represent
 How do you translate from real-world situations
and solve real-world and mathematical problems.
into mathematical equations and inequalities?
 Relationships between two quantities can be
 How do you determine if a situation is best
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2015-2016 Algebra II 2nd Quarter Mathematics Scope and Sequence
represented through the creation of equations in two
variables and graphed on coordinate axes with labels
and scales.
 Solutions are viable or not in different situations
depending upon the constraints of the given context.
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represented by an equation, an inequality, a
system of equations or a system of inequalities?
 Why would you want to create an equation or
inequality to represent a real-world problem?
 How are graphs of equations and inequalities
similar and different?
 How do you determine if a given point is a viable
solution to a system of equations or inequalities,
both on a graph and using the equations?
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