ASSE: AlgebraSeeing Structure in Expressions

A. Interpret the structure of
expressions

ASSE.A.2

Use the structure of an
expression to identify ways to rewrite it. For example,
see
as
, thus recognizing it as a
difference of squares that can be factored as
.
*I
can identify patterns of factoring.
*I
can factor a polynomial or rational expression.
*I
can classify expressions by method of factoring.

AAPR:
Arithmetic with Polynomials and Rational Expressions

B. Understand the
relationship between zeros and factors of polynomials

AAPR.B.2

Know and apply
the Remainder Theorem: For a polynomial p(x) and
a number a, the remainder on division by x– a is p(a),
so p(a) = 0 if and only if (x – a) is a factor of p(x).
*I
can define the remainder theorem.
*I
can use the remainder theorem to show the relationship between a factor and
a zero.

AAPR.B.3

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.
*I
can factor polynomials using any method.
*I
can sketch graphs of polynomials using zeroes and a sign chart.

C. Use polynomial
identities to solve problems

AAPR.C.4

Prove polynomial
identities and use them to describe numerical relationships. For example, the polynomial identity
can be used to generate Pythagorean triples.
*I
can prove polynomial identities.

D. Rewrite
rational expressions

AAPR.D.6

Rewrite simple rational expressions in different forms; write
in the form
, where
,
,
, and
are
polynomials with the degree of
less than the degree of
, using inspection, long division, or, for the more complicated
examples, a computer algebra system.
*I
can rewrite rational expressions using inspection or by long or synthetic
division.

AREI:
Reasoning with Equations and Inequalities

A. Understand
solving equations as a process of reasoning and explain the reasoning

AREI.A.1

Explain each step in solving a simple equation as following from
the equality of numbers asserted at the previous step, starting from the
assumption that the original equation has a solution. Construct a viable
argument to justify a solution method.
*I
can demonstrate that solving an equation means that the equation remains
balanced during each step.
*I
can recall the properties of equality.
*I
can explain why, when solving equations, it is assumed that the original
equation is equal.
*I
can determine if an equation has a solution.
*I
can choose an appropriate method for solving the equation.
*I
can justify solution(s) to equations by explaining each step in solving a
simple equation using the properties of equality, beginning with the
assumption that the original equation is equal.
*I
can construct a mathematically viable argument justifying a given, or
selfgenerated, solution method.

AREI.A.2

Solve simple rational and radical equations in one variable, and
give examples showing how extraneous solutions may arise.
*I
can determine the domain of a rational function.
*I
can determine the domain of a radical function.
*I
can solve radical equations in one variable.
*I
can solve rational equations in one variable.
*I
can give examples showing how extraneous solutions may arise when solving
rational and radical equations.

B. Solve
equations and inequalities in one variable

AREI.B.4b

Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection
(e.g., for
), taking square roots, completing the
square, the quadratic formula and factoring, as appropriate to the initial
form of the equation. Recognize when the quadratic formula gives complex
solutions and write them as
for real
numbers
and
.
*I
can solve quadratic equations by inspection (e.g., for
), taking square roots, completing
the square, the quadratic formula and factoring.
*I
can determine appropriate strategies (see first knowledge target listed) to
solve problems involving quadratic equations, as appropriate to the initial
form of the equation.
*I
can recognize when the quadratic formula gives complex solutions.

C. Solve systems
of equations

AREI.C.6

Solve systems of linear equations
exactly and approximately (e.g., with graphs), limited to systems of at
most three equations and three variables. With graphic solutions, systems
are limited to two variables.
*I
can solve systems of linear equations by any method.
*I
can justify the method used to solve systems of linear equations exactly
and approximately.
*I
can graph systems of two linear equations.

AREI.C.7

Solve
a simple system consisting of a linear equation and a quadratic equation in
two variables algebraically and graphically. For example, find the points of intersection between the line
and the circle
.
*I
can transform a simple system consisting of a linear equation and quadratic
equation in 2 variables so that a solution can be found algebraically and
graphically.
*I
can explain the correspondence between the algebraic and graphical
solutions to a simple system consisting of a linear equation and a
quadratic equation in 2 variables.

FBF:
Building Functions

B. Build new functions from existing
functions

FBF.B.3

Identify the effect on the graph of replacing
by
,
,
, and
for specific
values of
(both
positive and negative); find the value of
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.
*I
can perform transformation on functions which may involve simple radical,
rational, polynomial, exponential, logarithmic. (Simple
exponential and logarithmic functions were introduced in Algebra I).
*I
can identify the effect a single transformation will have on the function
(symbolic or graphic).
*I
can use technology to identify effects of single transformations on graphs
of functions.
*I
can describe the differences and similarities between a parent function and
the transformed function.
*I
can find the value of
, given the graphs of a parent function, f(x), and
the transformed function:
,
,
, and
.
*I
can recognize even and odd functions from their graphs and from their
equations.
*I
can identify transformations of a function on a graph.
*I
can describe the effects of transformations on parent functions.

FIF: Interpreting Functions

C. Analyze
functions using different representations

FIF.C.7c

Graph functions expressed symbolically and
show key features of the graph, by hand in simple cases and using
technology for more complicated cases.★
c. Graph
polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
*I
can graph polynomial, rational, and radical functions accurately.

NQ: Quantities

A. Reason
quantitatively and use units to solve problems.

NQ.A.2

Define
appropriate quantities for the purpose of descriptive modeling.
*I
can define descriptive modeling
*I
can determine appropriate quantities for the purpose of descriptive
modeling

NCN:
The Complex Number System

A. Perform
arithmetic operations with complex numbers

NCN.A.1

Know there is a
complex number
such that
, and every complex number has the form
with
and
real.
*I can define
as the square
root of
−1 or
.
*I can define
complex numbers.
*I can write
complex numbers in the form
with
and
being real numbers.

NCN.A.2

Use the relation
and the
commutative, associative, and distributive properties to add, subtract, and
multiply complex numbers.
*I
can recognize that the commutative, associative, and distributive
properties extend to the set of complex numbers over the operations of addition
and multiplication.
*I can use the
relation
to
simplify.

C. Use complex
numbers in polynomial identities and equations.

NCN.C.7

Solve quadratic equations with real coefficients that have complex
solutions.
*I can solve
quadratic equations that have complex solutions.

ACED:
Creating Equations

A. Create
equations that describe numbers or relationships

ACED.A.1

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.
*I can solve
equations specified in this standard in one variable.
*I can solve
inequalities specified in this standard in one variable.
*I can describe
the relationships between the quantities in the problem (for example, how
the quantities are changing or growing with respect to each other); express
these relationships using mathematical operations to create an appropriate
equation or inequality to solve.
*I can create
equations and inequalities specified in this standard in one variable and
use them to solve problems.
*I can create
equations and inequalities specified in this standard in one variable to
model realworld situations.
*I can compare
and contrast problems that can be solved by the different types of
equations specified in this standard.
