Advanced Math Essential Guide
This course is designed for those who have had a good math background. It is the intent of this course to bridge the gap between high school math and college work. The Advanced Math
course offered at Chamberlain Academy, Springfield Academy, McCrossan Boys Ranch, and Career Academy is based on the South Dakota Content Standards. This course offered at Elmore
Academy is based on Minnesota Academic Standards. The primary focus of this course includes the following:
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Understand and use trigonometric terminology;
Demonstrate an understanding of the relationships of the sides of a right triangle that describe the trigonometric ratios;
Use trigonometric functions to find lengths and angles of right triangles;
Graph trigonometric functions;
Solve problems using trigonometric laws and formulas;
Find logarithms of trigonometric functions;
Demonstrate an understanding of conic sections and their properties;
Solve and graph problems involving conic section equations;
Study progressions, series and binomial expansions;
Find permutations, combinations and probabilities.
Essential Question
How does writing and
solving equations and
systems o equations
enable me to solve
complex problems
graphically and
algebraically?
How can solving
quadratic equations
be used in real world
situations?
Why are exponents
and exponential
functions important
to simplifying and
Content
Points and lines
Slopes of lines
Finding equations of
lines
Linear functions and
models
Complex numbers
Solving quadratic
equations
Quadratic functions
and their graphs
Vocabulary
South Dakota State
Content Standards
9-12.A.1.1.A.
Equivalent forms:
Having the same
value when
evaluated.
Rational algebraic
expressions: A ratio
of two or more
algebraic expressions.
It is not an
equation.
Properties of real
numbers: A set of
mathematical rules or
laws that results in an
equivalent
expression.
9-12.A.1.1.A.
(Application)
Students are able to
write equivalent
forms of rational
algebraic expressions
using properties of
real numbers.
9-12.A.1.2.A.
(Application)
Students are able to
extend the use of real
number properties to
expressions involving
complex numbers.
9-12.A.2.1.A.
(Analysis) Students
South Dakota Skills:
Expectations of
learning
I can locate the
intersection of two
lines and determine
the length and
midpoint of a
segment.
I can solve for the
slope of a line.
I can write an
equation of a line
given certain
properties.
I can model real
world situations using
Minnesota Academic
Benchmarks
Minnesota Academic
Standards
Assessments
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
formative and
summative
assessments
* Textbook formative
and summative
assessments
* Performance – pre
and post test
solving many real
world problems
involving math and
science?
What does a slope
tell you about a
graph?
Quadratic models
9-12.A.1.2.A. Real
number properties: A
set of mathematical
rules or laws that
results in an
equivalent
expression.
Expression: A
mathematical
combination of
numbers, variables,
and operations. It is
not
an equation.
9-12.A.2.1.A.
Quadratic equation:
an equation
containing x2 , a
polynomial of degree
2 such that it can
be transformed into
ax2 + bx + c = 0, a ≠ 0
9-12.A.3.1.A.
Linear model: A
representation of a
problem that can be
expressed as an
equation in the
form y = mx + b
where m represents
the constant rate of
change, or slope, and
b
represents some
are able to determine
solutions of quadratic
equations.
9-12.A.3.1.A.
(Analysis) Students
are able to
distinguish between
linear, quadratic,
inverse variation, and
exponential models.
9-12.A.3.2.A.
(Synthesis) Students
are able to create
formulas to model
relationships that are
algebraic, geometric,
trigonometric, and
exponential.
9-12.A.4.1.A.
(Analysis) Students
are able to determine
the domain, range,
and intercepts of a
function.
9-12.A.4.3.A.
Students are able to
apply
transformations to
graphs and describe
the results.
9-12.A.4.5.A.
Students are able to
describe
characteristics of
nonlinear functions
and relations.
linear and quadratic
functions.
I can solve quadratic
equations and graph
quadratic functions.
fixed value, or the yintercept.
Quadratic model: A
representation of a
problem that can be
expressed as an
equation
containing x2 , a
polynomial of degree
2 such that it can be
transformed into
y = ax2 + bx + c, a ≠ 0.
9-12.A.3.2.A.
Formulas: Equations
that can be applied to
set of problems that
have common
parameter.
Algebraic: A relation
that can be classified
as linear, quadratic,
cubic, quartic,
absolute
value, square root,
rational or piecewise.
Trigonometric: A
function that can be
modeled with the six
trigonometric
functions.
Exponential: A
representation of a
problem that can be
expressed as
y = a ⋅ bx , a ≠ 0 &b ≠
9-12.N.1.1A.
(Comprehension)
Students are able to
describe the
relationship of the
real number system
to the complex
number system.
1. This also includes
logarithmic models,
log , 0, 1 a y = x a > a
≠.
Geometric: All of the
conic sections: circles,
parabolas, hyperbolas
and ellipses.
9-12.A.4.1.A.
Domain: The set of
inputs. The set of
possible values for x
or the independent
variable.
Range: The set of
outputs. The set of
possible values for y
or f(x) or the
dependent
variable.
Intercepts: The
value(s) where the
graph of a function
crosses the axes.
Function: A
mathematical
relation that
associates each
object in a set with
exactly one
value.
9-12.A.4.3.A.
Transformation: A
rule that sets up a
one to one
correspondence
between sets of
points.
How are polynomials
and factoring useful
in modeling real
world data?
Polynomials
Synthetic division:
the remainder and
factor theorems
Graphing polynomial
functions
Finding maximums
and minimums of
polynomial functions
Using technology to
approximate roots of
polynomial equations
Solving polynomial
equations by
factoring
9-12.N.1.1A.
Real Number System:
The set of numbers
consisting of the
union of rational and
irrational numbers.
Complex Number
System: The set of
numbers consisting of
the union of
imaginary and real
numbers.
9-12.A.4.2.A.
Polynomial: Sum of
two or more
monomials (i.e. ). In
this standard all
polynomials are
single variable.
Leading coefficient:
The coefficient of the
highest degree
monomial in a
polynomial.
Roots: The zeros of
the polynomial. It is
also the x-intercept if
the roots are real.
Degree: The
exponent of a single
variable polynomial.
9-12.N.1.2.A
9-12.A.3.1.A
9-12.A.3.2.A
9-12.A.4.1.A
9-12.A.4.3.A
9-12.A.3.1.A
9-12.A.4.5.A
(stated in previous
unit)
9-12.A.4.2.A.
(Analysis) Students
are able to describe
the behavior of a
polynomial, given the
leading coefficient,
roots, and degree.
9-12.N.1.2A.
Students are able to
apply properties and
axioms of the real
I can identify
polynomials.
I can apply synthetic
division to apply the
remainder and factor
theorems.
I can write
polynomial functions
for a given situation.
I can solve
polynomial equations
using various
methods.
I can apply general
theorems about
polynomial
equations.
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
formative and
summative
assessments
* Textbook formative
and summative
assessments
* Performance – pre
and post test
General results of
polynomial equations
Properties: A set of
mathematical rules or
laws that results in an
equivalent
expression.
Axiom: A basic
assumption about a
mathematical system
from which theorems
can be deduced.
Subset: A set that is
contained within
another set.
number system to
various subsets, e.g.,
axioms of order,
closure.
9-12.N.2.1A.
(Application)
Students are able to
add, subtract,
multiply, and divide
real numbers
including rational
exponents.
9-12.N.2.1A.
Real Number: Any
number that can be
graphed on the
number line. This
includes rational and
irrational numbers.
Why are inequalities
important to use in
representing some
real world situations?
Linear inequalities;
absolute value
Polynomial
inequalities in one
variable.
Polynomial
inequalities in two
variables.
Rational Exponent: A
power that can be
expressed as a
rational number.
9-12.A.2.2.A.
Solutions: value or
values of the
variable(s) that make
the statement true
Systems of equations:
two or more
equations
Systems of
inequalities: two or
9-12.A.3.2.A
9-12.A.4.1.A
(stated in previous
unit)
9-12.A.2.2.A.
(Application)
Students are able to
determine the
I can solve and graph
linear and polynomial
inequalities in one
variable.
I can graph
polynomial
inequalities in two
variables.
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
formative and
more inequalities
Linear programming.
9-12.A.2.3.A.
Absolute value
statement: an
equation or
inequality in which
the absolute value
contains the variable
9-12.A.4.6.A.
Linear inequality: A
comparison of two
first degree
expressions. The
comparisons can be
<, >, ≤, ≥ .
Why are graphs
important when
trying to find the
relationships in a
desired situation?
How can the
trigonometric graphs
be transformed?
Properties of
functions
Operations on
functions
Reflecting graphs:
symmetry
Periodic functions;
stretching and
translating functions
Inverse functions
Linear programproblems that can be
expressed in standard
form
9-12.S.1.2.
One-variable data
set: A collection of
numbers or
information
representing one
variable.
Range: The difference
between the greatest
and least values in a
data set.
Interquartile range:
The difference
between the values
solution of systems of I can graph the
equations and
solution of a system
systems of
of inequalities.
inequalities.
I can solve problems
9-12.A.2.3.A.
using linear
(Application)
programming.
Students are able to
determine solutions
to absolute value
statements.
summative
assessments
* Textbook formative
and summative
assessments
* Performance – pre
and post test
9-12.A.4.6.A.
(Application)
Students are able to
graph solutions to
linear inequalities.
9-12.A.3.1.A
9-12.A.3.2.A
9-12.A.4.1.A
9-12.A.4.2.A
9-12.A.4.3.A
9-12.A.4.5.A
(stated in previous
unit)
9-12.S.1.2.
(Comprehension)
Students are able to
compare multiple
one-variable data
I can determine the
domain, range, and
zeros of a function.
I can reflect graphs
using symmetry.
I can determine
periodicity and
amplitude from
graphs.
I can solve for the
inverse of a function.
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
formative and
summative
assessments
* Textbook formative
and summative
Functions in two
variables
Forming functions
from verbal
descriptions
Why are exponents
and exponential
functions important
to simplifying and
solving many real
world problems
involving math and
science?
How is sine, cosine,
Growth and decay:
- integral
exponents
- rational
exponents
Exponential functions
The number e and
the function e to the
x power.
of the third (upper)
and first (lower)
quartiles in a data
set.
Mean: The arithmetic
average which is the
sum of two or more
quantities divided by
the number of
quantities.
Mode: The value that
occurs most
frequently in a data
set.
Median: The quantity
designated the
central value in a set
of numbers. The
center number (or
the average of the
two central numbers)
of a list of data when
the numbers are
arranged in order
from least to
greatest.
9-12.A.4.4.A.
Trigonometric
Expression: An
expression that uses
one of three
trigonometric
functions (sine,
cosine, or tangent) or
their reciprocals
(cosecant, secant,
sets, using range,
interquartile range,
mean, mode, and
median.
9-12.A.1.1.A
9-12.A.3.1.A
9-12.A.3.2.A
9-12.A.4.2.A
9-12.A.4.5.A
9-12.N.2.1.A
(stated in previous
units)
9-12.A.4.4.A.
I can graph and solve
functions.
I can apply integral
and rational
exponents.
I can apply
exponential functions
and natural
exponential functions
to….
assessments
* Performance – pre
and post test
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
formative and
tangent, and their
cofunctions used to
determine
information about a
right triangle?
Logarithmic functions
Laws of logarithms
Exponential
equations:
Changing bases
What type of real
world problem would
use trigonometry to
help model and solve
it?
How is sine, cosine,
tangent, and their
cofunctions used to
determine
information about a
right triangle?
Measurement of
angles
Sectors of circles
The sine and cosine
functions
Evaluating the
graphing sine and
cosine
Trigonometric
functions
Inverse trigonometric
functions
How is sine, cosine,
tangent, and their
cofunctions used to
determine
information about a
right triangle?
How are the
trigonometric
Trigonometric
equations
Sine and cosine
curves
Modeling periodic
behavior
cotangent).
Exponential
Expression: Any
expression of the
form
Logarithmic
Expression: An
expression of the
form
9-12.A.1.1.A.
Equivalent forms:
Having the same
value when
evaluated.
Rational algebraic
expressions: A ratio
of two or more
algebraic expressions.
It is not an equation.
Properties of real
numbers: A set of
mathematical rules or
laws that results in an
equivalent
expression.
(Application)
Students are able to
apply properties and
definitions of
trigonometric,
exponential, and
logarithmic
expressions.
9-12.A.1.1.A.
(Application)
Students are able to
write equivalent
forms of rational
algebraic expressions
using properties of
real numbers.
I can apply
logarithms.
I can prove and apply
laws of logarithms.
I can solve
exponential
equations.
I can find the
measure of an angle
using degrees of
radians
I can determine the
value of
trigonometric
functions.
I can find the value of
inverse trigonometric
functions.
9-12.A.1.1.A.
9-12.A.3.2.A.
9-12.A.4.1.A.
9-12.N.1.2.A.
(stated in previous
units)
I can solve
trigonometric
equations and apply
them.
I can solve equations
of sine and cosine
curves.
summative
assessments
* Textbook formative
and summative
assessments
* Performance – pre
and post test
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
formative and
summative
assessments
* Textbook formative
and summative
assessments
* Performance – pre
and post test
* Rubrics
* Achievement Series
formative
assessments
* Student selfassessment – written
reflection
*Teacher-made
assessments
functions periodic
functions?
Relationships among
the functions
Solving more difficult
trigonometric
equations
I can apply
trigonometric
functions to model
periodic behavior.
I can simplify
trigonometric
expressions and
prove identities.
formative and
summative
assessments
* Textbook formative
and summative
assessments
* Performance – pre
and post test