Download Advanced Math Essential Guide

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 High Impact/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.
Minnesota Academic
Benchmarks
Minnesota Academic
Standards
Assessments
9.2.1.3 Find the
domain of a function
defined symbolically,
graphically or in a
real-world context.
9.2.1.5 Identify the
vertex, line of
I can solve for the
symmetry and
slope of a line.
intercepts of the
parabola
I can write an
corresponding to a
equation of a line
quadratic function,
given certain
using symbolic and
properties.
graphical methods,
when the function is
I can model real
expressed in the form
world situations using f (x) = ax2 + bx + c, in
9.2.1 Understand the
concept of function,
and identify
important features of
functions and other
relations using
symbolic and
graphical methods
where appropriate.
* 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
9.2.2 Recognize
linear, quadratic,
exponential and
other common
functions in realworld and
mathematical
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.
the form
situations; represent
these functions with
f (x) = a(x – h)2 + k , or tables, verbal
in factored form.
descriptions, symbols
9.2.1.7 Understand
and graphs; solve
the concept of an
problems involving
asymptote and
these functions, and
identify asymptotes
explain results in the
for exponential
original context.
functions and
reciprocals of linear
9.2.3 Generate
functions, using
equivalent algebraic
symbolic and
expressions involving
graphical methods.
polynomials and
9.2.2.1 Represent and radicals; use algebraic
solve problems in
properties to
various contexts
evaluate expressions.
using linear and
quadratic functions.
9.2.4 Represent realworld and
9.2.2.2 Represent and mathematical
solve problems in
situations using
various contexts
equations and
using exponential
inequalities involving
functions, such as
linear, quadratic,
investment growth,
exponential and nth
depreciation and
root functions. Solve
population growth.
equations and
9.2.2.3 Sketch graphs inequalities
symbolically and
of linear, quadratic
graphically. Interpret
and exponential
solutions in the
functions, and
original context.
translate between
graphs, tables and
symbolic
representations.
9.3.1 Calculate
measurements of
plane and solid
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.
Know how to use
graphing technology
to graph these
functions.
9.2.3.1 Evaluate
polynomial and
rational expressions
and expressions
containing radicals
and absolute values
at specified points in
their domains.
9.2.3.3 Factor
common monomial
factors from
polynomials, factor
quadratic
polynomials, and
factor the difference
of two squares.
9.2.3.5 Check
whether a given
complex number is a
solution of a
quadratic equation by
substituting it for the
variable and
evaluating the
expression, using
arithmetic with
complex numbers.
9.2.3.6 Apply the
properties of positive
and negative rational
geometric figures;
know that physical
measurements
depend on the choice
of a unit and that
they are
approximations.
9.3.4 Solve real-world
and mathematical
geometric problems
using algebraic
methods.
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
exponents to
generate equivalent
algebraic expressions,
including those
involving nth roots.
9.2.3.7 Justify steps in
generating equivalent
expressions by
identifying the
properties used. Use
substitution to check
the equality of
expressions for some
particular values of
the variables;
recognize that
checking with
substitution does not
guarantee equality of
expressions for all
values of the
variables.
9.2.4.1 Represent
relationships in
various contexts
using quadratic
equations and
inequalities. Solve
quadratic equations
and inequalities by
appropriate methods
including factoring,
completing the
square, graphing and
the quadratic
between sets of
points.
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.
formula. Find nonreal complex roots
when they exist.
Recognize that a
particular solution
may not be applicable
in the original
context. Know how to
use calculators,
graphing utilities or
other technology to
solve quadratic
equations and
inequalities.
9.2.4.2 Represent
relationships in
various contexts
using equations
involving exponential
functions; solve these
equations graphically
or numerically. Know
how to use
calculators, graphing
utilities or other
technology to solve
these equations.
9.2.4.3 Recognize
that to solve certain
equations, number
systems need to be
extended from whole
numbers to integers,
from integers to
rational numbers,
from rational
numbers to real
numbers, and from
real numbers to
complex numbers. In
particular, non-real
complex numbers are
needed to solve some
quadratic equations
with real coefficients.
9.2.4.7 Solve
equations that
contain radical
expressions.
Recognize that
extraneous solutions
may arise when using
symbolic methods.
9.3.1.1 Determine the
surface area and
volume of pyramids,
cones and spheres.
Use measuring
devices or formulas
as appropriate.
9.3.4.6 Use numeric,
graphic and symbolic
representations of
transformations in
two dimensions, such
as reflections,
translations, scale
changes and
rotations about the
origin by multiples of
90˚, to solve
problems involving
figures on a
coordinate grid.
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
General results of
polynomial equations
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.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
9-12.N.1.2.A
axioms of the real
Properties: A set of
number system to
mathematical rules or various subsets, e.g.,
laws that results in an axioms of order,
equivalent
closure.
expression.
9-12.N.2.1A.
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.
9.2.1.3.
9.2.3.1.
9.2.3.3.
9.2.3.6.
9.2.4.1.
9.2.4.3.
(stated in previous
unit)
9.2.3.2 Add, subtract
and multiply
polynomials; divide a
polynomial by a
polynomial of equal
or lower degree.
9.2.1 Understand the
concept of function,
and identify
important features of
functions and other
relations using
symbolic and
graphical methods
where appropriate.
9.2.2 Recognize
linear, quadratic,
exponential and
other common
functions in realworld and
9.3.2.1 Understand
mathematical
the roles of axioms,
situations; represent
definitions, undefined these functions with
terms and theorems
tables, verbal
in logical arguments. descriptions, symbols
and graphs; solve
9.4.1.3 Use
problems involving
scatterplots to
these functions, and
analyze patterns and explain results in the
describe relationships original context.
* 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
Axiom: A basic
assumption about a
mathematical system
from which theorems
can be deduced.
Subset: A set that is
contained within
another set.
9-12.N.2.1A.
Real Number: Any
number that can be
graphed on the
number line. This
includes rational and
irrational numbers.
Rational Exponent: A
power that can be
expressed as a
rational number.
(Application)
Students are able to
add, subtract,
multiply, and divide
real numbers
including rational
exponents.
between two
variables. Using
technology,
determine regression
lines (line of best fit)
and correlation
coefficients; use
regression lines to
make predictions and
correlation
coefficients to assess
the reliability of those
predictions.
9.2.3 Generate
equivalent algebraic
expressions involving
polynomials and
radicals; use algebraic
properties to
evaluate expressions.
9.2.4 Represent realworld and
mathematical
situations using
equations and
inequalities involving
linear, quadratic,
exponential and nth
root functions. Solve
equations and
inequalities
symbolically and
graphically. Interpret
solutions in the
original context.
9.3.2 Construct
logical arguments,
based on axioms,
definitions and
theorems, to prove
theorems and other
results in geometry.
9.4.1 Explain the uses
of data and statistical
thinking to draw
inferences, make
predictions and
justify conclusions.
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.
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
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
Properties of
9-12.A.3.2.A
9-12.A.4.1.A
(stated in previous
unit)
I can solve and graph
linear and polynomial
inequalities in one
variable.
I can graph
polynomial
inequalities in two
variables.
9-12.A.2.2.A.
(Application)
Students are able to
determine the
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.
9.2.3.1
9.2.3.2
9.2.3.3
(stated in previous
units)
9.2.4.7 Solve
equations that
contain radical
expressions.
Recognize that
extraneous solutions
may arise when using
symbolic methods.
9.2.3 Generate
equivalent algebraic
expressions involving
polynomials and
radicals; use algebraic
properties to
evaluate expressions.
9.2.4 Represent realworld and
mathematical
situations using
equations and
inequalities involving
linear, quadratic,
exponential and nth
root functions. Solve
equations and
inequalities
symbolically and
graphically. Interpret
solutions in the
original context.
* 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
9.4.1.1 Describe a
9.4.1 Explain the uses
* Rubrics
9.2.4.6 Represent
relationships in
various contexts
using absolute value
inequalities in two
variables; solve them
graphically.
9-12.A.4.6.A.
(Application)
Students are able to
graph solutions to
linear inequalities.
Linear programproblems that can be
expressed in standard
form
9-12.S.1.2.
9-12.A.3.1.A
I can determine the
important when
trying to find the
relationships in a
desired situation?
How can the
trigonometric graphs
be transformed?
functions
Operations on
functions
Reflecting graphs:
symmetry
Periodic functions;
stretching and
translating functions
Inverse functions
Functions in two
variables
Forming functions
from verbal
descriptions
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
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
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
sets, using range,
interquartile range,
mean, mode, and
median.
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.
I can graph and solve
functions.
data set using data
displays, including
box-and-whisker
plots; describe and
compare data sets
using summary
statistics, including
measures of center,
location and spread.
Measures of center
and location include
mean, median,
quartile and
percentile. Measures
of spread include
standard deviation,
range and interquartile range. Know
how to use
calculators,
spreadsheets or
other technology to
display data and
calculate summary
statistics.
9.4.1.2 Analyze the
effects on summary
statistics of changes
in data sets.
9.4.1.3 Use
scatterplots to
analyze patterns and
describe relationships
between two
variables. Using
technology,
of data and statistical
thinking to draw
inferences, make
predictions and
justify conclusions.
* 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
arranged in order
from least to
greatest.
Why are exponents
and exponential
functions important
to simplifying and
solving many real
world problems
involving math and
science?
How is sine, cosine,
tangent, and their
cofunctions used to
determine
information about a
right triangle?
Growth and decay:
- integral
exponents
- rational
exponents
Exponential functions
The number e and
the function e to the
x power.
Logarithmic functions
Laws of logarithms
Exponential
equations:
Changing bases
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,
cotangent).
Exponential
Expression: Any
expression of the
form
Logarithmic
Expression: An
expression of the
form
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.
(Application)
Students are able to
apply properties and
definitions of
trigonometric,
exponential, and
logarithmic
expressions.
I can apply integral
and rational
exponents.
I can apply
exponential functions
and natural
exponential functions
to….
I can apply
logarithms.
I can prove and apply
laws of logarithms.
I can solve
exponential
equations.
determine regression
lines (line of best fit)
and correlation
coefficients; use
regression lines to
make predictions and
correlation
coefficients to assess
the reliability of those
predictions.
9.3.4.1 Understand
how the properties of
similar right triangles
allow the
trigonometric ratios
to be defined, and
determine the sine,
cosine and tangent of
an acute angle in a
right triangle.
9.3.4.2 Apply the
trigonometric ratios
sine, cosine and
tangent to solve
problems, such as
determining lengths
and areas in right
triangles and in
figures that can be
decomposed into
right triangles. Know
how to use
calculators, tables or
other technology to
evaluate
9.3.4 Solve real-world
and mathematical
geometric problems
using algebraic
methods.
* 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
trigonometric ratios.
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?
How are the
trigonometric
functions periodic
functions?
Measurement of
angles
Sectors of circles
The sine and cosine
functions
Evaluating the
graphing sine and
cosine
Trigonometric
functions
Inverse trigonometric
functions
Trigonometric
equations
Sine and cosine
curves
Modeling periodic
behavior
Relationships among
the functions
Solving more difficult
trigonometric
equations
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
equivalent9
expression.
9-12.A.3.2.A.
9-12.A.4.1.A.
9-12.N.1.2.A.
(stated in previous
units)
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 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.
I can solve
trigonometric
equations and apply
them.
I can solve equations
of sine and cosine
curves.
I can apply
trigonometric
functions to model
periodic behavior.
I can simplify
trigonometric
expressions and
prove identities.
9.3.3.5
9.3.4.1
(Stated in previous
unit)
9.3.3.5 Know and
apply properties of
right triangles,
including properties
of 45-45-90 and 3060-90 triangles, to
solve problems and
logically justify
results.
9.3.4.3 Use
calculators, tables or
other technologies in
connection with the
trigonometric ratios
to find angle
measures in right
triangles in various
contexts.
9.3.3 Know and apply
properties of
geometric figures to
solve real-world and
mathematical
problems and to
logically justify results
in geometry.
9.3.4 Solve real-world
and mathematical
geometric problems
using algebraic
methods.
* 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