Download Course Title: Algebra 2

Course Title:
Algebra 2
Levels:
2, 3, 4, 5
Grade:
11
Length of Course: One Year
Credits:
5.0
Prerequisites:
Algebra
Description:
The purpose of the Algebra 2 course is to build a strong foundation in preparation for students to ultimately take Calculus.
Beginning with a review of Algebra 1 topics to enhance critical skills and concepts, students will move onto more
advanced topics. A real-world orientation is emphasized in guiding the approaches to exercises and problems, and
technology is integrated, wherever appropriate, in the form of graphing calculators and computer programs.
Building on a more intuitive foundation developed in Algebra 1, the course formalizes the idea of what makes a function
and emphasizes various families of functions. In addition to becoming proficient with linear, quadratic, and exponential
functions, students will begin to develop an understanding of logarithmic and polynomial functions as well. These topics
will be utilized both for their abstract ideas and for modeling real-world phenomena.
Evaluation:
Student performance will be measured using a variety of assessments, including projects, teacher-generated tests and
quizzes, and a common departmental Midterm and Final Exam. Assessments will emphasize how well key concepts
have been understood as well as the depth to which required skills have been mastered.
Text: Algebra 2 (Glencoe McGraw-Hill)
5/7/2017 4:31 AM
1
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Content Outline: Unit 1 –
Core Content Review
Learning Objectives
The student will …
1. Graph real world
phenomena and solve
problems that involve
variation.
NJCCCS 4.1 A3
4.3 B1, 2, 4, C1, 2, 3,
4.5 ABCDE
2. Apply and explain
methods for solving
problems involving
integers and rational
numbers.
1.
2.
3.
4.
5.
6.
7.
8.
Graphing data on a Cartesian plane:
A. Connect aspects of a situation with its graph
1. Identify the x axis and y axis
2. Graph and describe coordinate pairs
3. Place independent and dependent events
on correct axis
4. Analyze and explain the direction and
shape of the graph
B. Graph direct variation situations and
interpret their graphs
1. Identify the rate of change of the
variation
2. Properties of the graph (start at origin, is
linear, etc.)
C. Connect situation, recursive formula, & graph
Instructional Materials
Printed Materials:
District constructed supplemental
packets (Obj 1, 2, 3)
Investigations:
Technology:
Obj 2Bb Use TI-83 FRAC
function to check if a decimal can
be written as a fraction
Supplies: graph paper
1. A. Properties and operations with integers
9.
1. Working with negative numbers
2. Applying the Distributive property
B. Properties and operations with rational numbers
1. Recognize fractions as division
2. Simplify rational expressions
NJCCCS4.1 A1, 2, 3,
B1,2, C
4.3 D 3
3. Evaluate and simplify
polynomial expressions
NJCCCS 4.1 A3
4.3 D1, 3
Add & subtract polynomials
Multiply monomials and binomials
A. Distributing a monomial over a binomial
B. Multiplying 2 binomials using repeated
distribution (FOIL)
Divide a polynomial by a monomial
5/7/2017 4:31 AM
Notes
NJCCC Process
Standards are coded for
Objective 1, but are also
enacted for all other
objectives across this
curriculum.
Objectives 1-3 enable
students to review and
reinforce prior learning.
These objectives are
foundational in the
development of new
course content.
See pacing chart to
determine the
approximate time
required by level for this
review.
Vocabulary:
quadrant ordered pair
integer
rational numbers
monomial binomial
Distributive Property
2
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objectives
The student will…
Content Outline: Unit 2 –
Expressions, Equations, and Inequalities
Instructional materials
Notes
4. Use the properties of
real numbers to evaluate
expressions and
formulas, and solve
equations
A. Expressions and Formulas
1. Expressions vs. Equations: Differences and
commonalities
2. Variables as unknowns, varying quantities,
and in formulas
3. Evaluating Expressions & Using Formulas
4. Order of operations to evaluate expressions
5. Using formulas by substituting for the
independent variable and simplifying
B. Properties of Real Numbers
1. Classifying numbers: Rational, integers,
natural numbers, whole numbers, irrational
2. Properties: Commutative, Associative,
Distributive, Additive & multiplicative inverses
C. Solving Equations
1. Solving linear equations with one variable
2. Translating verbal & algebraic expressions
3. Reverse order of operations to solve equation
4. Properties of Equality: Reflexive, Symmetric,
Transitive, Substitution
5. Solving for a particular variable in a formula
D. Solving Absolute Value Equations
1. Solve for variables inside abs val brackets
2. Separate into 2 equations, find 2 solutions
E. Inequalities
1. Solve single/multi step inequalities
2. Recognize that the solution is a RANGE of
solutions rather than a single value
3. Graph the range of solutions on a number line
4. Solve compound & abs value inequalities
Printed Materials:
Algebra 2 Chapter 1 sections 1-6
See pacing chart to
determine the level of
development and
approximate time
required by level for these
and all subsequent
objectives.
NJCCCS 4.1 A3
4.3 B1, C1, D1,3
5. Solve absolute value
equations
NJCCCS4.3 C1, D1, 2, 3
6. Solve and graph
inequalities
NJCCCS 4.1 A3
4.3 D2, 3
Investigations:
Technology:
Supplies:
5/7/2017 4:31 AM
Definitions
variable
expression
formula
absolute value
real numbers
inverse
Properties:
Equality: Symmetric,
Reflexive, Transitive,
Substitution
3
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objectives
The student will …
7. Recognize, represent,
and use linear functions
to represent real world
phenomena and solve
problems.
NJCCCS 4.1 A3
4.3 B1, 2, C1, 2
8. Analyze and
determine the rate of
change using appropriate
graphing technologies.
NJCCCS 4.1 A3
4.3 C1, B1, 2, D3
9. Select and use
appropriate methods for
solving linear equations.
NJCCCS 4.1 A3
4.3 C1, D2, 3
Content Outline: Unit 3 –
Linear Relations and Functions
A. Relations and Functions
1. Definitions and Properties
2. Representations of types of functions
3. Determine whether a relation is a function by
looking at a table, graph, or equation
4. Vertical Line test for functions
5. Identify domain and range, by looking at a
table, graph, or equation
6. Distinguish independent & depend.variables
7. Identify & graph discrete real world data sets
8. Distinguish between discrete vs. continuous
domain & range in real data; graph both
C. Evaluating a function
1. Introduction to Euler’s notation
2. Substitute independent quantity into the fx
notation and simplifying
D. Forms of Linear Relations and Functions
1. Slope-intercept form (y=mx + b)
2. Standard form (Ax + By = C)
3. Graph a line by finding both x and y intercepts
4. Roots of equations and zeros of functions
5. Set the dependent var to zero (y=0) & solve
Rate of Change and Slope
1. Recognize properties of different slopes:
positive, negative, zero, or undefined
2. Using the slope formula
3. Finding the slope of a line from a graph
Writing Linear Equations (from given information)
1. Slope intercept. form y = mx + b
2. Point slope form
y – y1 = m(x – x1)
3. The relationship between slopes and intercepts
for parallel and perpendicular lines
Scatter Plots and Lines of Regression
1. Correlation-strong/weak,negative/positive/none
2. Find a line of best fit by hand & w/ith a TI-83+
Instructional Materials
Notes
For Slope
Printed Materials:
1.Show students that slope
Algebra 2 Chapter 2 sections 1-5 is best expressed as a
Investigations:
fraction. Let them discover
Relations
why slope as a decimal is
Determine whether a relation is a not as useful.
function by looking at a table,
2. Move a negative to the
graph, or equation
top of slope fraction. This
Domain and range
way Rise is up or down but
Identify domain and range, by
run is always to the right.
looking at a table, graph, or
3. Have students use a
equation
ruler to draw a random line
Application
Use function notation to write and across an entire sheet of
solve equations that model real- graph paper. Then have
them find the slope of that
world situations
line by finding the two
Distinguishing discrete vs.
“best” lattice points
continuous domain/range
&counting the rise and run.
Students compare their results to
explain the significant differences. 4. Simplify the rise/run to
lowest terms and see that
Scatterplots
the slope is still valid
1.Students use a graphing
5. Parallel & perpendicular
calculator to find correlation
Have students graph lines
coefficient, regression line slope,
with opposite-sign slopes.
and y-intercept.
Let them come to realize
2. Students use data from their
graphing calculator to create a y = that a negative (opposite)
mx + b prediction equation in order slope does not create a
to extrapolate what would happen perpendicular line. Rather
it must be an opposite and
if the trend continues.
reciprocal
Technology: TI-83+
Definitions
Supplies: graph paper
Domain & range
5/7/2017 4:31 AM
4
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objectives
The student will …
10. Analyze and explain
the general properties
and behavior of functions
of one variable, using
appropriate graphing
technologies.
Identify and compare the
properties of classes of
functions.
NJCCCS 4.1 A3
4.3 B2, 4, C1, D3
Content Outline: Unit 4 –
Parent Functions and Transformations
A. Parent Functions
1. Constant function
2. Identity function (fx = x)
3. Absolute value
4. Piecewise function
5. Simple quadratic function (centered parabola)
6. Compares properties of classes of functions
11. Perform
transformations on
commonly-used functions.
NJCCCS 4.1 A3
4.3 B2, 3, C1, D3
Transformations (on the functions listed above)
1. Translations
2. Reflections across the x & y intercepts
3. Dilations of graphs
4. Effects of parameter changes in equations
12. Graph linear and
absolute value
inequalities
NJCCCS 4.1 A3
4.3 B1, C1, D3
Graphing inequalities
1. Dashed vs solid boundary lines
2. Defining a half plane
3. Absolute value inequalities ad their graphs
Notes
Instructional Materials
Printed Materials:
Algebra 2 Chapter 2 sections 7-8
Investigations:
Translations
Have students graph each of the 4
families at the origin, then have
them draw an image of each graph
at a translation point of their
choice. Below the translation, they
must write a modified (translated)
version of the function’s equation.
Circulate among students to verify
their equations are correct, then let
volunteers present their creations
on the board.
Technology: TI-83+
Supplies: graph paper
5/7/2017 4:31 AM
5
COLUMBIA HIGH SCHOOL
Learning Objective
The student will..
13. Solve systems of
linear equations
NJCCCS 4.1 A3
4.3 A3, B2, C1, D3
ALGEBRA II CURRICULUM
Content Outline Unit 5 –
Systems of Equations and Inequalities
Instructional Materials
Solving Systems of Equations by Graphing
1. Solve a system using a table & note its limitations Printed Materials:
2. Solve a system by plotting two lines on a graph
Algebra 2 Chapter 3 sections 1-3
and estimating the coordinates of their point of
intersection (if any)
Investigations:
3. Classification of Systems: consistent, identity
Break-Even point analysis: When
(dependent), inconsistent (no intersection at all)
does one function “pass up”
another? Finding the Best Cellular
Solving Systems of Equations Algebraically *
Phone Plan: Given brochures for
Solving Systems by Substitution
1. Use when equations do not share the same form two companies’ cellular phone
plans and a list of family members
2. Choose substitution so that only one variable
with varying usage requirements,
remains in the resultant equation
students will plot a system of
3. Substitute back to find the remaining variable
piecewise linear functions that
Solving Systems by Elimination
1. Best used when equations are of the same form represent the two plans and
decide, based on their point(s) of
2. Recognize that multiplying both side of an
intersection which plan is less
equation doesn’t effect the equality
expensive for each family
3. Recognize that adding two correct equations
member’s usage requirements.
results in another correct equation.
4. Show how eliminating variables from a system
Technology: TI-83+
results in isolating one variable.
5) Use back substitution to find the remaining
variable (same as substitution)
Supplies: graph paper
Systems with No Solution and Infinite Solutions
Identifying identity, inconsistent, and consistent
systems algebraically.
Notes
Equations containing more
than one variable cannot
be solved conclusively.
There will always be
another variable in the
solution.
To solve an equation with
two variables, we seek a
set of values that will
simultaneously satisfy both
equations. This occurs at
the point of intersection of
the two functions on a
graph.
*Level 4 should also
solve 3x3 systems with
Substitution and
Elimination
Solving Systems of Inequalities by Graphing
1. Graph intersecting half-planes created by a
14. Solve real world
system of linear inequalities
problems using systems of
2.
Find the resultant “Feasible Region” of a
inequalities.
system of linear inequalities
NJCCCS 4.1 A3
3. Use of constraints such as x > 0, etc., to limit
4.3 B1, C1, D3
graph to specific quadrants in real-world examples
4. Find the vertices of an enclosed region
5. Find vertices (corners) of the feasible region by
Inspection, look for intercepts & lattice points
5/7/2017 4:31 AM
6
Columbia High School
ALGEBRA II CURRICULUM
Learning Objectives
The student will …
15. Use linear
programming to solve
real-world problems**
NJCCCS 4.1 A3, B 3, C1
4.3 B1, 2, C1, 2
16. Describe and
perform operations on
matrices. Solve systems
through matrix
multiplication, using
inverses.
Content Outline Unit 6 –
Linear Programming and Matrices
This is a Level 4 & 5 only unit
Linear programming
1. Find the maximum and minimum values of a
Function, given various constraints
2. Solve real world optimization problems using
linear programming
Notes
Instructional Materials
Printed Materials:
Algebra 2 Chapter 3 sections 4 -5 **Level 3 and 4 only
Chapter 4 sections 1, 3, 5, 6
Investigations:
Matrices
Technology: TI-83+
1. Describe a matrix according to its dimensions
2. Find the determinant & inverse of a 2x2 matrix
Supplies: graph paper
3. Use matrices to solve 2x2 systems by hand
Use Cramer’s Rule
Use inverse matrices
4. Use matrices to solve 3x3 systems with a TI-83+
NJCCCS 4.1 B 3
5/7/2017 4:31 AM
7
Columbia High School
ALGEBRA II CURRICULUM
Learning Objectives
The student will …
17. Recognize and use
connections among
significant values of a
quadratic function, points
on the graph of the
function, and its symbolic
representation.
NJCCCS 4.1 A3
4.3 B1, 2, C1,2, D2
Content Outline Unit 7 – Quadratic Functions
Instructional Materials
A. Graphing a quadratic function by using a table
1. Identify the quadratic term, linear term, constant
2. Make a table of values and graph the parabola
3. Identify the absolute maximum & minimum of
quadratic function (at the vertex)
B. Solving Quadratic Equations by Graphing
1. Examine the graph of a quadratic equation to
find its roots
2. Graph the quadratic function by making a table
of values
3. Estimate the roots by locating any x-intercepts
on the graph.
C. Solving quadratics equations by factoring*
Printed Materials:
Algebra 2 Chapter 5 sections 1-4
Notes
Standard form of a
Quadratic Function (
Investigations:
f ( x)  ax 2  bx  c )
Axis of symmetry -
b
), with
2a
vertex ( h, k ) , y-intercept c
(h  
Technology: TI-83+
Supplies: graph paper
18. Identify properties of
imaginary and complex
numbers and operate on
them.
Solve simple quadratic
equations with imaginary
solutions.
Imaginary number
1. Factor radicals to extract a negative & replace w/ i
2. Properties of imaginary numbers
3. Operations on imaginary numbers (+ - x ÷ )
4. Solve quadratic equations w/imaginary solutions
Complex Numbers (Level 2)
1. Definition of Complex Number in a  bi form.
2. Arithmetic with complex numbers (add / subtract)
NJCCCS – This objective 3. Multiplying complex numbers (using FOIL)
exceeds the range of math Complex conjugates
identified in the Standards 1. Divide complex numbers
for grades 9-12.
2. Divide a complex number by a constant:
3. Separate fractions
4. Divide a complex number by imaginary number
5. Divide 2 complex numbers -use complex conjugate
Complex Numbers (Levels 3-4)
1. Perform operations with complex numbers
2. Find complex roots for quadratic equations
3. Rationalize complex fractions by using the
complex conjugate
5/7/2017 4:31 AM
The number of possible
roots are:
Two real solutions (roots)
One real solution
No real solutions (any
algebraic roots would be
imaginary)
* Levels 3, 4, and 5 only
8
Columbia High School
Learning Objectives
The student will …
19. Solve quadratic
equations by completing
the square.
NJCCCS 4.3 B1,2,C1,
D2,3
20. Solving quadratic
equations using the
Quadratic Formula
NJCCCS 4.3 D2, 3
21. Write a quadratic
function in vertex form.
Transform graphs of
quadratic functions in
vertex form.
NJCCCS 4.1 B1, 4.3 B3
ALGEBRA II CURRICULUM
Content Outline Unit 7 –
Quadratic Functions
Instructional Materials
Completing the Square
Printed Materials:
1) Identify perfect squares of integers & monomials
Algebra 2 Chapter 5 sections 5-7
2) Recognize that perfect squares of binomials are
trinomials (prove with FOIL)
3) Completing the square of a perfect-square
trinomial (using c = (b/2)2
Investigations:
4) Solving equations by completing the square
5) Realize this strategy is best when lead a terms
equal to one, when b is even.
Technology: TI-83+
6) For lead terms not equal to 1, try to factor out a
The Quadratic Formula and the Discriminant
Supplies: graph paper
1) Derivation of the Quadratic Formula (optional)
2) The four stages of the Quadratic Formula:
a) Standardize: Put quadratic equation into
standard form equal to zero. Id.values of a, b,c
b) Substitute c) Simply
d) Split: Separate the plus/minus expression to
create (up to) 2 solutions
The Discriminant
1) Use discriminant to determine the nature of the
solutions to the equation, and number of xintercepts on the graph
Transformations with Quadratic Equations
1) Find the vertex of a parabola
2) Use completing the square to change a quadratic
equation to vertex form if practical: a  1 & b is even
3) Use h   b to change a quadratic equation
2a
into vertex form if completing the square is too
tedious (if a  1 or b is odd.)
Graphing equations in Vertex Form
1) Use the a, h, k, and the y-intercept to graph
2) Plot symmetrical points on either side of vertex (if
needed) to establish the dilation of the parabola.
3) Recognize the necessity if vertex is on the y-axis
5/7/2017 4:31 AM
Notes
Quadratic Formula
x
b  b2  4ac
2a
The Quadratic Formula is
a last resort when solving
a quadratic equation. It
can solve any quadratic
equation but is more time
consuming than other
methods such as factoring
and completing the
square.
Possible outcomes of a
QF: 0,1, or 2 real roots
The Discriminant
( b  4ac )
2
Vertex Form of a
Parabola
y  a ( x  h) 2  k
9
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objectives
The student will …
Content Outline Unit 8- Polynomial Functions
22. Classify and factor
polynomials
Classify polynomials by degree & number of terms
Multiply polynomial expressions
NJCCCS 4.3 B4, C1, D1,
2, 3
Factor polynomials
1. Recognize greatest common monomials factors
2. Factor binomial special cases:
a. difference of two squares,
b. difference of two cubes,
c. sum of two cubes
3. Factor quadratic trinomials
Instructional Materials
Notes
Printed Materials:
Algebra 2
Chapter 6 sections 1, 3, 4, 5, 7
Investigations:
Technology: TI-83+
Supplies: graph paper
23. Find the zeroes of
polynomials and graph
polynomial functions
NJCCCS 4.3 D2, 3
1. Use factoring and the Zero-Product Property to
solve for the zeroes of polynomials
2. Recognize two new factoring special cases:
Sum and Difference of Cubes
3. Use Descartes’ Rule of Signs to determine the
number and type of real zeroes
Graph polynomial functions
1. Sketch polynomial graphs using the zeroes of the
function
2. Identify double roots as tangent to the x-axis on
the graph
3. Identify the number of real roots and complex
Roots from a graph
5/7/2017 4:31 AM
10
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objectives
The student will …
Content Outline Unit 9 Composite Functions and Inverses
24. Write and evaluate
composite functions and
inverse functions
Composite functions
1. Basic operations with functions (+ - x ÷ )
2. Composition of functions
3. Domain/range implications of composing functions
4. Domains of compound functions
NJCCCS – This objective 5. Compose functions by evaluating in sequence 9.
exceeds the range of math 6* Calculate value of composite function at specific
identified in the Standards
values
for grades 9-12.
7*Find formula for composite function-all values of x
8. Create compound function from existing functions
9. Real-world examples of composition of functions
Inverse Functions and Relations
1. Finding the inverse of a function (various types)
a. Ordered pair functions
b. Table functions
c. Algebraic functions (by transposing
independent and dependent variables)
2. Only 1:1 functions have true inverse functions.
3. Horizontal line test to verify a function is 1:1
4. Finding the inverse of a graphed function
5. Horizontal line test - verify that inverse is function
6. Reflecting the graph of the function across
f ( x)  x to create the inverse
7. Verifying that two functions are inverses by
composition of functions
8* Apply the properties of inverse relations
Notes
Instructional Materials
Printed Materials:
Algebra 2 Chapter 7sections 1-2
Notations:
(f + g) (x), (f – g) (x),
(f . g) (x), (f / g) (x)
Investigations:
Car dealer offering a rebate and
percent discount on new cars:
Which is better to apply first?
Applying a percentage discount
before or after sales tax
Composition notation:
( f g )( x), f ( g ( x)), etc
* # 6 & 7 - level 4 only
Technology: TI-83+
Supplies: graph paper
5/7/2017 4:31 AM
Definition of inverse
functions
( f g )( x)  ( g f )( x)  x
* # 8 - level 4 only
11
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objective
The student will…
25. Use reciprocals to
solve equations using
exponents and radicals.
Graph radical functions.
NJCCCS 4.1 A 2, B2, 4
4.3 B4, C1, D2, 3
Content Outline Unit 10 –
Radical Functions and Relations
Properties of Exponents
1. Simplify expressions with exponents
Square root functions
1. Domain and range of square root functions
2. Set expression under radical > 0
3. Resultant inequality is the domain of the function
4. Find lower bound of a range by plugging in the
result from step 2 into the function
5. Graph square root functions, including those with
Transformations
Instructional Materials
Printed Materials:
Algebra 2 Chapter 7sections 3-5
Investigations:
Technology: TI-83+
Supplies: graph paper
6. Use f ( x)  a x  h  k as done w/ vertex form
7. Simplify expressions using radicals
8. Add, subtract, and multiply radical expressions
9. Rationalize a denominator containing a radical
10. Convert rational exponent to radicals& vice-versa
nth Roots
1. Work with nth roots
Notes
Even roots (of a positive
number, a) have two real
values n a and  n a
Even roots (of a negative
number, a)have no real val
Odd roots have one root
for any + or - number
(same sign as a)
n
0  0 for any n,odd-even
Make sure students
express the final answer
as an absolute value
since it is possible that
the variable could be a
negative number
4 4
a  a (not just “a”)
4
16( x  3)12  2 ( x  3)3
n
2. Properties of nth roots a
3. Introduction: radical sign √, index n, & radicand a
4. Generalize the value of roots of numbers with
various characteristics (See notes)
5. Simplify even root of an even power of a variable
6. Approximate radicals with a calculator
Make sure to express the
final answer as an abs.
value since it is possible
that the variable could be
a negative number
4
For irrational expressions, a calculator can be used
to approximate the decimal value, but it should be
understood that it is a rounded estimate and not an
exact value.
5/7/2017 4:31 AM
4
a4  a
(not just “a”)
16( x  3)12  2 ( x  3)3
12
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objective
The student will…
26. Perform operations
that contain radical
expressions.
NJCCC 4.1 A1, 2, 3, B1,
NJCCC 4.3 B1, 2, C1, 2
Content Outline Unit 10 –
Radical Functions and Relations
Instructional Materials
The Product Property of Radicals
1. Recognize radicals can be factored into
components of the same degree
2. Simplifying Expressions with the Product Property
For a root of degree n, factor the radicand into
power-of-a-power expressions based on power n.
3. Use the fact that the nth root is the inverse of the
nth power and cancel out all pairs of matching roots
and powers
Printed Materials:
Algebra 2 Chapter 7 section3-7
(selected parts)
Investigations:
Supplies: graph paper
2
2
3 2 3



3
3
3 3
n
x
For roots other than square roots, a radical b
can be rationalized by multiplying the numerator and
denominator by
n
Examples
Radicals can be factored
into components of the
same degree
n
Technology: TI-83+
The Quotient Property of Radicals
1. Radicals of a quotient can be written as a quotient
of radicals of the same degree
2. Rationalizing the denominator
3. When the denominator is a square root, it can be
rationalized by multiplying it times itself (in both the
numerator and denominator)
Notes
(assuming all roots are
defined)
2  8  16
Use the fact that the nth
root is the inverse of the
nth power and cancel out
all pairs of matching roots
and powers
4
5/7/2017 4:31 AM
16a 24b13  4 (24 )(a 6 )4 (b3 )4  b  2a 6 b3
Radicals of a quotient can
be written as a quotient of
radicals of the same
degree
n
bn x
2
2 3 34 2 3 81



5
3
3 5 3 3 34
ab  n a  n b
a na
(assuming all

b nb
roots are defined)
3
x6

8
3
x6 x2

3
2
8
13
4
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objective
The student will…
27. Evaluate
expressions with
exponents that are
negative and/or fractions.
NJCCC 4.1 A1, 2, 3, B1,
NJCCC 4.3 B1, 2, C1, 2
Content Outline Unit 10 –
Radical Functions and Relations
Instructional Materials
Rational Exponents
1. Radical vs. Exponential Form
2. Express radicals as rational exponents
3. Evaluate expressions with radical exponents
4. Solve equations with rational exponents
5. Simplify expressions with radical exponents
6. Simplify radical expressions
Printed Materials:
Algebra 2 Chapter 7 section6
Solve Radical Equations and Inequalities
1. Solve radical equations by raising both sides to a power
2. For an equations with a single radical: Isolate the
radical on one side of the equation and all other
3. Raise both sides of the equation to the power that
would cancel out the radical
4. For an equation with more than one radical of the
same degree:
a. Raise both sides of the equation to the power
that would cancel out the radicals
b. If a single radical remains as a result of FOIL,
follow steps above for solving an equation with
a single radical
Technology: TI-83+
Notes
Investigations:
Supplies: graph paper
5/7/2017 4:31 AM
14
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objective
The student will…
28. Solve and graph
exponential functions.
Solve exponential
equations.
Content Outline Unit 11 –
Exponential and Logarithmic Functions
This is a level 3, 4, and 5 only unit
Exponential equations
1. Write and solve exponential equations
2. Distinguish exponential growth and decay
3. Graph exponential functions
Fit exponential models to data
NJCCC 4.1 A1, 2, 3, B1,4, 1. Recognize when graphed data follows an
C1
exponential pattern
2. Find the exponential equation that satisfies 2 pts.
3. Distinguish exponential growth and decay
Logarithms
1. Learn logs base 10
29. Understand
2. Utilize the properties of logs of any base
properties of logarithms
(Product, Quotient, and Power properties)
and use natural logs
3.
Determine
values of natural logs
and e
4. Apply properties of natural logs
5. Use applications of e
NJCCC 4.3 B1, 2, C1, 2
6. Use log properties to solve exponential equations
7. Use the Change of Base Property to find the log
of any number using any base
8. Graph logarithmic functions
9. Convert exponential functions to logs &vice-versa
10. If given any two variables, solve for the missing
variables in the equation y = logb x
11. Utilize the Product, Quotient, and Power
properties of logs of any base
12. Use the Change of Base Property to find the log
of any number using any base
13. Use applications of e, including exponential
growth and decay problems
Notes
Instructional Materials
Printed Materials:
Algebra 2 Chapter 8 sections1-8
Investigations:
Technology: TI-83+
Supplies: graph paper
5/7/2017 4:31 AM
15
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Learning Objective
The student will…
30. Solve problems
involving rational
functions.
Content Outline Unit 12 –
Rational Functions, and Sequences & Series
Instructional Materials
Simplify rational expressions
1. Simplify rational expressions by dividing out
common factors from numerator & denominator
2. Identify restricted values
Printed Materials:
Algebra 2 Chapter 9 sections 1-6
Manipulate rational expressions with addition and
subtraction
1. Add and subtract rational expressions
2. Identify restricted values
Investigations:
Graph rational functions
1. Identify horizontal and vertical asymptotes
2. Identify holes in the graph
(note: textbook uses the term “point discontinuity”)
Supplies: graph paper
Notes
NJCCC 4.3 B1, 2, 4, C1
Technology: TI-83+
Solve rational equations
31. Evaluate arithmetic
and geometric sequences
and series.
Write sequences using
recursive formulas.
1. Describe geometric sequences by formula
2. Calculate the nth term of a geometric sequence
3. Use the Compound Interest formula
Printed Materials:
Algebra 2 Chapter 11 sections 1-5
NJCCCS 4.3 A1, 2, 3
5/7/2017 4:31 AM
16
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Critical Questions
Obj. 2 Number systems
Can you name a rational number that is not an integer?
An integer that is not rational?
Obj. 4 Expressions
Which operation comes first in the expression 36  (6 + 2 * 3) ?
Obj. 6 Inequalities
How does a test point help identify the feasible region?
Obj. 7 Linear Functions
Give an example of a function that is both “one-to-one” and “onto.” Change your example so that it satisfies only one, but not the
other.
In Standard Form, Ax + By = C, how can A and B be manipulated to create lines that are horizontal, vertical, or oblique?
Obj. 8 Slope
How are slopes of parallel and perpendicular lines related?
What kind of line has a y-intercept, but no x-intercept? What is the slope of such a line?
Obj. 10 General forms and properties of functions
In Standard Form, Ax + By = C, how can A and B be manipulated to create lines that are horizontal, vertical, or oblique?
What kind of line has a y-intercept, but no x-intercept? What is the slope of such a line?
Obj. 11 Translations of functions
How does the graph of y = –|x + 5| compare to the graph of y = |x| ?
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17
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Critical Questions- continued
Obj. 13-14 Systems of equations, inequalities
What do the graphs of consistent and inconsistent systems look like?
Given the system: 6x + 3y = 9 and 2x = 2y + 18, what would your first step be, if you wanted to solve with Substitution? With
Elimination?
How does a test point help identify the feasible region?
Obj. 16 Matrices
What does the determinant determine?
Give an example of a matrix that has no inverse.
Obj. 17 Quadratic functions
When will a parabola have a maximum value? A minimum value?
How is the axis of symmetry related to the vertex point?
Is there any difference between roots, zeroes, and solutions?
Obj. 20 The Quadratic Formula
What does the discriminant tell you about the factorability of an expression?
In what situations is each of the different methods for solving (graphing, factoring, completing square, Quadratic Formula)
appropriate?
Obj. 22 Factor polynomials
What does the discriminant tell you about the factorability of an expression?
5/7/2017 4:31 AM
18
COLUMBIA HIGH SCHOOL
ALGEBRA II CURRICULUM
Critical Questions- continued
Obj. 23 Graph polynomial functions
How do the graphs of f(x) = (x – 3)2 (x + 5) and g(x) = (x – 3)2 (x + 5)2 compare in regard to basic shape, end behavior, and
zeroes?
Obj. 25 Radical functions
Is the square root of x2 always equal to x?
How is the process of rationalizing a fraction with a radical in the denominator similar to a fraction with an Imaginary number in
the denominator?
What is an extraneous solution?
Obj. 28 Exponential functions
How does the value of “a” affect the graph of an exponential function?
Why is the x-axis an asymptote for an exponential function?
Obj. 29 Logarithms
Given log 4 = .6 and log 5 = .7, use the log properties to evaluate the following without a calculator: log 20, log 25, log 64, log
1/4, log 5/4 .
Obj. 30 Rational Expressions
What role does factoring play in simplifying rational expressions?
How do you determine that a common denominator is the lowest possible one, when dealing with rational expressions?
Obj. 31 Sequences and Series
How are arithmetic sequences similar to linear functions?
How are geometric sequences similar to exponential functions?
What must be true about the ratio for an infinite geometric series to converge to a sum?
5/7/2017 4:31 AM
19