Download Curriculum Coordinator - South Plainfield Public Schools

South Plainfield Public Schools
Curriculum Guide
Mathematics
Honors Algebra 2
Authors:
Anthony Emmons
John Greco
Curriculum Coordinator:
Paul C. Rafalowski
Board Approved on: June 13, 2012
1
Table of Contents
South Plainfield Public Schools Board of Education Members
and Administration
Page: 3
Recognitions
Page: 4
District Mission Statement
Page: 5
Index of Courses
Page: 6
Curriculum Guides
Page: 7-45
Mathematics Practice Standards
Page: 46-49
Common Core Standards
Page: 50-57
Resources for State Assessments
Page: 58
2
Members of the Board of Education
Jim Giannakis, President
Debbie Boyle, Vice President
Carol Byrne
John T. Farinella, Jr
Christopher Hubner
Sharon Miller
William Seesselberg
Joseph Sorrentino
Gary Stevenson
Central Office Administration
Dr. Stephen Genco, Superintendent of Schools
Dr. Frank Cocchiola, Interim Assistant Superintendent of Schools
Mr. James Olobardi, Board Secretary/ BA
Mrs. Laurie Hall, Supervisor of Student Personnel Services
Mr. Vincent Parisi, Supervisor of Math and Science
Mrs. Marlene Steele, Supervisor of Transportation
Mrs. Annemarie Stoeckel, Supervisor of Technology
Ms. Elaine Gallo, Director of Guidance
Mr. Al Czech, Director of Athletics
Mr. Paul Rafalowski, Curriculum Coordinator
3
Recognitions
The following individuals are recognized for their support in developing this Curriculum Guide:
Grade/Course
Writer(s)
Kindergarten:
Ms. Joy Czaplinski and Ms. Pat Public
Grade 1:
Ms. Patti Schenck-Ratti, Ms. Kim Wolfskeil
and
Ms. Nicole Wrublevski
Grade 2:
Ms. Cate Bonanno, Ms. Shannon Colucci
and
Ms. Maureen Wilson
Grade 3:
Ms. Cate Bonanno and Ms. Theresa Luck
Grade 4:
Ms. Linda Downey and Ms. Kathy Simpson
Grade 5:
Mr. John Orfan and Ms. Carolyn White
Grade 6:
Ms. Joanne Haus and Ms. Cathy Pompilio
Grade 7:
Ms. Marianne Decker and Ms. Kathy Zoda
Grade 8:
Ms. Marianne Decker and Ms. Donna Tierney
Algebra 1:
Ms. Donna Tierney and Ms. Kathy Zoda
Geometry:
Mr. Anthony Emmons and Ms. Kathy Zoda
Algebra 2:
Mr. Anthony Emmons and Mr. John Greco
Algebra 3/Trigonometry:
Ms. Anu Garrison and Mr. David Knarr
Senior Math Applications:
Mr. John Greco
Pre-Calculus:
Ms. Anu Garrison and Mr. David Knarr
Calculus:
Mr. David Knarr
Supervisors:
Supervisor of Mathematics and Science:
Mr. Vince Parisi
Curriculum Coordinator:
Mr. Paul C. Rafalowski
Supervisor of Technology:
Ms. Annemarie Stoeckel
4
South Plainfield Public Schools
District Mission Statement
To ensure that all pupils are equipped with essential skills necessary to acquire a
common body of knowledge and understanding;
To instill the desire to question and look for truth in order that pupils may become
critical thinkers, life-long learners, and contributing members of society in an
environment of mutual respect and consideration.
It is the expectation of this school district that all pupils achieve the New Jersey
Core Curriculum Content Standards at all grade levels.
Adopted September, 2008
NOTE: The following pacing guide was developed during the creation of these curriculum
units. The actual implementation of each unit may take more or less time. Time should also be
dedicated to preparation for benchmark and State assessments, and analysis of student results on
the same. A separate document is included at the end of this curriculum guide with suggestions
and resources related to State Assessments (if applicable). The material in this document should
be integrated throughout the school year, and with an awareness of the State Testing
Schedule. It is highly recommended that teachers meet throughout the school year to coordinate
their efforts in implementing the curriculum and preparing students for benchmark and State
Assessments in consideration of both the School and District calendars.
5
Index of Mathematics Courses
Elementary Schools
(Franklin, Kennedy, Riley, Roosevelt)
Kindergarten
Grade 1
Grade 2
Grade 3
Grade 4
Grant School
Grade 5
Grade 6
Honors Grade 6
Middle School
Grade 7
Honors Grade 7
Grade 8
Honors Grade 8
Grade 8 Algebra 1
High School
Algebra 1
Academic Algebra 1
Honors Algebra 1
Geometry
Academic Geometry
Honors Geometry
Algebra 2
Academic Algebra 2
Honors Algebra 2
Algebra 3/Trigonometry
Senior Math Applications
Pre-Calculus
Honors Pre-Calculus
Calculus
Calculus AB
Calculus BC
6
South Plainfield Public Schools Curriculum Guide
Content Area: Mathematics
Course Title: Honors Algebra 2
Grade Level: 9-11
Unit 1: Operations and Solving Linear
Equations
3 Weeks
Unit 2: Linear Functions & Inequalities
3 Weeks
Unit 3: Systems of Equations &
Inequalities
3 Weeks
Unit 4: Quadratic Functions &
Equations
3 Weeks
Unit 5: Polynomial Functions &
Equations
3 Weeks
Unit 6: Sequences, Series, & Matrices
3.5 Weeks
Board Approved on: June 13, 2012
7
South Plainfield Public Schools Curriculum Guide
Content Area: Mathematics
Course Title: Honors Algebra 2
Grade Level: 9-11
Unit 7: Probability & Statistics
3.5 Weeks
Unit 8: Rational Functions & Equations
3.5 Weeks
Unit 9: Powers, Roots, & Radicals
3.5 Weeks
Unit 10: Exponential & Logarithmic
Functions
3.5 Weeks
Unit 11: Trigonometric Functions
3.5 Weeks
Board Approved on: June 13, 2012
8
Unit 1 Overview
Content Area – Mathematics
Unit Title 1: Operations and Solving Linear Equations
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to follow the order of operations; identify
like- and non-like terms; apply inverse operations to solve linear equations and inequalities
algebraically; graph and solve single-variable absolute value equations
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
Smart Board
21st Century Themes –
Financial Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
Domain Standards:
A.CED.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
A.CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations.
A.SSE.1: Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
9
A.REI.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. (Review from Algebra 1)
A.REI.3: Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters. (Review from Algebra 1)
F.IF.4: For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
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
Unit Essential Questions
 What is absolute value and how can it be applied to global situations?
 How does modeling equations facilitate problem solving?
Terminology:
 Variable
 Inverse Operations
 Solution Set
Unit Enduring Understandings
 Order of Operations to Evaluate/Simplify
 Order of Operations in Reverse to Solve
Goals/Objectives
Students will be able
to solve single variable
linear equations
recognize appropriate
use of inverse
operations
Recommended Learning
Activities/Instructional Strategies




construct absolute
value graphs for
simple equations and
compound equations
and inequalities
(single variable)
Instructional Strategies
Whole Group Lessons
Small Group Explorations
- In-class group work
Independent Practice
- Homework
Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student
Teaching Cooperatively With Students
of Varied Levels of Understanding
3.Think Pair Share
Evidence of Learning
(Formative &
Summative)
Quiz – Apply Order of
Operations Correctly to
Simplify Various
Algebraic Expression
Quiz – Apply Order of
Operations and
Knowledge of Inverse
Operations to Solve Linear
Equations
Test – Cumulative
Assessment of First Two
Quizzes
Alternative or projectbased assessments will be
evaluated using a teacherselected or created rubric
or other instrument
10
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
11
Unit 2 Overview
Content Area – Mathematics
Unit 2: Linear Functions & Inequalities
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students should be able to understand and interpret characteristics
of linear functions using a table of values, a graph, and/or function rule. Specific to Linear Functions,
students should be able to identify intercepts and slope/rate-of-change, as well as differentiate
between vertical & horizontal, positive & negative slopes. Additionally, students will be able to
graph linear inequalities and have the understanding that each and every ordered pair in the shaded
region represents a solution to the inequality.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
Smart Board
21st Century Themes –
Financial Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
12
Domain Standards:
A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an
expression, such as terms, factors, and coefficients.
A.CED.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
A.CED.2: Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
A.REI.3: Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters. (Review from Algebra 1)
A.REI.10: Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line). (Review from Algebra
1)
A.REI.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the
boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes. (Review from
Algebra 1)
F.IF.1: Understand that a function from one set (called the domain) to another set (called the range)
assigns to each element of the domain exactly one element of the range. If f is a function and x is an
element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is
the graph of the equation y = f(x). (Review from Algebra 1)
F.IF.4: For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
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.6: Calculate and interpret the average rate of change of a function (presented symbolically or as
a table) over a specified interval. Estimate the rate of change from a graph
F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k 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.
Unit Essential Questions
 What is slope and what does it represent?
 How can linear functions be used to model and
interpret real-world applications?
Unit Enduring Understandings
 The slope of a linear function is the same
between any two points on the line.
 Slopes of Parallel Lines are Equal; Slopes of
Perpendicular Lines are Opposite Reciprocals
Key Concepts/Vocabulary/Terminology/Skills:
Solution Set, Shaded Region, Slope, Rate of Change, Parallel, Perpendicular, Horizontal, Vertical
13
Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
create a table of values
from a function rule
plot ordered pairs on the
coordinate plane and
sketch a graph of the
function




interpret key features of
graphs and tables in terms
of the quantities, and
sketch graphs showing
key features given a
verbal description of the
relationship
Instructional Strategies
Whole Group Lessons
Small Group Explorations
- In-class group work
Independent Practice
- Homework
Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student
Teaching Cooperatively With Students
of Varied Levels of Understanding
3.Think Pair Share
Evidence of Learning
(Formative &
Summative)
Quiz – graphing linear
functions, writing
equations of lines
Quiz – using test
points, solving linear
inequalities
algebraically,
graphing linear
inequalities
Test – Cumulative
Assessment on both
quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
graph a function given the
rule in slope-intercept
form and standard form
write the equation of a
line in point-slope or
slope-intercept form
given multiple
descriptions of the line
graph solutions to a twovariable linear inequality
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
14
Unit 3 Overview
Content Area – Mathematics
Unit 3: Systems of Equations & Inequalities
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to construct a system of equations and
distinguish between appropriate methods of solving systems: develop a cost and income equation
using linear programming and apply to global economy: selecting correct coordinates to check
inequalities and their graphs; graph and solve two-variable absolute value inequalities
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Graphing calculators to illustrate intersecting points: Scientific
Calculators When Deemed Absolutely Necessary, Overhead, SMART Board
21st Century Themes –Financial, Economic,
Business, and Entrepreneurial Literacy
Global Awareness
21st Century Skills – Creativity and
Innovation, Critical Thinking and Problem
Solving, Communication and Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
15
Domain Standards:
N.Q.2: Define appropriate quantities for the purpose of descriptive Modeling (Review from Algebra
1)
A.CED.2: Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on
combinations of different foods
A.REI.5: Prove that, given a system of two equations in two variables, replacing one equation by the
sum of that equation and a multiple of the other produces a system with the same solutions (Review
from Algebra 1)
A.REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing
on pairs of linear equations in two variables (Review from Algebra 1)
A.REI.10: Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line). (Review from Algebra
1)
A.REI.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and
y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g.,
using technology to graph the functions, make tables of values, or find successive approximations.
Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.★
A.REI.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the
boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes. (Review from
Algebra 1)
Unit Essential Questions
 What is the best way to solve a linear system?
 Why would a business owner need to understand
linear programming?
 Why would inequalities be more useful than
equations in illustrating systems of inequalities?
Unit Enduring Understandings
 Ability to solve systems of equations in
multiple ways
 To be able to interpret inequalities and what
the graph mean
 Use linear programming to illustrate business
profit and loss
Terminology: Systems, Linear Programming, Substitution, Elimination
16
Goals/Objectives
Students will be able to:
Recommended Learning
Activities/Instructional Strategies
solve two linear equations
with multiple variables
Instructional Strategies
 Whole Group Lessons
 Small Group Explorations
- In-class group work
 Independent Practice
- Homework
 Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student Teaching
Cooperatively With Students of Varied Levels
of Understanding
3.Student based lesson- Have students
decipher word problem and explain set up of
equations to other students
evaluate output values in
cost/expense ratio
formulas
sketch inequalities in two
variables and
appropriately identify
solutions
Evidence of Learning
(Formative &
(Summative)
Quiz- Apply different
methods to solving
systems of equations
Quiz- Graph systems
of inequalities and use
knowledge to evaluate
linear programming
max and min values
Test- Cumulative
assessment f first two
quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
17
Unit 4 Overview
Content Area – Mathematics
Unit 4: Quadratic Functions & Equations
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students should be able to differentiate between linear and quadratic
functions, as well as be able to choose the more appropriate type to model a real world problem.
Students will use the vertex of a parabola to determine the maximum/minimum, and interpret its
coordinates in the context of a real world problem. Also in this unit, students will extend the domain
and range to the set of complex numbers, and apply their understanding of number properties and
operations to this set.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
Smart Board, and Graphing Calculators can be used by the teacher, via overhead, to demonstrate the
effects that changing the values of a b and c (in the quadratic function) have on the graph.
21st Century Themes –
21st Century Skills –
Financial, Economic, Business, and Entrepreneurial Critical Thinking & Problem Solving
Literacy, Environmental Literacy
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
18
Domain Standards:
N.CN.1: Know there is a complex number i such that i2 = −1, and every complex number has the form
a + bi with a and b real
N.CN.2: Use the relation i2 = –1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers
N.CN.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of
complex numbers
N.CN.7: Solve quadratic equations with real coefficients that have complex solutions
N.CN.8: Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x +
2i)(x – 2i)
N.CN.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials
A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an
expression, such as terms, factors, and coefficients.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as
(x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2).
(Review from Algebra 1)
A.SSE.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1),
and use the formula to solve problems. For example, calculate mortgage payments
F.IF.4: For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
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.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and
showing end behavior.
F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros,
extreme values, and symmetry of the graph, and interpret these in terms of a context.
F.IF.9: Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger maximum
F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k 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.
Unit Essential Questions
 How can the properties of a quadratic function
be used to model, analyze, and solve real world
problems?
Unit Enduring Understandings
 The graph of a quadratic function is a parabola.
 A table of values can always be used to graph a
function.
19
 Why are relations and functions represented in
multiple ways?
 Solution sets to both quadratic equations and
inequalities can be displayed using a graph of
the function.
Key Concepts/Vocabulary/Terminology/Skills:
Increasing, Decreasing, Average Rate of Change, Vertex, Axis of Symmetry, Quadratic Formula,
Roots, Factors, Zeros, Parabola, Square Root Property
Goals/Objectives
Students will be able to solve quadratic equations
using inverse operations
(square root property)
solve quadratic equations
by factoring and by
applying the quadratic
formula
graph quadratic functions
by completing a table of
values
interpret key features of
graphs and tables in terms
of the quantities, and
sketch graphs showing key
features given a verbal
description of the
relationship
Recommended Learning
Activities/Instructional Strategies




Instructional Strategies
Whole Group Lessons
Small Group Explorations
- In-class group work
Independent Practice
- Homework
Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student
Teaching Cooperatively With Students
of Varied Levels of Understanding
3.Think Pair Share
4. Written Paper detailing the different
methods for solving a quadratic
equation, discussing the pros/cons,
likes/dislikes, etc.
simplify and perform
operations on complex
numbers
20
Evidence of Learning
(Formative &
Summative)
Quiz – solving
quadratic equations
Quiz – graphing
quadratic functions
Test – Cumulative
Assessment on first
two quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment. Modifications
may also be made as they relate to the special needs of students in accordance with their
Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These
may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources- Glencoe McGraw Hill Algebra 2, 2010
21
Unit 5 Overview
Content Area – Mathematics
Unit 5: Polynomial Functions & Equations
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to identify polynomial equations by name and
identify specific terms in a polynomial: interpret the fundamental theorem of algebra to solve higher
order polynomial equations: factor higher order polynomial functions using known patterns
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
SMART Board
21st Century Themes – Global awareness
Financial, Economic, Business and Entrepreneurial
Literacy
21st Century Skills – Creativity and
Innovation Critical thinking and
Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
Domain Standards:
A.SSE.1: Interpret expressions that represent a quantity in terms of its
context.★
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For
example, interpret P (1+r)n as the product of P and a factor not depending on P.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example: see x4 – y4
22
as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 +
y2).
A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they are
closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply
polynomials
A.APR.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).
A.APR.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.
A.APR.4: Prove polynomial identities and use them to describe numerical relationships. For
example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
A.APR.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +
r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree
of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra
system
F.BF.1: Write a function that describes a relationship between two quantities.★
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
b. Combine standard function types using arithmetic operations. For example, build a function
that models the temperature of a cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a
function of height, and h (t) is the height of a weather balloon as a function of time, then T
(h (t)) is the temperature at the location of the weather balloon as a function of time.
Unit Essential Questions
 How are operations with polynomials different
than operations with real numbers?
 How do you determine the amount of solutions a
given polynomial will have?
Unit Enduring Understandings
 Demonstrate a knowledge of how to solve
polynomial equations to different degrees
 Identify terms and degrees with the
appropriate terminology
 Apply the remainder theorem appropriately
and understand the concept behind the
theorem
Key Concepts/Vocabulary/Terminology/Skills: Polynomial, Fundamental Theorem of Algebra
23
Goals/Objectives
Students will be able to organize polynomials in
standard form and name
by degree and term
find possible solutions to
polynomial equations and
use irrational and
imaginary root theorems
to understand possibilities
Recommended Learning
Activities/Instructional Strategies




Instructional Strategies
Whole Group Lessons
Small Group Explorations
- In-class group work
Independent Practice
- Homework
Class Discussion
Activities
solve polynomial
equations using remainder
theorem, factoring, and
previously learned
methods
students will use end
behavior to analyze
polynomial functions and
determine zeros
perform operations on
polynomials (addition,
subtraction,
multiplication, division,
composition)
determine domain and
range of a polynomial
function
determine extrema of a
function
identify intervals of the
domain in which a
function is
increasing/decreasing
1.Textbook Activities
2.Group Math Tutor – Strong Student Teaching
Cooperatively With Students of Varied Levels
of Understanding
3. Think-Pair-Share. Using synthetic division
and the remainder theorem each individual pair
tries different roots to find first root. Students
are able to work independently while also using
their partner to check work and find solutions
at a quicker pace
4. Telephone – A student begins with a
polynomial function, finds all of the roots. The
roots are passed on to a second student, who
uses the roots to find the ‘original function.’
The function is then passed on to a third
student, who finds the roots all over again. This
continues on with a set number of students, the
object of the activity being to have the same
function in the end that you had in the
beginning.
5. Communication – A student seated with their
back to the board describes a function to a
student at the board. The object is for the seated
student to give a sufficiently detailed
explanation of the function graph so the student
at the board is able to accurately sketch the
same function.
24
Evidence of Learning
(Formative &
Summative)
Quiz- Identify
polynomial names and
understanding
synthetic division
Quiz- Solve
polynomial equations
using remainder
theorem and apply
factoring methods to
larger order
polynomials
Test- Cumulative
assessment f first two
quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
25
Unit 6 Overview
Content Area – Mathematics
Unit 6: Sequence, series, and matrices
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to identify patterns in order to develop
appropriate formulas and apply to certain problems. They will also be able to compute sums of series
and use derived formulas to complete the task. Finally, students will be able to sketch and organize
simple matrices and demonstrate an ability to complete simple mathematical operations using
matrices
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
SMART Board. Graphing calculator can be used by teacher to demonstrate how to enter a matrix into
a graphing calculator
21st Century Themes –
Financial, Economic, Business, and
Entrepreneurial Literacy
Environmental literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
26
Domain Standards:
A.SSE.4: Derive the formula for the sum of a finite geometric series (when the common ratio is not
1), and use the formula to solve problems. For example, calculate mortgage payments.★
F.IF.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers. For example, the Fibonacci sequence is defined recursively by f (0) = f (1) = 1,
f (n+1) = f (n) + f (n-1) for n ≥ 1.
F.BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between the two forms.★
N.VM.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network (HSPA)
N.VM.7: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a
game are doubled. (HSPA)
N.VM.8: Add, subtract, and multiply matrices of appropriate dimensions (HSPA)
Unit Essential Questions
 How do sequences and series model realworld problems and their solutions?
 How can matrices help visualize and
organize data for use with other
applications?
Unit Enduring Understandings
 Sequences and series are discrete
functions whose domain is the set of
whole numbers
 Matrices are commonly used to organize
data while comparing and contrasting
information
Terminology: Scalars, Commutative property, Sequence, Series, Infinite, Arithmetic, Geometric,
Fibonacci
Goals/Objectives
Students will be able to sketch matrices of different
sizes and organize data
appropriately
use matrix properties to
add, subtract, and multiply
by a scalar with matrices
identify a sequence and
design a formula based on
the sequence
evaluate sum of arithmetic
and geometric series
perform matrix
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
 Whole Group Lessons
 Small Group Explorations
- In-class group work
 Independent Practice
- Homework
 Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student
Teaching Cooperatively With Students of
Varied Levels of Understanding
3. Research. A) Find a newspaper article
where matrices were used to compare and
27
Evidence of Learning
(Formative &
Summative)
Quiz- Complete simple
operations with
matrices
Quiz- Identify patterns
and understand
sequence and series
Quiz- Recognize sum
formulas and apply
appropriately
Test- Cumulative
multiplication, understand
it is not commutative,
determine whether matrices
are inverses (emphasize the
recurring idea of inverses
and identity elements)
contrast data. B) Real-life series (Choose
real-life situations which use arithmetic or
geometric sequences and series. You must
have at least one of each type for this project.)
After data is acquired teacher can adapt
project to fit idea in lesson and develop and
change formulas. Also can use for computer
programming if applicable.
assessment of first
three quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment. Modifications
may also be made as they relate to the special needs of students in accordance with their
Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL). These
may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
28
Unit 7 Overview
Content Area – Mathematics
Unit 7: Probability & Statistics
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to calculate probabilities, identify types of
samples, find measures of central tendency, calculate standard deviation, make statistical decisions
based on the normal curve, and use the binomial theorem to expand polynomials and calculate
probability
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
SMART Board. Graphing calculator can be used to demonstrate how calculator can obtain statistical
data in a much simpler way
21st Century Themes –
Financial, Economic, Business, and Entrepreneurial
Literacy
Global awareness
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Creativity and Innovation
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
29
Domain Standards:
A.APR.5: Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y
for a positive integer n, where x and y are any numbers, with coefficients determined for example by
Pascal’s Triangle
S.ID.4: Use the mean and standard deviation of a data set to fit it to a normal distribution and to
estimate population percentages. Recognize that there are data sets for which such a procedure is not
appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
S.IC.1: Understand statistics as a process for making Inferences about population parameters based on
a random sample from that population.
S.IC.2: Decide if a specified model is consistent with results from a given data-generating process,
e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5.
Would a result of 5 tails in a row cause you to question the model?
S.IC.3: Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.
S.IC.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin
of error through the use of simulation models for random sampling.
S.IC.5: Use data from a randomized experiment to compare two treatments; use simulations to decide
if differences between parameters are significant.
S.IC.6: Evaluate reports based on data
S.MD.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
S.MD.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game
Unit Essential Questions
 How can probability be applied in realworld situations and used to make
predictions?
 How can the curve of best fit help predict
trends of data?
 How can probability help make
unemotional decisions?
Unit Enduring Understandings
 More advanced concepts of probability
can be used to solve real-word problems.
 Data sets can be analyzed to form
hypotheses and make predictions using
different types of statistical models
 Measures of central tendency can be
understood more clearly and used to
relate data accordingly
Terminology: Mean, median, mode, standard deviation, z-score, probability, conditional probability,
normal curve, outlier, Binomial Theorem
30
Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
evaluate mean, median,
and mode
Instructional Strategies
 Whole Group Lessons
 Small Group Explorations
evaluate standard
- In-class group work
deviation and make
 Independent Practice
decisions using data
- Homework
calculate probabilities and
 Class Discussion
understand the
Activities
ramifications of certain
1.Textbook Activities
probabilities
2. Think-pair-share. Students should use
use standard normal curve partners to check answers when calculating
statistical parameters and assigning
to approximate range of
appropriately.
data and identify outliers
3. Statistical experiment. Students should
expand polynomials using conduct an experiment with appropriate
sampling method to obtain data and then fit
Binomial Theorem and
data into a normal curve explaining how
Pascal’s triangle
values were obtained and the meaning of
use Binomial Theorem to standard deviation. Students can present
findings in a report or as a class project to the
calculate binomial
entire class. This may also be done as a group
probabilities
project
4. Group project. Students will use probability
identify specific types of
to obtain data to help explain why people
samples and select
make the decisions they do in real world
appropriate sampling
situations. Simple hypothesis testing may be
method to obtain data
introduced if time permits. Examples may be
blackjack strategy, safety ratings on cars,
baseball statistics, etc.
Evidence of Learning
(Formative &
Summative)
Quiz- Probability
models and simple
sampling
Quiz- Standard
deviation, the standard
normal curve, and the
Binomial theorem
Test- Cumulative
assessment of first two
quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
31
Unit 8 Overview
Content Area – Mathematics
Unit 8: Rational Functions & Equations
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Building upon their understanding of other function types,
students will identify key characteristics of rational functions such as asymptotes, domain, and
range. Necessary skills include factoring polynomials as well as performing operations on fractions
and polynomials.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
Smart Board. Graphing calculator can be used by the teacher to enhance the presentation and
understanding of graphs of functions and the corresponding table of values.
21st Century Themes –
Financial, Economic, Business, and Entrepreneurial
Literacy
Environmental Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
32
Domain Standards:
A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of
an expression, such as terms, factors, and coefficients.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as
(x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2).
A.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +
r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree
of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra
system
A.APR.7: Understand that rational expressions form a system analogous to the rational numbers,
closed under addition, subtraction, multiplication, and division by a nonzero rational expression;
add, subtract, multiply, and divide rational expressions
A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing
how extraneous solutions may arise
A.REI.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x)
and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,
e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions
F.IF.4: For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal
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.5: Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives the number of person-hours it takes
to assemble n engines in a factory, then the positive integers would be an appropriate domain for the
function
F.IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as
a table) over a specified interval. Estimate the rate of change from a graph
Unit Essential Questions
 How are the properties of functions and
functional operations useful?
 How are rational functions related to rational
numbers?
Unit Enduring Understandings
 Performing Operations on Rational
Expressions is Exactly Like Performing
Operations on Fractions
 Not being able to divide by zero is the reason
for restrictions in the domain of a rational
function.
Key Concepts/Vocabulary/Terminology/Skills:
Asymptote, Limit, Removable and Non-Removable Discontinuity, Increasing, Decreasing, End
Behavior, Domain, Range
33
Goals/Objectives
Students will be able to extend knowledge of
operating on fractions to
operating on rational
expressions
Recommended Learning
Activities/Instructional Strategies



identify domain and range
of a rational function
factor polynomials to
identify greatest common
factor / common
denominator
students will reduce
complex fractions and
understand how a
complex fraction is
simplified
graph rational
expressions

Instructional Strategies
Whole Group Lessons
Small Group Explorations
- In-class group work
Independent Practice
- Homework
Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student
Teaching Cooperatively With Students
of Varied Levels of Understanding
3.Think Pair Share
4. Communication – A student seated
with their back to the board describes a
function to a student at the board. The
object is for the seated student to give
a sufficiently detailed explanation of
the function graph so the student at the
board is able to accurately sketch the
same function.
determine extrema of a
function
identify intervals of the
domain in which a
function is
increasing/decreasing
identify and understand
occurrence of asymptotes,
points of removable
discontinuity (holes)
understand why a
function can cross a
horizontal or slant
asymptote but not a
34
Evidence of Learning
(Formative &
Summative)
Quiz – Performing
Operations on
Rational Expressions,
Solving Rational
Equations
Quiz – Graphing
Rational Functions
Test – Cumulative
Assessment on first
two quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
vertical asymptote
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
35
Unit 9 Overview
Content Area – Mathematics
Unit 9: Powers, Roots, & Radicals
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to evaluate and simplify expressions
involving powers and radicals. They will have the understanding that powers and radicals have an
inverse relationship and can be used to solve equations. Building upon their understanding of square
roots, students will interpret, simplify, operate on, and solve equations involving radical expressions
with indices greater than two.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
SMART Board. Graphing calculator can be used by teacher to demonstrate via graph and table of
values the inverse relationship between powers and radicals.
21st Century Themes –
Financial, Economic, Business, and Entrepreneurial
Literacy
Environmental Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
36
Domain Standards:
A.SSE.1: Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of
an expression, such as terms, factors, and coefficients.
A.SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4
as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 +
y2).
A.APR.5: Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y
for a positive integer n, where x and y are any numbers, with coefficients determined for example by
Pascal’s Triangle
A.REI.2: Solve simple rational and radical equations in one variable, and give examples showing
how extraneous solutions may arise
F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k 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.
F.BF.4: Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write
an expression for the inverse. For example,
f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that the function has
an inverse
Unit Essential Questions
Unit Enduring Understandings
 What’s the most efficient way to simplify a
radical expression?
 How are expressions and functions involving
radicals and exponents related?
 What are the strengths and weaknesses of radical
expressions? of exponential expressions?
 Power and radical functions add to our
arsenal of inverse functions and operations
and can be used to model and solve real
world applications.
Terminology: Prime Factorization, Exponent, Index Number, Radical, Radicand, Rational
Exponent, Properties of Exponents
37
Goals/Objectives
Students will be able to extend properties of
integer exponents to
rational exponents
convert to and from
exponential form/radical
form, understand
appropriateness and
benefits of each form
evaluate radical
expressions
solve radical equations
Recommended Learning
Activities/Instructional Strategies




Instructional Strategies
Whole Group Lessons
Small Group Explorations
- In-class group work
Independent Practice
- Homework
Class Discussion
Activities
1.Textbook Activities
2.Group Math Tutor – Strong Student
Teaching Cooperatively With
Students of Varied Levels of
Understanding
3.Think Pair Share
perform binomial
expansions using Pascal’s
Triangle
Evidence of Learning
(Formative &
Summative)
Quiz – simplifying
exponential and radical
expressions, binomial
expansion
Quiz – solving
exponential and radical
equations
Test – Cumulative
Assessment
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
38
Unit 10 Overview
Content Area – Mathematics
Unit 10: Exponential and Logarithmic Functions
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Students will be able to extend their knowledge of exponential
functions from to include real number powers. Properties and graphs, including transformed graphs, of
exponential functions are studied. The number e is explored as the natural base of exponential
functions. Applications include exponential growth and decay, as well as modeling of exponential
functions from data. Logarithmic functions are introduced as inverses of exponential functions. The
properties and graphs of logarithmic functions are studied, including transformed graphs. Students
solve exponential and logarithmic equations by equating bases and by using the inverse relationship
between logarithms and exponents.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
SMART Board. Graphing calculator can be used by teacher to illustrate exponential and logarithmic
graphs
21st Century Themes –
Financial, Economic, Business, and Entrepreneurial
Literacy
Global awareness
Health Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Creativity & Innovation
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
39
Domain Standards:
N.RN.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. (Review from Algebra 1)
N.RN.2: Rewrite expressions involving radicals and rational exponents using the properties of
exponents (Review from Algebra 1)
F.IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.★
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.
F.IF.8: Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
b. Use the properties of exponents to interpret expressions for exponential functions. For example,
identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y =
(1.2)t/10, and classify them as representing exponential growth or decay.
F.LQE.3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds
a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F.LQE.4: For exponential models, express as a logarithm the solution to a bct = d where a, c, and d are
numbers and the base b is 2, 10, or e; evaluate the logarithm using technology
Unit Essential Questions
 How do exponential functions model real-world
problems and their solutions?
 How do logarithmic functions model real-world
problems and their solutions?
 How are expressions involving exponents and
logarithms related?
Unit Enduring Understandings
 The characteristics of exponential and
logarithmic functions and their representations
are useful in solving real-world problems
 Translating equations from logarithmic form to
exponential form can be a useful tool for
completing appropriate operations
Terminology: Logarithmic, Exponential, Interest, Decay, Growth, Asymptote, Natural Logs,
Continuously Compounding Interest
40
Goals/Objectives
Students will be able to model exponential growth
and decay
to introduce the number e
to apply constants to the
equation y=abx
to evaluate logarithmic
expressions
to sketch exponential and
logarithmic graphs
to understand and apply
the properties of
logarithms
to solve exponential
equations
to solve logarithmic
equations
to introduce and evaluate
natural logarithms
to solve equations using
natural logarithms
Recommended Learning
Activities/Instructional Strategies
Instructional Strategies
 Whole Group Lessons
 Small Group Explorations
- In-class group work
 Independent Practice
- Homework
 Class Discussion
Activities
1.Textbook Activities
2. Think-pair-share. Students should use
partners to check answers when calculating
statistical parameters and assigning
appropriately.
3. Think of one real situation that involves
exponential growth and that involves
exponential decay. for each example, your
project should include the following:
* Paragraph - Briefly explain the situation.
You may make up your own information, but
make it realistic. Include the facts needed to
write an equation.
* Equation - Model the situation with an
exponential equation.
* Graph - Make a graph of the exponential
function. Be sure to label what each axis
represents and use an appropriate scale.
to explain inverse
variation and use to solve
equations
solve real-life problems
including but not limited
to compound interest,
half-life, logistic
representations of
population growth
41
Evidence of Learning
(Formative &
Summative)
Quiz- Exponential
growth/decay and the
exponential equation
Quiz- Log functions
and solving
exponential/logarithmi
c equations
Test- Cumulative
assessment of first two
quizzes
Alternative or projectbased assessments will
be evaluated using a
teacher-selected or
created rubric or other
instrument
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
42
Unit 11 Overview
Content Area – Mathematics
Unit 11: Trigonometric Functions
Target Course/Grade Level – Honors Algebra 2 for Grades 9 through 11
Unit Summary and Rationale – Introducing Pre-Calculus concepts, students will be introduced to
the unit circle, radian/degree conversions, moving on to the concept of periodic and oscillating
functions. The graphs of trigonometric functions will be explored and used to analyze and solve real
world problems. They will also use their knowledge of 30-60-90’s and 45-45-90’s from geometry to
prove trigonometric identities.
Interdisciplinary Connections – Science and Language Arts: journal writing; open-ended extended
response questions.
Technology standard(s) – Scientific Calculators When Deemed Absolutely Necessary, Overhead,
Smart Board
21st Century Themes –
Financial, Economic, Business, and Entrepreneurial
Literacy, Environmental Literacy
21st Century Skills –
Critical Thinking & Problem Solving
Communication & Collaboration
Productivity & Accountability
Initiative & Self-Direction
Learning Targets
Content and Practice Standards:
MP.1: Make sense of problems and persevere in solving them.
MP.2: Reason abstractly and quantitatively.
MP.3: Construct viable arguments and critique the reasoning of others.
MP.4: Model with mathematics.
MP.5: Use appropriate tools strategically.
MP.6: Attend to precision.
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
43
Domain Standards:
F.TF.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended
by the angle.
F.TF.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise
around the unit circle
F.TF.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for π /3,
π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x,
and 2π-x in terms of their values for x, where x is any real number
F.TF.5: Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline
F.TF.8: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it find sin(θ), cos(θ), or tan(θ)
given sin(θ), cos(θ), or tan(θ) and the quadrant
Unit Essential Questions
 Why are triangles so important?
 How are trigonometric functions, Pythagorean
Theorem, and other properties of triangles
related?
 What natural phenomena can be modeled using
trigonometric functions?
Unit Enduring Understandings
 Definition of Sine Cosine and Tangent
 Reference Angles are based upon 30-60-90’s
and 45-45-90’s and their sine and cosine
values are reflected over the x- and y- axes
to obtain the sine and cosine values in the
unit circle.
 Triangle Similarity is the reason we are able
to use trigonometric ratios to solve triangles.
Key Concepts/Vocabulary/Terminology/Skills:
Reference Angles, Unit Circle, Radian, Co-terminal Angles, Amplitude, Period, Midline, Frequency
Goals/Objectives
Students will be able to -
Recommended Learning
Activities/Instructional Strategies
recall properties of
special right triangles and
use them to derive the
unit circle
Instructional Strategies
using coordinates from
the unit circle, extend
trigonometric functions to
all real numbers and
graph trigonometric
functions

Whole Group Lessons
Evidence of Learning
(Formative &
Summative)
Quiz – derive the unit
circle, identifying trig
ratios

Small Group Explorations
- In-class group work
Quiz – graphing
trigonometric functions

Independent Practice
- Homework
Test – Cumulative
Assessment on Quizzes

Class Discussion
Alternative or project-
44
interpret key properties of
trigonometric function
Activities
based assessments will be
evaluated using a teacherselected or created rubric
or other instrument
1.Textbook Activities
2.Group Math Tutor – Strong
Student Teaching Cooperatively
With Students of Varied Levels of
Understanding
3.Think Pair Share
Diverse Learners (ELL, Special Ed, Gifted & Talented)- Differentiation strategies may include,
but are not limited to, learning centers and cooperative learning activities in either heterogeneous or
homogeneous groups, depending on the learning objectives and the number of students that need
further support and scaffolding, versus those that need more challenge and enrichment.
Modifications may also be made as they relate to the special needs of students in accordance with
their Individualized Education Programs (IEPs) or 504 plans, or English Language Learners (ELL).
These may include, but are not limited to, extended time, copies of class notes, refocusing strategies,
preferred seating, study guides, and/or suggestions from special education or ELL teachers.
Resources - Glencoe McGraw Hill Algebra 2, 2010
45
Mathematics: Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on
important “processes and proficiencies” with longstanding importance in mathematics
education. The first of these are the NCTM process standards of problem solving, reasoning and
proof, communication, representation, and connections. The second are the strands of
mathematical proficiency specified in the National Research Council’s report Adding It Up:
adaptive reasoning, strategic competence, conceptual understanding (comprehension of
mathematical concepts, operations and relations), procedural fluency (skill in carrying out
procedures flexibly, accurately, efficiently and appropriately), and productive disposition
(habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a
belief in diligence and one’s own efficacy).
1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem
and looking for entry points to its solution. They analyze givens, constraints, relationships, and
goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous
problems, and try special cases and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their progress and change course if
necessary. Older students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get the information
they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on
using concrete objects or pictures to help conceptualize and solve a problem. Mathematically
proficient students check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?” They can understand the approaches of
others to solving complex problems and identify correspondences between different
approaches.
2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of the quantities and their relationships in
problem situations. Students bring two complementary abilities to bear on problems involving
quantitative relationships: the ability to decontextualize—to abstract a given situation and
represent it symbolically and manipulate the representing symbols as if they have a life of their
own, without necessarily attending to their referents—and the ability to contextualize, to pause
as needed during the manipulation process in order to probe into the referents for the symbols
involved. Quantitative reasoning entails habits of creating a coherent representation of the
problem at hand; considering the units involved; attending to the meaning of quantities, not
46
just how to compute them; and knowing and flexibly using different properties of operations
and objects.
3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and
previously established results in constructing arguments. They make conjectures and build a
logical progression of statements to explore the truth of their conjectures. They are able to
analyze situations by breaking them into cases, and can recognize and use counterexamples.
They justify their conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that take into account
the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct, even though
they are not generalized or made formal until later grades. Later, students learn to determine
domains to which an argument applies. Students at all grades can listen or read the arguments
of others, decide whether they make sense, and ask useful questions to clarify or improve the
arguments.
4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace. In early grades, this might be as simple as
writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high
school, a student might use geometry to solve a design problem or use a function to describe
how one quantity of interest depends on another. Mathematically proficient students who can
apply what they know are comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to identify
important quantities in a practical situation and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical
results in the context of the situation and reflect on whether the results make sense, possibly
improving the model if it has not served its purpose.
5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their
grade or course to make sound decisions about when each of these tools might be helpful,
47
recognizing both the insight to be gained and their limitations. For example, mathematically
proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can
enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are
able to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to
explore and deepen their understanding of concepts.
6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use
clear definitions in discussion with others and in their own reasoning. They state the meaning of
the symbols they choose, including using the equal sign consistently and appropriately. They
are careful about specifying units of measure, and labeling axes to clarify the correspondence
with quantities in a problem. They calculate accurately and efficiently, express numerical
answers with a degree of precision appropriate for the problem context. In the elementary
grades, students give carefully formulated explanations to each other. By the time they reach
high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young
students, for example, might notice that three and seven more is the same amount as seven
and three more, or they may sort a collection of shapes according to how many sides the
shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression x2 + 9x + 14, older
students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed
of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a
square and use that to realize that its value cannot be more than 5 for any real numbers x and
y.
8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for
general methods and for shortcuts. Upper elementary students might notice when dividing 25
by 11 that they are repeating the same calculations over and over again, and conclude they
have a repeating decimal. By paying attention to the calculation of slope as they repeatedly
check whether points are on the line through (1, 2) with slope 3, middle school students might
abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when
48
expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the
general formula for the sum of a geometric series. As they work to solve a problem,
mathematically proficient students maintain oversight of the process, while attending to the
details. They continually evaluate the reasonableness of their intermediate results.
49
Common Core Standards
(Bold Apply to Algebra 2)
The Complex Number System N-CN
Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i2 = −1, and every complex number has
the form a + bi with a and b real.
2. Use the relation i2 = –1 and the commutative, associative, and distributive properties to
add, subtract, and multiply complex numbers.
3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of
complex numbers.
Represent complex numbers and their operations on the complex plane.
4. (+) Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar forms of
a given complex number represent the same number.
5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation.
For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
6. (+) Calculate the distance between numbers in the complex plane as the modulus of the
difference, and the midpoint of a segment as the average of the numbers at its endpoints.
Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4
as (x + 2i)(x – 2i).
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.
Seeing Structure in Expressions A-SSE
Interpret the structure of expressions.
1. Interpret expressions that represent a quantity in terms of its context.★
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1+r)n as the product of P and a factor not
depending on P.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 –
y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2
– y2)(x2 + y2).
Write expressions in equivalent forms to solve problems.
3. Choose and produce an equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.★
a.
Factor a quadratic expression to reveal the zeros of the function it defines.
50
Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to
reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
4. Derive the formula for the sum of a finite geometric series (when the common ratio is
not 1), and use the formula to solve problems. For example, calculate mortgage
payments.★
b.
Arithmetic with Polynomials and Rational Expressions A-APR
Perform arithmetic operations on polynomials.
1. Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
Understand the relationship between zeros and factors of polynomials.
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).
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.
Use polynomial identities to solve problems.
4. Prove polynomial identities and use them to describe numerical relationships. For
example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers
of x and y for a positive integer n, where x and y are any numbers, with coefficients
determined for example by Pascal’s Triangle.1
Rewrite rational expressions.
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form
q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)
less than the degree of b(x), using inspection, long division, or, for the more
complicated examples, a computer algebra system.
7. (+) Understand that rational expressions form a system analogous to the rational
numbers, closed under addition, subtraction, multiplication, and division by a nonzero
rational expression; add, subtract, multiply, and divide rational expressions.
Reasoning with Equations and Inequalities A-REI
Understand solving equations as a process of reasoning and explain the reasoning.
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.
1
The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.
51
2. Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
Solve equations and inequalities in one variable.
3. Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula
from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), 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 a ± bi for real numbers a and b.
Solve systems of equations.
5. Prove that, given a system of two equations in two variables, replacing one equation by
the sum of that equation and a multiple of the other produces a system with the same solutions.
6. Solve systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
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 y = –3x and the circle x2 + y2 = 3.
8. (+) Represent a system of linear equations as a single matrix equation in a vector
variable.
9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear
equations (using technology for matrices of dimension 3  3 or greater).
Represent and solve equations and inequalities graphically.
10. Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line).
11.Explain why the x-coordinates of the points where the graphs of the equations y =
f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.★
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding
the boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Trigonometric Functions F-TF
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
2. Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle.
52
3.
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π
/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x,
π+x, and 2π-x in terms of their values for x, where x is any real number.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.★
6. (+) Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context.★
Prove and apply trigonometric identities.
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it find sin(θ), cos(θ), or
tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and
use them to solve problems.
Creating Equations★ A-CED
Create equations that describe numbers or relationships.
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.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Interpreting Functions F-IF
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set (called the range)
assigns to each element of the domain exactly one element of the range. If f is a function and x is
an element of its domain, then f(x) denotes the output of f corresponding to the input x. The
graph of f is the graph of the equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1)
= 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context.
53
4. For a function that models a relationship between two quantities, interpret key features
of graphs and tables in terms of the quantities, and sketch graphs showing key features
given a verbal 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.★
5. Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.★
6. Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph.★
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph, by hand
in simple cases and using technology for more complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showing period, midline, and amplitude.
8. Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms
of a context.
b. Use the properties of exponents to interpret expressions for exponential functions.
For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t,
y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For example,
given a graph of one quadratic function and an algebraic expression for another, say which
has the larger maximum.
Building Functions F-BF
Build a function that models a relationship between two quantities.
★
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
b. Combine standard function types using arithmetic operations. For example, build a
function that models the temperature of a cooling body by adding a constant function
to a decaying exponential, and relate these functions to the model.
54
c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere
as a function of height, and h(t) is the height of a weather balloon as a function of time,
then T(h(t)) is the temperature at the location of the weather balloon as a function of
time.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula,
use them to model situations, and translate between the two forms.★
Build new functions from existing functions.
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
for specific values of k (both positive and negative); find the value of k 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.
4.
Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and
write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠
1.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that the
function has an inverse.
d. (+) Produce an invertible function from a non-invertible function by restricting
the domain.
5. (+) Understand the inverse relationship between exponents and logarithms and use this
relationship to solve problems involving logarithms and exponents.
Linear, Quadratic, and Exponential Models★ F-LQE
Construct and compare linear, quadratic, and exponential models and solve problems.
1. Distinguish between situations that can be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval
relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences,
given a graph, a description of a relationship, or two input-output pairs (include reading these
from a table).
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a
quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
4. For exponential models, express as a logarithm the solution to a bct = d where a, c, and
d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.
55
Interpreting Categorical and Quantitative Data S-ID
Summarize, represent, and interpret data on a single count or measurement variable.
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
2. Use statistics appropriate to the shape of the data distribution to compare center (median,
mean) and spread (interquartile range, standard deviation) of two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting
for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal distribution and
to estimate population percentages. Recognize that there are data sets for which such a
procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas
under the normal curve.
Summarize, represent, and interpret data on two categorical and quantitative variables.
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative
frequencies in the context of the data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the data.
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables
are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of
the data. Use given functions or choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.
7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the
context of the data.
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
9. Distinguish between correlation and causation.
Making Inferences and Justifying Conclusions S-IC
Understand and evaluate random processes underlying statistical experiments.
1. Understand statistics as a process for making inferences to be made about
population parameters based on a random sample from that population.
2. Decide if a specified model is consistent with results from a given data-generating
process, e.g., using simulation. For example, a model says a spinning coin falls heads up
with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Make inferences and justify conclusions from sample surveys, experiments, and
observational studies.
3. Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.
4. Use data from a sample survey to estimate a population mean or proportion;
develop a margin of error through the use of simulation models for random sampling.
5. Use data from a randomized experiment to compare two treatments; use
simulations to decide if differences between parameters are significant.
6. Evaluate reports based on data.
56
Using Probability to Make Decisions
S-MD
Calculate expected values and use them to solve problems.
1. (+) Define a random variable for a quantity of interest by assigning a numerical value to
each event in a sample space; graph the corresponding probability distribution using the same
graphical displays as for data distributions.
2. (+) Calculate the expected value of a random variable; interpret it as the mean of the
probability distribution.
3. (+) Develop a probability distribution for a random variable defined for a sample space in
which theoretical probabilities can be calculated; find the expected value. For example, find
the theoretical probability distribution for the number of correct answers obtained by guessing
on all five questions of a multiple-choice test where each question has four choices, and find
the expected grade under various grading schemes.
4. (+) Develop a probability distribution for a random variable defined for a sample space in
which probabilities are assigned empirically; find the expected value. For example, find a
current data distribution on the number of TV sets per household in the United States, and
calculate the expected number of sets per household. How many TV sets would you expect to
find in 100 randomly selected households?
Use probability to evaluate outcomes of decisions.
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values
and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings
from a state lottery ticket or a game at a fast-food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a
high-deductible versus a low-deductible automobile insurance policy using various, but
reasonable, chances of having a minor or a major accident.
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random
number generator).
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
Resources for State Assessments
57