Download Building on their work with linear, quadratic, and exponential

Algebra 2 Syllabus
School Year: 2013-2014
Certificated Teacher: Ashlee Munsey
Desired Results
Course Title: Algebra 2 A and B
Credit: ____ one semester (.5)
_ two semesters (1.0)
Prerequisites and/or recommended preparation: Completion of Geometry
Estimate of hours per week engaged in learning activities:
5 hours of class work per week per 18 week semester
Instructional Materials:
All learning activity resources and folders are contained within the student online
course. Online course is accessed via login and password assigned by student’s school
(web account) or emailed directly to student upon enrollment, with the login website.
Other resources required/Resource Costs:
This course requires a MathXL account which will be provided by your course instructor. Holt
McDougal Geometry 2011 – online videos, examples, and activities.
Course Description:
Building on their work with linear, quadratic, and exponential functions, students extend their
repertoire of functions to include polynomial, rational, and radical functions. Students work
closely with the expressions that define the functions, and continue to expand and hone their
abilities to model situations and to solve equations, including solving quadratic equations over
the set of complex numbers and solving exponential equations using the properties of logarithms.
The process standards; problems solving, communication and connections apply throughout this
course. Through the content and process standards, students will experience mathematics as a
coherent, useful, and logical subject that makes use of their ability to make sense of problem
situations. Use of the graphing calculator is an integral part of this course.
Enduring Understandings for Course (Performance Objectives):
We can use a variety of mathematical tools to describe our world and help solve daily problems.
Course Learning Goals (including WA State Standards, Common Core Standards, National Standards):
First Semester Work Units 1-4 Second Semester Units 5 – 8
Algebra 2 Unit 1 Understanding Transformations
Target 1A
Understanding transformations.
1.
2.
3.
4.
5.
I can identify transformation(s) when given a function.
I can identify transformation(s) when given a table.
I can identify transformation(s) when given a graph.
I can apply transformation(s) to graphs.
I can apply transformation(s) to functions. (quadratic, exponential, absolute value, and piecewise
functions).
Washington State PEs
A2.5.A Construct new functions using the
transformations
f(x – h), f(x) + k, cf(x), and by adding and
subtracting functions, and describe the
effect on the original graph(s).
College Readiness Standards
(italicized text expectations beyond basic)
8.2.c Use simple transformations (horizontal
and vertical shifts, reflections about axes,
shrinks and stretches) to create the graphs of
new functions using linear, quadratic, and/or
absolute value functions. cubic, quartic,
exponential,
logarithmic, square root, cube root, absolute
value, piecewise, and rational functions of the
type f(x) = 1/(x-a) .
Common Core State Standards
HSF-BF.B.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.
Algebra 2 Unit 2 Extending Linear Models
Target 2A
Understanding the use of linear programming in
solving problems.
6. I can create and interpret constraint
equations and their graphs.
7. I can optimize an objective function given a
set of constraints.
Washington State PEs
A2.7.A Solve systems of three equations
with three variables.
College Readiness Standards
(italicized text indicates expectations beyond
basic)
7.4 Demonstrate an understanding of
matrices and their application.
8.4 Model situations and relationships using a
variety of basic functions (linear, quadratic,
logarithmic, exponential, and reciprocal) and
piecewise-defined functions.
Target 2B
Understanding linear equations in three
dimensions.
1. I can write a system of equations in three
dimensions.
2. I can create and interpret matrix equations for
linear equations in three dimensions.
3. I can use matrices to solve a system of linear
equations in three dimensions.
4. I can identify the solution of a 3x3 system of
equations as intersecting planes, parallel
planes, or coplanar figures.
Common Core State Standards
HSA-REI.C.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.
HSA-REI.C.6 Solve systems of linear equations
exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two
variables.
Algebra 2 Unit 3 Extending Exponential Models
3A
Understanding the application of
exponential models to real world
situations.
8. I can identify the rate of change and
initial value of an exponential model.
9. I can model real-world situations with
exponential functions.
Washington State PEs
A2.4.A Know and use basic properties of exponential
and logarithmic functions and the inverse relationship
between them.
A2.4.C Solve exponential and logarithmic
equations.
College Readiness Standards
(italicized text indicates expectations beyond
basic)
7.3.h Solve exponential equations in one variable
(numerically, graphically and algebraically)
8.4 Model situations and relationships using a
variety of basic functions (linear, quadratic,
logarithmic, exponential, and reciprocal) and
piecewise-defined functions.
Target 3B
Understanding the Application of Logarithms to Solve
Exponential Equations.
5. I can use logarithms to solve exponential equations.
6. I can apply transformations to exponential
functions.
Common Core State Standards
HSF-BF.B.4 Find inverse functions.
HSF-BF.B.4a 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.
HSF-BF.B.4b (+) Verify by composition that
one function is the inverse of another.
HSF-BF.B.4c (+) Read values of an inverse
function from a graph or a table, given that
the function has an inverse.
HSF-BF.B.4d (+) Produce an invertible function
from a non-invertible function by restricting the
domain.
HSF-BF.B.5 (+) Understand the inverse
relationship between exponents and logarithms
and use this relationship to solve problems
involving logarithms and exponents.
Algebra 2 Unit 4 Extending Exponential Models
Target 4A
Understanding the complex
number system.
1. I can perform the three
mathematical operations
addition, subtraction and
multiplication on non-real
numbers.
Target 4B
Understanding complex solutions to
quadratic equations.
7. I can solve quadratic equations which
have non-real solutions.
8. I can identify whether the roots of a
quadratic function are real or nonreal given an equation or a graph.
Washington State PEs
A2.2.A. Explain how whole, integer, rational, real and
complex numbers are related, and identify the number
systems within which a given algebraic equation can be
solved
A2.3.A.trasnslate between the standard form a a
quadratic function, the vertex form, and the factored
form; graph and interpret the meaning of each form.
Target 4C
Understanding vertex form of
quadratic equations.
1. I can write quadratic functions in
vertex form.
2. I can graph quadratic functions in
vertex form.
3. I can apply transformations to
quadratic functions.
Common Core State Standards
HSA-SSE.B.3a Factor a quadratic expression to reveal
the zeros of the function it defines.
HSA-SSE.B.3b Complete the square in a quadratic
expression to reveal the maximum or minimum value
of the function it defines.
A2.3.B Determine the number and nature of the roots of a
quadratic function.
HSF-IF.C.8a 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.
A2.3.C Solve quadratic equations and inequalities,
including equations with complex roots.
HSN-CN.C.7 Solve quadratic equations with real
coefficients that have complex solutions.
College Readiness Standards
7.2.c Factor quadratic polynomials with integer
coefficients into a product of linear terms.
HSN-CN.C.8 (+) Extend polynomial identities to the
2
complex numbers. For example, rewrite x + 4 as (x +
2i)(x – 2i).
7.3.f Use a variety of strategies to solve quadratic
equations including those with irrational solutions and
recognize when solutions are non-real. Simplify complex
solutions and check algebraically. Solve quadratic
equations by completing the square and by taking roots.
8.3.d Recognize and sketch, without the use of
technology, the graphs of the following families of
functions: linear, quadratic, cubic quartic, exponential,
logarithmic, square root, cube root, absolute value, and
rational functions of the type f(x)=1/(x-a).
8.4 Model situations and relationships using a variety of
basic functions (linear, quadratic, logarithmic,
exponential, and reciprocal) and piecewise-defined
functions.
HSA-REI.B.4a Use the method of completing the
square to transform any quadratic equation in x into
2
an equation of the form (x – p) = q that has the
same solutions. Derive the quadratic formula from
this form.
HSA-REI.B.4b Solve quadratic equations by
2
inspection (e.g., for x = 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.
Algebra 2 Unit 5 Higher Degree Polynomial Functions
Target 5A
Understanding the equivalence of polynomial
expressions
2. I can use the remainder and factor theorems.
3. I can factor higher order polynomials.
4. I can use arithmetic operations (add, subtract,
multiply, divide) on functions to create
equivalent forms.
Washington State PEs
A2.5.D Plot points, sketch, and describe the graphs
of cubic polynomial functions of the form f(x) =
ax3+d as an example of higher order polynomials
and solve related equations.
College Readiness Standards
(italicized text indicates expectations beyond basic)
7.2.a Find the sum, difference, or product of two
polynomials, then simplify the result.
7.2.b Factor out the greatest common factor from
polynomials of any degree.
7.2.c Factor quadratic polynomials with integer
coefficients into a product of linear terms.
8.2.b Describe relationship between the algebraic
features of a function and the features of its graph
and/or its tabular representation.
8.2.g Sketch the graph of a polynomial given the
degree, zeros, max/min values, and /or initial
conditions.
8.3.e Understand the relationship between the
degree of a polynomial and the number of roots;
interpret the multiplicity of roots graphically.
Target 5B
Understanding the relationship between zeros,
analytical or graphical representations of polynomial
functions
9. I can determine a function rule for a function
when given the graph.
10. I can determine the number of real and non-real
zeros when given a function.
11. I can apply transformations to higher degree
polynomial functions.
Common Core State Standards
HSA-APR.B.3 Identify zeros of polynomials
when suitable factorizations are available, and
use the zeros to construct a rough graph of
the function defined by the polynomial.
HSF-IF.C.7c Graph polynomial functions,
identifying zeros when suitable factorizations
are available, and showing end behavior.
Algebra 2 Unit 6 Rational Expressions and Functions
Target 6A Understanding the equivalence of
Target 6B Understanding characteristics of
1. I can apply properties of exponents to simplify
expressions with rational exponents.
2. I can simplify rational expressions.
3. I can use arithmetic operations (add, subtract,
multiply, divide) on rational expressions to create
equivalent forms.
4. I can solve rational equations in the form of
proportions.
1. I can model problem situations with rational
functions.
2. I can identify the domain of a rational function
3. I can graph rational functions.
4. I can determine the zeros of a rational
function.
rational expressions.
Washington State PEs
A2.1.A Select and justify functions and equations to model and
solve problems.
A2.1.E Solve problems that can be represented by inverse
2
variations of the forms f(x) = (a/x) + b, f(x) = (a/x ) + b and f(x) =
a/(bx+c).
A2.2.B Use laws of exponents to simplify and evaluate numeric
and algebraic expressions that contain rational exponents.
A2.5.C Plot points, sketch, and describe the graphs of functions
2
of the form f(x) = (a/x) + b, f(x) = (a/x ) + b and f(x) = a/(bx+c),
and solve related equations.
A2.2.C Add, subtract, multiply, divide and simplify rational
and more general algebraic expressions.
College Readiness Standards
(italicized text indicates expectations beyond basic)
rational functions.
Common Core State Standards
HSA-REI.A.2 Solve simple rational and radical equations
in one variable, and give examples showing how
extraneous solutions may arise.
HSF-IF.C.7d (+) Graph rational functions, identifying
zeros and asymptotes when suitable factorizations are
available, and showing end behavior.
HSN-RN.A.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
1/3
exponents. For example, we define 5 to be the cube
1/3 3
(1/3)3
to hold, so
root of 5 because we want (5 ) = 5
1/3 3
(5 ) must equal 5.
HSN-RN.A.2 Rewrite expressions involving radicals and
rational exponents using the properties of exponents.
7.2.d Simplify quotients of polynomials given in factored form,
or in a form which can be factored.
HSA-SSE.B.3c Use the properties of exponents to
transform expressions for exponential functions.
7.2.e Add, subtract, multiply, and divide two rational
expressions of the form, a/(bx+c) where a, b, and c are real
numbers and B is non-zero and of the form p(x)/q(x), where p(x)
and q(x) are polynomials.
HSA-APR.D.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.
7.2.f Simplify products and quotients of single-term expressions
with rational exponents (rationalizing denominators not
necessary).
7.3.i Solve rational equations in one variable that can be
transformed into an equivalent linear or quadratic equation
(limited to monomial or binomial denominators).
8.1.b Determine the domain and range of a function.
HSA-APR.D.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.
Algebra 2 Unit 7 Extending Trigonometric Models
Target 7A
Understanding the unit circle.
5. I can use both radian and
degree measure fluently.
6. I can identify trigonometric
values of common angles on
the unit circle fluently.
Target 7B
Understanding the application of
Target 7C
Understanding graphs of
trigonometric functions.
trigonometric functions.
12. I can use trigonometric
1. I can graph the Sine and
functions to solve non-right
Cosine parent functions
triangles.
without technology.
13. I can model real world situations 2. I can identify transformations
with trigonometry
on trigonometric functions.
Washington State PEs
College Readiness Standards
(italicized text indicates expectations beyond basic)
8.6.a Represent and interpret trig functions using the
unit circle.
Common Core State Standards
HSF-TF.A.1 Understand radian measure of an angle as
the length of the arc on the unit circle subtended by the
angle.
8.6.b Demonstrate an understanding of radians and
degrees by converting between units, finding areas of
sectors, and determining arc lengths of circles.
HSF-TF.A.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.
8.6.c Find exact values (without technology) of sine,
cosine and tangent ratios for (common) unit circle
angles; and distinguish between exact and
approximate values when evaluating trig
ratios/functions.
HSF-TF.B.5 Choose trigonometric functions to model
periodic phenomena with specified amplitude,
frequency, and midline.★
8.6.d Sketch graphs of sine, cosine, and tangent
functions, without technology; identify the domain,
range, intercepts, and asymptotes.
HSF-TF.B.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.
8.6.e Use transformations (horizontal and vertical
shifts, reflections about axes, period and amplitude
changes) to create new trig functions (algebraic,
tabular, and graphical).
8.6.i Use trig and inverse trig functions to solve
application problems.
HSF-TF.B.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.
Algebra 2 Unit 8 Probability and Statistics
Target 8A Understand Independence and
Conditional Probability
1. I can solve problems involving the
fundamental counting principle.
2. I can solve problems involving
permutations and combinations.
3. I can find the theoretical probability
of an event.
4. I can find the experimental
probability of an event.
Washing State PE’s
A.2.1.F Solve problems involving combinations
and permutations.
A.2.6.A Apply the fundamental counting principle
and the ideas of order and replacement to
calculate probabilities in situations arising form
two-stage experiments.
A.2.6.B Given a finite sample space consisting of
equally likely outcomes and containing events A
and B, determine whether A and B are
independent or dependent, and find conditional
probability of A given B.
College Readiness Standards
(italicized text indicates expectations beyond
basic)
6.1 Use empirical/experimental and theoretical
probability to investigate, represent, solve, and
interpret the solutions to problems involving
uncertainty(probability) or counting techniques.
Target 8B Understand and use the rules of probability to
compute probabilities of compound events in a uniform
probability model.
1. I can determine whether events are independent or
dependent.
2. I can find the probability of independent and
dependent events.
3. I can find the probability of mutually exclusive events.
4. I can find the probability of inclusive events.
Common Core State Standards
HSS-CP.A.1 Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes, or
as unions, intersections, or complements of other events (“or,”
“and,” “not”).
HSS-CP.A.2 Understand that two events A and B are independent if
the probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are
independent.
HSS-CP.A.3 Understand the conditional probability of A given B as
P(A and B)/P(B), and interpret independence of A and B as saying
that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the
same as the probability of B.
HSS-CP.A.4 Construct and interpret two-way frequency tables of data
when two categories are associated with each object being classified.
Use the two-way table as a sample space to decide if events are
independent and to approximate conditional probabilities.
HSS-CP.A.5 Recognize and explain the concepts of conditional
probability and independence in everyday language and everyday
situations.
HSS-CP.B.6 Find the conditional probability of A given B as the
fraction of B’s outcomes that also belong to A, and interpret the
answer in terms of the model.
HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and
B), and interpret the answer in terms of the model.
HSS-CP.B.8 (+) Apply the general Multiplication Rule in a uniform
probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and
interpret the answer in terms of the model.
HSS-CP.B.9 (+) Use permutations and combinations to compute
probabilities of compound events and solve problems.
Evidence of Assessment
Performance Tasks
Units are arranged so that students work through MathXL to master many of the objectives. Additional
exercises are provided to develop application of newly learned concepts. Each unit concludes with both
a skills-based assessment and one with applications.
Other Evidence (self-assessments, observations, work samples, quizzes, tests and so on):
•
•
•
Work in Math XL
Quizzes and Unit tests in Math XL
Weekly blogging on unit-specific topics
Types of Learning Activities
Each unit follows the same basic pattern:
1.
Work in MathXL
2.
Weekly Blog Entries
3.
Target Activities (including additional work if needed to meet standard)
4.
Unit Assessments
Online office hours are available for students to get real-time help with their learning.
Direct Instruction
Indirect Instruction Experiential
Independent Study Interactive
Learning
Instruction
_x___Structured
__x__Problem____ Virt. Field
____Essays
_x___Discussion
Overview
based
Trip
_x__Self-paced
____Debates
___Case Studies
____Experiments computer
___Role Playing
____Mini
____Inquiry
____Simulations __x__Journals
____Panels
presentation
____Games
____Learning Logs ____Peer Partner
__x__Drill & Practice ____Reflective
____Field
____Reports
Learning
____Demonstrations Practice
____Project
Observ.
____Directed
__x__Other (List)
____Project team
____Paper
___Role-playing
Study
____Laboratory
MathXL
____Concept
____Model Bldg. ____Research
Groups
Mapping
____Surveys
Projects
____Think, Pair,
____Other (List)
____Other (List) ____Other (List)
Share
____Cooperative
Learning
____Tutorial Groups
____Interviewing
____Conferencing
____Other (List)
Other:
Learning Activities
These learning activities are aligned with the successful completion of the course learning goals and
progress towards these learning activities will be reported monthly on a progress report.
1st Semester Algebra 2
Unit 1 Transformation of Functions
Duration: 1-2 weeks
Enduring Understandings
Students will identify and apply transformations both algebraically and graphically to the family of
functions previously explored in Algebra 1. Students will continue to studying transformations
throughout this course as they extend their understanding of functions.
Essential Questions:
1. How can we identify and apply transformations to the family of functions.
Student Learning Targets:
Target 1A: Understanding Transformations
This target focuses on identifying, describing and sketching the results of transformations. The
family of functions studied in Algebra 1 will be used to illustrate transformations. Ultimately,
students should be able to work with transformations for the general case.
Target 1B: Understanding Applications of Transformations
This target focuses on identifying, describing and sketching the results of transformations. The
family of functions studied in Algebra 1 will be used to illustrate transformations. Ultimately,
students should be able to work with transformations for the general case.
Learning Activities:
Unit 1 Target 1A
Task #1 Exploring Transformations
Task #2 Parent Functions
Task #3 Target 1A Quiz
Unit 1 Target 1B
Task #1 Transforming Functions
Task #2 Target 1B
Unit 1 Assessment
Unit 2 Extending Linear Models
Duration: 4 weeks
Enduring Understandings
Students will extend their understanding of linear functions from Algebra 1 to include linear
programming and linear equations in three dimensions.
Essential Questions:
1. How can linear models be used to solve real world problems and make predictions?
Student Learning Targets:
Target 2A: Understanding the use of linear programming in solving problems.
Students will graph constraint equations both with and without technology; determine lattice
points both algebraically and graphically; and evaluate the objective function. Shading should
be used as a tool to identify the feasible region.
Target 2B: Understanding linear equations in three dimensions.
Students should understand that a 3x3 system of linear functions models three planes in space.
They should be able to interpret the solution to a 3x3 system in the context of intersecting
planes by identifying that the planes intersect, are parallel or are coplanar. Once students have
modeled the situation with a matrix equation in a vector variable, technology is used to solve.
Algebraic techniques for solving matrices will be studied in a course after Algebra 2.
Learning Activities:
Unit 2 Target 2A
Task #1 Linear Inequalities
Task #2 Systems of Linear Inequalities
Task #3 Linear Programming
Task #4 Target 2A Quiz
Unit 2 Target 2B
Task #1 Linear Functions in Three Dimensions
Task #2 Solving Systems Using Matrices
Task #3 Target 2B Quiz
Unit 2 Assessment
Unit 3 Extending Exponential Models
Duration: 4 weeks
Enduring Understandings
Students will extend their understanding of exponential functions from Algebra 1 to include the use of
logarithms.
Essential Questions:
1. How can exponential models be used to solve real world problems and make predictions?
Student Learning Targets:
Target 3A: Understanding the application of exponential models to real world situations.
Students leave Algebra 1 with a good understanding of exponential models. They may not have
had any experiences with exponential functions since their time in Algebra 1. Time spent
recalling prior knowledge sets them up for the application of logarithms. This learning target is
about bringing back prior knowledge. The new leaning for this unit is embodied in the next
learning target.
Target 3B: Understanding the application of logarithms to solve exponential equations.
The point of this target (and unit) is to have students leaving Algebra 2 with confidence and
knowledge that (1) a logarithm is an exponent (2) logarithms are used to solve exponential
equations and (3) logarithmic functions are inverse functions of exponential functions. This will
open the door to classroom discussions around the concept of inverse functions. Formal study
of inverse functions will occur in courses beyond Algebra 2.
Learning Activities:
Unit 3 Target 3A
Task #1 Exponential Functions Growth and Decay
Task #2 Curve Fitting with Exponential Models
Task #3 Target 3A Quiz
Unit 3 Target 3B
Task #1 Logarithmic Functions
Task #2 Properties of Logarithms
Task #3 Exponential and Logarithmic Equations
Unit 3 Target 3B
Task #4 The Natural Base, e
Task #5 Target 3B Quiz
Unit 3 Assessment
Unit 4 Extending Quadratic Models
Duration: 5 weeks
Enduring Understandings
Students will complete their study of the complex number system by extending their understanding of
quadratic functions from Algebra 1 to include non-real solutions. The vertex form of a quadratic
function is used to help students analyze quadratic functions and their graphs.
Essential Questions:
1. How use the non-real number system in working with quadratic functions.
2. How can I use the vertex form of a quadratic function to analyze its graph.
Student Learning Targets:
Target 4A. Understanding the complex number system.
During Algebra 1 students completed their study of the real number system. During Algebra 2
they will extend that number system to include non-real numbers, thus completing the complex
number system. Students will leave this unit with an understanding of the entire complex
number system and what it means to be “closed”.
This learning target should not be taught in isolation. Students should see the operations
addition, subtraction and multiplication on non-real complex numbers of the form a + bi where
b is nonzero embedded and motivated from the application of zeros to quadratic equations.
This target suggests that students should be able to perform these operations within and
outside of context. Students should learn from this target (unit) that any quadratic equation
with real coefficients will have solutions within the complex numbers (ie the complex field is
closed).
Target 4B. Understanding complex solutions to quadratic equations.
This target outlines the motivation for studying non-real complex numbers. Students should be
able to determine the nature of solutions as real or non-real given an equation or a graph. They
should leave this unit understanding what it means for a quadratic equation to be solvable over
the complex numbers.
Students should also leave this unit understanding what it means for a quadratic expression to
be factorable over the set of real numbers and connect this understanding to the graph of a
quadratic function.
Target 4C Understanding vertex form of quadratic equations.
This form was not studied during Algebra 1 and should be emphasized as a useful form for graphing.
Students should be able to identify the vertex or note the location of a maximum or minimum along the
graph. Students should be able to convert from standard form to vertex and vice versa. Vertex form
also gives students an alternate method of solution.
Learning Activities:
Unit 4 Target 4A
Task #1 Complex Numbers and Roots
Task #2 Operations with Complex Numbers
Task #3 Target 4A Quiz
Unit 4 Target 4B
Task #1 The Quadratic Formula
Task #2 Target 4B Quiz
Unit 4 Target 4C
Task #1 Graphing Quadratic Equations in Vertex Form
Task #2 Writing Quadratic Functions in Vertex Form
Unit 4 Target 4c
Task #3 Graphing Quadratics Using Transformations
Task #4 Target 4C Quiz
Unit 4 Assessment
Second Semester Work
Unit 5 Higher Degree Polynomial Functions
Duration: 4 weeks
Enduring Understandings
Students will extend their understanding of functions and their zeros to include higher degree
polynomial functions. Equivalence of polynomial expressions will be explored by using factor and
remainder theorems.
Essential Questions:
1. How does my understanding of functions and their zeros expand to include higher degree
polynomial functions?
Student Learning Targets:
Target 5A Understanding the equivalence of polynomial expressions.
Emphasis is on equivalence when factoring and manipulating polynomial expressions. Students
should work with grouping techniques (up to 3rd degree), the remainder and factor theorems
and synthetic division. This should be done within the context of finding zeros and should be
woven throughout the unit.
Target 5B Understanding the relationship between zeros, analytical or graphical representations of
polynomial functions.
Students should see the connection between the degree of a polynomial and the number of
roots. Further analysis should allow them to distinguish the type of roots (real vs. non-real).
Reconnect students to the complex number system developed in unit 4.
The degree of the polynomial (including odd/even) should be studied to help students analyze
possible roots. This is a good time to use the concept of end behavior with students as it
becomes a focal point in courses beyond Algebra 2.
Be efficient with the Rational Root theorem by having students narrow the choice of roots by
using technology. Keep the focus on the connection of degree with the number of roots and
their nature. Ultimately, students should leave this unit understanding the relationship
between the degree of higher degree polynomials and their roots both algebraically and
graphically as well as being able to apply algebraic techniques to determine zeros.
Learning Activities:
Unit 5 Target 5A
Task #1 Polynomials
Task #2 Multiplying Polynomials
Task #3 Dividing Polynomials
Task #4 Factoring Polynomials
Task #5 Finding Root of Polynomial Functions
Task #6 Target 5A Quiz
Unit 5 Target 5B
Task #1 Writing Polynomials Functions Given a Graph
Task #2 Investigating Graphs of Polynomial Functions
Task #3 Transforming Polynomial Functions
Task #4 Target 5B Quiz
Unit 5 Assessment
Unit 5 Rational Expressions and Functions
Duration: 2 weeks
Enduring Understandings
A central theme of this unit is that the arithmetic operations on rational expressions are governed by
the same rules as the arithmetic operations on rational numbers. Students will apply the concept of
equivalence to simplify rational expressions and solve rational equations. The exploration of the zeros
of rational functions provides an introduction to the asymptotic behavior of functions.
Essential Questions:
1. How can we use arithmetic operations on rational expressions?
2. How can we use equivalence to simplify rational expression and solve rational equations?
Student Learning Targets:
Target 6A: Understanding the equivalence of rational expressions.
This learning target emphasizes that rational expressions are an extension of rational numbers.
Students will simplify expressions revisit and extend properties of exponents from Algebra 1 and apply
the four arithmetic operations on rational expressions.
During Algebra 1, students used properties of exponents to simplify expressions with integer exponents
(see Prior Knowledge in program guide). During Algebra 2 students should review these properties and
extend exponents to include non-integer exponents.
While applying the four operations to rational expressions, the focus of this target is expressions of the
form p(x)/q(x) where p(x) and q(x) are in factored form or can be factored.
Students should be solving rational equations both in and out of context. Build on their prior knowledge
of proportional reasoning while considering context. Rational equations should not go beyond students
having to solve quadratic equations.
Target 6B: Understanding characteristics of rational functions.
The purpose for studying these functions is on domain, understanding graphical representation and
zeros. When working with rational functions for this target, focus on those that illustrate removable
discontinuities (“hole in graph”), vertical asymptotes and horizontal asymptote of y = 0. Formal study of
range, horizontal asymptotes, limits and end behavior will occur in courses beyond Algebra 2.
Problems involving Inverse variation provide good context for modeling with rational functions.
Learning Activities:
Unit 6 Target 6A
Task #1 Radical Expressions and Rational Exponents
Task #2 Multiplying and Dividing Rational Expressions
Task #3 Adding and Subtracting Rational Expressions
Task #4 Solving Rational Expressions
Task #5 Target 6A Quiz
Unit 6 Target 6B
Task #1 Rational Functions
Task #2 Finding the Zeros of Rational Functions
Task #3 Target 6B Quiz
Unit 6 Assessment
Unit 7 Extending Trigonometric Models
Duration: 4 weeks
Enduring Understandings
Students will extend their understanding of trigonometry as ratios from Geometry to include
trigonometric functions. In addition to becoming fluent with the unit circle and the Pythagorean
identity, students will graph and transform trigonometric functions to model and solve problems.
Essential Questions:
1. How can I expand my understanding of trigonometry to include trigonometric functions?
Student Learning Targets:
Target 7A Understanding the unit circle - (radian and degree measure, common angles, and the
Pythagorean identity).
Students should see the unit circle as a tool for efficiently identifying ratios for the sine and
cosine functions. Use the unit circle to help students make the connection between the
trigonometric ratios and trigonometric functions. Don’t get hung up having students memorize
the unit circle as it is far more helpful for them to see the special right triangles within the unit
circle and make use of their properties. Fluency does not equate to speed and memorization.
Beyond using the unit circle for identifying ratios of special angles, students should use it make
generalizations about sine and cosine ratios and in the use of estimation. The unit circle also
serves as a nice demonstration of the Pythagorean identity.
Target 7B Understanding the application of trigonometric functions.
Students solved right triangles in geometry by using the sine, cosine and tangent ratios (see
prior knowledge in program guide). This target includes the Law of Sines and Cosines. Be sure
to have students work both within and outside of context. It is not expected that a thorough
study of inverse functions should be addressed at this time. They should see (and have seen in
geometry) the inverse operations on sine, cosine and tangent as a “tool” for determining an
angle. A deeper study of inverse functions will occur in courses following Algebra 2.
Target 7C Understanding trigonometric functions.
Students should be able to graph the sine, cosine and tangent parent graphs accurately without
technology. They should also be able to identify (algebraically and graphically) and graph
changes in amplitude and phase shift. Students should be fluent with amplitude and period
change. Don’t dwell on phase shift as students will revisit this in courses beyond Algebra 2.
Don’t stress all 3 at a sacrifice for the other two.
Inverse functions should be used as a method for determining and angle when given a ratio.
Students should be able to identify the domain and range for these 3 trig functions. Keep your
eye on the purpose of this target which is to extend the 3 trig ratios to functions.
Learning Activities:
Unit 7 Target 7A
Task #1 Angles of Rotation
Task #2 The Unit Circle
Task #3 Target 7A Quiz
Unit 7 Target 7B
Task #1 Graphs of Sine and Cosine
Task #2 Target 7B Quiz
Unit 7 Target 7C
Task #1 Right Triangle Trigonometry
Task #2 Law of Sines
Task #3 Law of Cosines
Unit 7 Target 7C
Task #4 Target 7C Quiz
Unit 7 Assessment
Unit 8 Probability and Statistics
Enduring Understandings
Students will extend their understanding of probability and statistics.
Essential Questions:
1. How does my understanding of functions and their zeros expand to include higher degree
polynomial functions?
Student Learning Targets
Target 8A. Understand independence and conditional probability and use them to interpret data.
Target 8B. Understand and use the rules of probability to compute probabilities of compound events in
a uniform probability model.
Learning Activities
Unit 8 Target 8A
Task #1 Permutations and Combinations
Task #2 Theoretical and Experimental Probability
Task #3 Target 8A Quiz
Unit 8 Target 8B
Task #1 Independent and Dependent Probability
Task #2 Compound Events
Task #3 Target 8B Quiz
Unit 8 Assessment