ALGEBRA II/TRIGONOMETRY CURRICULUM UIDE · The GUIDE is the lead document for planning, assessment and curriculum work. It is a summarized ... Benchmark Assessment AII/T.19 Data Analysis

ALGEBRA II/TRIGONOMETRY

CURRICULUM GUIDE DRAFT - 2012-2013

Please Note: The Mathematics Office is still vetting and editing this document for typos and errors. The sequencing and general pacing will not change.

Loudoun County Public Schools

Complete scope, sequence, pacing and resources are available on the LCPS Intranet.

INTRODUCTION TO LOUDOUN COUNTY’S MATHEMATICS CURRICULUM GUIDE

This CURRICULUM GUIDE is a merger of the Virginia Standards of Learning (SOL) and the Mathematics Achievement Standards for

Loudoun County Public Schools. The CURRICULUM GUIDE includes excerpts from documents published by the Virginia Department of

Education. Other statements, such as suggestions on the incorporation of technology and essential questions, represent the professional

consensus of Loudoun’s teachers concerning the implementation of these standards. In many instances the local expectations for

achievement exceed state requirements. The GUIDE is the lead document for planning, assessment and curriculum work. It is a summarized

reference to the entire program that remains relatively unchanged over several student generations. Other documents, called RESOURCES,

are updated more frequently. These are published separately but teachers can combine them with the GUIDE for ease in lesson planning.

Mathematics Internet Safety Procedures

1. Teachers should review all Internet sites and links prior to using it in the classroom.

During this review, teachers need to ensure the appropriateness of the content on the site,

checking for broken links, and paying attention to any

inappropriate pop-ups or solicitation of information.

2. Teachers should circulate throughout the classroom while students are on the

internet checking to make sure the students are on the appropriate site and

are not minimizing other inappropriate sites.

Teachers should periodically check and update any web addresses that they have on their

LCPS web pages.

3. Teachers should assure that the use of websites correlate with the objectives of

lesson and provide students with the appropriate challenge.

4. Teachers should assure that the use of websites correlate with the objectives

of the lesson and provide students with the appropriate challenge.

Algebra II/Trigonometry SemesterOverview

1st Semester 2cd Semester

Number Sense AII.1

AII.3

AII.4

Roots, Radicals, and Exponents AII.1

AII.4

Functions AII.7

AII.6

AII.7

Quadratic Equations and Systems AII.1

AII.4

AII.8

AII.9

AII.5

Benchmark Assessment

Data Analysis AII.11

AII.12

AII.9

AII.10

Polynomial Functions AII.8

AII.6

AII.7

Rational Expressions and Equations AII.1

AII.4

AII.7

AII.6

Logarithms and Exponentials AII.6

AII.7

AII.9

Sequences and Series AII.2

Unit Circle Trigonometry AII/T.13

AII/T.14

AII/T.15

AII/T.16

Trigonometric graphing AII/T.18

AII/T.19

Trigonometric Identities AII/T.17

AII/T.18

Trigonometric Equations

AII/T.20

Applications of Trigonometry AII/T.21

Algebra II/Trigonometry Semester 1 page 4

Number of

Blocks

Topics and Essential

Understandings

Standards of Learning & Essential Knowledge

and Skills

4 blocks

Unit 1: Number Sense

Operations with rational,

algebraic expressions

Complex number system

Solving and graphing absolute

value equations and

inequalities

AII/T.3 Essential

Understandings

Complex numbers are

organized into a hierarchy of

subsets.

A complex number multiplied

by its conjugate is a real

number.

Equations having no real

number solutions may have

solutions in the set of complex

numbers.

Field properties apply to

complex numbers as well as

real numbers.

All complex numbers can be

written in the form a+bi where

a and b are real numbers and i

is 1 .

SOL AII/T.3 The student will perform operations

on complex numbers, express the results in simplest

form using patterns of the powers of I, and identify

field properties that are valid for complex numbers.

AII/T.3 Essential Knowledge and Skills

Recognize that the square root of –1 is

represented as i.

Determine which field properties apply to

the complex number system.

Simplify radical expressions containing

negative rational numbers and express in

a+bi form.

Simplify powers of i.

Add, subtract, and multiply complex

numbers.

Place the following sets of numbers in a

hierarchy of subsets: complex, pure

imaginary, real, rational, irrational, integers,

whole, and natural.

Write a real number in a+bi form.

Write a pure imaginary number in a+bi

form.

NO CALCULATORS IN

THIS UNIT

Stress mastery of fractions

Do not include AII.1d at

this time

Resources:

http://education.ti.com/educatio

nportal/activityexchange/Activit

y.do?cid=US&aId=10887

http://www.ditutor.com/natural

_number/types_numbers.html

Unit 1 Summary Sheet

Discussion Questions: Compare and contrast

the different number

systems

What does it mean to

“solve” an equation or

inequality?

What does “absolute value”

mean? Give an example of a

real-world situation that

involves the concept of absolute

value.

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=10887



http://www.ditutor.com/natural_number/types_numbers.html

http://www.ditutor.com/natural_number/types_numbers.html


AII/T.4 Essential

Understandings

The definition of absolute

value (for any real numbers a

and b, where b 0, if a b ,

then a = b or a = - b) is used in

solving absolute value

equations and inequalities.

Absolute value inequalities

can be solved graphically or

by using a compound

statement.

Real-world problems can be

interpreted, represented, and

solved using equations and

inequalities.

Equations can be solved in a

variety of ways.

Set builder notation may be

used to represent solution sets

of equations and inequalities.

SOL AII/T.4 a The student will solve,

algebraically and graphically,

a. absolute value equations and inequalities;…


Solve absolute value equations and

inequalities algebraically and graphically.

Apply an appropriate equation to solve a

real-world problem.


5 blocks

Unit 2 : Roots, Radicals, and

Exponents

Integer and rational exponents

(rationalizing the

denominator)

Simplifying algebraic

expressions containing

radicals in the denominator

Solve radical equations

AII/T.1 Essential

Understandings

Radical expressions can be

written and simplified using

rational exponents.

Only radicals with a common

radicand and index can be

added or subtracted.

AII/T.4 Essential

Understandings




inequalities.

The process of solving radical

or rational equations can lead

to extraneous solutions.


variety of ways.




SOL AII/T.1 b, c The student will…

b. add,. subtract, multiply, divide, and

simplify radical expressions containing

…rational exponents;

c. write radical expressions as expressions

containing rational exponents and vice

versa;….


Simplify radical expressions containing

positive rational numbers and variables.

Convert from radical notation to exponential

notation, and vice versa.

Add and subtract radical expressions.

Multiply and divide radical expressions not

requiring rationalizing the denominators.

SOL AII/T.4 b The student will solve,

algebraically and graphically, …

a. equations containing radical expressions.


Solve an equation containing a radical

expression algebraically and graphically.

Verify possible solutions to an equation

containing rational or radical expressions.


real-world problem.

Include the absolute value

piece with even numbered

roots

Simplifying radicals should

not contain anything greater

than fifth roots

Resources:


Discussion Questions: Explain the relationship

between rational

exponents and thn roots.

What is an extraneous

root and how does it

impact the solution to an

equation?

5 blocks

Unit 3: Functions

Domain and range

Parent functions and

SOL AII/T.7 a The student will investigate and

analyze functions algebraically and graphically.

Key concepts include

No rational functions at

this time

Use context pieces for


transformations on parent

functions including the

following: linear, quadratic,

cubic, absolute value, step,

square root, cube root,

piecewise

Composition

Inverses (linear and quadratic)

AII/T.7 Essential

Understandings

The domain and range of a

function may be restricted

algebraically or by the real-

world situation modeled by the

function.

If (a, b) is an element of a

function, then (b, a) is an

element of the inverse of the

function.

AII/T.6 Essential Understandings

The graphs/equations for a

family of functions can be

determined using a

transformational approach.

Transformations of graphs

include translations,

reflections, and dilations.

A parent graph is an anchor

graph from which other graphs

are derived with

transformations.

a. domain and range, including limited and

discontinuous domains and ranges;…


Identify the domain, range, zeros, and

intercepts of a function presented

algebraically or graphically.

Describe restricted/discontinuous domains

and ranges.

SOL AII/T.6 The student will recognize the

general shape of function families and will convert

between graphic and symbolic forms of functions.

A transformational approach to graphing will be

employed.


Recognize graphs of parent functions.

Given a transformation of a parent function,

identify the graph of the transformed

function.

Given the equation and using a

transformational approach, graph a function.

Given the graph of a function, identify the

parent function.


transformations that map the preimage to

the image in order to determine the equation

of the image.

Using a transformational approach, write the

equation of a function given its graph.

SOL AII/T.7 g, h The student will investigate and


parent functions (real-

world examples for each

type of function)

Graph the inverse of a

function and

algebraically verify

inverses of functions

using composition.

Resources:

http://www.regentsprep.org/Reg

ents/math/algtrig/ATP9/funcres

ource.htm

http://www.purplemath.com/mo

dules/fcntrans.htm

http://illuminations.nctm.org/Le

ssonDetail.aspx?ID=L725

http://dnet01.ode.state.oh.us/IM

S.ItemDetails/LessonDetail.asp

x?id=0907f84c80531456

X:\Algebra 2\Using Models to

Build an Understanding of

Functions.pdf


Discussion Questions: Explain how the

graphical

transformations of a

given parent function

are evident in the

equation of the function.

Compare and contrast

the domain and range of

http://www.regentsprep.org/Regents/math/algtrig/ATP9/funcresource.htm



http://www.purplemath.com/modules/fcntrans.htm

http://www.purplemath.com/modules/fcntrans.htm

http://illuminations.nctm.org/LessonDetail.aspx?ID=L725


http://dnet01.ode.state.oh.us/IMS.ItemDetails/LessonDetail.aspx?id=0907f84c80531456




Key concepts include …

g. inverse of a function; and

h. composition of multiple functions.


Find the inverse of a function.

Graph the inverse of a function as a

reflection across the line y = x.

the parent functions

discussed.

10 blocks

Unit 4: Quadratics and Systems

Factoring (ALL factoring,

including quadratic, difference

of squares, sum and difference

of cubes, grouping, GCF, and

special patterns).

Solving quadratic equations,

including a discussion of the

following: quadratic formula,

completing the square,

discriminant, complex

solutions, zeros, graphing

quadratics in all forms

Systems of equations and

inequalities

Matrices – basic operations

and using matrices to solve

systems of 3 equations in 3

unknowns

Non-linear systems of

equations – finding solutions

algebraically and graphically


The complete factorization of

polynomials has occurred

when each factor is a prime

polynomial.

SOL AII/T.1a The student, given rational, radical,

or polynomial expressions, will

a. factor polynomials completely ….


Factor polynomials by applying general

patterns including difference of squares,

sum and difference of cubes, and perfect

square trinomials.

Factor polynomials completely over the

integers.

Verify polynomial identities including the

difference of squares, sum and difference of

cubes, and perfect square trinomials.

SOL AII/T.4 b The student will solve,


b. quadratic equations over the set of complex

numbers;…


Solve a quadratic equation over the set of

complex numbers using an appropriate

strategy.

Calculate the discriminant of a quadratic

equation to determine the number of real

Hit all topics but do not

spend too much time here.

Students will eventually be

coming in with an

understanding of quadratics

so only the complex

solutions will need to be

covered.

Be sure that students can

convert between the

different forms of quadratic

equations.

Resources: http://www.webgraphing.com/q

uadraticequation_quadraticform

ula.jsp

X:\Algebra 2\Quadratic CBR

Exploration.docx


Discussions Questions:

What is the difference

between a factor and a

zero?

What is the importance

of finding the

http://www.webgraphing.com/quadraticequation_quadraticformula.jsp




Pattern recognition can be

used to determine complete

factorization of a polynomial

AII/T.4 Essential

Understandings

A quadratic function whose

graph does not intersect the x-

axis has roots with imaginary

components.

The quadratic formula can be

used to solve any quadratic

equation.

The value of the discriminant

of a quadratic equation can be

used to describe the number of

real and complex solutions.




inequalities.


variety of ways.




AII/T.8 Essential

Understandings

The Fundamental Theorem of

Algebra states that, including

complex and repeated

solutions, an nth

degree

polynomial equation has

exactly n roots (solutions).

The following statements are

and complex solutions.


real-world problem.

Recognize that the quadratic formula can be

derived by applying the completion of

squares to any quadratic equation in

standard form.

SOL AII/T.8 The student will investigate and

describe the relationships among solutions of an

equation, zeros of a function, x-intercepts of a

graph, and factors of a polynomial expression.


Describe the relationships among solutions

of an equation, zeros of a function, x-

intercepts of a graph, and factors of a

polynomial expression.

Define a polynomial function, given its

zeros.

Determine a factored form of a polynomial

expression from the x-intercepts of the

graph of its corresponding function.

For a function, identify zeros of multiplicity

greater than 1 and describe the effect of

those zeros on the graph of the function.

Given a polynomial equation, determine the

number of real solutions and nonreal

solutions.

SOL AII/T.9 The student will collect and analyze

data, determine the equation of the curve of best fit,

make predictions, and solve real-world problems

using mathematical models.

discriminant?

Identify all forms of a

quadratic equation and

explain the advantages

and disadvantages to

graphing the function

from each form.

How can real life

problem situations be

modeled using

quadratics?


the different methods of

solving a quadratic

equation.


equivalent:

– k is a zero of the

polynomial function f;

– (x – k) is a factor of f(x);

– k is a solution of the

polynomial equation f(x)

= 0; and

k is an x-intercept for the graph of

y = f(x).

AII/T.9 Essential

Understandings

Data and scatterplots may

indicate patterns that can be

modeled with an algebraic

equation.

Graphing calculators can be

used to collect, organize,

picture, and create an algebraic

model of the data.

Data that fit polynomial (1

1 1 0( ) ...n n

n nf x a x a x a x a

, where n is a nonnegative

integer, and the coefficients

are real numbers), exponential

(xy b ), and logarithmic (

logby x ) models arise from

real-world situations.

AII/T.5 Essential

Understandings


Collect and analyze data.

Investigate scatterplots to determine if

patterns exist and then identify the patterns.

Find an equation for the curve of best fit for

data, using a graphing calculator. Models

will include polynomial, exponential, and

logarithmic functions.

Make predictions, using data, scatterplots,

or the equation of the curve of best fit.

Given a set of data, determine the model

that would best describe the data.

SOL AII/T.5 The student will solve nonlinear

systems of equations, including linear-quadratic

and quadratic-quadratic, algebraically and

graphically.


Predict the number of solutions to a

nonlinear system of two equations.

Solve a linear-quadratic system of two

equations algebraically and graphically.

Solve a quadratic-quadratic system of two

equations algebraically and graphically.


Solutions of a nonlinear

system of equations are

numerical values that satisfy

every equation in the system.

The coordinates of points of

intersection in any system of

equations are solutions to the

system.



solved using systems of

equations.

5 blocks

Unit 5: Data Analysis

Collect and analyze real-world data

using the following:

Normal distribution

z-scores

Standard deviations

Standard normal probability

Combinatorics (permutations,

combinations, counting

principle)

Regression – include linear,

quadratic, cubic, and

exponential/logarithmic

Variation – direct, inverse, and

joint


A normal distribution curve is

a symmetrical, bell-shaped

curve defined by the mean and

the standard deviation of a

data set. The mean is located

on the line of symmetry of the

SOL AII/T.11 The student will identify properties

of a normal distribution and apply those properties

to determine probabilities associated with areas

under the standard normal curve.


Identify the properties of a normal

probability distribution.

Describe how the standard deviation and the

mean affect the graph of the normal

distribution.

Compare two sets of normally distributed

data using a standard normal distribution

and z-scores.

Represent probability as area under the

curve of a standard normal probability

distribution.

Use the graphing calculator or a standard

normal probability table to determine

probabilities or percentiles based on z-

scores.

SOL AII/T.12 The student will compute and

Z-scores will be covered

in the Algebra 1

curriculum in the future,

but students have not yet

seen it.

Be sure to include

correct notation,

including and .

The 10 days noted does

not include the days for

exam review and

BMA’s..

Resources:

http://www.regentsprep.org/Reg

ents/math/algtrig/math-

algtrig.htm#m9

X:\Algebra 2\Life

Expectancy.doc


Discussion Questions:

http://www.regentsprep.org/Regents/math/algtrig/math-algtrig.htm#m9




curve.

Areas under the curve

represent probabilities

associated with continuous

distributions.

The normal curve is a

probability distribution and the

total area under the curve is 1.

For a normal distribution,

approximately 68 percent of

the data fall within one

standard deviation of the

mean, approximately 95

percent of the data fall within

two standard deviations of the

mean, and approximately 99.7

percent of the data fall within

three standard deviations of

the mean.

The mean of the data in a

standard normal distribution is

0 and the standard deviation is

1.

The standard normal curve

allows for the comparison of

data from different normal

distributions.

A z-score is a measure of

position derived from the

mean and standard deviation

of data.

A z-score expresses, in

standard deviation units, how

far an element falls from the

mean of the data set.

A z-score is a derived score

distinguish between permutations and combinations

and use technology for applications.


Compare and contrast permutations and

combinations.

Calculate the number of permutations of n

objects taken r at a time.

Calculate the number of combinations of n

objects taken r at a time.

Use permutations and combinations as

counting techniques to solve real-world

problems.

SOL AII/T.9 the student will collect and analyze

data, determine the equation of the curve of best fit,

make predictions, and solve real-world problems

using mathematical models.


















Explain when you

would use each

statistical measurement

in analyzing data.


the different statistical

measurements discussed

in this unit.

When looking at a

graphical display of a

data set, how do you

determine which

regression model is the

best fit for the data?


from a given normal

distribution.

A standard normal distribution

is the set of all z-scores.


The Fundamental Counting

Principle states that if one

decision can be made n ways

and another can be made m

ways, then the two decisions

can be made nm ways.

Permutations are used to

calculate the number of

possible arrangements of

objects.

Combinations are used to

calculate the number of

possible selections of objects

without regard to the order

selected.


Data and scatterplots may

indicate patterns that can be

modeled with an algebraic

equation.

Graphing calculators can be

used to collect, organize,

picture, and create an algebraic

model of the data.

Data that fit polynomial (1

1 1 0( ) ...n n

n nf x a x a x a x a

, where n is a nonnegative

integer, and the coefficients







SOL AII/T.10 The student will identify, create,

and solve real-world problems involving inverse

variation, joint variation, and a combination of

direct and inverse variations.


Translate “y varies jointly as x and z” as y =

kxz.

Translate “y is directly proportional to x” as

y = kx.

Translate “y is inversely proportional to x”

as y = k

x .

Given a situation, determine the value of the

constant of proportionality.

Set up and solve problems, including real-

world problems, involving inverse variation,

joint variation, and a combination of direct

and inverse variations.


are real numbers), exponential

(xy b ), and logarithmic (

logby x ) models arise from

real-world situations



modeled and solved by using

inverse variation, joint

variation, and a combination

of direct and inverse

variations.

Joint variation is a

combination of direct

variations.

6 blocks

Unit 6: Polynomial Functions

Fundamental Theorem of

Algebra

Synthetic division and long

division

Rational Root Theorem

Factor Theorem

End Behavior

Polynomial Models

Zeros

AII.8 Essential Understandings

The Fundamental Theorem of

Algebra states that, including

complex and repeated

solutions, an nth

degree

polynomial equation has

exactly n roots (solutions).

The following statements are

SOL AII.8 The student will investigate and

describe the relationships among solutions of an

equation, zeros of a function, x-intercepts of a

graph, and factors of a polynomial expression.

AII.8 Essential Knowledge and Skills

Describe the relationships among solutions

of an equation, zeros of a function, x-

intercepts of a graph, and factors of a

polynomial expression.

Define a polynomial function, given its

zeros.

Determine a factored form of a polynomial

expression from the x-intercepts of the

graph of its corresponding function.

For a function, identify zeros of multiplicity

greater than 1 and describe the effect of

those zeros on the graph of the function.

Given a polynomial equation, determine the

Do not do AII.7e until the

next unit.

Make sure you discuss the

multiplicity of roots.

Resources: http://algebralab.org/lessons/les

son.aspx?file=algebra_poly_gra

phs.xml

http://illuminations.nctm.org/Le

ssonDetail.aspx?ID=L282



What does the

Fundamental Theorem

of Algebra tell us about

a polynomial function?

http://algebralab.org/lessons/lesson.aspx?file=algebra_poly_graphs.xml






equivalent:

– k is a zero of the

polynomial function f;

– (x – k) is a factor of f(x);

– k is a solution of the

polynomial equation f(x)

= 0; and

– k is an x-intercept for the

graph of y = f(x).


The graphs/equations for a

family of functions can be

determined using a


Transformations of graphs

include translations,

reflections, and dilations.

A parent graph is an anchor

graph from which other graphs

are derived with

transformations.


Functions may be used to

model real-world situations.

The domain and range of a

function may be restricted

algebraically or by the real-

world situation modeled by the

function.

A function can be described on

an interval as increasing,

decreasing, or constant.

Asymptotes may describe both

number of real solutions and nonreal

solutions.

SOL AII.6 The student will recognize the

general shape of function (absolute value,

square root, cube root, rational, polynomial,

exponential, and logarithmic) families and

will convert between graphic and symbolic

forms of functions. A transformational

approach to graphing will be employed.

Graphing calculators will be used as a tool to

investigate the shapes and behaviors of these

functions.



Given a transformation of a parent function,

identify the graph of the transformed

function.


transformational approach, graph a function.


parent function.



the image in order to determine the equation

of the image.

Using a transformational approach, write the

equation of a function given its graph.

SOL AII.7 The student will investigate and



a) domain and range, including limited and

discontinuous domains and ranges;

b) zeros;

How do you determine

the end behavior of an thn degree polynomial

function?

How can real-life

problem situations be

modeled by polynomial

functions?


local and global behavior of

functions.

End behavior describes a

function as x approaches

positive and negative infinity.

A zero of a function is a value

of x that makes ( )f x equal

zero.

If (a, b) is an element of a

function, then (b, a) is an

element of the inverse of the

function.

c) x- and y-intercepts;

d) intervals in which a function is increasing or

decreasing;

e) asymptotes;

f) end behavior;

g) inverse of a function; and

h) composition of multiple functions.


assist in investigation of functions.






and ranges.

Given the graph of a function, identify

intervals on which the function is increasing

and decreasing.

Find the equations of vertical and horizontal

asymptotes of functions.

Describe the end behavior of a function.




Find the composition of two functions.

Use composition of functions to verify two

functions are inverses.

4 blocks Enrichment, Assessment, and

Remediation

Algebra/Trigonometry II Semester 2

Number

of Blocks

Topics and Essential Understandings Standards of Learning

Essential Knowledge and Skills

Additional Instructional

Resources / Comments

7 blocks

Unit 7: Rational Expressions and Equations

Add, subtract, multiply, and divide rational

expressions.

Simplify complex fractions

Solve rational equations

Graph rational functions

Domain and range

Asymptotes and discontinuity


Computational skills applicable to

numerical fractions also apply to rational

expressions involving variables.

Pattern recognition can be used to

determine complete factorization of a

polynomial.

SOL AII/T.1 The student, given rational,

radical, or polynomial expressions, will

a) add, subtract, multiply, divide, and simplify

rational algebraic expressions;

b) add, subtract, multiply, divide, and simplify

radical expressions containing rational

numbers and variables, and expressions

containing rational exponents;…

d) factor polynomials completely.


Add, subtract, multiply, and divide rational

algebraic expressions.

Simplify a rational algebraic expression

with common monomial or binomial

factors.

Recognize a complex algebraic fraction,

and simplify it as a quotient or product of

simple algebraic fractions.

Factor polynomials by applying general

patterns including difference of squares,

sum and difference of cubes, and perfect

The SOL refers to

solving rational

equations with

monomial and

binomial

denominators only.

Discuss horizontal

and vertical

asymptotes only –

no slant

asymptotes.

Make sure the

students can graph

rational functions

in any form.

Resources: http://www.analyzemath.c

om/Graphing/GraphRation

alFunction.html


Discussion Questions: What does

discontinuity mean

and how does it

affect the graph of

a function?

Compare and

contrast

polynomial and

rational functions.

http://www.analyzemath.com/Graphing/GraphRationalFunction.html





A quadratic function whose graph does not

intersect the x-axis has roots with

imaginary components.

The quadratic formula can be used to solve

any quadratic equation.

The value of the discriminant of a quadratic

equation can be used to describe the

number of real and complex solutions.

Real-world problems can be interpreted,

represented, and solved using equations

and inequalities.

The process of solving radical or rational

equations can lead to extraneous solutions.

Equations can be solved in a variety of

ways.

Set builder notation may be used to

represent solution sets of equations and

inequalities.

square trinomials.

Factor polynomials completely over the

integers.

Verify polynomial identities including the

difference of squares, sum and difference

of cubes, and perfect square trinomials.†

SOL AII/T.4 The student will solve,


b) quadratic equations over the set of complex

numbers;

c) equations containing rational algebraic

expressions; and ….

Graphing calculators will be used for solving

and for confirming the algebraic solutions.


Solve absolute value equations and

inequalities algebraically and graphically.

Solve a quadratic equation over the set of

complex numbers using an appropriate

strategy.

Calculate the discriminant of a quadratic

equation to determine the number of real

and complex solutions.

Solve equations containing rational

algebraic expressions with monomial or

binomial denominators algebraically and

graphically.

Verify possible solutions to an equation

containing rational or radical expressions.


real-world problem.

How can real-life

problem situations

be modeled by

rational functions?



Functions may be used to model real-world

situations.

The domain and range of a function may be

restricted algebraically or by the real-world

situation modeled by the function.

A function can be described on an interval

as increasing, decreasing, or constant.

Asymptotes may describe both local and

global behavior of functions.

End behavior describes a function as x

approaches positive and negative infinity.

A zero of a function is a value of x that

makes ( )f x equal zero.

If (a, b) is an element of a function, then (b,

a) is an element of the inverse of the

function.


The graphs/equations for a family of

functions can be determined using a


Transformations of graphs include






b) zeros;



decreasing;

e) asymptotes;

f) end behavior;










and ranges.


intervals on which the function is

increasing and decreasing.

Find the equations of vertical and

horizontal asymptotes of functions.


SOL AII/T.6 The student will recognize the

general shape of function (absolute value, square

root, cube root, rational, polynomial,…) families

and will convert between graphic and symbolic

forms of functions. A transformational approach to


translations, reflections, and dilations.

A parent graph is an anchor graph from

which other graphs are derived with

transformations.

graphing will be employed. Graphing calculators

will be used as a tool to investigate the shapes and

behaviors of these functions.



Given a transformation of a parent

function, identify the graph of the

transformed function.


transformational approach, graph a

function.


parent function.



the image in order to determine the

equation of the image.

Using a transformational approach, write

the equation of a function given its graph.

4 blocks

Unit 8: Logs and Exponentials

Log and exponential equations

Convert between logarithmic and

exponential form


The graphs/equations for a family of

functions can be determined using a


Transformations of graphs include

translations, reflections, and dilations.

A parent graph is an anchor graph from

which other graphs are derived with

SOL AII/T.6 The student will recognize

the general shape of function (absolute

value, square root, cube root, rational,

polynomial, exponential, and logarithmic)

families and will convert between graphic

and symbolic forms of functions. A

transformational approach to graphing will

be employed. Graphing calculators will be

used as a tool to investigate the shapes and

behaviors of these functions.



Do not do AII/T.7e

until the next unit.

Make sure you

discuss the

multiplicity of

roots.

Resources: http://www.regentsprep.or

g/Regents/math/algtrig/AT

P8b/indexATP8b.htm

X:\Algebra 2\M & M

Decay.doc

http://www.regentsprep.org/Regents/math/algtrig/ATP8b/indexATP8b.htm




transformations.


Functions may be used to model real-world

situations.

The domain and range of a function may be

restricted algebraically or by the real-world

situation modeled by the function.

A function can be described on an interval

as increasing, decreasing, or constant.

Asymptotes may describe both local and

global behavior of functions.

End behavior describes a function as x

approaches positive and negative infinity.

A zero of a function is a value of x that

makes ( )f x equal zero.

If (a, b) is an element of a function, then (b,

a) is an element of the inverse of the

function.

Exponential (xy a ) and logarithmic (

logay x ) functions are inverses of each

other.

Functions can be combined using

composition of functions.


Data and scatterplots may indicate patterns

that can be modeled with an algebraic

equation.

Graphing calculators can be used to collect,

organize, picture, and create an algebraic

model of the data.

Given a transformation of a parent

function, identify the graph of the

transformed function.


transformational approach, graph a

function.


parent function.



the image in order to determine the

equation of the image.

Using a transformational approach, write

the equation of a function given its graph.






b) zeros;



decreasing;

e) asymptotes;

f) end behavior;








X:\Algebra 2\starbucks

expansion.pdf

X:\Algebra 2\Who wants

to be a millionaire.doc


Discussion Questions: Explain how the

logarithmic and

exponential

functions are

inverses of each

other.

How can real life

problem situations

be modeled by

exponential and

logarithmic

functions?


Data that fit polynomial 1

1 1 0( ) ...n n

n nf x a x a x a x a where n is

a nonnegative integer, and the coefficients

are real numbers), exponential (xy b ),

and logarithmic ( logby x ) models arise

from real-world situations.



and ranges.


intervals on which the function is

increasing and decreasing.

Find the equations of vertical and

horizontal asymptotes of functions.





Investigate exponential and logarithmic

functions, using the graphing calculator.

Convert between logarithmic and

exponential forms of an equation with

bases consisting of natural numbers.

Find the composition of two functions.

Use composition of functions to verify two

functions are inverses.

SOL AII/T.9 The student will collect and

analyze data, determine the equation of the curve

of best fit, make predictions, and solve real-

world problems, using mathematical models.

Mathematical models will include polynomial,

exponential, and logarithmic functions.





Find an equation for the curve of best fit

for data, using a graphing calculator.


Models will include polynomial,

exponential, and logarithmic functions.





3 blocks

Unit 9: Sequences and Series

Arithmetic and geometric sequences and

series, including infinite geometric series

thn terms and sums of series

Explicit and recursive sequences


Sequences and series arise from real-world

situations.

The study of sequences and series is an

application of the investigation of patterns.

A sequence is a function whose domain is

the set of natural numbers.

Sequences can be defined explicitly and

recursively.

SOL AII/T.2 The student will investigate

and apply the properties of arithmetic and

geometric sequences and series to solve

real-world problems, including writing the

first n terms, finding the nth

term, and

evaluating summation formulas. Notation

will include and an.


Distinguish between a sequence and a

series.

Generalize patterns in a sequence using

explicit and recursive formulas.

Use and interpret the notations , n, nth

term, and an.

Given the formula, find an (the nth

term) for

an arithmetic or a geometric sequence.

Given formulas, write the first n terms and

find the sum, Sn, of the first n terms of an

arithmetic or geometric series.

Given the formula, find the sum of a

convergent infinite series.

Model real-world situations using

sequences and series.

Resources: http://teachers.henrico.k12.

va.us/math/hcpsalgebra2/7

-7.htm



Explain how to

classify sequences

and series as

arithmetic,

geometric, or

neither.

How can real-life

problem situations

be modeled using

sequences and

series?

http://teachers.henrico.k12.va.us/math/hcpsalgebra2/7-7.htm




22 blocks

Trigonometry

Unit Circle Trigonometry


Triangular trigonometric function

definitions are related to circular

trigonometric function definitions.

Both degrees and radians are units for

measuring angles.

Drawing an angle in standard position

will force the terminal side to lie in a

specific quadrant.

A point on the terminal side of an angle

determines a reference triangle from

which the values of the six trigonometric

functions may be derived.


If one trigonometric function value is

known, then a triangle can be formed to

use in finding the other five trigonometric

function values.

Knowledge of the unit circle is a useful

tool for finding all six trigonometric

values for special angles.


Special angles are widely used in

mathematics.

Unit circle properties will allow special

angle and related angle trigonometric

values to be found without the aid of a

calculator.

Unit Circle Trigonometry

SOL AII/T.13 The student, given a point

other than the origin on the terminal side of

the angle, will use the definitions of the six

trigonometric functions to find the sine,

cosine, tangent, cotangent, secant, and

cosecant of the angle in standard position.

Trigonometric functions defined on the unit

circle will be related to trigonometric

functions defined in right triangles.


Define the six triangular trigonometric

functions of an angle in a right triangle.

Define the six circular trigonometric

functions of an angle in standard

position.

Make the connection between the

triangular and circular trigonometric

functions.

Recognize and draw an angle in standard

position.

Show how a point on the terminal side of

an angle determines a reference triangle.

SOL AII/T.14 The student, given the value

of one trigonometric function, will find the

values of the other trigonometric functions,

using the definitions and properties of the

trigonometric functions.


Degrees and radians are units of angle

measure.

A radian is the measure of the central

angle that is determined by an arc whose

length is the same as the radius of the

circle.


The trigonometric function values of any

angle can be found by using a calculator.

The inverse trigonometric functions can

be used to find angle measures whose

trigonometric function values are known.

Calculations of inverse trigonometric

function values can be related to the

triangular definitions of the trigonometric

functions.


Given one trigonometric function value,

find the other five trigonometric function

values.

Develop the unit circle, using both

degrees and radians.

Solve problems, using the circular

function definitions and the properties of

the unit circle.

Recognize the connections between the

coordinates of points on a unit circle and

– coordinate geometry;

– cosine and sine values; and

– lengths of sides of special right

triangles (30 -60 -90 and 45 -45 -

90 ).

SOL AII/T.15 The student will find, without

the aid of a calculator, the values of the

trigonometric functions of the special angles

and their related angles as found in the unit

circle. This will include converting angle

measures from radians to degrees and vice

versa.


Find trigonometric function values of

special angles and their related angles in

both degrees and radians.

Apply the properties of the unit circle

without using a calculator.

Use a conversion factor to convert from

radians to degrees and vice versa without

using a calculator.


Trigonometric graphing


The domain and range of a trigonometric

function determine the scales of the axes

for the graph of the trigonometric

function.

The amplitude, period, phase shift, and

vertical shift are important characteristics

of the graph of a trigonometric function,

and each has a specific purpose in

applications using trigonometric

equations.

The graph of a trigonometric function can

be used to display information about the

periodic behavior of a real-world

situation, such as wave motion or the

motion of a Ferris wheel.


Restrictions on the domains of some inverse

trigonometric functions exist.

SOL AII/T.16 The student will find, with the

aid of a calculator, the value of any

trigonometric function and inverse

trigonometric function.


Use a calculator to find the trigonometric

function values of any angle in either

degrees or radians.

Define inverse trigonometric functions.

Find angle measures by using the inverse

trigonometric functions when the

trigonometric function values are given.

Trigonometric graphing

SOL AII/T.18 The student, given one of the six

trigonometric functions in standard form, will

a) state the domain and the range of the

function;

b) determine the amplitude, period, phase

shift, vertical shift, and asymptotes;

c) sketch the graph of the function by using

transformations for at least a two-period interval;

and

d) investigate the effect of changing the

parameters in a trigonometric function on the

graph of the function.


Determine the amplitude, period, phase

shift, and vertical shift of a trigonometric

function from the equation of the


Trigonometric Identities


Trigonometric identities can be used to

simplify trigonometric expressions,

equations, or identities.

Trigonometric identity substitutions can

function and from the graph of the

function.

Describe the effect of changing A, B, C,

or D in the standard form of a

trigonometric equation {e.g., y = A sin

(Bx + C) + D or y = A cos [B(x + C)] +

D}.

State the domain and the range of a

function written in standard form {e.g., y

= A sin (Bx + C) + D

or y = A cos [B(x + C)] + D}.

Sketch the graph of a function written in

standard form {e.g.,

y = A sin (Bx + C) + D or y = A cos [B(x +

C)] + D} by using transformations for at

least one period or one cycle.

SOL AII/T.19 The student will identify the

domain and range of the inverse trigonometric

functions and recognize the graphs of these

functions. Restrictions on the domains of the

inverse trigonometric functions will be included.


Find the domain and range of the inverse


Use the restrictions on the domains of the

inverse trigonometric functions in finding

the values of the inverse trigonometric

functions.

Identify the graphs of the inverse


Trigonometric Identities


help solve trigonometric equations, verify

another identity, or simplify

trigonometric expressions.



Solutions for trigonometric equations will

depend on the domains.

A calculator can be used to find the

solution of a trigonometric equation as

the points of intersection of the graphs

when one side of the equation is entered

in the calculator as Y1 and the other side

is entered as Y2.

Applications of Trigonometry


A real-world problem may be solved by using one

of a variety of techniques associated with

SOL AII/T.17 The student will verify basic

trigonometric identities and make substitutions,

using the basic identities.


Use trigonometric identities to make

algebraic substitutions to simplify and

verify trigonometric identities. The basic

trigonometric identities include

– reciprocal identities;

– Pythagorean identities;

– sum and difference identities;

– double-angle identities; and

– half-angle identities.


SOL AII/T.20 The student will solve

trigonometric equations that include both infinite

solutions and restricted domain solutions and solve

basic trigonometric inequalities.


Solve trigonometric equations with

restricted domains algebraically and by

using a graphing utility.

Solve trigonometric equations with

infinite solutions algebraically and by

using a graphing utility.

Check for reasonableness of results, and

verify algebraic solutions, using a

graphing utility.


triangles. Applications of Trigonometry

SOL AII/T.21 The student will identify,

create, and solve real-world problems

involving triangles. Techniques will include

using the trigonometric functions, the

Pythagorean Theorem, the Law of Sines,

and the Law of Cosines.


Write a real-world problem involving

triangles.

Solve real-world problems involving

triangles.

Use the trigonometric functions,

Pythagorean Theorem, Law of Sines, and

Law of Cosines to solve real-world

problems.

Use the trigonometric functions to model

real-world situations.

Identify a solution technique that could

be used with a given problem.

Prove the addition and subtraction

formulas for sine, cosine, and tangent and

use them to solve problems.†

5 blocks

Assessment, Enrichment, and Remediation

Documents

ALGEBRA II/TRIGONOMETRY CURRICULUM UIDE · The GUIDE is the lead document for planning, assessment and curriculum work. It is a summarized ... Benchmark Assessment AII/T.19 Data Analysis