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Algebra II/Trigonometry Unit Plan 2009-10
Unit
Aims/Lesson Objectives
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Trig
Applications
(Triangles)
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Pythagorean Theorem and
SOHCAHTOA
Law of Sines
Law of Cosines
Using Law of Sines and Cosines to
find angle measure
Finding the area of a triangle
Law of sines and cosines
applications (triangles with shared
sides, vector problems)
The ambiguous case
Time
9/9 – 10/16
(End of First
Marking
Period)
NYS Standards
A2.A.73 Solve for an unknown side or
angle, using the Law of Sines or the Law
of Cosines
A2.A.74 Determine the area of a triangle or
a parallelogram, given the measure
of two sides and the included angle
A2.A.75 Determine the solution(s) from the
SSA situation (ambiguous case)
Facts
180° in a triangle.
Right triangle use
SOHCAHTOA,
Pythagorean
Theoream.
Any triangle use
Law of
Sines/Cosines.
Area equation
works for any
triangle, provided
you have two
sides and the
included angle.
Essential
Questions
How do we solve
for missing sides
and angles in
triangles?
How do we adapt
what we know
about right
triangles to work
for all triangles
(angles, sides,
and area)
Big Idea/Enduring
Understanding
“The students will
understand that…”
Law of Sines and
Cosines are like the
big brother to
SOHCAHTOA and
Pythagorean Theorem,
and work to find the
missing sides and
angles of any triangle.
Assessments/Projects
TV Project – what’s
the biggest television
that will fit in a 42”
space? (Resources:
chart paper, meter
sticks)
Double Triangle
Problem – Measure
the height of the
building, the distance
of the East River by
taking two different
angle measurements a
known distance apart.
(Resources: chart
paper,
sextant/inclinometer,
meter
sticks/measuring tape)
Past Regents
questions on finding
missing sides and
angles, especially for
triangles with shared
sides and vector
problems, and area
problems.
Tests, quizzes, and
nightly homework.
Statistics
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Gathering/organizing data
Measures of central tendency
Measures of central tendency for
grouped data
Measures of dispersion
Variance and Standard Deviation.
Normal distribution
Scatterplots, regression lines,
correlation
Non-linear regression
Summations
10/19 – 11/6
A2.S.1 Understand the differences among
various kinds of
studies (e.g., survey, observation, controlled
experiment)
A2.S.2 Determine factors which may affect
the outcome of a survey
A2.S.3 Calculate measures of central
tendency with group
frequency distributions
A2.S.4 Calculate measures of dispersion
(range, quartiles, interquartile range,
standard deviation, variance) for both
samples and populations
Measure average
using mean,
median, or mode
Standard
deviation
measures how
spread out the
data are
If a set up
numbers are
normally
distributed, the
normal curve can
be used to make
How do we
classify and
describe trends in
statistics?
Mean and standard
deviation summarize
characteristics of a set
of numbers.
The line of best fit is
found using the TI-83
graphing calculator,
and that equation can
then be used to make
predictions
(interpolation).
Coin Flip experiment
– find mean and
standard deviation for
every student flipping
a fair coin 20 times,
use that data to make
predictions using the
normal curve.
(Resources: chart
paper, quarters for
every student,
graphing calculators)
Analyzing attendance
and grade data to
introduce correlation
Algebra II/Trigonometry Unit Plan 2009 - 10
predictions
A2.S.5 Know and apply the characteristics
of the normal distribution
A2.S.6 Determine from a scatter plot
whether a linear, logarithmic,
exponential, or power regression model is
most appropriate
A2.S.7 Determine the function for the
regression model, using appropriate
technology, and use the regression function
to interpolate and
extrapolate from the data

Building
Blocks of
Algebra
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Whole Numbers, Integers, and the
Number line
Writing and Solving Number
Sentences
Solving Absolute Value Equations
Solving inequalities and compound
inequalities
Solving Absolute Value Inequalities
Graphing Absolute Value
Inequalities
Adding Polynomials
Multiplying Polynomials
Factoring Polynomials
Quadratic Equations with Integral
Roots.
11/9 – 11/17
A2.S.8 Interpret within the linear regression
model the value of the correlation
coefficient as a measure of the strength of
the relationship
A2.N.3 Perform arithmetic operations with
polynomial expressions containing
rational coefficients
A2.A.1 Solve absolute value equations and
inequalities involving
linear expressions in one variable
A2.A.7 Factor polynomial expressions
completely, using any Expressions
combination of the following techniques:
common factor
extraction, difference of two perfect squares,
quadratic
trinomials
You can use TI83 to make
regression
equations and
find the line of
best fit based on
given data or a
scatterplot
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Roots and radicals. What do the
parts of a radical expression (index,
radical, radicand) represent?
Simplifying radicals, fractional
radicals, and radicals with indexes
greater than 2
Adding and subtracting radicals.
Multiplying radicals
Multiplying polynomials with
radicals
Dividing radicals
Rationalizing a denominator
Solving radical equations
11/18 – 11/25
Absolute value is
distance to zero,
“make it
positive”
Inequalities have
infinite solutions,
written as a
range, not just
one or two
Perfect squares
can be “factored
out” from
underneath the
radical
A2.N.4 Perform arithmetic operations on
irrational expressions
Radicals combine
like any other
like terms
A2.A.13 Simplify radical expressions
A2.A.14 Perform addition, subtraction,
multiplication and division of radical
Tests, quizzes, and
nightly homework.
How is solving an
inequality or
absolute value
equation similar
to solving
traditional
algebraic
equations?
Ranges of solutions
can be represented
using inequalities
Why is there
more than one
answer for both?
The solutions are on
the graph.
Coefficients and
exponents are
shorthand ways of
representing variables
Practice Regents
questions involving
inequalities and
absolute value.
Tests, quizzes, and
nightly homework..
How does a
coefficient differ
from an
exponent?
A2.N.2 Perform arithmetic operations
(addition, subtraction, multiplication,
division) with expressions containing
irrational numbers in radical
form
A2.N.5 Rationalize a denominator
containing a radical expression
Past Regents
questions based on
finding regression
equations and
standard
deviation/normal
distribution problems.
The correlation
coefficient tells
you the
relationship
between x and y,
if there is one.
A2.A.23 Solve rational equations and
inequalities
Radicals
and regression and as
a reminder of
classroom
expectations.
(Resources:
attendance data,
graphing calculators)
A square root of
a number times
itself is just the
number itself
How do we
simplify radicals?
Find the largest factor
that is a perfect square
Factor out the largest
perfect square from
under a radical
Add, subtract,
multiply, and divide
radicals
Rationalize a
denominator
Practice Regents
questions involving
radicals.
Locker Problem to
remind students of
factors/perfect
squares.
(Resources: visual
representation of
lockers)
Tests, quizzes, and
nightly homework.
Multiplying by
Algebra II/Trigonometry Unit Plan 2009 - 10
Expressions
A2.A.15 Rationalize denominators
involving algebraic radical expressions
A2.A.22 Solve radical equations
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Exponential
Functions
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Exponent rules, zero exponents,
negative exponents, fractional
exponents
Exponential functions and their
graphs
Solving equations with exponents
Applications (compound interest)
11/30 – 12/11
(End of
Second
Marking
Period)
A2.N.1 Evaluate numerical expressions with
negative and/or fractional
exponents, without the aid of a calculator
(when the answers are
rational numbers)
A2.A.6 Solve an application which results in
an exponential function
A2.A.8 Apply the rules of exponents to
simplify expressions involving
negative and/or fractional exponents
A2.A.9 Rewrite algebraic expressions that
contain negative exponents using
only positive exponents
the conjugate will
rationalize an
expression
Square root of a
fraction can be
thought of as the
square root of the
numerator over
the square root of
the denominator
Fractional
exponents can be
rewritten as
radicals, and vice
versa
Treat negative
exponents like
positives, but in
the denominator.
How do
investments
grow?
Simplify expressions
using the exponent
rules.
What are some
real-world nonlinear functions,
and how do we
model and
analyze them?
What the exponential
curve looks like at its
extremes.
Exponent rules
from algrebramultiplying,
dividing, raising
to a power
If you can get the
bases the same in an
equation, the
exponents can then be
set equal.
You can analyze and
exponential equation
to see if it is going to
increase (growth) or
decrease (decay).
A2.A.10 Rewrite algebraic expressions with
fractional exponents as radical
Expressions
A2.A.11 Rewrite algebraic expressions in
radical form as expressions with
fractional exponents
A2.A.27 Solve exponential equations with
and without common bases
Relations &
Functions
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What is a function? Domain, range
Transformations of linear functions
Quadratic functions/transformations
of quadratic functions
Roots of polynomial functions
How to find the vertex, axis of
symmetry
Higher degree polynomial functions
Using the graphing calculator to find
the roots of polynomial functions
12/14 – 12/23
Tests, quizzes, and
nightly homework.
Domain is the x
values, range is
the y’s.
A2.A.37 Define a relation and function
A2.A.38 Determine when a relation is a
function
A2.A.39 Determine the domain and range of
a function from its equation
Investment project to
see that finding
percent growth yearto-year is the same as
using the interest
equation.
(Resources: latest
market rates for CD,
Money Market, etc)
Practice Regents
questions involving
exponential regression
and equations, getting
bases the same, and
exponent rules.
A2.A.12 Evaluate exponential expressions,
including those with base e
A2.A.53 Graph exponential functions of the
form y = bx for positive values of
b, including b = e
A2.A.5 Use direct and inverse variation to
solve for unknown values
Skittles Lab
demonstrating
exponential growth
and decay, first nonlinear regression on
the graphing
calculator.
(Resources: bags of
candy, paper plates,
paper cups, graph
paper)
Roots are when
the equation
equals zero.
Functions have
an input and an
How does
transforming an
function on the
graph reflect in
the equation
form?
How do we find
the roots of an
equation?
Functions can be
transformed just like
points and shapes can.
To find the roots, look
for when the entire
equation is equals
zero.
Compositions of
Buried Treasure
activity to review
transformations.
(Resources: mini
candy bars)
Practice Regents
questions involving
manipulating
functions, looking at
Algebra II/Trigonometry Unit Plan 2009 - 10
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Function arithmetic
Composition of functions
Inverse functions
Review of equation of a circle
A2.A.40 Write functions in functional
notation
A2.A.41 Use functional notation to evaluate
functions for given values in the
Domain
output, and
functions can be
strung together as
composites.
functions work from
the inside out.
Inverse means switch
x and y in the equation
AND graph form.
A circle has a
center and a
radius.
their graphs,
composite functions,
and inverse functions.
Tests, quizzes, and
nightly homework.
A2.A.42 Find the composition of functions
A2.A.43 Determine if a function is one-toone, onto, or both
A2.A.44 Define the inverse of a function
A2.A.45 Determine the inverse of a function
and use composition to justify the
Result
A2.A.46 Perform transformations with
functions and relations: f (x + a) ,
f(x)+ a, f (−x), − f (x), af (x)
A2.A.47 Determine the center-radius form
for the equation of a circle in standard form
A2.A.48 Write the equation of a circle,
given its center and a point on the circle
A2.A.49 Write the equation of a circle from
its graph
A2.A.50 Approximate the solution to
polynomial equations of higher
degree by inspecting the graph
A2.A.51 Determine the domain and range of
a function from its graph
A2.A.52 Identify relations and functions,
using graphs

Quadratic
Functions &
Complex
Numbers
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Solving quadratic equations by
completing the square
Solving quadratic equations by the
quadratic formula
The discriminant
What is an imaginary number? What
is a complex number?
Adding and subtracting complex
numbers
Multiplying and dividing complex
numbers
Complex roots of a quadratic
1/4 – 1/22
(End of Third
Marking
Period)
A2.N.6 Write square roots of negative
numbers in terms of i
Roots are when
the equation
equals zero.
A2.N.7 Simplify powers of i
A2.N.8 Determine the conjugate of a
complex number
A2.N.9 Perform arithmetic operations on
complex numbers and write the
answer in the form a + bi. Note: This
includes simplifying expressions
Standard form is
y = ax2 + bx + c.
The discriminant
is what’s under
the radical in the
quadratic
formula.
How are the
discriminant, the
roots, and the
graph of the
quadratic
equation related?
If an equations has an
x2, chances are its
quadratic and must be
put into standard form.
Also there will be two
answers.
How do we find
the roots of a
quadratic
equation?
To find the roots, use
the quadratic formula.
To find the type of
roots, don’t use
Practice Regents
problems involving
finding the roots of a
quadratic, and the
roots can be rational,
irrational, double, or
imaginary.
Practice Regents
problems involving
imaginary and
complex numbers.
Algebra II/Trigonometry Unit Plan 2009 - 10
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equation
Solving higher degree polynomial
equations
Solving quadratic-linear systems of
equations (algebraically vs.
graphically)
with complex denominators.
A2.A.2 Use the discriminant to determine
the nature of the roots of a quadratic
Equation
A2.A.3 Solve systems of equations
involving one linear equation and one
quadratic equation algebraically Note: This
includes rational equations that result in
linear equations with extraneous roots
.
A2.A.4 Solve quadratic inequalities in one
and two variables, algebraically and
Graphically
Complex
numbers combine
like any other
like terms
How do we take
the square root of
a negative
number?
quadratic formula, just
look at the
discriminant.
Tests, quizzes, and
nightly homework.
The discriminant can
be determined based
on the graph.
Patterns in the
powers of i
You can take the
square root of a
negative number, it
just turns out
imaginary
TI-83 does have
an ‘i’ value
A2.A.20 Determine the sum and product of
the roots of a quadratic
equation by examining its coefficients
A2.A.21 Determine the quadratic equation,
given the sum and product of its
Roots
A2.A.24 Know and apply the technique of
completing the square
A2.A.25 Solve quadratic equations, using
the quadratic formula
Logarithmic
Functions
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Rational
Expressions
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What is a logarithm?
Why is the function logbx=y the
inverse of the by=x
Properties of logarithms
Base 10 (common) logs vs. natural
logs
Rewriting exponential equations as
logarithmic equations
Solving logarithmic equations
Review what makes a number
rational
What is a rational expression? When
is it undefined? How do we
simplify?
Multiplying and Dividing Rational
Expressions
2/1 – 2/12
A2.A.26 Find the solution to polynomial
equations of higher degree that can be
solved using factoring and/or the quadratic
formula
A2.A.18 Evaluate logarithmic expressions
in any base
A2.A.19 Apply the properties of logarithms
to rewrite logarithmic expressions
in equivalent forms
A2.A.28 Solve a logarithmic equation by
rewriting as an exponential equation
2/22 – 3/5
A2.A.54 Graph logarithmic functions, using
the inverse of the related
exponential function
A2.A.16 Perform arithmetic operations with
rational expressions and rename to
lowest terms
A2.A.17 Simplify complex fractional
expressions
There are 5 skills
to simplifying
fractions
How do we use
skills we know to
simplify rational
expressions?
Rational
expressions are
two expressions,
Algebra II/Trigonometry Unit Plan 2009 - 10
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
one a num, one a
denom
Adding and Subtracting Rational
Expressions
Simplifying Complex Rational
Expressions
Solving Rational Equations
Undefined it
denom=0
Long division
can be used with
expressions
Sequences &
Series
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Finite vs. infinite sequences
Arithmetic sequences, sigma
notation, and arithmetic series
Geometric sequences and series
Infinite series
A2.N.9 Know and apply sigma notation
3/8 – 3/19
(End of Third
Marking
Period)
A2.A.29 Identify an arithmetic or geometric
sequence and find the
formula for its nth term
A2.A.30 Determine the common difference
in an arithmetic sequence
A2.A.31 Determine the common ratio in a
geometric sequence
A2.A.32 Determine a specified term of an
arithmetic or geometric sequence
A2.A.33 Specify terms of a sequence, given
its recursive definition
A2.A.34 Represent the sum of a series,
using sigma notation
Trigonometric
Functions
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The unit circle, sine, and cosine
Trig trainer
The tangent function. Why is
tangent equivalent to sine/cos
The reciprocal trig functions
Function values of special angles
Inverse sine/cosine to find measure
of an angle
Reference angles (for angles in
quadrants II, III and IV
3/22 – 4/9
A2.A.35 Determine the sum of the first n
terms of an arithmetic or geometric
Series
A2.A.55 Express and apply the six
trigonometric functions as ratios Functions
of the sides of a right triangle
Paper plate spinners
A2.A.56 Know the exact and approximate
values of the sine, cosine, and
tangent of 0º, 30º, 45º, 60º, 90º, 180º, and
270º angles
A2.A.57 Sketch and use the reference angle
for angles in standard position
A2.A.58 Know and apply the co-function
and reciprocal relationships between
trigonometric ratios
A2.A.59 Use the reciprocal and co-function
relationships to find the value of
the secant, cosecant, and cotangent of 0º,
30º, 45º, 60º, 90º, 180º, and
270º angles
Algebra II/Trigonometry Unit Plan 2009 - 10
A2.A.60 Sketch the unit circle and represent
angles in standard position
A2.A.61 Determine the length of an arc of a
circle, given its radius and the
measure of its central angle
A2.A.62 Find the value of trigonometric
functions, if given a point on the
terminal side of angle θ
A2.A.63 Restrict the domain of the sine,
cosine, and tangent
functions to ensure the existence of an
inverse function
A2.A.64 Use inverse functions to find the
measure of an angle, given its sine,
cosine, or tangent
A2.A.66 Determine the trigonometric
functions of any angle, using technology
Trig Functions
II
Graphs of Trig
Functions
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Radian measure
Trig function values and radian
measure
Pythagorean identities
Domain and range of trig functions
Cofunctions
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Graph of sine function
Graph of cosine function
Amplitude, period, and phase shift
Graph of the tangent function
4/12 – 4/23
A2.M.1 Define radian measure
Measurement
Radian to Degree
foldables
A2.M.2 Convert between radian and degree
measures
A2.A.67 Justify the Pythagorean identities
4/26 – 4/30
A2.A.65 Sketch the graph of the inverses of
the sine, cosine, and tangent
functions
Spaghetti Sine graphs
Hours of Daylight vs.
Latitude graphs
A2.A.69 Determine amplitude, period,
frequency, and phase shift, given the
graph or equation of a periodic function
A2.A.70 Sketch and recognize one cycle of
a function of the form y = Asin Bx
or y = Acos Bx
A2.A.71 Sketch and recognize the graphs of
the functions y = sec(x) ,
y = csc(x), y = tan(x), and y = cot(x)
A2.A.72 Write the trigonometric function
that is represented by a given periodic
graph
Algebra II/Trigonometry Unit Plan 2009 - 10
Trig Identities
Trig Equations
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Basic identities
Proving an identity
Cosine (A – B)
Cosine (A + B)
Sine (A – B)
Sine (A + B)
Functions of 2A
Functions of ½ A
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
1st degree trig equations
Using factoring to solve 2nd degree
trig equations
Using the quadratic formula to solve
2nd degree trig equations
Using substitution to solve trig
equations
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Probability &
the Binomial
Theorem
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The counting principle
Permutations
Combinations
Probability
Probability with 2 outcomes
Binomial probability and the normal
curve (area under a normal curve)
The binomial theorem (Pascal’s
triangle)
5/3 – 5/14
(End of Fifth
Marking
Period)
5/17 – 5/28
6/1 – 6/14
(End of Sixth
Marking
Period)
A2.A.76 Apply the angle sum and
difference formulas for trigonometric
functions
A2.A.77 Apply the double-angle and halfangle formulas for trigonometric
functions
A2.A.68 Solve trigonometric equations for
all values of the variable from 0º to
360º
A2.A.36 Apply the binomial theorem to
expand a binomial and determine a
specific term of a binomial expansion
A2.S.9 Differentiate between situations
requiring permutations and those
requiring combinations
A2.S.10 Calculate the number of possible
permutations ( ) n r P of n items taken r
at a time
A2.S.11 Calculate the number of possible
combinations ( ) n r C of n items taken
r at a time
A2.S.12 Use permutations, combinations,
and the Fundamental Principle of
Counting to determine the number of
elements in a sample space and a
specific subset (event)
A2.S.13 Calculate theoretical probabilities,
including geometric applications
A2.S.14 Calculate empirical probabilities
A2.S.15 Know and apply the binomial
probability formula to events involving
the terms exactly, at least, and at most
A2.S.16 Use the normal distribution as an
approximation for binomial
probabilities
Algebra II/Trigonometry Unit Plan 2009 - 10
Algebra II/Trigonometry Unit Plan 2009 - 10