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Curriculum Design Template
Content Area: Mathematics
Course Title: Pre Calculus Grade Level: 11
Functions and Graphs
Polynomials and Rational Functions
Exponential and Logarithmic Functions
Marking Period 1
Introduction to Trigonometry
Trigonometric Graphs
Marking Period 2
Solving Trigonometric Equations
Trigonometric Identities and Proofs
Trigonometric Applications
Marking Period 3
Analytic Geometry
Limits and Continuity
Marking Period 4
Date Created: May 2012
Board Approved on: August 27, 2012
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Course Title: Pre-Calculus
Grade Level: 11th & 12th
Overarching Essential Questions
What are Functions?
What types of functions exist?
How can we solve all polynomial equations?
How can we use symmetry and transformations to graph polynomial
functions?
How can we graph rational functions?
What are exponential and logarithmic functions?
How can exponential and logarithmic functions be used in
real-world applications?
What is trigonometry and how is it used?
What do the graphs of the trigonometric functions look like?
How can we solve trigonometric equations?
What are the trigonometric identities and how can we verify
them?
How can we use trigonometry to solve triangles?
How does trigonometry relate to vectors?
What are the conic sections?
How can we write equations to describe the conic sections?
How can we convert from rectangular coordinates to polar
coordinates?
What is a limit and what does it tell us about the behavior of a
function?
Overarching Enduring Understandings
Students in Pre-Calculus will learn about various types of
functions: polynomial, inverse,
piece-wise, exponential, logarithmic, and trigonometric.
Emphasis will be placed on
graphical as well as algebraic knowledge of each type of
function. Real-world applications
will be infused throughout the curriculum as appropriate.
Additionally, students will learn
about the conic sections and vectors as well as the concept of
limits.
Course Description
This course is designed for students who have successfully
completed algebra 2 and/or
who wish to pursue higher level mathematics. In addition to
those topics normally covered
in a pre-calculus course, concepts such as conic sections and
limits will be explored.
Graphing calculators are used on a daily basis and it would be
beneficial to have one's own
so that homework assignments may be completed more easily. This
course is designed for
the college bound student who intends to attend a 4-year college
and/or a STEM career.
Non-seniors who take and pass pre-calculus are eligible to take
calculus as a senior if they
choose to do.
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Technology and 21st-Century Life and Careers Standards
8.1.12 A. Basic Computer Skills and Tools 3. Construct a
spreadsheet, enter data, use mathematical or logical functions to
manipulate and process data, generate charts and graphs, and
interpret the results. 8.1.12 A. Basic Computer Skills and Tools 5.
Produce a multimedia project using text, graphics, moving images,
and sound. 8.1.12 B. Application of Productivity Tools 9. Create
and manipulate information, independently and/or collaboratively,
to solve problems and design and develop products. 8.1.12 B.
Application of Productivity Tools 11. Identify a problem in a
content area and formulate a strategy to solve the problem using
brainstorming, flowcharting, and appropriate resources. 8.2.12 B.
Design Process and Impact Assessment 3. Develop methods for
creating possible solutions, modeling and testing solutions, and
modifying proposed design in the solution of a technological
problem using hands-on activities.
9.1.12.C.5 Assume a leadership position by guiding the thinking
of peers in a direction
that leads to successful completion of a challenging task or
project
9.1.4.F.2 Establish and follow performance goals to guide
progress in assigned areas of
responsibility and accountability during classroom projects and
extra-curricular activities.
9.2.12.A.1 Analyze the relationship between various careers and
personal earning goals.
9.2.12.A.2 Identify a career goal and develop a plan and
timetable for achieving it,
including educational/training requirements, costs, and possible
debt.
9.2.12.A.5 Evaluate current advances in technology that apply to
a selected occupational
career cluster.
9.3.12.C.2 Characterize education and skills needed to achieve
career goals, and take steps to prepare for postsecondary options,
including making course selections, preparing for and taking
assessments, and participating in extra-curricular activities.
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Functions and Graphs
Essential Questions
What is a function? How can we determine the domain and range of
a function? How can we apply transformations to parent graphs to
graph more complicated
functions? What are the characteristics of certain functions?
Can we create new functions from old functions? What are inverses
and are they necessarily also functions?
Key Terms
Function, domain, range, piece-wise function, vertical line
test, increasing, decreasing,
maxima, minima, interval notation, concavity, point of
inflection, parent graphs,
transformation, symmetry, composite function, inverse function,
horizontal line test, one-
to-one, domain restriction
Objectives
Students will be able to:
Determine if a relation or graph is a function.
Find the domain and range of a given function.
Evaluate piece-wise defined functions and greatest integer
functions.
Define and recognize parent functions.
Graph transformations of the parent graphs.
Determine if a graph has x-axis, y-axis or origin symmetry.
Determine if a function is even, odd or neither.
Form sum, difference, product, quotient and composite functions
and find their
domains.
Define and find inverse functions from tables, graphs and
equations.
Determine if an inverse is a function using composition of
functions.
Standards associated with objectives
F-IF.1. Understand that a function from one set (called the
domain) to another set (called the range) assigns to each element
of the domain exactly one element of the range. If f is a function
and x is an element of its domain, then f(x) denotes the output of
f corresponding to the input x. The graph of f is the graph of the
equation y = f(x). F-IF.2. Use function notation, evaluate
functions for inputs in their domains, and interpret statements
that use function notation in terms of a context. F-IF.4. For a
function that models a relationship between two quantities,
interpret key features of graphs and tables in terms of the
quantities, and sketch graphs showing key features given a
verbal
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description of the relationship. Key features include:
intercepts; intervals where the function is increasing, decreasing,
positive, or negative; relative maximums and minimums; symmetries;
end behavior; and periodicity. F-IF.5. Relate the domain of a
function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives
the number of person-hours it takes to assemble n engines in a
factory, then the positive integers would be an appropriate domain
for the function. F-IF.7. Graph functions expressed symbolically
and show key features of the graph, by hand in simple cases and
using technology for more complicated cases. a. Graph linear and
quadratic functions and show intercepts, maxima, and minima. b.
Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions. c. Graph
polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior. F-IF.8.
Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties of the
function. a. Use the process of factoring and completing the square
in a quadratic function to show zeros, extreme values, and symmetry
of the graph, and interpret these in terms of a context. F-BF.1.
Write a function that describes a relationship between two
quantities
Combine standard function types using arithmetic operations. For
example, build a function that models the temperature of a cooling
body by adding a constant function to a decaying exponential, and
relate these functions to the model.
(+) Compose functions. For example, if T(y) is the temperature
in the atmosphere as a function
of height, and h(t) is the height of a weather balloon as a
function of time, then T(h(t)) is the temperature at the location
of the weather balloon as a function of time.
F-BF.3. Identify the effect on the graph of replacing f(x) by
f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k
(both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects
on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
F-BF.4. Find inverse functions. 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.
(+) Verify by composition that one function is the inverse of
another. (+) Read values of an inverse function from a graph or a
table, given that the function has an
inverse. (+) Produce an invertible function from a
non-invertible function by restricting the domain.
Suggested Lesson Activities
Give students characteristics of a graph: when it is increasing,
decreasing, concave up, concave down, where the extrema are
located, where points of inflection are located and have them
sketch the graph.
Have students collect data from the most recent census. Have
them build a scatter
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plot of the data, find its inverse, find the domain and range of
both functions. Have students create a composite function on an
index card. Students must them
swap cards and figure out the component functions of the
composite functions. Differentiation /Customizing learning
(strategies)
Have students (in a group) develop a lesson on piecewise
functions. They may present their lesson in any way they choose:
blackboard, Smartboard, video, Jing, etc.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
Polynomial and Rational Functions
Essential Questions
How can we simplify rational expressions? Is there a limit to
the number of roots a function has? Can a polynomial be simplified
and what is the best way to do so? Can we sketch a graph of a
polynomial function of degree greater than 2 without a
graphing calculator? What are asymptotes and what do they tell
us about the behavior of a graph? What is the Fundamental Theorem
of Algebra If given the roots of a polynomial function, can we
determine the shape of the graph
and a general equation for the function? What do polynomial
functions looks like if they have complex roots?
Key Terms
Polynomial, constant term, degree, linear, quadratic, cubic,
quartic, synthetic division,
zeros, factors, Remainder Theorem, Factor Theorem, irreducible,
continuous, end behavior,
multiplicity, vertical asymptote, horizontal asymptote, oblique
asymptote, discontinuity,
holes, Big-Little Concept, complex roots, complex conjugate,
Fundamental Theorem of
Algebra
Objectives
Students will be able to:
Divide polynomials using synthetic division and apply both the
Remainder Theorem
and the Factor Theorem.
Determine the maximum number of zeros a polynomial can have.
Factor a polynomial completely.
Recognize the shape of basic polynomial functions, including
behavior at the x-
intercepts of specific multiplicities, local extrema, and end
behavior as well as write
the general equation of the polynomial function if given the
graph.
Find the domain, intercepts, and asymptotes of rational
functions and graph them.
Apply the Fundamental Theorem of Algebra.
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Find complex conjugates, the number of zeros of a polynomial and
give complete
factorizations of a polynomial expression.
Standards associated with objectives
F-IF.7. Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using technology
for more complicated cases. d. (+) Graph rational functions,
identifying zeros and asymptotes when suitable factorizations are
available, and showing end behavior. A-APR.2. Know and apply the
Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if
(x – a) is a factor of p(x). A-APR.3. Identify zeros of polynomials
when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
A-APR.4. Prove polynomial identities and use them to describe
numerical relationships. For example, the polynomial identity (x2 +
y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean
triples. A-APR.5. (+) Know and apply the Binomial Theorem for the
expansion of (x + y)n in powers of x and y for a positive integer
n, where x and y are any numbers, with coefficients determined for
example by Pascal’s Triangle.1 N-CN.9. (+) Know the Fundamental
Theorem of Algebra; show that it is true for quadratic
polynomials.
Suggested Lesson Activities
Give students the roots of a polynomial function. Have them
sketch the general shape of the function and then determine a
polynomial equation that describes the graph.
Have students determine the size square that should be cut from
each corner of a rectangular-shaped piece of paper to create an
open-top box with a given value. Students will need to determine a
polynomial to determine the solution.
Have students do a complete graph of a rational function,
complete with intercepts, asymptotes, and extrema.
Differentiation /Customizing learning (strategies) Any student
wishing to explore complex numbers further may want to explore
the
Mandelbrot Set, which is the set of complex numbers c such that
the orbit of 0 under the function does not approach infinity.
Examine each diagonal of Pascal’s Triangle. Explain why the nth
term of each diagonal represented by a polynomial. Find examples of
patterns in the triangle that can be described by linear,
quadratic, cubic, quartic, and quintic polynomials. Present
findings in a variety of ways individual to each student/group.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
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Exponential and Logarithmic Functions
Essential Questions
What are rational exponents? How do we simplify radical and
rational exponents? What are exponential and logarithmic functions?
Can we transform exponential and logarithmic functions as we do
polynomial
functions? Is there a real-world purpose to exponential and
logarithmic functions? How can we solve exponential and logarithmic
equations?
Key Terms
Rational exponent, radical exponent, rationalizing,
irrational,
Exponential function, exponential growth and decay, e,
compounded interest, common
logarithm, regular logarithm, natural logarithm, natural
exponential function, properties of
logarithms
Objectives
Students will be able to:
Define and apply both rational and irrational exponents.
Simplify expressions containing radicals and rational
exponents.
Graph and identify transformations of exponential functions.
Use exponential functions to solve real-world applications.
Evaluate common, regular and natural logarithmic functions both
with and without
a calculator.
Convert flexibly between exponential and equation
equivalents.
Graph and identify transformations of logarithmic functions.
Apply the properties of logarithms to simplify expressions
involving logarithms.
Solve exponential and logarithmic equations as they apply to
real-world problems.
Standards associated with objectives
N-RN.1. Explain how the definition of the meaning of rational
exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for radicals in
terms of rational exponents. For example, we define 51/3 to be the
cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so
(51/3)3 must equal 5. N-RN.2. Rewrite expressions involving
radicals and rational exponents using the properties of exponents.
F-IF.7. Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using technology
for more complicated cases. e. Graph exponential and logarithmic
functions, showing intercepts and end behavior, and trigonometric
functions, showing period, midline, and amplitude. F-BF.3. Identify
the effect on the graph of replacing f(x) by f(x) + k, k f(x),
f(kx), and f(x + k) for specific
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values of k (both positive and negative); find the value of k
given the graphs. Experiment with cases and illustrate an
explanation of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and algebraic
expressions for them. F-LE.1. Distinguish between situations that
can be modeled with linear functions and with exponential
functions.
Prove that linear functions grow by equal differences over equal
intervals, and that exponential functions grow by equal factors
over equal intervals.
Recognize situations in which one quantity changes at a constant
rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a
constant percent rate per unit interval relative to another.
F-LE.2. Construct linear and exponential functions, including
arithmetic and geometric sequences, given a graph, a description of
a relationship, or two input-output pairs (include reading these
from a table). F-LE.3. Observe using graphs and tables that a
quantity increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more generally) as a
polynomial function. F-LE.4. For exponential models, express as a
logarithm the solution to abct = d where a, c, and d are numbers
and the base b is 2, 10, or e; evaluate the logarithm using
technology. F-LE.5. Interpret the parameters in a linear or
exponential function in terms of a context. A-APR.6. Rewrite simple
rational expressions in different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x) less than the degree of b(x),
using inspection, long division, or, for the more complicated
examples, a computer algebra system. A-APR.7. (+) Understand that
rational expressions form a system analogous to the rational
numbers, closed under addition, subtraction, multiplication, and
division by a nonzero rational expression; add, subtract, multiply,
and divide rational expressions.
Suggested Lesson Activities
Have students build a chart that compares the interest gained on
a principal amount at a fixed interest rate for a fixed number of
years but with varying compounding periods. Have a discussion on
what compounding period seems better and would it be best to
compound continuously? Would the money grow without bound?
Have students graph both and . Have a discussion on the
comparison of these two graphs (domain, range, increasing,
decreasing, concavity, symmetry, etc.)
Have students practice converting from exponential form to
logarithmic form and vice versa. This is an important skill when
solving both types of equations.
Differentiation /Customizing learning (strategies)
Have students collect data from the most recent census or from
another population source (biology data, death rates, etc.). Have
them build a scatter plot of the data and determine what type of
model best fits the population growth or decay: linear,
exponential, logarithmic, etc. then, have them determine a
regression equation describing the data.
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At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
Introduction to Trigonometry
Essential Questions
What is a trigonometric ratio? What is a reference triangle? How
can we use the special right triangles from geometry to find
trigonometric
values? Is there another way to measure angles other than
degrees? How are degrees and
radians related? Can angles have a negative measure? What does
this mean? What is a unit circle? What are trigonometric identities
and how are they used?
Key Terms
Angle, vertex, sides, degrees, minutes, seconds, hypotenuse,
adjacent, opposite,
trigonometric ratios, special angles, SOHCAHTOA, angle of
elevation, angle of depression,
initial side, terminal side, coterminal, unit circle, radian,
linear speed, angular speed,
cosine, sine, tangent, secant, cosecant, cotangent, exact value,
quotient identities, reciprocal
identities, Pythagorean identities, period, periodicity
identities, negative angle identities
Objectives
Students will be able to:
Define the six trigonometric ratios of an acute angle in terms
of a right triangle.
Evaluate trigonometric ratios using reference triangles,
calculator, and special right
triangles.
Solve right triangles using trigonometric ratios and apply
trigonometric ratios to
real-world problems.
Define radian measure and convert flexibly between radians and
degrees.
Extend the definition of angle measure to negative angles and
angles greater than
180 degrees.
Define the trigonometric ratios in the coordinate plane and in
terms of the unit
circle.
Develop and apply basic trigonometric identities.
Standards associated with objectives
F-TF.1. Understand radian measure of an angle as the length of
the arc on the unit circle subtended by the angle.
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F-TF.2. Explain how the unit circle in the coordinate plane
enables the extension of trigonometric functions to all real
numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle. F-TF.3. (+) Use special
triangles to determine geometrically the values of sine, cosine,
tangent for π/3, π/4 and π/6, and use the unit circle to express
the values of sine, cosines, and tangent for x, π + x, and 2π – x
in terms of their values for x, where x is any real number. F-TF.4.
(+) Use the unit circle to explain symmetry (odd and even) and
periodicity of trigonometric functions. G-SRT.6. Understand that by
similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric
ratios for acute angles. G-SRT.7. Explain and use the relationship
between the sine and cosine of complementary angles. G-SRT.8. Use
trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
Suggested Lesson Activities
Using geometer sketchpad 5.0, show examples of graphs of sine
functions and show visually their number of cycles
Use geometer sketchpad to measure arc length, radian measure,
and radius of any given circle.
Show using the unit circle where tangent functions are
undefined, and find its asymptotes
Build a unit circle using reference triangles and the special
right triangle ratios. Differentiation /Customizing learning
(strategies)
Allow students to work with pictures of graphs of non whole
coefficients and determine the properties of a trigonometric
function
Have students review the unit circle and see if they can
interpolate values for non special angles
Work with real-world word problems to show how trigonometric
functions are used and how they will have more than one answer
Students who are struggling should review the unit circle and
learn to use a graphing calculator to aid in their knowledge of
trigonometry
Reinforce oscillations using geometer sketchpad and show the
period and amplitude of basic trigonometric functions.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
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Trigonometric Graphs
Essential Questions
What do the graphs of the trigonometric functions look like?
What are the characteristics of the graphs of the trigonometric
functions?
What are period, amplitude and phase shift?
Can we graph trigonometric transformations in the same manner as
polynomial,
exponential, and logarithmic functions?
Key Terms
Even function, odd function, Even-Odd identities, period,
amplitude, phase shift
Objectives
Students will be able to:
Graph sine, cosine and tangent functions as well as their
transformations.
Recognize the graphs of the secant, cosecant and cotangent
functions.
State the period, amplitude, and phase shift of the
trigonometric functions.
Standards associated with objectives
F-BF.3. Identify the effect on the graph of replacing f(x) by
f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k
(both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects
on the graph using technology. Include recognizing even and odd
functions from their graphs and algebraic expressions for them.
F-TF.5. Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and midline.
Suggested Lesson Activities
Given a trigonometric function with transformations, have
students list what the transformations are, graph the parent
trigonometric function, and then transform the parent function for
the final graph.
Have students graph the trigonometric functions by hand (making
an x-y table) and then compare to the graphs on the graphing
calculator. Have students list as many characteristics as they can
abut each graph (intercepts, roots, maximums, minimums, asymptotes,
etc,)
Differentiation /Customizing learning (strategies)
Have students look up what a sinusoid graph is and report back,
using whatever
method they choose, as to what one is and how they can be
applied.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
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Solving Trigonometric Equations
Essential Questions
Can we solve equations that involve trigonometric functions? How
do we recognize that we have all solutions to a trigonometric
equation? What are inverse trigonometric functions?
Key Terms
Basic trigonometric equation, inverse trigonometric function,
arcsine, arcosine, arctangent,
arcsecant, arccosecant, arccotangent,
Objectives
Students will be able to
Solve trigonometric equations graphically and algebraically.
Recognize and find all solutions to a trigonometric
equation.
Standards associated with objectives
F-TF.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. F-TF.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.
Suggested Lesson Activities
Have students sketch the trigonometric functions and determine
if they are one-to-one. If they are not, have them determine the
domain restriction on each function that will make it
one-to-one.
Review solving simple algebraic equations and relate them to
solving trigonometric equations, stressing that one must use the
inverse of a trigonometric function when solving for an angle.
Stress the importance of changing the mode of the calculator to
degrees or radians as appropriate. Do an activity with them to show
how not being in the correct mode CAN affect the outcome of the
solution of the trigonometric equation.
Differentiation /Customizing learning (strategies)
Have students research simple harmonic motion and sound waves
and report on how trigonometry applies in these areas.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
-
Trigonometric Identities and Proofs
Essential Questions
Can we use the basic identities to prove more complicated
identities? What are the cofunction identities? What are the
double-angle and half-angle identities? How can we use identities
to help solve complicated trigonometric equations?
Key Terms
Cofunction identities, double-angle and half-angle
identities
Objectives
Students will be able to:
Identify possible identities by using graphs.
Apply algebraic strategies to prove identities.
Use the cofunction identities.
Use the double-angle and half angle identities.
Use trigonometric identities to solve trigonometric
equations.
Standards associated with objectives
F-TF.8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and
use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or
tan(θ) and the quadrant of the angle. F-TF.9. (+) Prove the
addition and subtraction formulas for sine, cosine, and tangent and
use them to solve problems.
Suggested Lesson Activities
Have students graph both sides of a given identity on the
graphing calculators to see visually that the left side does indeed
equal the right side.
Review the following algebraic maneuvers BEFORE trying to prove
trigonometric identities: Dividing fraction, adding and subtracting
fractions, working with conjugates, multiplying binomials.
Have students work in partners, at first, to solve some of the
more basic trigonometric identities.
Differentiation /Customizing learning (strategies)
Tell students that there are three sets of two parent functions
each whose product results in another parent function. There is
also a set of three parent functions whose product is another
parent functions. Ask them to see how many of these four sets they
can find, then prove the identity.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
-
Trigonometric Applications
Essential Questions
How can we solve non-right triangles? Is there a way to find the
area of a triangle if we do not know its height but only the
lengths of the tree sides? What is a vector? Can we perform
operations on vectors?
Key Terms
Oblique triangles, Law of Cosines, Law of Sines, Heron’s
Formula, vector, initial point,
terminal point, length, magnitude, equivalent vectors,
components, scalar multiplication,
vector addition, vector subtraction, zero vector, unit vector,
resultant
Objectives
Students will be able to:
Solve oblique triangles by using the Law of Cosines.
Solve oblique triangles using the Law of Sines.
Use area formulas to find the areas of triangles.
Find the components and magnitude of a vector.
Perform scalar multiplication of vectors, vector addition, and
vector subtraction.
Perform operations with vectors, determine direction angle of a
vector, and
determine resultant forces in real-world applications.
Standards associated with objectives
G-SRT.9. (+) Derive the formula A = 1/2 ab sin(C) for the area
of a triangle by drawing an auxiliary line from a vertex
perpendicular to the opposite side. G-SRT.10. (+) Prove the Laws of
Sines and Cosines and use them to solve problems. G-SRT.11. (+)
Understand and apply the Law of Sines and the Law of Cosines to
find unknown measurements in right and non-right triangles (e.g.,
surveying problems, resultant forces). N-CN.1. Know there is a
complex number i such that i2 = –1, and every complex number has
the form a + bi with a and b real. N-CN.2. Use the relation i2 = –1
and the commutative, associative, and distributive properties to
add, subtract, and multiply complex numbers. N-CN.3. (+) Find the
conjugate of a complex number; use conjugates to find moduli and
quotients of complex numbers. N-CN.4. (+) Represent complex numbers
on the complex plane in rectangular and polar form (including real
and imaginary numbers), and explain why the rectangular and polar
forms of a given complex number represent the same number. N-CN.5.
(+) Represent addition, subtraction, multiplication, and
conjugation of complex numbers geometrically on the complex plane;
use properties of this representation for computation. For example,
(-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument
120°.
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N-CN.7. Solve quadratic equations with real coefficients that
have complex solutions. N-CN.8. (+) Extend polynomial identities to
the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x –
2i). N-VM.1. (+) Recognize vector quantities as having both
magnitude and direction. Represent vector quantities by directed
line segments, and use appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v). N-VM.2. (+) Find the
components of a vector by subtracting the coordinates of an initial
point from the coordinates of a terminal point. N-VM.3. (+) Solve
problems involving velocity and other quantities that can be
represented by vectors. N-VM.4. (+) Add and subtract vectors.
Add vectors end-to-end, component-wise, and by the parallelogram
rule. Understand that the magnitude of a sum of two vectors is
typically not the sum of the magnitudes.
Given two vectors in magnitude and direction form, determine the
magnitude and direction of their sum.
Understand vector subtraction v – w as v + (–w), where –w is the
additive inverse of w, with the same magnitude as w and pointing in
the opposite direction. Represent vector subtraction graphically by
connecting the tips in the appropriate order, and perform vector
subtraction component-wise.
N-VM.5. (+) Multiply a vector by a scalar.
Represent scalar multiplication graphically by scaling vectors
and possibly reversing their direction; perform scalar
multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
Compute the magnitude of a scalar multiple cv using ||cv|| =
|c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the
direction of cv is either along v (for c > 0) or against v (for
c < 0).
Suggested Lesson Activities
Use real-world applications for the Law of Cosines ( the heading
of 2 planes and the resultant distance between them, and finding
the distance across a lake from a fixed point with known distances
and included angle) and the Law of Sines (an airplane flying at a
fixed altitude and finding its distance from a landing area).
Review the distance between two points formula and the slope
between two points formula and relate them to the equivalent
vectors formula. Have a discussion about these first two formulas
help us when we “reposition” vectors at the origin.
There are 9 properties of vector addition and scalar
multiplication. Have pairs of students take one property an explain
it to the class.
Differentiation /Customizing learning (strategies)
Research the Dot Product of two vectors and present findings.
Determine how the Mandelbrot Set relates to fractal images. At the
end of each chapter in the Holt Pre-Calculus: A Graphing Approach
there is a
“can do calculus” section. Have students who are ahead of the
curve in the main lesson work on these projects to gain a stronger
foundation for Calculus.
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Analytic Geometry
Essential Questions
What are the conic sections? How can we represent an ellipse, a
hyperbola, and a circle? What are the characteristics of the conic
sections?
Key Terms
Circle, center, radius, conic sections, ellipse and ellipse
terms: center, major axis, minor
axis, foci, vertices; hyperbola and hyperbola terms: distance
difference, foci, asymptotes,
center, focal axis, vertices, auxiliary rectangle; parabola and
parabola terms: focus,
directrix, axis, vertex; polar coordinates, origin, pole, polar
axis, polar equatin, cardioids,
eccentricity,
Objectives
Students will be able to:
Define a circle and write its equation.
Define and graph an ellipse.
Write the equation of an ellipse.
Identify characteristics of an ellipse.
Define and graph a hyperbola.
Write the equation of a hyperbola.
Identify characteristics of a hyperbola.
Standards associate with objectives
G-GMD.4. Identify the shapes of two-dimensional cross-sections
of three-dimensional objects, and identify three-dimensional
objects generated by rotations of two-dimensional objects.
Suggested Lesson Activities
Using actual physical models as well as images on Geometer’s
Sketchpad, show how to generate each of the conic sections and how
they are related.
Have students research how conic sections are used in the
development of telescopes.
Review solving systems briefly in order that students will be
better able to understand this concept when finding the general
form of each of the conic sections.
Differentiation /Customizing learning (strategies)
There are some REALLY interesting projects at the end of Chapter
12 in the old, green University of Chicago School Mathematics
Project book Functions, Statistics, and Trigonometry. These would
be really good to use for differentiated instruction.
At the end of each chapter in the Holt Pre-Calculus: A Graphing
Approach there is a “can do calculus” section. Have students who
are ahead of the curve in the main lesson work on these projects to
gain a stronger foundation for Calculus.
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Limits and Continuity
Essential Questions
What is a limit? What techniques can be used to calculate a
limit? What is the difference between a one-sided limit and a
two-sided limit? How is calculating a limit related to the
continuity of a function? What do limits tell us about the
asymptotes of the graph of a function?
Key Terms
Limit, vertical asymptote, horizontal asymptote, one-sided
limit, two-sided limit, end
behavior, continuity
Objectives
Students will be able to:
Have a basic understanding of the intuitive idea of a limit.
Understand the difference between a one-sided limit and a
two-sided limit.
Calculate simple limits using the properties of limits.
Calculate limits at infinity and describe the end behavior of a
graph.
Determine if a given function is continuous at a point or on an
interval.
Standards associated with objectives
No standards specifically relate to these topics.
Suggested Lesson Activities
Have students sketch graphs with given specific characteristics.
Use the graphing calculator to analyze tables and graphs of
difficult functions to
determine a function’s limit. Complete the continuity worksheet
from the University of Delaware AP Calculus
course binder. Use released College Board AP Exam questions.
Differentiation /Customizing learning (strategies)
It is very difficult to differentiate in a course of this nature
since the material is brand new to everyone. One possible way to
differentiate within the course is to allow students to sign out
reference material such as Five Steps to a Five or AP Calculus AB
Flash Cards in order for them to work at their own individual
pace.