HOLMDEL TOWNSHIP PUBLIC SCHOOLS
CURRICULUM GUIDE
HONORS PRE-CALCULUS
Ms. Mary Beth Currie
Assistant Superintendent.
Curriculum and Instruction
Ms. Danielle Bellavance
Ms. Ann Wright
Holmdel High School
Table of Contents
Course Description...........................................................................................................................................................................................
Course Philosophy ...........................................................................................................................................................................................
Course Goals ....................................................................................................................................................................................................
Enduring Understandings.................................................................................................................................................................................
Scope and Sequence
Unit #1
Review of Essentials .............................................................................................................................................................
Unit #2
Functions and Graphs ...........................................................................................................................................................
Unit #3
Exponential and Logarithmic Functions ................................................................................................................................
Unit #4
Sequences and Series ............................................................................................................................................................
Unit #5
Combinatorics and Probability .............................................................................................................................................
Unit #6
Basic Trigonometric Functions and Graphs...........................................................................................................................
Unit #7
Trigonometric Equations and Applications ..........................................................................................................................
Unit #8
Triangle Trigonometry ...........................................................................................................................................................
Unit #9
Triangle Formulas and Applications ......................................................................................................................................
Unit #10 Polar Coordinates and Complex Numbers ............................................................................................................................
Unit #11 Limits .....................................................................................................................................................................................
Unit #12 Derivatives ................................................................................................................................................................................
Unit #13 Applications of Derivatives ..................................................................................................................................................
Required Instructional Resources ....................................................................................................................................................................
Evaluation and Grading ...................................................................................................................................................................................
New Jersey Core Curriculum Content Standards - Mathematics.....................................................................................................................
New Jersey Core Curriculum Content Standards – 21st Century Life & Career Skills....................................................................................
New Jersey Core Curriculum Content Standards – Technology Literacy .......................................................................................................
Scope and Sequence Overview ........................................................................................................................................................................
Career Infusion Lesson ....................................................................................................................................................................................
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12
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30
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47
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PROPOSED COURSE OF STUDY
HOLMDEL TOWNSHIP PUBLIC SCHOOLS
Course Title: Honors Pre-Calculus
Curriculum Area:
Mathematics
Length of Course: Full Year
New Course
Credits:
X
5
Half Year
Revision of Existing Course
X
Course Pre-Requisites: Honors Algebra 2 (≥ 80) or Advanced Algebra 2 (≥ 90) and Geometry (≥ 90)
Course Description: The curriculum for Honors Pre-Calculus emphasizes conceptual and graphical understanding of functions. It reinforces
critical thinking and keeps pace with the changes in mathematics and its applications. The content of this course includes the study of a variety of
function types (polynomial, exponential, logarithmic, periodic, rational), as well as function transformations and function inverses, the Binomial
Theorem, mathematical induction, sequences and series, combinatorics and probability, trigonometry, complex numbers, polar coordinates and
topics at the onset of Calculus: limits and derivatives.
Course Philosophy The course provides students with the foundation necessary for the rigors of future mathematics courses, including Calculus.
This course also prepares students well for the SAT Subject Test Mathematics Level 2.
Course Goals: This course is fast-paced with coverage of Pre-Calculus topics in depth, including problem-solving in most units. Since Honors
Pre-Calculus is a college-level course, many of its topics extend beyond the K-12 Mathematics standards. For those topics within the scope, of the
K-12 Mathematic Standards there is alignment with the New Jersey Core Curriculum Content Standards.
3
Enduring Understandings:
Unit #1 Linear and quadratic functions can be used to effectively model real-world situations. Real and imaginary numbers are subsets of the
complex number system. The Rational Root Theorem can be used to solve polynomial equations. Polynomial and rational inequalities
are solved by finding critical values and testing intervals.
Unit #2 Functions can be inverted to create new functions. Two functions can be combined to produce functions that are more complex. Such
combinations impact the graph, domain, and range of the functions. Functions can be transformed through both rigid and non-rigid
transformations. Symmetry and periodicity of functions can be exploited in generating their graphs. Rational functions include
domain issues and their graphs display asymptotic behavior. Models involving more than one variable can be transformed into a
function in one variable by expressing one variable in terms of the other
Unit #3
Exponential and logarithmic functions are inverses of each other. Properties of exponents and logarithms are closely related and can be
used to simplify expressions and solve equations. Exponential equations can be solved by rewriting them in quadratic form. Rational
exponents represent the roots of numbers. Many applications can be modeled using exponential or logarithmic functions.
Unit #4
Sequences can be defined explicitly or recursively. Summation formulas are available for finite arithmetic and geometric series.
Limits can be used to find the last term of infinite sequences. Infinite geometric series may converge or diverge. Mathematical
induction can be used to prove statements that are incrementally defined.
Set theory can be used to classify and count objects. Counting principles and combination and permutation formulas can also be used.
Binomial coefficients are linked to combinations and can be used to expand binomials raised to a power. Probability is a value
between 0 and 1 indicating the likelihood of an event. Decision trees are useful in understanding the sample space of experiments and
determining conditional probability. Expected value is the payoff for an experiment or game where gains or losses are involved.
Angles can be measured in revolutions, degrees or radians. The length of an arc and the area of a sector are most easily determined
when central angles are measured in radians. The six trigonometric functions can be extended to angles of any measure using the
circular definition and periodic nature of trigonometric functions. Restricting the domains of trigonometric functions allows us to
define inverse trigonometric functions.
Trigonometric functions can be transformed by translation, reflection or non-rigid stretches or shrinks. Non-rigid transformations
result in period or amplitude changes whereas horizontal translations are called phase shifts. Fundamental trigonometric identities
together with algebraic techniques are used to simplify expressions, prove identities and solve equations. When trigonometric
equations are solved an angle measure is the result.
Unit #5
Unit #6
Unit #7
Unit #8
Trigonometry is extended using the Law of Sines and Law of Cosines to solve or find the area of oblique (non right) triangles.
Because the sine values for acute and obtuse angles are positive, there can be ambiguity in applying the Law of Sines. Navigation and
surveying are common applications of trigonometry.
Unit #9
Angle summation and difference formulas can be proven for all six trigonometric functions. The double angle and power reduction
formulas are derived from the summation/difference formulas. These formulas can be used to find new trigonometric values or to
solve more complex trigonometric equations.
4
Unit #10 Simple polar equations can produce beautiful polar graphs including circles, limacons, roses, and spirals. Complex numbers can be
plotted on a rectangular, complex plane (a.k.a. Argand Diagram). Using the polar representation of these points, a complex number
can be represented in polar form. Complex numbers are much easier to manipulate in polar form.
Unit #11 Calculus is the mathematics of change. Limits are the fundamental process that converts Pre-calculus mathematics to Calculus. A limit
is taken at a specific input value and, if defined, is a real number. As a result, limits behave like real numbers. Limits can be evaluated
graphically, numerically, or analytically.
Unit #12 Differentiation is one of the two fundamental operations of Calculus. The derivative is the instantaneous rate of change of a curve and
is the limit as the denominator goes to zero of the difference quotient (i.e., slope). The tangent line at a point on a curve has a slope
equal to the derivative at that point. Several differentiation rules can be proven and then used to derive expressions or find the
derivative of functions. The chain rule is an important construct and is the foundation of implicit differentiation and related rates.
Unit #13 A sketch of the graph of a function can be determined using Calculus. Limits at infinity (end behavior) together with relative extrema
and roots can be used. The first derivative test identifies relative maximums, minimums, and intervals of increase or decrease. The
second derivative test can identify extrema as well as points of inflection.
5
Scope and Sequence
Duration: 1 week
Unit 1: Review of Essentials
Standards:
NJCCCS: 4.1A1, 4.1A3, 4.1B1, 4.2C1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Linear and quadratic functions can be used
to effectively model real-world situations. Real and imaginary numbers
are subsets of the complex number system. The Rational Root Theorem
can be used to solve polynomial equations. Polynomial and rational
inequalities are solved by finding critical values and testing intervals.
Essential Question(s): How does the rate of change relate to the slope
of a linear function? How can the maximum or minimum of quadratic
and cubic functions be used to answer real-world problems? How can
the roots and end-behavior of a polynomial be used to sketch its graph?
What is the Fundamental Theorem of Algebra and why is it important?
How can a system of equations be used to find the root of a complex
number? What is the meaning of absolute value?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
1.
2.
3.
4.
6
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
1.
2.
3.
4.
Skills: Students will be able to …
Review the Summer Assignment (Chapters 1-3)
Find the Square Roots of a Complex Number
Solve and Graph Polynomial Functions
Solve and Graph Inequalities
1. Find the square roots of a complex number.
2. Review four techniques to solve quadratics, focusing on complete
the square.
3. Use the discriminant to determine number and type of roots.
4. Find the roots of polynomials.
5. Sketch polynomials using end behavior and roots.
6. Use linear and polynomial functions to model real world problems.
7. Solve and graph compound inequalities, absolute value inequalities,
rational inequalities and polynomial inequalities.
INSTRUCTIONAL PLAN
Unit #1 Sequence of instructional Topics*
Topic #1: Review the Summer Assignment
Resources
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 1 – 116
Teacher-generated materials
Graphing Calculator
Topic #2: Find the Square Roots of a Complex Same
Number
Suggested Options for Differentiation
PowerPoint presentations
Have students summarize in words how to
solve a quadratic equation using
complete the square.
Have students develop a chart summarizing
the meaning of the discrimant (value /
number of solutions / type of solutions).
Have students in groups come up with real
world applications of positive, negative
and zero slope; step functions, quadratic
functions, etc.
Have students work in pairs to find the square
root of a complex number.
7
Topic #3: Solve and Graph Polynomial
Functions
Same
Function aerobics
Guess the number and type of real roots from a
graph.
Create a table comparing degree of a
polynomial, number and nature of roots
and end-behavior.
Topic #4: Solve and Graph Inequalities
Same
Visual board graphs and charts.
8
Scope and Sequence
Duration: 4 weeks
Unit 2: Functions and Graphs
Standards:
NJCCCS: 4.2C1, 4.3B1, 4.3B2, 4.3B3, 4.3B4, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Functions can be inverted to create new
functions. Two functions can be combined to produce functions that
are more complex. Such combinations impact the graph, domain, and
range of the functions. Functions can be transformed through both rigid
and non-rigid transformations. Symmetry and periodicity of functions
can be exploited in generating their graphs. Rational functions include
domain issues and their graphs display asymptotic behavior. Models
involving more than one variable can be transformed into a function in
one variable by expressing one variable in terms of the other.
Essential Question(s): What is a function? What is its domain and
range? What is the impact on the domain and range when functions are
combined arithmetically or via composition? How are the graphs of
inverse functions related? How can functions be transformed via
translation, reflection, stretches or shrinks? How can function attributes
such as symmetry or periodicity be determined and used in
understanding the behavior of the graph of a function? What does it
mean for a function to be even or odd? How is a rational function
graphed? How are graphing calculators used to visualize and verify
results?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Other Evidence of Mastery (Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
1.
2.
3.
4.
9
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1.
2.
3.
4.
1. Highlight key functions on Ti89, focus on function graphing and
tables.
2. Identify key function types, including piece-wise and step
functions.
3. Determine the domain, range, and zeros of a function, and graph a
function.
4. Perform operations on functions and determine the domains of the
resulting functions
5. Reflect graphs horizontally and vertically.
6. Use symmetry to sketch graphs. Identify even and odd functions.
7. Determine the periodicity and amplitude from graphs. Stretch and
shrink graphs both vertically and horizontally. Translate graphs.
8. Find the inverse of a function, when the inverse exists.
9. Form a function of one variable from a verbal description and
determine the minimum and maximum values of the function
analytically of via a graphing calculator.
10. Find domain issues, vertical asymptotes, holes, horizontal and/or
slant asymptotes for a rational function (Supplemental Material).
11. Graph rational functions (Supplemental Material).
Ti89 Tutorial
Properties of Functions
Graphs and Inverses of Functions
Applications of Functions
10
INSTRUCTIONAL PLAN
Unit 2 Sequence of instructional Topics*
Topic #1 Ti89 Tutorial
Suggested Options for Differentiation
Visual representations of graph.
Observe impact of window parameters on
graphs.
Use trace, and other Ti89 built-in functions.
Topic #2: Properties of Functions
Resources
McDougal Littell, Inc., Advanced
Mathematics, 1997, Brown, p 119-167
D. C. Heath & Company, Precalculus, 1993,
by Larson and Hostetler, p 241-257
Teacher-generated materials
Graphing Calculator
Same
Topic #3: Graphs and Inverses of Functions
Same
Create tables or charts of x/y values.
Use board graphs.
Practice worksheets
Visualize inverse functions by graphing.
Topic #4: Applications of Functions
Same
Solve real world problems with multiple
variables.
11
PowerPoint presentations.
Interactive web displays (Shodor website)
Have students graphically show the addition,
subtraction and composition of functions.
Start with a basic function like y = |x| and
working in pairs write equations that
stretch, shrink, translate and reflect the
function.
Have students generate functions that are not
defined at specific values.
Scope and Sequence
Duration: 2.5 weeks
Unit 3: Exponential and Logarithmic Functions
Standards:
NJCCCS: 4.1B2, 4.1B4, 4.2C1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.2.C2, 9.4O
TL:
8.1.F.1
Enduring Understanding: Exponential and logarithmic functions are
inverses of each other. Properties of exponents and logarithms are
closely related and can be used to simplify expressions and solve
equations. Exponential equations can be solved by rewriting them in
quadratic form. Rational exponents represent the roots of numbers.
Many applications can be modeled using exponential or logarithmic
functions.
Essential Question(s): What is the meaning of rational exponents? How
can population growth, compound interest and radioactive decay be
modeled using exponential functions? What is the natural base? What is a
logarithm? How are exponential and logarithmic functions related? What
basic operations apply to logarithms?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in the
Evaluation Section.
1.
2.
3.
4.
12
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
1. Exponential Expressions, Functions and Applications
2. Logarithmic Expressions, Functions and Applications
Skills: Students will be able to …
1. Define and apply integral exponents.
2. Define and apply rational exponents. Convert between radical and
exponential form.
3. Define and use exponential functions; understand the domains and
ranges of these functions.
4. Solve problems relating to compound interest and growth/decay
applications.
5. Define and apply the natural exponential functions.
6. Define and apply logarithms.
7. Prove and apply laws of logarithms.
8. Apply the change of base formula.
9. Define and use logarithmic functions; understand the domains and
ranges of these functions.
10. Solve problems relating to logarithmic changes (e.g., magnitude of
earthquakes).
11. Solve exponential equations.
12. Solve logarithmic equations.
INSTRUCTIONAL PLAN
Unit #3
Sequence of instructional Topics*
Resources
Suggested Options for Differentiation
Topic #1: Exponential Expressions, Functions
and Applications
McDougal Littell Inc., Advanced
Mathematics, 1997, Brown, p 169-209
Teacher-generated materials
Graphing Calculator
Group students to review laws of integer and
rational exponents.
Use graphing calculators to visualize exponential
functions.
Explore compound interest and growth/decay
applications.
Use a table or chart to explore the value of ‘e’.
Topic #2: Logarithmic Expressions, Functions
and Applications
Same
Use graphing calculators to visualize logarithmic
functions.
13
Use graph paper to plot logarithmic functions.
Use board graphs.
Pair students to discuss applications of logarithms.
14
Scope and Sequence
Duration: 2.5 weeks
Unit 4: Sequences and Series
Standards:
NJCCCS: 4.2C1, 4.3A1, 4.3A2, 4.3A3, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3C3, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2,
4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Sequences can be defined explicitly or
recursively. Summation formulas are available for finite arithmetic and
geometric series. Limits can be used to find the last term of infinite
sequences. Infinite geometric series may converge or diverge.
Mathematical induction can be used to prove statements that are
incrementally defined.
Essential Question(s): How can recursion be used to define a
sequence or iterate a function? What is sigma notation? What are the
proofs for the summation formulas for finite arithmetic and finite and
infinite geometric series? How can induction be used to prove
summation and other types of statements?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery (Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
15
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1. Finite Sequences and Series
2. Infinite Sequences and Series
3. Mathematical Induction
1. Identify an arithmetic or geometric sequence and find a formula for
its nth term.
2. Use sequences defined recursively to solve real-world problems.
3. State an explicit and/or recursive definition for a sequence.
4. Explore function iteration [Optional].
5. Define the sum of the first n terms of an arithmetic or a geometric
series.
6. Derive the finite summation formulas [Optional].
7. Find or estimate the limit of an infinite sequence or determine that
the limit does not exist.
8. Determine whether an infinite series converges or diverges.
9. Find the sum of a convergent infinite series using a formula or the
limit of the sequence of its partial sums.
10. Represent series using sigma notation.
11. Use mathematical induction to prove a statement is true.
INSTRUCTIONAL PLAN
Unit #4 Sequence of instructional Topics*
Topic #1 Finite Sequences and Series
Resources
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 473-515
Teacher-generated materials
Graphing Calculator
16
Suggested Options for Differentiation
Motivate the section: Give students a few
minutes to see if they can come up with
the sum of the first 100 counting numbers.
Without a calculator. Apparently Gauss
was able to do this in a matter of seconds!
(reference text p486).
Guess the next term!
Identify formulas from example sequences.
Worksheets / Puzzles
Give each group of three a different recursive
definition of a sequence. Each group
should find and prove the resulting
explicit definition. Pair the groups and
have them exchange papers and critique
the work of the other group. (reference
text p513).
Topic #2: Infinite Sequences and Series
Same
Visualize infinite terms graphically.
Demonstrate limits numerically and
graphically.
Use sigma notation and other functions on the
Ti89 that apply to sequences and series.
Topic #3: Mathematical Induction
Same
Work induction proofs in teams.
17
Scope and Sequence
Duration: 4 weeks
Unit 5: Combinatorics and Probability
Standards:
NJCCCS: 4.2C1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.4B1-6, 4.4C1-4, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Set theory can be used to classify and
count objects. Counting principles and combination and permutation
formulas can also be used. Binomial coefficients are linked to
combinations and can be used to expand binomials raised to a power.
Probability is a value between 0 and 1 indicating the likelihood of an
event. Decision trees are useful in understanding the sample space of
experiments and determining conditional probability. Expected value is
the payoff for an experiment or game where gains or losses are
involved.
Essential Question(s): How can Venn diagrams be used to count
items? What is the Inclusion-Exclusion Principle? What counting
techniques can be used to answer “how many” type questions? What is
the Binomial Theorem? How can a decision tree help to answer
questions about conditional probability? Since probability can be
determined using properties or counting techniques, which strategy is
best to determine the probability of a particular event? How can one
determine whether a game is fair?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
18
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
1.
2.
3.
4.
Skills: Students will be able to …
Counting Principles
Permutations and Combinations
The Binomial Theorem
Finding Probabilities
1. Use Venn diagrams to illustrate the intersections and unions of
sets and to use the Inclusion-Exclusion Principle to solve
counting problems.
2. Use the multiplication, addition, and complement principles to
solve counting problems.
3. Solve problems involving permutations and combinations.
Know the factorial definitions of nPr and nCr.
4. Solve counting problems that involve permutations with
repetition of elements, and circular permutation.
5. Use the Binomial Theorem and Pascal’s Triangle to expand a
binomial raised to a power.
6. Find the sample space of an experiment. Understand the
difference between theoretical and empirical probability. Be
able to distinguish between an outcome and an event.
7. Find the probability of an event, or the probability of either of
two events occurring.
8. Find the probability of events occurring together and determine
whether two events are independent.
9. Sketch a tree diagram to represent the sample space of an
experiment.
10. Use combinations to solve probability problems.
11. Solve problems involving conditional probability.
12. Find expected value in situations involving gains and losses and
determine if a game is fair.
19
INSTRUCTIONAL PLAN
Unit #5 Sequence of instructional Topics*
Topic #1: Counting Principles
Suggested Options for Differentiation
PowerPoint presentations
Use tree diagrams to illustrate sample space.
Worksheets
Topic #2: Permutations and Combinations
Resources
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 565-637
Teacher-generated materials
Graphing Calculator
Same
Topic #3: The Binomial Theorem
Same
Work in pairs to expand binomials.
Worksheets
Topic #4: Finding Probabilities
Same
Games involving dice, spinners, cards,
marbles, coins.
Discuss the “birthday problem”.
Investigate real world problems relating to
conditional probability. Use tree diagrams
to represent the problem.
20
PowerPoint presentations.
Use graphing calculator to evaluate nPr and
nCr functions.
Relate combinatorics to computer applications
(reference text p577, problem #30).
Use group method called “jigsawing” to
complete a set of mixed combinatorics
problems (reference text p587).
Scope and Sequence
Unit 6: Basic Trigonometric Functions and Graphs
Duration: 4 weeks
Standards:
NJCCCS: 4.2C1, 4.2C3, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Angles can be measured in revolutions,
degrees or radians. The length of an arc and the area of a sector are
most easily determined when central angles are measured in radians.
The six trigonometric functions can be extended to angles of any
measure using the circular definition and periodic nature of
trigonometric functions. Restricting the domains of trigonometric
functions allows us to define inverse trigonometric functions.
Essential Question(s): What are the domains, ranges, and periods of
the six trigonometric functions? What is the unit circle and how is it
used to determine the trigonometric values for key angles? What are
reference angles? Why do the domains of the trigonometric functions
need to be restricted before the inverse functions can be found? How do
you evaluate the compositions of trigonometric functions and their
inverses?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
21
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1.
2.
3.
4.
1. Find the measure of an angle in number of revolutions, degrees (DD
or DMS) and/or radians.
2. Sketch an angle in standard position or as a central angle. Find
coterminal angles.
3. Find the arc length and the area of a sector of a circle; Solve
problems involving apparent size.
4. Use the definitions of sine and cosine to find values of key acute
angles.
5. Extend sine and cosine to non-acute angles using circular
definitions of these functions. Map these circular definitions to the
unit circle.
6. Use reference angles, calculators, and key angles to find values of
the sine and cosine functions.
7. Use the Quotient and Reciprocal Identities to find the values of the
tangent, cotangent, secant, and cosecant functions.
8. Sketch the graphs of the six trigonometric functions. Understand
their periods, domains and ranges, and whether they are even or odd
functions.
9. Find values of the inverse trigonometric functions. Understand why
the textbook refers to the inverse functions with leading capital
letters (e.g., y = Tan-1(x)).
10. Find values of compositions of trigonometric functions and their
inverses. Understand when this can be done without the assistance
of a calculator.
11. Use trigonometry to explore great circle problems given the latitude
of points on the earth. [Optional]
Angles, Arcs and Sectors
The Trigonometric Functions – Circular Definitions
The Trigonometric Functions – Graphs
Trigonometric Applications [Optional]
22
INSTRUCTIONAL PLAN
Unit #6
Sequence of instructional Topics*
Resources
Suggested Options for Differentiation
Topic #1: Angles, Arcs and Sectors
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 257-293
Teacher-generated materials
Graphing Calculator
Explore in groups the meaning of a radian
using pipe cleaners.
Introduce the topic of arcs by discussing great
circle routes that are traveled by airplanes.
Use graphing calculator functions or
programmed solutions to convert between
types of angle measures.
Link linear motion to radian angle measures.
Topic #2: The Trigonometric Functions –
Circular Definitions
Same
Develop trigonometric values from standard
triangles and map to the unit circle.
Have students a create table of key angles in
degree and radian, with their
corresponding sine and cosine values.
Worksheets (unit circle)
Have students summarize in words how to use
a reference angle to find an angle’s
trigonometric value.
Topic #3: The Trigonometric Functions –
Graphs
Same
Use applet-based, interactive web resources to
visualize the sine wave.
Form small groups to discuss restrictions on
the domains and ranges of the six
trigonometric functions. Create a
summary table including this data, along
with periodicity, even/odd, and a sketch of
the graph.
For each of the six trigonometric functions,
construct tables of values to do board
graphing of the functions and its inverses.
23
Topic #4: Trigonometric Applications
[Optional]
Same
Group students, assign two application
problems and present their solutions to
the class (reference text p281).
Review
PowerPoint Jeopardy Game
Trigonometry Puzzles
24
Scope and Sequence
Unit 7: Trigonometric Equations and Applications
Duration: 3 weeks
Standards:
NJCCCS: 4.2C1, 4.2E1, 4.3B1, 4.3B2, 4.3B3, 4.3B4, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Trigonometric functions can be
transformed by translation, reflection or non-rigid stretches or shrinks.
Non-rigid transformations result in period or amplitude changes
whereas horizontal translations are called phase shifts. Fundamental
trigonometric identities together with algebraic techniques are used to
simplify expressions, prove identities and solve equations. When
trigonometric equations are solved an angle measure is the result.
Essential Question(s): How can standard transformations be applied
to trigonometric graphs? How can new trigonometric identities be
proven using a set of fundamental trigonometric identities? How can
trigonometric equations be solved within a specific interval or across all
real numbers?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
25
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1. Fundamental Trigonometric Identities
2. Equations and Applications of Sinusoidal Waves
1. Solve simple and more difficult trigonometric equations using
fundamental trigonometric identities.
2. Use fundamental trigonometric identities to simplify trigonometric
expressions and prove additional trigonometric identities.
3. Use fundamental trigonometric identities to find the values of
trigonometric functions give one trigonometric value and the
quadrant of the angle.
4. Relate the tangent of the angle of inclination to the slope of a line.
5. Graph transformed sine and cosine curves. Understand the impact
on amplitude, period, and any horizontal or vertical shifts
(horizontal shift = phase shift).
6. Find equations of transformed sine and cosine curves.
7. Use trigonometric functions to model periodic behavior.
INSTRUCTIONAL PLAN
Unit #7 Sequence of instructional Topics*
Topic #1: Fundamental Trigonometric
Identities
Resources
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 295-329
Teacher-generated materials
Graphing Calculator
26
Suggested Options for Differentiation
PowerPoint presentations
Use the unit circle and trigonometric graphs to
visualize key fundamental identities.
Have students work in groups to verify new
identities and present their findings to the
entire class.
Group students to review solving polynomial
equations. Then make mathematical
connections to solving similar
trigonometric equations.
Have students write a paragraph about why it
does not make sense to discuss the
amplitude of the tangent, cotangent, secant
or cosecant functions.
Topic #2: Equations and Applications of
Sinusoidal Waves
Same
PowerPoint presentations
Use graphing calculators to visualize changes
in amplitude, frequency and phase shifts.
Use reverse analysis to generate function
equations from graphs.
Worksheets
Relate sine or cosine waves to AM/FM radios,
applications in electronics, music, and
various topics in physics, including simple
harmonic motion.
27
Scope and Sequence
Duration: 2 weeks
Unit 8: Triangle Trigonometry
Standards:
NJCCCS: 4.2C1, 4.2E1, 4.2E2, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Trigonometry is extended using the Law
of Sines and Law of Cosines to solve or find the area of oblique (non
right) triangles. Because the sine values for acute and obtuse angles
are positive, there can be ambiguity in applying the Law of Sines.
Navigation and surveying are common applications of trigonometry.
Essential Question(s): What cases are insufficient or ambiguous for
solving a triangle? When do we use the Law of Sines to solve an
oblique triangle? When do we use Law of Cosines? How are these
laws applied in navigation and surveying applications? How can we
prove the Law of Sines and the Law of Cosines?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
28
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1. Solving a Right Triangle
2. The Law of Sines and Cosines
3. Applications of Trigonometry
1. Use trigonometry to find the unknown sides or angles of a right
triangle.
2. Apply trigonometry to solve real-world problems, including ones
involving angle of elevation and angle of depression.
3. Use the Law of Sines to find the unknown parts of an oblique
triangle.
4. Use the Law of Cosines to find the unknown parts of an oblique
triangle.
5. Use trigonometry to find the area of an oblique triangle (include
Heron’s Formula).
6. Use trigonometry to solve navigation and surveying problems.
INSTRUCTIONAL PLAN
Unit #8 Sequence of instructional Topics*
Topic #1: Solving a Right Triangle
Topic #2 Law of Sines and Cosines
Resources
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 331-365
Teacher-generated materials
Graphing Calculator
Same
Topic #3: Applications of Trigonometry
Same
29
Suggested Options for Differentiation
Introduce applications in engineering (e.g.,
bridge struts).
Role play angle of elevation and depression.
PowerPoint presentations
Develop understanding of SSA ambiguous
case through a compass/ruler activity
(reference text p346).
Worksheets / Puzzles
Work in teams to solve complex navigation
and surveying problems. Present results to
the entire class.
Scope and Sequence
Duration: 1.5 weeks
Unit #9: Triangle Formulas and Applications
Standards:
NJCCCS: 4.1A1, 4.1A3, 4.1B1, 4.2C1, 4.2E1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2,
4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Angle summation and difference formulas
can be proven for all six trigonometric functions. The double angle and
power reduction formulas are derived from the summation/difference
formulas. These formulas can be used to find new trigonometric values
or to solve more complex trigonometric equations.
Essential Question(s): How do you prove the sine, cosine and tangent
sum and difference formulas? How are they extended to the double
angle and power reduction formulas? How can these formulas be used
to find new trigonometric values? How can they be used to solve
trigonometric equations where different angle arguments are involved?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
30
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1. Trigonometric Sum, Difference, Double Angle, and Power
Reduction Formulas
2. Applications of Trigonometric Formulas
1. Derive and apply sum and difference formulas for sine, cosine, and
tangent.
2. Derive and apply double angle formulas.
3. Derive and apply half angle formulas. [Optional]
4. Derive and apply power reduction formulas (Supplemental
Material).
5. Use these formulas to solve trigonometric equations and
inequalities.
INSTRUCTIONAL PLAN
Unit #9 Sequence of instructional Topics*
Topic #1: Trigonometric Sum, Difference,
Double Angle, and Power Reduction Formulas
Resources
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 369-393
D. C. Heath & Company, Precalculus, 1993,
by Larson and Hostetler, p 461
Teacher-generated materials
Graphing Calculator
Suggested Options for Differentiation
Visualize the framework of formula
derivations.
Work in pairs to develop the full complement
of trigonometric formulas from the
framework.
To discourage common errors, have students
work in groups to give examples of why
tan(x + y) ≠ tan(x) + tan(y), etc.
Topic #2: Applications of Trigonometric
Formulas
Same
Use formulas to solve more complex
trigonometric equations.
Visualize solutions to multiple trigonometric
inequalities by graphing.
31
Scope and Sequence
Duration: 1.5 weeks
Unit #10: Polar Coordinates and Complex Numbers
Standards:
NJCCCS: 4.2C1, 4.2E1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Simple polar equations can produce
beautiful polar graphs including circles, limacons, roses, and spirals.
Complex numbers can be plotted on a rectangular, complex plane
(a.k.a. Argand Diagram). Using the polar representation of these points,
a complex number can be represented in polar form. Complex numbers
are much easier to manipulate in polar form.
Essential Question(s): How is a point in two dimensional space
represented using polar coordinates? How are polar graphs generated?
What is the form of a polar equation for a circle, limacon, spiral or
straight line? How can a complex number be represented in polar
form? How can complex numbers in polar form be multiplied, raised to
a power or its nth root taken? What is DeMoivre’s Theorem?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
32
Quizzes
Test
Problem-solving
Oral Presentations
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1. Polar Coordinates and Graphs
2. Complex Numbers
1.
2.
3.
4.
5.
6.
7.
8.
Plot points in polar form.
Graph polar equations by hand or using a graphing calculator.
Convert between polar and rectangular representations of points
and equations.
Graph complex numbers on the complex plane (Argand Diagram)
or on polar graph paper.
Convert complex numbers from rectangular form (a + bi) to polar
form (r cis θ).
Find products of complex numbers using polar form.
Use De Moivre’s theorem to find powers of complex numbers.
Find the roots of complex numbers.
INSTRUCTIONAL PLAN
Unit #10
Sequence of instructional
Topics*
Topic #1: Polar Coordinates and Graphs
Topic #2: Complex Numbers
Resources
Suggested Options for Differentiation
McDougal Littell Inc., Advanced Mathematics,
1997, Brown, p 395-417
Teacher-generated materials
Graphing Calculator
Use a graphing calculator to visualize polar
equations.
Discuss symmetry vis-à-vis polar graphing.
Discuss applications of polar graphs to
engineering and other disciplines.
PowerPoint presentations.
Same
Visualize complex numbers on a coordinate
plane.
Relate the extension to DeMoivre’s Theorem
to the method used in Chapter 1 for
finding square roots of a complex number.
Worksheets.
Research mathematicians who first theorized
complex numbers. Present results to the
class.
33
Research applications of complex numbers.
Present results to the class.
34
Scope and Sequence
Duration: 3 weeks
Unit 11: Limits
Standards:
NJCCCS: 4.2C1, 4.2E1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Calculus is the mathematics of change.
Limits are the fundamental process that converts Pre-calculus
mathematics to Calculus. A limit is taken at a specific input value and,
if defined, is a real number. As a result, limits behave like real
numbers. Limits can be evaluated graphically, numerically, or
analytically.
Essential Question(s): How can a limit be found graphically?
numerically? analytically? When does a limit not exist? What are the
properties of limits? What is the Squeeze Theorem? What important
trigonometric limits are defined through the Squeeze Theorem? How is
continuity defined in terms of limits? How can one-sided limits be used
to extend continuity to a closed interval? What is the Intermediate
Value Theorem? What are infinite limits?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
35
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1. Limits
2. Continuity
1.
2.
3.
4.
5.
6.
Understand how the limit process is the foundation of Calculus.
Develop basic properties of limits.
Find a limit both graphically and numerically.
Evaluate a limit analytically.
Find limits at infinity (x → ±∞) and infinite limits (y → ±∞).
Find special trigonometric limits derived from the Squeeze
Theorem.
7. Explore the definition and properties of continuity and their
relationship to limits.
8. Apply the Intermediate Value Theorem.
INSTRUCTIONAL PLAN
Unit #11
Sequence of instructional
Topics*
Topic #1: Limits
Resources
Houghton Mifflin, Calculus of a Single
Variable, 1998, by Larson, Roland E, R. P.
Hostetler, B. H. Edwards and D. E. Heyd, p
39-88
D.C.Heath and Company, Precalculus, 1993,
Larson and Hosteltler, p 587-654
Teacher-generated materials
Graphing Calculator
36
Suggested Options for Differentiation
PowerPoint presentations.
Compare solution methods for limits:
graphical, numerical and analytic
approaches.
Use graphing calculators to perform numerical
evaluation of limits.
Use graphing calculator to confirm results
obtained from graphical, numerical and
analytic approaches.
Make mathematical connections between the
end behavior of rational and exponential
function graphs and the limits of these
expressions at infinity.
Make mathematical connections between the
behavior of rational functions at vertical
asymptotes and infinite limits of these
expressions at these values.
Limits Sudoku Puzzle
Worksheets
Topic #2: Continuity
Same
Discuss the progression of limits and
continuity.
Pair & Share: Discuss the formal definition of
continuity and why all three aspects of
that definition are required.
37
Scope and Sequence
Duration: 3 weeks
Unit 12: Derivatives
Standards:
NJCCCS: 4.2C1, 4.2E1, 4.3B1, 4.3B2, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1
Enduring Understanding: Differentiation is one of the two
fundamental operations of Calculus. The derivative is the instantaneous
rate of change of a curve and is the limit as the denominator goes to
zero of the difference quotient (i.e., slope). The tangent line at a point
on a curve has a slope equal to the derivative at that point. Several
differentiation rules can be proven and then used to derive expressions
or find the derivative of functions. The chain rule is an important
construct and is the foundation of implicit differentiation and related
rates.
Essential Question(s): What is the limit-based definition of a
derivative? How can it be used to prove other differentiation rules?
What is the chain rule? Why is it so important? When is implicit
differentiation used? How can related rate problems be solved?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery(Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
38
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1.
2.
3.
4.
1. Understand the geometric interpretation of the derivative and its
application to the tangent line.
2. Use the derivative to find the equation of the tangent line at a point
on a curve.
3. Find the derivative of functions using the limit-based definition of
the derivative.
4. Analyze the graphs of a function and its derivative.
5. Understand differentiability and continuity.
6. Develop basic properties of the derivative as a rate of change.
7. Derive and use the derivatives of algebraic and trigonometric
functions.
8. Understand and be able to apply the chain rule.
9. Find higher level derivatives of a function.
10. Use applications of the derivative as a rate of change, especially
related to velocity and acceleration problems.
11. Implicitly differentiate equations.
12. Solve Related Rate problems.
Definitions of Derivative, Differentiability and Continuity
Differentiation Rules
Rates of Change, Velocity and Acceleration
Implicit Differentiation and Related Rates
INSTRUCTIONAL PLAN
Unit #12
Sequence of instructional
Topics*
Topic #1: Definitions of Derivative,
Differentiability and Continuity
Resources
Houghton Mifflin, Calculus of a Single
Variable, 1998, by Larson, Roland E, R. P.
Hostetler, B. H. Edwards and D. E. Heyd, p
91-152
Teacher-generated materials
Graphing Calculator
39
Suggested Options for Differentiation
PowerPoint presentations
Use a graphing calculator to provide insight
into the interpretation of a derivative
(tangent line to the curve).
Visualize graphs of functions and their
derivatives.
Apply differentiation to physics problems
including the Doppler effect, Newton’s
Law of Cooling, etc.
Discuss the progression of limits, continuity
and differentiability.
Topic #2: Differentiation Rules
Same
Practice worksheets.
Compare results when problems are solved
with and without the chain rule.
Distinguish scenarios that require the chain
rule from those that do not.
Derivative Sudoku Puzzle
Topic #3: Rates of Change, Velocity and
Acceleration
Same
Pair students to apply differentiation to
velocity and acceleration problems.
Topic #4: Implicit Differentiation and Related
Rates
Same
Work in groups to apply related rates to real
world problems.
In teams, discuss how the chain rule provides
the foundation of implicit differentiation
and related rates. Report results to the
class.
40
Scope and Sequence
Duration: 2 weeks
Unit 13: Applications of Derivatives
Standards:
NJCCCS: 4.2C1, 4.2E1, 4.3B1, 4.3B2, 4.3B4, 4.3C1, 4.3C2, 4.3D1, 4.3D2, 4.3D3, 4.5A1-5, 4.5B1-4, 4.5C1-6, 4.5D1-6, 4.5E2, 4.5F4
21st CLCS: 9.1A.1, 9.4O
TL:
8.1.F.1.4
Enduring Understanding: A sketch of the graph of a function can be
determined using Calculus. Limits at infinity (end behavior) together
with relative extrema and roots can be used. The first derivative test
identifies relative maximums, minimums, and intervals of increase or
decrease. The second derivative test can identify extrema as well as
points of inflection.
Essential Question(s): What is Rolles Theorem? What is the Mean
Value Theorem? How is Calculus used to find the relative extrema of a
function? How is Calculus used to determine the end behavior of a
function? What is the First Derivative Test? What is the Second
Derivative Test? What is a point of inflection? What is meant by the
concavity of a curve?
EVIDENCE OF STUDENT LEARNING
Performance Tasks: Activities to provide evidence for student
learning of content and cognitive skills.
1.
2.
3.
4.
5.
6.
7.
Problem-based learning
Teacher directed
Cooperative groups
Technology
Participation and discussion
Homework
Classwork
Other Evidence of Mastery (Summative): Student proficiency (for a
specific unit or multiple units) is defined for the individual at 80% or
better; for the class: 80% of the students attain the established minimum
standard; an exemplar or rubric should be referenced and included in
the Evaluation Section.
1.
2.
3.
4.
41
Quizzes
Test
Problem-solving
Oral Presentation
KNOWLEDGE AND SKILLS
Knowledge: Students will know…
Skills: Students will be able to …
1.
2.
3.
4.
5.
1. Find absolute extrema on a closed interval.
2. Find relative extrema on an open interval.
3. Apply Rolle’s Theorem and the Mean Value Theorem to identify
points of zero or average slope on an interval.
4. Use the first derivative test to find extrema and identify intervals
where a function increases or decreases.
5. Use the Second Derivative Test to understand concavity and apply
it to finding extrema and points of inflection.
6. Find horizontal asymptote(s) of rational or non-rational functions
using their limits at infinity.
7. Sketch functions using calculus (limits and derivatives).
Extrema on an Interval
The First Derivative Test
The Second Derivative Test
Limits at Infinity
Curve Sketching
INSTRUCTIONAL PLAN
Unit #13
Sequence of instructional
Topics*
Topic #1: Extrema on an Interval
Resources
Topic #2: The First Derivative Test
Houghton Mifflin, Calculus of a Single
Variable, 1998, by Larson, Roland E, R. P.
Hostetler, B. H. Edwards and D. E. Heyd, p
155-195
Teacher-generated materials
Graphing Calculator
Same
Topic #3: The Second Derivative Test
Same
42
Suggested Options for Differentiation
Use a graphing calculator to confirm extrema
results obtained analytically.
Visualize the graphical interpretations of
Rolle’s Theorem and the Mean Value
Theorem.
Relate pre-calculus techniques for finding the
maximum or minimum of polynomial
functions to using calculus. What are the
short comings of the pre-calculus
techniques?
Use a graphing calculator to confirm results
obtained analytically.
In groups, use a graphing calculator to graph
several cubics, and discuss why a) the
value of the constant term does not affect
the first derivative and b) the value of the
coefficient of the linear term does not
affect the value of the second derivative.
Visualize extrema and points of inflection
from standard curves. From the curves,
determine whether the second derivative
at those points would be positive, negative
or zero.
Practice worksheets / Puzzles
Topic #4: Limits at Infinity
Same
Make mathematical connections between
limits at infinity and prior practice in
determining the end behavior of
functions.
Practice worksheets
Topic #5: Curve Sketching
Same
In groups, describe the important features of a
graph. Use a graphing calculator to find a
viewing region that includes all of these
features. Describe your reasoning.
Create student teams to compare curve
sketching methods pre-calculus, vs. curve
sketching using calculus techniques.
43
Required Instructional Resources
CORE MATERIAL
McDougal Littell, Inc, Advanced Mathematics, 1997, by Brown, Richard G.
Houghton Mifflin, Calculus of a Single Variable, 1998, by Larson, Roland E, R. P. Hostetler, B. H. Edwards and D. E. Heyd
Ti89 or Ti-Inspire Graphing calculator
SUPPLEMENTAL TEXTS
D. C. Heath & Company, Precalculus, 1993, by Larson, Roland E. and Robert P. Hostetler
Addison-Wesley, Pre-Calculus Mathematics, 1976, Shanks, Merrill E, C.F. Brumfiel, C.R. Fleenor and R. E. Eicholz
Key Curriculum Press, Precalculus with Trigonometry, 2007, Paul A Foerster
Houghton Mifflin, Algebra 2 and Trigonometry, 1992, by Dolciani, Graham, Swanson and Sharron
Evaluation and Grading
The marking period grade will be determined as follows:
90% Tests and Quizzes
10% Homework, classwork, and projects
New Jersey Core Curriculum Content Standards - Mathematics
Standard
Strand
Standard
Strand
4.1 (Number And Numerical Operations) All Students Will Develop Number Sense And Will Perform Standard Numerical
Operations And Estimations On All Types Of Numbers In A Variety Of Ways.
A. Number Sense
B. Numerical Operations
C. Estimation
4.2 (Geometry And Measurement) All Students Will Develop Spatial Sense And The Ability To Use Geometric Properties,
Relationships, And Measurement To Model, Describe And Analyze Phenomena.
A. Geometric Properties
B. Transforming Shapes
C. Coordinate Geometry
44
Standard
Strand
Standard
Strand
Standard
Strand
D. Units of Measurement
E. Measuring Geometric Objects
4.3 (Patterns And Algebra) All Students Will Represent And Analyze Relationships Among Variable Quantities And Solve
Problems Involving Patterns, Functions, And Algebraic Concepts And Processes.
A. Patterns
B. Functions and Relationships
C. Modeling
D. Procedures
4.4 (Data Analysis, Probability, And Discrete Mathematics) All Students Will Develop An Understanding Of The Concepts
And Techniques Of Data Analysis, Probability, And Discrete Mathematics, And Will Use Them To Model Situations,
Solve Problems, And Analyze And Draw Appropriate Inferences From Data.
A. Data Analysis (Statistics)
B. Probability
C. Discrete Mathematics – Systematic Listing and Counting
D. Discrete Mathematics – Vertex-Edge Graphs and Algorithms
4.5 (Mathematical Processes) All Students Will Use Mathematical Processes Of Problem Solving, Communication,
Connections, Reasoning, Representations, And Technology To Solve Problems And Communicate Mathematical Ideas.
A. Problem Solving
B. Communication
C. Connections
D. Reasoning
E. Representations
F. Technology
New Jersey Core Curriculum Content Standards - 21st-Century Life and Career Skills
Standard
Strand
9.1 21st-Century Life & Career Skills: All students will demonstrate the creative, critical thinking, collaboration,
and problem-solving skills needed to function successfully as both global citizens and workers in diverse ethnic and
organizational cultures.
A. Critical Thinking and Problem Solving
B. Creativity and Innovation
45
Standard
Strand
Standard
Strand
Standard
Strand
Strand
Strand
Strand
Strand
Strand
Strand
Strand
C. Collaboration, Teamwork, and Leadership
D. Cross-Cultural Understanding and Interpersonal Communication
E. Communication and Media Fluency
F. Accountability, Productivity, and Ethics
9.2 Personal Financial Literacy: All students will develop skills and strategies that promote personal and financial
responsibility related to financial planning, savings, investment, and charitable giving in the global economy.
A. Income and Careers
B. Money Management
C. Credit and Debt Management
D. Planning, Saving, and Investing
E. Becoming a Critical Consumer
F. Civic Financial Responsibility
G. Risk Management and Insurance
9.3 Career Awareness, Exploration, and Preparation: All students will apply knowledge about and engage in the
process of career awareness, exploration, and preparation in order to navigate the globally competitive work
environment of the information age.
A. Career Awareness
B. Career Exploration
C. Career Preparation
9.4 Career and Technical Education: All students who complete a career and technical education program will
acquire academic and technical skills for careers in emerging and established professions that lead to technical skill
proficiency, credentials, certificates, licenses, and/or degrees. (For descriptions of the 16 career clusters, see the
Career Clusters Table.)
A. Agriculture, Food, & Natural Resources Career Cluster
B. Architecture & Construction Career Cluster
C. Arts, A/V Technology, & Communications Career Cluster
D. Business, Management & Administration Career Cluster
E. Education & Training Career Cluster
F. Finance Career Cluster
G. Government & Public Administration Career Cluster
H. Health Science Career Cluster
46
Strand
Strand
Strand
Strand
Strand
Strand
Strand
Strand
I. Hospitality & Tourism Career Cluster
J. Human Services Career Cluster
K. Information Technology Career Cluster
L. Law, Public Safety, Corrections, & Security Career Cluster
M. Manufacturing Career Cluster
N. Marketing Career Cluster
O. Science, Technology, Engineering, & Mathematics Career Cluster
P. Transportation, Distribution, & Logistics Career Cluster
New Jersey Core Curriculum Content Standards – Technology Literacy
Standard
Strand
Standard
Strand
8.1 Educational Technology: All students will use digital tools to access, manage, evaluate, and synthesize information in
order to solve problems individually and collaboratively and to create and communicate knowledge.
A. Technology Operations and Concepts
B. Creativity and Innovation
C. Communication and Collaboration
D. Digital Citizenship
E. Research and Information Literacy
F. Critical Thinking, Problem Solving, and Decision-Making
8.2 Technology Education, Engineering, and Design: All students will develop an understanding of the nature and impact of
technology, engineering, technological design, and the designed world, as they relate to the individual, global society, and the
environment.
A. Nature of Technology: Creativity and Innovation
B. Design: Critical Thinking, Problem Solving, and Decision-Making
C. Technological Citizenship, Ethics, and Society
D. Research and Information Fluency
E. Communication and Collaboration
F. Resources for a Technological World
G. The Designed World
47
Scope and Sequence Overview:
1
2
3
Review of
Essentials
(1w)
5
6
11
Sequences and
Series, cont
(2.5w)
12
13
14
15
20
21
22
Trigonometric Equations and Applications
(3w)
29
30
31
Limits (3w)
8
16
23
24
25
Triangle Trigonometry (2w)
32
17
18
Derivatives (3w)
Submitted by: Danielle Bellavance & Ann Wright ___
26
Triangle Formulas
and Applications
(1.5w)
33
34
Polar Coordinates
and Complex
Numbers (1.5w)
35
July 19, 2012
Board of Education Curriculum and Instruction Committee:
Approved
Date:
August 15, 2012
Board of Education:
Approved
Date:
August 29, 2012
48
Midterms
(1w)
27
Applications of Derivatives
(2w)
Date:
9
Sequences and Series
(2.5w)
Basic Trigonometric Functions and Graphs
(4w)
Combinatorics and Probability (4w)
19
7
Exponential and Logarithmic
Functions (2.5w)
Functions and Graphs (4 w)
10
Basic Trig
Functions and
Graphs, cont
(4w)
28
4
36
Finals (1w)
CAREER INFUSION LESSON
Department:
Mathematics
Course:
Honors Pre-Calculus
Focus:
Full Class
Title of Activity:
Using Trigonometry to Survey Land
Purpose of Activity:
To acquaint students with possible careers based upon mathematics
Time Needed to Complete Activity: One full period
Description of Activity:
Obtain the surveyor’s report for the parcel of land on which the high school resides. Using
trigonometric techniques learned in this course, determine the parcel land area (to the nearest 100th of
an acre) and determine the line of sight distance from the NE corner of the property to the SW corner.
Evaluation:
Oral report
Resources:
Teacher, textbook, online access, and survey data.
49