Trigonometry
Curriculum
McCann Technical School
70 Hodges Cross Road
North Adams, MA 01247
Kozik
Page 1 of 17
December 2010
Course Philosophy
In order to fulfill our mission of graduating “technically skilled, academically prepared,
and socially responsible individuals ready to meet the challenges of the 21st century”
(McCann mission statement), it is important that our graduates have achieved
mathematical competence in many areas. This trigonometry course provides the
opportunity to delve beyond the curriculum into emerging mathematical topics. The
goals for grade 12 students enrolled in Trigonometry include:
•
•
•
•
•
•
To introduce concepts and applications of Trigonometry
To express mathematical ideas coherently both verbally and in writing
To explore the connections that exist within mathematics and with other
disciplines
To develop critical thinking and problem solving skills
To demonstrate understanding of more advanced math concepts
To identify and dispel common math misconceptions
Course Description
Trigonometry topics in this year-long, ninety-minute course (meeting on alternating
weeks) include Pythagorean relationships, functions and their graphs, trigonometric
functions, right triangle trigonometry, angles of rotation and radian measure, graphs of
trigonometric functions, trigonometric formulas and identities, and polar coordinates.
Projects typically require students to research topics beyond the scope of the textbook as
well as across the curriculum. These projects expand the students’ abstraction, reflect
upon their understanding, and practice technical writing in anticipation of senior project
presentations in the spring of senior year.
Kozik
Page 2 of 17
December 2010
Course Syllabus
Instructional Philosophy
This course will allow students to explore and experience mathematics through a variety
of activities and real world applications. Emphasis will be placed on students’
understanding of key concepts and the ability of students to demonstrate their learned
knowledge through exams, projects, discussions and written work. Students will be
encouraged to inquire, discuss, analyze, and question the various topics presented
throughout the course in order to promote complete mastery of topics.
Major Course Projects and Activities
•
•
•
•
Kozik
Assignments
o A variety of assignments will be given to students throughout the
course to help reinforce learning objectives, are graded on
completeness; solutions being reviewed upon student request.
Notebook/Portfolio
o Students are required to maintain a course notebook which will include
all class notes, homework assignments, writing assignments, and
handouts.
o Quizzes and projects are kept on file in the classroom and are
accessible to students in order to review, maintain a general student
portfolio, and to help prepare for make-up quizzes.
o The notebook grade is based on completeness.
Projects
o These will serve as extensions to the material learned in class.
o Students may be asked to work individually, with partners, or in
groups and may complete such assignments as webpages, PowerPoint
presentations, research reports, posters, models, diagrams, etc.
Attendance/Participation
o Daily attendance, preparation, and participation are expected, and
will be recorded. This is in accordance with McCann’s Attendance
Policy as detailed in the Student/Parent Handbook.
o Attendance/participation grades are based on student being present and
prepared for class, cooperation, successful progress towards
completing class work, participation in daily activities.
Page 3 of 17
December 2011
COURSE ASSESSMENT PLAN
The following assessment plan is applied for the mathematics students at McCann
Technical School:
GRADING POLICY:
“Student assessment and grade reporting is considered a positive tool to measure growth,
progress, and the development of the student. Report cards are issued four times each year.
In addition, progress reports are issued at the mid-point of each quarter.” (2010-2011
Student/Parent Handbook)
A+ 100-97
B+ 89-87
C+ 79-77
D+ 69-67
A
96-94
B 86-84
C 76-74
D
66-65
A- 93-90
B- 83-80
C- 73-70
F
64-0
ACADEMIC GRADING POLICY:
70% Tests, quizzes, projects, portfolios, laboratory experiments, research papers,
and oral presentations
30% Attendance, participation, class assignments, homework, notebook, effort
Extra Help – Homework club – Tuesdays and Thursdays and with teacher by
appointment.
Kozik
Page 4 of 17
December 2010
Timeline:
• First Quarter
o Coordinate Geometry Review
• Cartesian Plane
• Pythagorean Theorem
• Distance Formula
• Definition of a function
o Trigonometric Functions
• Forming angles by rotation
• Six trigonometric functions
• Reference angles and triangles
• Solving right triangles
• Second Quarter
o Right triangle applications
Angles of Elevation/Depression
Applications: Surveying
Applications: Navigation
Applications: Construction
o Radians and Rotary motion
• Radian Measure
• Degree to radian conversion
• Angular Displacement
• Angular Velocity and Applications
•
Third Quarter
o Graphs of Trigonometric Functions
• Properties of sine and cosine
• Reduction formulas
• Graphs of sine and cosine
o Transformations of Trigonometric Functions
• Amplitude
• Period
• Phase shift
• Vertical shift
o Trigonometric Identities
• Fundamental Identities
• Manipulating Identities
• Proving Identities
Fourth Quarter
o Right Triangle Laws and Formulas
• Law of Sines
• Law of Cosines
• Hero’s Formula
• Area of a Triangle
o Complex Numbers
Polar Coordinates
Kozik
Page 5 of 17
December 2010
Standards
Massachusetts Mathematics Curriculum Framework
Learning Standards for Grades 11-12 (November 2000)
Course Curriculum Topic
Standard
Linear Functions
10.P.2 Demonstrate an understanding of the relationship
between various representations of a line. Determine a line's
slope and x- and y-intercepts from its graph or from a linear
equation that represents the line. Find a linear equation
describing a line from a graph or a geometric description of
the line, e.g., by using the "point-slope" or "slope yintercept" formulas. Explain the significance of a positive,
negative, zero, or undefined slope. (Learning Standards for
Grades 9-10 (November 2000))
Exponential & Logarithmic 12.P.4 Demonstrate an understanding of the trigonometric,
Functions
exponential, and logarithmic functions.
12.P.5 Perform operations on functions, including
composition. Find inverses of functions.
12.P.6 Given algebraic, numeric and/or graphical
representations, recognize functions as polynomial, rational,
logarithmic, exponential, or trigonometric.
12.P.11 Solve everyday problems that can be modeled using
polynomial, rational, exponential, logarithmic,
trigonometric, and step functions, absolute values, and
square roots. Apply appropriate graphical, tabular, or
symbolic methods to the solution. Include growth and decay;
joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G
(m1m2)/d2) variation, and periodic processes.
Kozik
Page 6 of 17
December 2010
Rational Expressions &
Functions
12.P.5 Perform operations on functions, including
composition. Find inverses of functions.
12.P.6 Given algebraic, numeric and/or graphical
representations, recognize functions as polynomial, rational,
logarithmic, exponential, or trigonometric.
12.P.11 Solve everyday problems that can be modeled using
polynomial, rational, exponential, logarithmic,
trigonometric, and step functions, absolute values, and
square roots. Apply appropriate graphical, tabular, or
symbolic methods to the solution. Include growth and decay;
joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G
(m1m2)/d2) variation, and periodic processes.
Kozik
Page 7 of 17
December 2010
Periodic Functions &
Trigonometry
12.P.4 Demonstrate an understanding of the trigonometric,
exponential, and logarithmic functions.
12.P.6 Given algebraic, numeric and/or graphical
representations, recognize functions as polynomial, rational,
logarithmic, exponential, or trigonometric.
12.P.8 Solve a variety of equations and inequalities using
algebraic, graphical, and numerical methods, including the
quadratic formula; use technology where appropriate.
Include polynomial, exponential, logarithmic, and
trigonometric functions; expressions involving absolute
values; trigonometric relations; and simple rational
expressions.
12.P.11 Solve everyday problems that can be modeled using
polynomial, rational, exponential, logarithmic,
trigonometric, and step functions, absolute values, and
square roots. Apply appropriate graphical, tabular, or
symbolic methods to the solution. Include growth and decay;
joint (e.g., I = Prt, y = k(w1 + w2)) and combined (F = G
(m1m2)/d2) variation, and periodic processes.
12.G.1 Define the sine, cosine, and tangent of an acute
angle. Apply to the solution of problems.
12.G.2 Derive and apply basic trigonometric identities (e.g.,
sin2θ + cos2θ = 1, tan2θ + 1 = sec2θ) and the laws of sines
and cosines.
12.M.1 Describe the relationship between degree and radian
measures, and use radian measure in the solution of
problems, in particular, problems involving angular velocity
and acceleration.
Conic Sections
Kozik
12.G.4 Relate geometric and algebraic representations of
lines, simple curves, and conic sections.
Page 8 of 17
December 2010
Polar Coordinates
PC.N.1 Plot complex numbers using both rectangular and
polar coordinates systems.
Represent complex numbers using polar coordinates, i.e., a
+ bi= r(cosθ+ isinθ).
Apply DeMoivre’s theorem to multiply, take roots, and raise
complex numbers to
a power.
PC.P.1 Use mathematical induction to prove theorems and
verify summation formulas
.
PC.P.2 Relate the number of roots of a polynomial to its
degree. Solve quadratic equations
with complex coefficients.
PC.P.3 Demonstrate an understanding of the trigonometric
functions (sine, cosine, tangent,
cosecant, secant, and cotangent). Relate the functions to their
geometric definitions.
PC.P.4 Explain the identity sin2q+ cos2q= 1. Relate the
identity to the Pythagorean
theorem.
PC.P.5 Demonstrate an understanding of the formulas for the
sine and cosine of the sum or
the difference of two angles. Relate the formulas to
DeMoivre’s theorem and use
them to prove other trigonometric identities. Apply to the
solution of problems.
PC.P.6 Understand, predict, and interpret the effects of the
parameters a, b, and c on
the graph of y = asin((x - b)) + c; similarly for the cosine and
tangent. Use to model
periodic processes. (12.P.13)
PC.P.9 Relate the slope of a tangent line at a specific point
on a curve to the instantaneous
rate of change. Explain the significance of a horizontal
tangent line. Apply these
concepts to the solution of problems.
Kozik
Page 9 of 17
December 2010
Vocational/Technical Education Curriculum Frameworks
Strands 1, 4, 5, and 6
Strand 4: Employability
4.b Develop employability skills to secure and keep employment in chosen field
4.B.01a
4.B.03a
Apply strategies to enhance effectiveness of all types of communications in
the workplace
Locate information from books, journals, magazines, and the Internet
4.B.06a
Explain information presented graphically
4.B.07a
Use writing/publishing/presentation applications
4.B.08a
Apply basic skills for work-related oral communication
4.c Solve problems using critical thinking
4.C.01a
Demonstrate skills used to define and analyze a given problem
4.C.04a
Explain strategies used to formulate ideas, proposals and solutions to
problems
Select potential solutions based on reasoned criteria
4.C.05a
Strand 6: Underlying Use of Technology
6.c Demonstrate ability to use technology for research, problem solving, and
communication
6.C.03a Demonstrate the use of appropriate electronic sources to conduct research
6.C.04a
6.C.05a
6.C.06a
(e.g., Web sites, online periodical databases, and online catalogs)
Demonstrate proper style (with correct citations) when integrating electronic
research results into a research project
Collect, organize, analyze, and graphically present data using the most
appropriate tools
Present information, ideas, and results of work using any of a variety of
communications technologies (e.g., multimedia presentations, Web pages,
videotapes, desktop-published documents)
Vocational/Technical Education Curriculum Frameworks
Strand 3:Embedded Academics
Automotive Technology
Kozik
Page 10 of 17
December 2010
3.B.19c
3.B.22c
3.B.24c
3.B.26c
Kozik
12.D.6 Use combinatorics (e.g.,
11/12 Data Analysis,
Probability and
"fundamental counting principle,"
permutations, and combinations) to
Statistics
solve problems, in particular, to
compute probabilities of compound
events. Use technology as
appropriate.
12.M.1 Describe the relationship between
11/12 Measurement
degree and radian measures, and use
radian measure in the solution of
problems, in particular, problems
involving angular velocity and
acceleration.
12.P.11 Solve everyday problems that can be 11/12
modeled using polynomial, rational,
exponential, logarithmic,
trigonometric, and step functions,
absolute values, and square roots.
Apply appropriate graphical, tabular,
or symbolic methods to the solution.
Include growth and decay; joint (e.g.,
I = Prt, y = k(w1 + w2)) and
combined (F = G(m1m2)/d2)
variation, and periodic processes.
11/12
12.P.8 Solve a variety of equations and
inequalities using algebraic,
graphical, and numerical methods,
including the quadratic formula; use
technology where appropriate.
Include polynomial, exponential,
logarithmic, and trigonometric
functions; expressions involving
absolute values; trigonometric
relations; and simple rational
expressions.
Page 11 of 17
Patterns,
relations,
algebra
Patterns,
relations,
algebra
December 2010
Carpentry/Cabinetry
3.B.13
10.P.2
3.B.17
12.P.8
3.B.20
12.G.4
Kozik
Demonstrate an understanding of the 9/10
relationship between various
representations of a line. Determine a
line's slope and x- and y-intercepts
from its graph or from a linear
equation that represents the line. Find
a linear equation describing a line
from a graph or a geometric
description of the line, e.g., by using
the "point-slope" or "slope yintercept" formulas. Explain the
significance of a positive, negative,
zero, or undefined slope.
11/12
Solve a variety of equations and
inequalities using algebraic,
graphical, and numerical methods,
including the quadratic formula; use
technology where appropriate.
Include polynomial, exponential,
logarithmic, and trigonometric
functions; expressions involving
absolute values; trigonometric
relations; and simple rational
expressions.
Relate geometric and algebraic
11/12
representations of lines, simple
curves, and conic sections.
Page 12 of 17
Patterns,
relations,
algebra
Patterns,
relations,
algebra
Geometry
December 2010
Computer Assisted Drafting:
12.P.8
3.B.13c
12.P.11
3.B.14c
Solve a variety of equations and
11/12
inequalities using algebraic, graphical,
and numerical methods, including the
quadratic formula; use technology where
appropriate. Include polynomial,
exponential, logarithmic, and
trigonometric functions; expressions
involving absolute values; trigonometric
relations; and simple rational expressions.
Solve everyday problems that can be
11/12
modeled using polynomial, rational,
exponential, logarithmic, trigonometric,
and step functions, absolute values, and
square roots. Apply appropriate graphical,
tabular, or symbolic methods to the
solution. Include growth and decay; joint
(e.g., I = Prt, y = k(w1 + w2)) and
combined (F = G(m1m2)/d2) variation,
and periodic processes.
Patterns,
relations,
algebra
Patterns,
relations,
algebra
Electricity:
10.P.2
3.B.15c
Kozik
9/10 Patterns,
Demonstrate an understanding of the
relationship between various
relations,
representations of a line. Determine a line's
algebra
slope and x- and y-intercepts from its graph
or from a linear equation that represents the
line. Find a linear equation describing a
line from a graph or a geometric description
of the line, e.g., by using the "point-slope"
or "slope y-intercept" formulas. Explain the
significance of a positive, negative, zero, or
undefined slope.
Page 13 of 17
December 2010
Information Technology:
3.B.27c
12.P.8
Machine Technology:
3.B.13c
12.P.8
Kozik
Solve a variety of equations and
11/12 Patterns,
inequalities using algebraic, graphical,
relations,
and numerical methods, including the
algebra
quadratic formula; use technology where
appropriate. Include polynomial,
exponential, logarithmic, and
trigonometric functions; expressions
involving absolute values; trigonometric
relations; and simple rational
expressions.
11/12 Patterns,
Solve a variety of equations and
relations,
inequalities using algebraic,
algebra
graphical, and numerical methods,
including the quadratic formula; use
technology where appropriate.
Include polynomial, exponential,
logarithmic, and trigonometric
functions; expressions involving
absolute values; trigonometric
relations; and simple rational
expressions.
Page 14 of 17
December 2010
12.P.11
3.B.14c
12.M.1
3.B.16
Solve everyday problems that can be 11/12
modeled using polynomial, rational,
exponential, logarithmic,
trigonometric, and step functions,
absolute values, and square roots.
Apply appropriate graphical, tabular,
or symbolic methods to the solution.
Include growth and decay; joint (e.g.,
I = Prt, y = k(w1 + w2)) and
combined (F = G(m1m2)/d2)
variation, and periodic processes.
Describe the relationship between
11/12
degree and radian measures, and use
radian measure in the solution of
problems, in particular, problems
involving angular velocity and
acceleration.
Patterns,
relations,
algebra
Measurement
Metal Fabrication:
12.P.8
3.B.13c
12.P.11
3.B.14c
Kozik
Solve a variety of equations and
11/12
inequalities using algebraic, graphical,
and numerical methods, including the
quadratic formula; use technology where
appropriate. Include polynomial,
exponential, logarithmic, and
trigonometric functions; expressions
involving absolute values; trigonometric
relations; and simple rational expressions.
11/12
Solve everyday problems that can be
modeled using polynomial, rational,
exponential, logarithmic, trigonometric,
and step functions, absolute values, and
square roots. Apply appropriate graphical,
tabular, or symbolic methods to the
solution. Include growth and decay; joint
(e.g., I = Prt, y = k(w1 + w2)) and
combined (F = G(m1m2)/d2) variation,
and periodic processes.
Page 15 of 17
Patterns,
relations,
algebra
Patterns,
relations,
algebra
December 2010
Performance Standards
In the Math Department at McCann, performance standards focus on inquiry-based
learning, which include problem solving, research papers, and following the steps of the
order of operations. Tests and quizzes are essential. Rubrics are utilized whenever possible
to help students understand the goals of the assignment and to aid in keeping grading
consistent. Weekly progress reports are shown to each student to allow them to keep track
of any missed assignments or low test grades. Students are expected to actively participate
in all classroom activities and daily attendance/performance is an integral part of all
students’ grades.
Competency Reporting Systems
Math teachers at McCann will be using the school’s database system, X2, which
includes an electronic rank book, for tracking student progress. Mid-quarter progress
reports and end of quarter report cards will be issued to students and parents through
utilization of this system.
Utilization of the high school web site (http://www.mccanntech.org) will provide
students and parents with the expected course requirements. The web site will be updated
biweekly to inform students of their responsibilities. We encourage parents to frequently
visit this site to help students make progress towards their goals.
Instructional Activities
Several methods of instruction are used throughout the course. Students are exposed to
the material through lecture, practice, cooperative learning groups, projects, and
presentations. Beyond the textbook, there are quarterly projects that are completed
throughout the year. The students should become familiar enough with the information
to speak fluently and answer questions about the topics being explored.
Kozik
Page 16 of 17
December 2010
Resources
Coxford, Arthur F. Trigonometry. Harcourt, Brace, Jovanovich, Orlando, FL:1987
Blatner, David. The Joy of Pi. Walker Publishing, New York, NY: 1997.
Devlin, Keith. The Math Instinct. Thunder’s Mouth Press, New York, NY: 2005.
Eames, Charles and Ray. Powers of Ten. W. H. Freeman, New York, NY: 1994.
Fadiman, Clifton. Fantasia Mathematica. Simon & Shuster, New York, NY: 1958
Korner, T.W. The Pleasures of Counting. Cambridge University Press, Cambridge, UK:
2002
Pickover, Clifford A. Wonders of Numbers. Oxford University Press, New York, NY:
2003.
Sweltz, Frank and Hartzler, J.S. Mathematical Modeling in the Secondary Curriculum.
National Council of Teachers of Mathematics. Reston, Virginia: 1991.
Kozik
Page 17 of 17
December 2010