Download Trigonometry (Semester) - Albuquerque Public Schools

APS
DISTRICT HIGH SCHOOL MATHEMATICS CURRICULUM FRAMEWORK
Course Title: Trigonometry (Semester)
Course Number:
Department:
ADS Number: 20434131
Mathematics
38020
Prerequisites: Successful completion of Algebra II or equivalent
Length of Course: One Semester
Credit/PRI Area: .50 per Semester/Mathematics
Grade Level(s): 10 - 12
Important Notes:
This course requires student access to a graphing calculator.
COURSE DESCRIPTION: Trigonometry is an advanced mathematics course devoted to the study of the six trigonometric functions and a basic introduction to polar
coordinates. The student solves triangles, trigonometric equations, finds their values, graphs, inverses, and recognizes the relationships to one another. Other areas of study are
global mathematics processes, number sense and operations, geometry, spatial sense, and measurement, and patterns, functions, and algebraic concepts.
References in parentheses following each performance standard align with the National Council of Teachers of Mathematics Standards (NCTM), the State of New Mexico
Mathematics Standards (NM), and the Albuquerque Public Schools Mathematics Standards (APS).
TRIGONOMETRY SEMESTER
12.1.8
Albuquerque Public Schools 03/04
STRATEGIES:
The “Illustrations” column in the Program of Studies provides exemplars of the performance standards, strategies, and best practices suggested by mathematics teachers in the
Albuquerque Public Schools (APS).
ASSESSMENTS:
Assessments may include: authentic and performance-based assessment, cooperative learning, teacher observations, checklists, tests and exams, formal and informal writing, small
group and full class discussions, oral and multimedia presentations, projects, demonstrations, and portfolios. Assessments are based on appropriate rubrics.
SUGGESTED TEXTBOOKS AND INSTRUCTIONAL MATERIALS:
 Current state adopted mathematics textbooks
 Supplemental materials
 Graphing calculators
 Computers
 Computer software (e.g., Geometer’s Sketchpad, TI Calbri, Green Globs)
SUGGESTED TITLES/AUTHORS WEB SITES:
Approved by HSCA:
TRIGONOMETRY SEMESTER
12.2.8
Albuquerque Public Schools 03/04
STRAND I: GLOBAL MATHEMATICAL PROCESSES
CONTENT STANDARD: The student understands and uses mathematical processes.
BENCHMARK: The student uses problem solving, reasoning and proof, communication, connections, and representations as appropriate in all mathematical experiences.
GRADE
10-12
PERFORMANCE STANDARDS
ILLUSTRATIONS
NOTE: Illustrations include suggested activities for attaining each
performance standard. A check for () refers to a key feature to look for
while assessing student performance.
1.
2.
3.
4.
5.
6.
7.
8.
Develops resourcefulness and perseverance in problem solving in
mathematics and other disciplines (APS – I.1).
Recognizes when to use previously learned strategies to solve new
problems (APS – I.2).
Develops and uses strategies (e.g., breaking complex problems into
simpler parts) for solving given problems (APS – I.3).
Monitors, discusses, and reflects on the process of mathematical problem
solving (APS – I.4).
Develops a logical sequence of arguments leading to a valid conclusion
or solution to a problem (APS – I.7).
Works in teams to share ideas, to develop and coordinate group
approaches to problems, and to communicate findings (APS – I.8).
Communicates mathematical thinking coherently and clearly to others
(APS – I.9).
Analyzes and evaluates mathematical thinking and strategies of others
(APS – I.10).
9.
Relates applications to mathematical language in various modalities
(APS – I.11).
10. Identifies and connects functions with real-world applications
(APS – I.12).
11. Makes connections among mathematical concepts (APS – I.12).
12. Identifies how seemingly different mathematical situations may be
essentially the same (APS – I.13).
TRIGONOMETRY SEMESTER
12.3.8
1 – 8, 12. The student, working in small groups, times a projectile from the
ground into the air and back to the ground again. Also, the group moves away
approximately 90 feet from the launch area and estimates the angle at which
the highest altitude is reached and compares it to an algebraic model.
S = 1/2at2
tan  = h
90 ft
The student compares algebraic vs trigonometric values, finds sources of error
and produces a percentage error analysis.
h
student





90 ft.
team work
accuracy
comparisons
analysis
9 – 11, 13. The student responds to the following scenario: Pollution Problem –
In a large city the amount of sulfur dioxide released into the atmosphere due to
the burning of coal and oil for heating purposes varies seasonally. Suppose
the number of tons of pollutant released into the atmosphere during the xth
week after January 1 is given approximately by P(x) = 1.1 + cos πx
26
Albuquerque Public Schools 03/04
GRADE
10-12
PERFORMANCE STANDARDS
ILLUSTRATIONS
13. Uses technology tools appropriately and understands their purposes
(APS – II.6E, II.3L).
TRIGONOMETRY SEMESTER
12.4.8
Using a graphing calculator, the student graphs this function for a two year
period starting January 1 and determines in which weeks the pollution
being released is the greatest and the least.
 use of technology
 accuracy
 connections
Albuquerque Public Schools 03/04
STRAND II: NUMBER SENSE AND OPERATIONS
CONTENT STANDARD: The student demonstrates number sense through experiences with meaningful mathematical problems that focus on number meaning, number
relationships, place value concepts, relative effects of operations, and multiple representations to communicate sound mathematical thinking.
BENCHMARK: The student understands rational and real numbers and uses a variety of means, including technology, as appropriate, to solve problems in these
number systems.
GRADE
10-12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Defines the trig functions as ratios and in terms of the unit circle
(APS – II.8E; NM – VB.FS).
2.
Changes between radians and degree measures (APS – II.11E;
NM – IID.5, VB.FS).
3.
Calculates values for the trig functions by various means (APS – II.3L;
NM – IC.2).
4.
Defines polar coordinates (APS – V.7E, V.8E; NM – VB.FS).
1 – 4. The student completes a project that presents relevant data concerning the
six trigonometric functions as they relate to the unit circle. The project is to
be constructed on a scientific display board (e.g., cardboard, foam core, mat
board), must be at least 22 x 28, and presented neatly, accurately, and in
color. The project must include all of the following:
 Large unit circle divided into increments of π/6 (30), π/4 (45), and π/3
(60) with (x,y) coordinates labeled for all points of the circle, degree
and radian measures indicated for all points on the circle, all points color
coded according to their reference angles, and the exact values of all six
trigonometric functions calculated for every point on the circle.
 A “Graph Characteristics” chart for all six trigonometric functions that
lists their domain and range and, if applicable, all points where the
function is undefined, whether the function’s symmetry is odd, even, or
neither, the function’s period, x-and y-intercepts, the maximums and
minimums and where they occur, and, if the function has no maximum
or minimum, determine the location of all asymptotes.
 Construction of three separate graphs of the trigonometric functions that
meet the following requirements:
- the primary trig function is to be graphed on the same set of
axes as its reciprocal function,
- the domain of each graph is to be -2π ≤x ≤ 2π and marked off
in units corresponding to those of the unit circle, and
- all asymptotes, maximums, and minimums clearly indicated
on each graph.
 Name, period, and course name PRINTED IN INK on the front and
back of the project.
Note: Let the student know that arrangements can be made for financial
hardships, that the project is not trivial and requires significant planning,
layout, and time in order to complete it adequately, and that it will be scored
according to a teacher-designed rubric. The rubric is shown to the student
TRIGONOMETRY SEMESTER
12.5.8
Albuquerque Public Schools 03/04
GRADE
10-12
PERFORMANCE STANDARDS
ILLUSTRATIONS
before the project is started so that there are no “surprises” upon receipt of
the final grade.
 all required components
 comprehension of all the trig functions
 correct labeling
 creativity
 effective presentation
 adherence to rubric
TRIGONOMETRY SEMESTER
12.6.8
Albuquerque Public Schools 03/04
STRAND III: GEOMETRY, SPATIAL SENSE, AND MEASUREMENT
CONTENT STANDARD: The student demonstrates an understanding of concepts, properties, and relationships of geometry and measurement through experiences with
meaningful mathematical problems that focus on identifying, describing, classifying, visualizing, comparing, estimating, and measuring various
aspects of shapes and objects.
BENCHMARK: The student probes theorems, explores and tests several logical reasoning methods, and uses trigonometric relationships and Cartesian coordinates to represent
objects in the plane. The student uses formulas for solving measurement problems and uses scaling as appropriate.
GRADE
10-12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Verifies trigonometric identities (NM – VB.FS).
1. The student verifies the accuracy of a variety of identities and equations
(e.g., sin 2x = 2sin x cos x) including double-angle, half-angle, and productsum identities.
 accuracy
 verifications
2.
Solves trigonometric equations by analytical and graphical methods
(APS – III.8L; NM – VB.FS).
2. The student is given the following scenario to respond to: A musical tone is
described by y = 0.04 sin 200πt – 0.03 cos 200πt. He/She writes the equation
in the form y = A sin(Bt + C), computes C to three decimal places so that C
is minimum, graphs the equation, and indicates the amplitude, period,
frequency, and phase shift.
 all required components
 accuracy
 connections
3.
Solves triangles and problems using the law of sines and law of cosines
(APS – III.12L, III.13L; NM – IID.6, VB.FS).
3. The student solves the following: A fire at F is spotted from two fire lookout
stations A and B, which are located 10.3 mi apart. If station B reports the fire
at  ABF = 52.6, and station A reports the fire at  BAF = 25.3, how far is
the fire from station A? from station B? The student verifies work including
a sketch that supports the work.
 accuracy
 application of law of sines and cosines
 documentation of work
TRIGONOMETRY SEMESTER
12.7.8
Albuquerque Public Schools 03/04
STRAND IV: PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS
CONTENT STANDARD: The student demonstrates an understanding of algebraic skills and concepts through experiences with meaningful mathematical problems that focus
on discovering, describing, modeling, and generalizing patterns and functions, representing and analyzing relationships, and finding and
supporting solutions.
BENCHMARK: The student represents patterns with relations and functions, investigates rates of change, and symbolically manipulates numbers.
GRADE
10-12
PERFORMANCE STANDARDS
ILLUSTRATIONS
1.
Graphs the trig functions and their inverses (APS – V.7L; NM – VB.FS).
2.
Identifies the amplitude, period, phase shift, and vertical shift of a graph
of a trigonometric function from its equations (APS – V.7L;
NM – VB.FS).
3.
Graphs more complex trig functions using knowledge of period, phase
shift, vertical shift, and amplitude (APS – V.6L; NM – VB.FS).
4.
Works with everyday problems and applications including integration
with other subject areas studied at the same grade level (APS – V.7L,
V.8L; NM – VB.FS).
5.
Graphs polar equations (APS – V.7L; NM – VB.FS).
1, 2. See illustration for Strand II and illustration #2, Strand III.
3, 4. See Strand III, illustration #2.
5.
Using either point-by-point sketching or rapid sketching techniques, the
student graphs a cardioid. He/She submits the sketch along with a table of
values used to determine the cardioid. Using a graphing calculator, the
student “checks” his figure for accuracy.
Extension: Using a software program at school or home, the student prints
out the cardioid and compares the rough sketch to the computer sketch.
 table of values
 two figures (e.g., computer, rough sketch)
 accuracy
 understanding of polar equations
6.
Transforms equations from Cartesian to polar form (APS – V.8L;
NM – VB.FS).
TRIGONOMETRY SEMESTER
12.8.8
6. The student converts x2 + y2 - 2y = 0 to polar form and r + 3 sin  = 0 to
rectangular form and documents his/her work.
 accuracy
 understanding of concepts
 documentation
Albuquerque Public Schools 03/04