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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