Dover-Sherborn High School Mathematics Curriculum Pre-Calculus Level 2/CP

Transcription

1 Mathematics Curriculum A. DESCRIPTION This is a precalculus course that expands the student s knowledge of some Algebra 2 topics and covers many of the topics found in a traditional college preparatory precalculus course. It prepares the student for a precalculus course (or a calculus course depending on the curriculum) at the college level. B. OBJECTIVES The student will: 1. develop an understanding of functions; 2. understand, apply and graph the six trigonometric 3. functions and their inverses; 4. prove some trigonometric identities; 5. apply the Law of Sines and the Law of Cosines; 6. solve trigonometric equations; 7. study complex numbers; 8. graph equations in rectangular form; 9. solve problems involving sequences and series; 10. use the graphical calculator appropriately, and C. OUTLINE 1. Trigonometry a. Trigonometry of the right triangle i. definition of the trig functions [Patterns, Relations, and Algebra: PC.P.3] ii. use of trig tables and calculators [Measurement: Apply appropriate techniques, tools, and formulas to determine measurements. ] iii. solution of right triangles [Geometry: PC.G.1; Patterns, Relations, and Algebra: PC.P.3] iv. word problems using right triangles including angles of elevation and depression [Patterns, Relations and Algebra: PC.P.3; Geometry:PC.G.3] v. reciprocal functions and co-functions [Patterns, Relations, and Algebra] vi. trig functions of special angles [Geometry] b. General Angles i. extended domains of trig functions [Patterns, Relations, and Algebra:PC.P.6] ii. negative angles and their trig functions [Patterns, Relations, and Algebra: PC.P.3] iii. quadrantal angles [Patterns, Relations, and Algebra: PC.P.7; Geometry: PC.G.3; Measurement: PC.M.1, PC.M.2] c. Radian Measure i. measuring angles in degrees and radians [Measurement: PC.M.1] ii. conversion from degrees to radians and vice versa [Measurement: PC.M.1, PC.M.2] iii. length of an arc [Geometry: PC.G.3; Measurement: PC.M.1] 11/30/2005 9:12:53 PM Page 1 of 11

2 Mathematics Curriculum d. Solution of Oblique Triangles i. Law of Sines [Geometry: PC.G.1] ii. Law of Cosines [Geometry: PC.G.1] iii. the ambiguous case [Geometry: PC.G.1] iv. area of a triangle [Geometry: PC.G.1] e. Review and Expansion of Knowledge of Functions i. domain, range, inverse, operations on functions [Patterns, Relations, and Algebra: PC.P.3, PC.P.6] ii. interval notation [Patterns, Relations, and Algebra] iii. piecewise functions [Patterns, Relations, and Algebra] iv. graphical analysis of functions [Patterns, Relations, and Algebra; Geometry: Use visualization to solve problems. ] f. Graphs of Trig Functions i. graphs of functions, their domain, range, period and amplitude [Patterns, Relations, and Algebra: PC.P.6] ii. graphs of functions, changing amplitude and period, displacement [Patterns, Relations, and Algebra: PC.P.6] g. Inverse Trig Functions i. review of definitions of relations and functions [Patterns, Relations, and Algebra] ii. review of finding the inverse of a function [Patterns, Relations, and Algebra; Geometry] iii. finding the inverse of the trig functions [Patterns, Relations, and Algebra] iv. principal range and principal value [Patterns, Relations, and Algebra] h. Trig Identities and Trig Equations i. review the difference between an identity and an equation [Patterns, Relations, and Algebra] ii. basic trig identities [Patterns, Relations, and Algebra: PC.P.5] iii. Pythagorean identities and their derivations [Patterns, Relations, and Algebra: PC.P.4] iv. verifying identities using specific angles [Patterns, Relations, and Algebra] v. proving trig identities [Patterns, Relations, and Algebra; Geometry] vi. solutions of trig equations [Patterns, Relations, and Algebra] i. General Trig Relationships i. functions of the sum and difference of two angles [Patterns, Relations, and Algebra: PC.P.5] ii. double angle and half angle formulas [Patterns, Relations, and Algebra] iii. sum and difference of two different functions [Patterns, Relations, and Algebra; Geometry] iv. solving trig equations with functions of the double angle and half angle [Patterns, Relations, and Algebra] 2. Sequences and Series (arithmetic and geometric) [Patterns, Relations, and Algebra] 3. Finite sequences and series [PC.P.1] 4. Infinite sequences [PC.P.1] 5. Exponential and Logarithmic Functions [Patterns, Relations, and Algebra] 6. Rational exponents 11/30/2005 9:12:53 PM Page 2 of 11

3 Mathematics Curriculum 7. Logarithmic equations and graphs 8. Exponential equations and graphs 9. Applications of exponential and logarithmic functions D. TEXT Trigonometry, Hayden and Hall - Prentice Hall, 1990 ISBN E. SUPPLEMENTARY MATERIALS 1. Computer programs for demonstration and experimentation 2. Worksheets prepared by teacher 3. Trigonometry, Isidore Dressler & Barnett Rich, AMSCO, Graphical calculators 6. Mathematics Teacher 7. CBLs 11/30/2005 9:12:53 PM Page 3 of 11

4 Course Title: Grade: 11/12 Review and Expand Knowledge of Functions What is a function? What are the key characteristics of various functions? Identify domain, range, and inverse of a function. Perform basic operations on functions including evaluating functions. Classify functions. 11/30/2005 9:12:53 PM Page 4 of 11

5 Course Title: Grade: 11/12 Trigonometry of Right Angles How can one find the trig value of an acute angle? How can one find any missing values from a right triangle? Use co-functions, given trig value or side lengths, reciprocal functions, identities. Evaluate trig functions for special angles (30, 45, 60 ) using special triangles. Solve right triangles including word problems such as angles of depression or elevation. Use calculator to find trig values. Clinometer Project: Constructing a device used to measure angles of elevation and calculate height. 11/30/2005 9:12:53 PM Page 5 of 11

6 Course Title: Grade: 11/12 Trigonometry of General Angles How can one use his/her knowledge of acute angles to find the trig value of any angle? Evaluate trig values of quadrantal angles. Use complementary angles and knowledge of the sign of trig functions depending on quadrants to evaluate trig functions. Use calculators to find trig values. Utilize identities such as Pythagorean Identities, double-angle formulas, halfangle formulas, sum and difference formulas. 11/30/2005 9:12:53 PM Page 6 of 11

7 Course Title: Grade: 11/12 Radian Measure How can one sketch an angle? In what various methods, can one use to give an angle measurement? Convert degree measurement to radian measure and vice versa. Convert decimal degrees to degrees, minutes, seconds and vice versa. Sketch an angle in standard position. Find the length of an arc using central angle measure. 11/30/2005 9:12:53 PM Page 7 of 11

8 Course Title: Grade: 11/12 Solutions to Oblique Triangles How can one solve oblique triangles to find any missing parts? Be able to distinguish when to use Law of Sines verses Law of Cosines. Use Law of Sines. Solve ambiguous case triangles and identify solutions to none, one, or two possible triangles. Use Law of Cosines. Find the area of a triangle using Heron s formula and right triangle trigonometry. 11/30/2005 9:12:53 PM Page 8 of 11

9 Course Title: Grade: 11/12 Graphs of Trigonometric Functions What are the characteristics of the graphs of the trigonometric functions? Determine domain, range, intercepts, amplitude, period, vertical shift, phase shift of the graph of a function when given an equation. Determine the equation when given a graph. Biorhythm Project: Use days lived to predict and evaluate the accuracy of the biorhythm Theory. 11/30/2005 9:12:53 PM Page 9 of 11

10 Course Title: Grade: 11/12 Trig Identities and Equations How does one know that an identity is valid? How can one solve an equation involving trigonometric functions? Verify identities using special angles. Derive various forms of the fundamental identities such as the Pythagorean Identities. Prove identities. Solve trig equations using graphs of trig functions, techniques for solving quadratics (such as factoring), and using calculators. 11/30/2005 9:12:53 PM Page 10 of 11

11 Course Title: Grade: 11/12 Exponential and Logarithmic Functions Why do logarithms exist? How can I solve an exponential or logarithmic equation? Solve exponential equations by getting the same base and equating powers or by using logs Use the properties of logarithms to condense/expand expressions and solve equations. Solve application problems such as growth, decay, logistic models, interest, Newton s Law of Cooling. Graph exponential and logarithmic functions. Use the laws of exponents to simplify an expression. PASCO demonstration of cooling 11/30/2005 9:12:53 PM Page 11 of 11