PreCalculus Pen Argyl Area High School 2012

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1 PreCalculus is a study in advanced algebra and trigonometry. Emphasis is on the analysis of various functions and their applications to both real life problems and higher-level mathematics. This course is for superior students. Topics include the language and notation of functions and an in-depth study of linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions. Additionally, students are introduced to sequences, series, counting principles, probability and the limit process. The graphing calculator (TI-83 or TI-84 Plus) is an integral part of the analysis of functions. Credits: 1.0 Prerequisites: Honors Algebra 2 and Honors Geometry (or 85% or higher in College Prep) 1

2 Unit 1: Prerequisites Unit Outcomes: Students will review pertinent material from Algebra 2 that is needed to continue the study of advanced Algebra and functions. Initial emphasis will be on simplifying expressions using the laws of exponents and polynomial factoring techniques. Students will review equation and inequality solving skills involving both linear and quadratic equations that extend to polynomial, radical, absolute value and equations involving fractions. Students will explore graphical solutions and begin the fundamentals of graphing by looking at intercepts, symmetry, circles and lines. Students will write and use linear models to solve real-life problems. Essential Outcomes and Related Standards: A. Simplify expressions using Laws of Exponents including negative and rational exponents. PA M11.A Simplify /evaluate expressions involving positive and negative exponents, roots and/or absolute values. PA M11.A Simplify expressions involving multiplying with exponents (x 6 *x 7 =x 13 ), powers of powers ((x 6 ) 7 =x 42 ), and powers of products (2x 2 ) 3 =8x 6 ). B. Factor polynomials. PA M11.A Factor algebraic expressions including differences of squares and trinomials. C. Simplify rational expressions. PA M11.A Simplify algebraic fractions. D. Solve equations linear, quadratic, factorable polynomials, radical, absolute value, and equations involving fractions. PA M11.D.2.1 Solve and/or graph linear equations and inequalities using various methods PA N Solve linear, quadratic and exponential equations both symbolically and graphically. E. Solve inequalities, both linear and quadratic; graph solution and name solution interval. PA M11.D.2.1 Solve and/or graph linear equations and inequalities using various methods. F. Find intercepts and symmetry of graphs from equations. PA T Analyze and categorize functions by their characteristics. G. Find center and radius of circles, write equations of circles, sketch circles. PA E E. Use equations to represent curves (e.g., lines, circles, ellipses, parabolas, hyperbolas). H. Write equations of lines and use linear models in problem solving. 2

3 PA L Write the equation of a line when given the graph of the line, two points on the line, or the slope of the line and a point on the line. PA E Demonstrate mathematical solutions to problems (e.g., in the physical sicences). Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 3

4 Unit 2: Functions and Their Graphs Unit Outcomes: Students will master the basics of function analysis and apply the principles to graphing, both with and without a calculator. Students will graph basic functions without a calculator using transformations. Students will analyze functions using the graphing calculator to set the stage for advanced study in Calculus. Additionally, students will combine functions with addition, subtraction, multiplication, division and composition. Students will also study inverse functions both algebraically and graphically. Essential Outcomes and Related Standards: A. Evaluate function notation including piecewise and step functions. B. Identify key features of functions domain, range, zeros, relative extrema, and intervals of increasing/decreasing. PA M11.D Identify domain, range or inverse of a relations. PA T Analyze and categorize functions by their characteristics. C. Identify even and odd functions and their symmetry. PA T Analyze and categorize functions by their characteristics. D. Graph common functions by transformation (without calculators). E. Graph step and piecewise functions. F. Combine functions by addition, subtraction, multiplication, division, and composition. G. Find and graph inverse functions. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Re-teaching Activities Extra worksheets Pre-test study guides 4

5 Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 5

6 Unit 3: Polynomial and Rational Functions Unit Outcomes: Students will study the unique features of polynomial functions, including zeros, extrema and end behavior. Initial focus will be the quadratic function, explored in depth both algebraically and graphically, as well as through modeling real-life problems involving maxima and minima. Students will work with higher order polynomial functions and learn techniques for finding zeros, both real and complex. Lastly, students will study the rational function with emphasis on domain, asymptotes and graphing. Essential Outcomes and Related Standards: A. Identify vertex, axis of symmetry, and x-intercepts of a quadratic function and graph the function without a calculator. PA R Create and interpret functional models. B. Apply quadratic functions to problems involving maximum and minimum. PA N Solve linear, quadratic and exponential equations both symbolically and graphically. PA A Determine maximum and minimum values of a function over a specified interval. PA B Interpret maximum and minimum values in problem situations. C. Sketch polynomial functions using zeros, multiplicity and end behavior. D. Find zeros of polynomial functions by factoring. E. Divide polynomials with long division and synthetic division. F. Simplify, add, subtract, multiply and divide complex numbers. PA A Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms). G. Apply the remainder and factor theorems to express polynomials as the product of linear factors. PA T Analyze and categorize functions by their characteristics. H. Use the rational zeros theorem to find real and complex zeros. PA Q Represent functional relationships in tables, charts and graphs. ` PA T Analyze and categorize functions by their characteristics. I. Find vertical, horizontal and slant asymptotes and graph rational functions. 6

7 Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 7

8 Unit 4: Exponential and Logarithmic Functions Unit Outcomes: Students will study exponential and logarithmic functions numerically, algebraically, and graphically. Graphically, students will expand previous concepts in transformations and asymptotes to analyze a function and create its graph. Algebraically, students will work with laws of logarithms to simplify expressions. Numerically, students will evaluate logarithmic functions without the aid of a calculator. Students will study the inverse nature of exponential and logarithmic functions and use inverses and algebraic skill to solve equations. Students will solve real-life models involving applications in compound interest and other growth models. Essential Outcomes and Related Standards: A. Graph exponential equations using basic graph and transformation. PA S Analyze properties and relationships of functions (e.g., linear, polynomial, rational, trigonometric, exponential, logarithmic). PA T Analyze and categorize functions by their characteristics. B. Solve application problems involving exponential functions, including compound interest. PA C Graph and interpret rates of growth/decay. C. Evaluate logarithms without the aid of a calculator. PA A Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms.) D. Simplify logarithmic expressions using properties to expand and condense. E. Solve exponential and logarithmic equations algebraically and graphically. PA N Solve linear, quadratic and exponential equations both symbolically and graphically. F. Solve application problems involving logarithms. PA N Solve linear, quadratic and exponential equations both symbolically and graphically. PA C Graph and interpret rates of growth/decay. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections 8

9 Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 9

10 Unit 5: Sequences, Series and Probability Unit Outcomes: An introduction to discrete mathematics, this unit explores new notations for sequences, series, factorials, permutations, and combinations. Students will study both arithmetic and geometric sequences and series, including domain and mathematical modeling. Students will use patterns and Pascal s Triangle to develop and use the Binomial Theorem to expand powers of binomials. Lastly, students will study basic combinatorics to find permutations, combinations and probability and odds. Essential Outcomes and Related Standards: A. Find terms of sequences from explicit and recursive formulas. PA D Determine sums of finite sequences of numbers and infinite geometric series. PA A Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically. B. Evaluate factorial notation. PA A Use operations (e.g., opposite, reciprocal, absolute value, raising to a power, finding roots, finding logarithms). C. Find sums of sequences sigma notation, nth partial sums, and infinite sums. PA D Determine sums of finite sequences of numbers and infinite geometric series. D. Recognize, write, and evaluate arithmetic sequences and series. PA D Determine sums of finite sequences of numbers and infinite geometric series. E. Recognize, write, and evaluate geometric sequences and series. PA D Determine sums of finite sequences of numbers and infinite geometric series. F. Solve application problems involving sequences and series. PA C Use patterns, sequences and series to solve routine and non-routine problems. G. Use the binomial theorem to expand binomials and find specific terms. State Standards(s): H. Solve simple counting problems. PA M.11.E.3.2 Apply counting techniques in problem-solving settings. 10

11 I. Find combinations and permutations and apply them to counting problems. State Standards(s): PA M11.E Determine the number of permutations and/or combinations. J. Find probability and odds of events. PA A Compare odds and probability. PA E Solve problems involving independent simple and compound events. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 11

12 Unit 6: Trigonometry Angles, Right Triangles, Trigonometric Functions of Any Number Unit Outcomes: Building on a basic introduction to sine, cosine, and tangent from triangles in Geometry, students will expand their knowledge to the six trigonometric functions. Angles will be measured in degrees and radians with an understanding of coterminal angles, complements, supplements, and reference angles. The six trigonometric functions will be viewed through triangles, the coordinate plane, and the unit circle. Emphasis will be on learning the ratios for the special angles without a calculator, but also full calculator use for evaluating the functions for all real numbers will be done as the functions are applied to real life problems including surveying, navigation, and other triangle applications. Essential Outcomes and Related Standards: A. Define and evaluate the six trigonometric functions using right triangles. PA B Identify, create, and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem. PA MA.A Simplify square roots. B. Solve application problems involving right triangles. PA B Identify, create, and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem. PA F: Demonstrate skills for using computer spreadsheets and scientific and graphing calculators. PA E: Demonstrate mathematical solutions to problems (e.g., in the physical sciences). C. Draw standard position angles in radians and degrees and identify by quadrant. Find coterminal angles, complements, and supplements in both radians and degrees. PA2.3.11B Measure and compare angles in degrees and radians. D. Define and evaluate the six trigonometric functions using the unit circle. PA M11.A Identify and/or use proportional relationships in problem solving settings. E. Find exact values of six trigonometric functions for special angles PA M11.C.1.2 Recognize and/or apply properties of angles, triangles, and quadrilaterals. F. Find a reference angle and understand its use in evaluating trigonometric functions for any real number. 12

13 F. Use a calculator to evaluate trigonometric functions in both radians and degrees, as well as be able to find an angle measurement given a particular trigonometric function value. PA F: Demonstrate skills for using computer spreadsheets and scientific and graphing calculators. PA2.3.11B Measure and compare angles in degrees and radians. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 13

14 Unit 7: Trigonometry Graphs and Inverses Unit Outcomes: Students will view the trigonometric functions from a graphic perspective, identifying domain, range, amplitude, period, and asymptotes. From a prior knowledge of transformations, students will develop the basic graphs and analyze how multiplying and adding constants affects the graph and the behavior of the function in general. Graphing will be done both by hand and with a graphing calculator, but the emphasis will be in the analyzing of the features and recognizing the basic shape of each function. Graphs will then be used to introduce the inverse trigonometric functions and students will evaluate inverses and apply both graphs and inverses to real-life problems. Essential Outcomes and Related Standards: A. Graph sine and cosine functions without the aid of a calculator and find amplitude, period, phase and vertical shifts. Note: A graphing calculator will be used to aid in the discovery of features, but students will learn to analyze the equation and sketch its graph by hand. PA Q Represent functional relationships in tables, charts, and graphs. PA T Analyze and categorize functions by their characteristics. PA A Use graphing calculators to display periodic and circular functions; describe properties of graphs. B. Analyze and graph tangent, cotangent, secant, and cosecant functions by finding period and asymptotes. PA Q Represent functional relationships in tables, charts, and graphs. PA T Analyze and categorize functions by their characteristics. PA A Use graphing calculators to display periodic and circular functions; describe properties of graphs. C. Define and evaluate inverse trigonometric functions both with and without a calculator. D. Use periodic functions and inverses to solve problems including, but not limited to, triangle applications and simple harmonic motion. PA E Demonstrate mathematical solutions to problems (e.g., in the physical sciences). PA I Model situations geometrically to formulate and solve problems. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections 14

15 Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 15

16 Unit 8: Analytic Trigonometry Identities and Equations Unit Outcomes: Students will learn the fundamental trigonometric identities and use them to simplify expressions and verify other identities. Algebraic skills with trigonometry will be developed. Students will solve trigonometric equations both with and without calculators. Students will use Sum and Difference, Multiple-Angle, and Product-Sum Formulas to evaluate and simplify expressions, verify identities and solve equations. A formula sheet will be provided for assessments. Essential Outcomes and Related Standards: A. Recognize, memorize, and use fundamental identities reciprocal, ratio, and Pythagorean. B. Use trigonometric identities to simplify expressions and verify identities. C. Solve trigonometric equations both with and without calculator. D. Develop and use Sum and Difference, Multiple Angle, and Product-Sum Formulas. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Textbook Computer Graphing Calculators 16

17 Unit 9: Trigonometry Law of Sines and Law of Cosines Unit Outcomes: Students will learn and use the Law of Sines and the Law of Cosines to find measurements in oblique (non-right) triangles. Many real-life application problems will follow, including problems with navigation, surveying, and other triangle applications. Students will then develop formulas for finding the area of oblique triangles. The unit culminates with an application project where students develop and solve triangles to find measurements for construction projects around the school. Essential Outcomes and Related Standards: A. Solve triangles using the Law of Sines, including the ambiguous case. B. Solve triangles using the Law of Cosines. C. Solve application problems involving oblique triangles. D. Find area of oblique triangles and apply to real-life problems. Content and Instructional Strategies: Lecture Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Assessment Criteria: Homework book and worksheets Teacher created quizzes, tests, and open-ended questions Project 17

18 Resources and Materials: Textbook Computer Graphing Calculators Clinometers Measuring Tapes 18

Syllabus ENGR 190 Introductory Calculus (QR)

Syllabus ENGR 190 Introductory Calculus (QR) Catalog Data: ENGR 190 Introductory Calculus (4 credit hours). Note: This course may not be used for credit toward the J.B. Speed School of Engineering B. S.