Honors PreCalculus Pen Argyl Area High School 2017 · Honors PreCalculus Pen Argyl Area High School...

Honors PreCalculus Pen Argyl Area High School 2017

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Honors PreCalculus is a study in advanced functions 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 and the limit process. The graphing calculator (TI-83 or TI-84 Plus) is an integral part of the analysis of functions, however students will be required to graph and calculate without the aid of any calculator, in preparation for the study of AP Calculus, where calculators are limited. Credits: 1.0 Prerequisites: Honors Algebra 2 and Honors Geometry (or 85% or higher in College Prep) Note: This course may be taken concurrently with Honors Geometry for a student who wishes to double up and prepare for Calculus in senior year.


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Unit 1: Functions and Their Graphs Unit Outcomes: Students will master the basics of function analysis numerically, algebraically, and graphically. Students will analyze functions using the graphing calculator to set the stage for advanced study in Calculus, but will also analyze functions using analytic methods. Students will graph basic functions without a calculator using transformations. 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 the greatest integer functions.

B. Identify key features of functions – domain, range, intercepts, relative extrema, and intervals of increasing/decreasing.

C. Identify even and odd functions and their symmetry.

D. Graph common functions by transformation (without calculators).

E. Graph piecewise functions.

F. Combine functions by addition, subtraction, multiplication, division, and composition.

G. Find and graph inverse functions.

PA Common Core Standards: CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations CC.2.2.HS.C.4 Interpret the effect transformations have on functions and find the inverses of functions. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides


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Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 2: 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. Analyze and transform quadratic functions in different forms – standard form, completed square form, factored form – to find axis of symmetry, vertex, and intercepts.

B. Apply quadratic functions to problems involving maximum and minimum.

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.

G. Apply the remainder and factor theorems to express polynomials as the product of linear

factors.

H. Use the rational zeros theorem to find real and complex zeros.

I. Find vertical, horizontal and slant asymptotes and graph rational functions. PA Common Core standards: CC.2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multi-step problems. CC.2.2.HS.F.6 Extend the knowledge of arithmetic operations and apply to complex numbers. CC.2.2.HS.F.7 Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems. CC.2.2.HS.D.1 Interpret the structure of expression to represent a quantity in terms of its context. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.3 Extend the knowledge of arithmetic operations and apply to polynomials. CC.2.2.HS.D.4 Understand the relationship between zeros and factors of polynomials to make generalizations about functions and their graphs. CC.2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between the different representations.


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Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 3: 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 and decay models. Essential Outcomes and Related Standards:

A. Graph exponential equations using basic graph and transformation.

B. Solve application problems involving exponential functions, including compound interest.

C. Evaluate logarithms without the aid of a calculator.

D. Simplify logarithmic expressions using properties to expand and condense.

E. Solve exponential and logarithmic equations algebraically and graphically.

F. Solve application problems involving logarithms.

G. Explore the logistic growth model and compare to exponential growth. PA Common Core Standards: CC.2.1.HS.F.1 Apply and extend the properties of exponents to solve problems with rational exponents. CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formlas, graphs, and data displays. CC.2.2.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections


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Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Exponential and Logarithmic application project Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 4: 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. Essential Outcomes and Related Standards:

A. Find terms of sequences from explicit and recursive formulas.

B. Evaluate factorial notation.

C. Find sums of sequences – sigma notation, nth partial sums, and infinite sums.

D. Recognize, write, and evaluate arithmetic sequences and series.

E. Recognize, write, and evaluate geometric sequences and series.

F. Solve application problems involving sequences and series.

G. Use the binomial theorem to expand binomials and find specific terms.

H. Solve simple counting problems.

I. Find combinations and permutations and apply them to counting problems.

J. Find probability of compound events. PA Common Core Standards: CC.2.2.HS.D.1 Interpret the structure of expressions to represent a quantity in terms of its context. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model. Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections


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Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 5: Trigonometry – Angles, Right Triangles, Trigonometric Ratios 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, including minutes and seconds, and radians with an understanding of coterminal angles and reference angles. Students will view the six trigonometric ratios 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 and apply 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.

B. Solve application problems involving right triangles.

C. Draw standard position angles in radians and degrees and identify by quadrant. Find coterminal angles in both radians and degrees.

D. Define and evaluate the six trigonometric functions using the unit circle.

E. Find a reference angle and understand its use in evaluating trigonometric functions for any

real number.

F. Find exact values of six trigonometric functions for quadrant angles, 30⁰, 45⁰, and 60⁰ as well as their radian counterparts and in all quadrants.

G. 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 Common Core Standards: CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurements when reporting quantities. CC.2.2.HS.C.7 Apply radian measure of an angle and the unit circle to analyze trigonometric functions. CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.


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Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 6: 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 with defined ranges 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.

B. Analyze and graph tangent, cotangent, secant, and cosecant functions by finding period and

asymptotes.

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, periodic behaviors, and simple harmonic motion.

PA Common Core Standards: CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.C.2 Graph and analyze functions and use their properties to make connections between different representations. CC.2.2.HS.C.4 Interpret the effects transformations have on functions and find the inverses of functions. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.2.HS.C.8 Choose trigonometric functions to model periodic phenomena and describe the properties of the graphs. Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides


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Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 6: 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, Double Angle, and Half Angle 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, Double Angle, and Half Angle Identities. PA Common Core Standards: CC.2.2.HS.D.1 Interpret the structure of expressions to represent a quantity in terms of its context. CC.2.2.HS.D.2 Write expressions in equivalent forms to solve problems. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions


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Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board


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Unit 8: 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. PA Common Core Standards: CC.2.2.HS.D.9 Use reaoning to solve equations and justify the solution method. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles. Content and Instructional Strategies: Lecture Investigations and Discovery Visual Aids/Power point TI-83 Graphing Calculator display Text-based questions Real life problems/connections Remediation: Websites Re-teaching Activities Extra worksheets Pre-test study guides Enrichment: Enrichment/challenge worksheets Websites Assessment Criteria: Homework – ebook and worksheets Teacher created quizzes, tests, and open-ended questions Project


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Resources and Materials: Chrome Book e-book Graphing Calculators Smart Board Clinometers Measuring Tapes

Honors PreCalculus Pen Argyl Area High School 2017 · Honors PreCalculus Pen Argyl Area High School...

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