Oley Valley School District Planned Course of Instruction Algebra 3/Trigonometry Submitted by: Gary J. McManus June 13, 2016 1
Oley Valley School District - Planned Course Instruction Cover Page Title of Planned Instruction: Algebra 3/Trigonometry Grade: 6-12 Subject area: Mathematics Date: June 13, 3016 Periods per week: 5 Length of period: 41 minutes Length of course: 1 school year Credits: 1 Text(s) and/or major resources required: N/A Names of teacher(s) designing planned course instruction: Gary J. McManus, High School Assistant Principal Approved by: Approval Date Approved by: Curriculum Committee Chair Date Approved by: Superintendent/Assistant Superintendent Date 2
Course: Algebra 3/Trigonometry Unit 1 Functions Primary PDE Standards Assessed in this Unit: 2.4 Mathematical Reasoning and Connections A, B 2.5 Mathematics Problem Solving and Communication A, B 2.8 Algebra and Functions B, C, D, E, F Common Core State Standards Assessed in this Unit: 3
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Unit Overview: The primary focus of this unit is the understanding of functions. Students begin with exploring linear functions and their graphs. They use scatter plots and graphing technology to find linear models for sets of data. Quadratic, polynomial, and rational functions are analyzed, sketched, graphed, and solved. Graphing technology is used to find quadratic models for data. Students extend their knowledge to recognize, evaluate, and graph exponential and logarithmic functions. Properties of logarithms are used to evaluate, rewrite, expand, or condense logarithmic expressions. Exponential growth and decay models are used to solve real-life problems. Students will also explore the concepts of data and curve fitting. 7
Enduring Understandings What big ideas are the focus? What are the essential questions? Functions are widely used to model phenomena in the physical world. Polynomial Functions can be used to model many different types of real world situations in which data changes from increasing to decreasing or from decreasing to increasing more than once. Rational functions are used to model situations involving two variables in which they vary inversely. Exponential and logarithmic functions are valuable when studying and predicting information relating to growth and decay. Assessments to be Used to Address Student Understanding How will you assess the students ability to know/do the big ideas in a meaningful way? Summative/Formative Quizzes Projects Chapter Tests Essential Content Functions and Their Graphs Lines in the Plane Shifting, Reflecting, and Stretching Graphs Inverse Functions Data Exploration Polynomial and Rational Functions Zeros of polynomial functions Rational Functions and Asymptotes Graphs of Rational Functions Quadratic Models Exponential and Logarithmic Functions Properties of Functions and Their Graphs Solving Exponential and Logarithmic Equations Exponential, Logarithmic, and Nonlinear Models 8
Course: Trigonometry/Analysis Unit 2 Trigonometry Primary PDE Standards Assessed in this Unit: 2.4 Mathematical Reasoning and Connections A, B 2.5 Mathematics Problem Solving and Communication A, B 2.8 Trigonometry and Functions C, D, E, F 2.11 Concepts of Calculus A Common Core State Standards Assessed in this Unit: 9
Unit Overview: In this unit, students will be introduced to trigonometry. At first, they will convert between degree and radian measures, find arc lengths and areas of sectors of circles, and investigate, evaluate, and graph the six trigonometric functions and the inverse trigonometric functions. They will then solve trigonometric equations and apply them to problems from analytic geometry. They will explore changes in the sine and cosine curves and use these changes to develop models of real-world periodic phenomena. They will also prove a variety of trigonometric identities and use them to simplify and solve more difficult trigonometric expressions and equations. Ultimately, they will use trigonometry to find the unknown sides or angles of a triangle, develop a formula to find the area of a triangle in terms of the sine of one of the angles, and use the law of sines and the law of cosines to solve problems from navigation and surveying. 10
Enduring Understandings An angle is represented as a rotation about a point and is measured in degrees or radians. The sine and cosine functions are defined in terms of the coordinates of a point on a circle. The definitions of the other trig functions involve restrictions on either the x or y coordinate of a point on the terminal ray of an angle. Trig functions are periodic and can experience transformations. An identity is true for all values of the variable for which the statement is meaningful. The Law of Sines and the Law of Cosines can be used to find the lengths of sides and the measures of angles in a triangle. In surveying applications, the angles measures are east or west of the north-south line. In navigation applications, the bearing of the ship or airplane is measured clockwise from the northsouth line. Assessments to be Used to Address Student Understanding How will you assess the students ability to know/do the big ideas in a meaningful way? Summative/Formative Quizzes Projects Chapter Tests Essential Content Trigonometric Functions Measurement of Angles Sectors of Circles The Sine and Cosine Functions Evaluating and Graphing Sine and Cosine The Other Four Trig Functions The Inverse Trig Functions Trig Equations and Applications Solving Simple Trig Equations Sine and Cosine Curves Modeling Periodic Behavior Relationships Among the Functions Solving More Difficult Trig Equations Triangle Trigonometry Solving Right Triangles The Area of a Triangle The Law of Sines The Law of Cosines Applications of Trig to Navigation and Surveying 11
Course: Trigonometry/Analysis Unit 3 Analytic Geometry, Sequences, and Series Primary PDE Standards Assessed in this Unit: 2.4 Mathematical Reasoning and Connections A, B 2.5 Mathematics Problem Solving and Communication A, B 2.7 Probability and Predictions A, C, E 2.8 Algebra and Functions B, C, D, E, F Common Core State Standards Assessed in this Unit: 12
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Unit Overview: This unit begins with an exploration of conic sections. The equations for conic sections are derived from the definition of conic sections and from the distance formula. Students then use rectangular equations to graph the conic sections. The terms and sums of sequences are written, and students learn to recognize, write, and use arithmetic and geometric sequences. Mathematical induction is used to prove statements involving a positive integer n. The Binomial Theorem and Pascal s Triangle are used to calculate binomial coefficients and write binomial expansions. Counting problems are solved using the Fundamental Counting Principle, permutations, and combinations. The unit concludes by finding the probabilities of events and their complements. 14
Enduring Understandings What big ideas are the focus? What are the essential questions? Conic sections can be applied to real world applications (i.e. navigation, astronomy, communications). Sequences and Series can be described in terms of arithmetic and geometric types. They are useful in modeling sets of values in order to identify a pattern. Assessments to be Used to Address Student Understanding How will you assess the students ability to know/do the big ideas in a meaningful way? Summative/Formative Quizzes Projects Chapter Tests Essential Content Analytic Geometry Equations of Circles Ellipses Hyperbolas Parabolas Sequences, Series, and Probability Arithmetic and Geometric Sequences and Series Mathematical Induction and Counting Principles Probability 15
Course: Trigonometry/Analysis Unit 4 PreCalculus Primary PDE Standards Assessed in this Unit: 2.11 Concepts of Calculus A, B, C Common Core State Standards Assessed in this Unit: n/a Unit Overview: The main focus of this unit is to give students a first look at concepts of Calculus with the topic of limits. Students use the definition and properties of limits and direct substitution to evaluate limits. Students are given an in-depth look at the techniques used for evaluating limits. Tangent lines, limits at infinity, and sequences conclude this unit of study. 16
Enduring Understandings What big ideas are the focus? What are the essential questions? The notion of a limit is a fundamental concept of Calculus. Techniques for evaluating limits such as division and rationalization are used. Limits are useful in applications involving maximization and positions functions. Assessments to be Used to Address Student Understanding How will you assess the students ability to know/do the big ideas in a meaningful way? Summative/Formative Quizzes Projects Chapter Tests Essential Content Limits and an Introduction to Calculus Introduction to and Techniques for Evaluating Limits The Tangent Line Problem Limits at Infinity 17
PDE Academic Standards Course: Trigonometry/Analysis 2.4.11.A: Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use of counter-examples, truth tables, etc.) to validate conjectures or arguments. 2.4.11.B: Use statements, converses, inverses, and contrapositives to construct valid arguments or to validate arguments. 2.5.11.A: Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts. 2.5.11.B: Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas, and results. 2.7.11.A: Use probability to predict the likelihood of an outcome in an experiment. 2.7.11.C: Compare odds and probability. 2.7.11.E: Use probability to make judgments about the likelihood of various outcomes. 18
2.8.11.B: Evaluate and simplify algebraic expressions and solve and graph linear, quadratic, exponential, and logarithmic equations and inequalities, and solve and graph systems of equations and inequalities. 2.8.11.C: Recognize, describe, and generalize patterns using sequences and series to predict long-term outcomes. 2.8.11.D: Demonstrate an understanding and apply properties of functions (domain, range, inverses) and characteristics of families of functions (linear, polynomial, rational, trigonometric, exponential, logarithmic). 2.8.11.E: Use combinations of symbols and numbers to create expressions, equations, and inequalities in two or more variables, systems of equations and inequalities, and functional relationships that model problem situations. 2.8.11.F: Interpret the results of solving equations, inequalities, systems of equations, and inequalities in the context of the situation that motivated the model. 2.11.11.A: Determine and interpret maximum and minimum values of a function over a specified interval. 2.11.11.B: Analyze and interpret rates of growth/decay. 2.11.11.C: Estimate areas under curves using sums of areas. 19