Syllabus
Precalculus B Course Overview Calculus is a mathematical study of how quantities change. It is used to solve complex problems that are continuously evolving and would otherwise be unsolvable with only algebra and geometry. Precalculus is a course designed to prepare you for future work in calculus. This course will strengthen your current knowledge of mathematics and build on that knowledge by introducing some more advanced concepts. Precalculus B represents your first introduction to calculus—it gives you the basic tools you need to take a limit and teaches you how to find a limit.
Course Goals By the end of this course, you will be able to do the following: Work with trigonometric identities. State the laws of cosines and sines. Graph vectors. Graph points using polar coordinates. Solve systems of linear equations both algebraically and by using Gauss-Jordan Elimination. Perform addition, subtraction, and multiplication on matrices. Find the inverse of a matrix. Work with conic sections and graph ellipses. Work with parametric equations. Understand sequences and series and how they relate to mathematical induction. Find the limit of a function.
Math Skills Algebra and Geometry are prerequisites for Precalculus. Before beginning this course, you should be able to do the following: Solve problems involving equations, inequalities, and functions. Understand how to visually represent a function with a graph. Work with different types of equations and functions including linear, quadratic, exponential, and absolute value.
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General Skills To participate in this course, you should be able to do the following: Complete basic operations with word processing software, such as Microsoft Word or Google Docs. Understand the basics of spreadsheet software, such as Microsoft Excel or Google Spreadsheets, but having prior computing experience is not necessary. Perform online research using various search engines and library databases. Communicate through email and participate in discussion boards. For a complete list of general skills that are required for participation in online courses, refer to the Prerequisites section of the Plato Student Orientation document, found at the beginning of this course.
Credit Value Precalculus B is a 0.5-credit course.
Course Materials Notebook Graphing calculator, recommend TI-83 or equivalent Computer with Internet connection and speakers or headphones Microsoft Word or equivalent Microsoft Excel or equivalent
Course Pacing Guide This course description and pacing guide is intended to help you keep on schedule with your work. Note that your course instructor may modify the schedule to meet the specific needs of your class.
Unit 1: Trigonometric Identities and Conditional Equations Summary This unit provides the conceptual understanding of trigonometric identities and conditional equations. It begins by examining the basic trigonometric identities and using them for simplifying expressions or solving equations. It explores identities that describe some of the relationships between functions and angles, and then expands on the relationships between multiple angles within trigonometric functions. This unit also looks at the productsum and sum-product identities. The unit concludes by exploring techniques used to solve trigonometric equations.
2
Day
Activity/Objective
Type
1 day: 1
Syllabus and Plato Student Orientation Review the Plato Student Orientation and Course Syllabus at the beginning of this course.
1 day: 2
Basic Trigonometric Identities Examine and apply the basic trigonometric identities.
Tutorial
2 days: 3–4
Sum, Difference, and Co Function Identities Examine and apply the sum, difference, and co function identities.
Tutorial
2 days: 5–6
Double-Angle and Half-Angle Identities Use the double-angle and half-angle identities.
Tutorial
2 days: 7–8
Product-Sum and Sum-Product Identities Examine and apply the product-sum and sum-product identifies.
Tutorial
2 days: 9–10
Solving Trigonometric Equations Solve trigonometric equations.
Tutorial
2 days: 11–12
Unit Activity and Discussion—Unit 1
Unit Activity Discussion
1 day: 13–14
Posttest—Unit 1
Assessment
3
Course Orientation
Unit 2: Applications of Trigonometry Summary This unit covers additional topics in trigonometry and their applications. The unit starts with the laws of sine and cosine and their applications. It then introduces vectors and applications of vectors to real-life problems. This unit also explores the polar coordinate system and shows how to graph complex numbers. It concludes by explaining conversion of complex numbers to trigonometric forms and using DeMoivre's theorem for finding roots of complex numbers. Day
Activity/Objective
Type
2 days: 15–16
The Law of Sines Examine the law of sines.
Tutorial
2 days: 17–18
The Law of Cosines Examine the law of cosines.
Tutorial
2 days: 19–20
Vectors in a Plane Demonstrate an understanding of vectors in a plane.
Tutorial
2 days: 21–22
Vectors and Dot Products Add vectors in a plane and calculate dot products.
Tutorial
2 days: 23–24
Polar Coordinates in Graphs Examine polar coordinates in graphs.
Tutorial
2 days: 25–26
Complex Numbers and DeMoivre's Theorem Perform advanced operations with complex numbers, including the use of DeMoivre’s theorem.
Tutorial
2 days: 27–28
Unit Activity and Discussion—Unit 2
Unit Activity Discussion
1 day: 29
Posttest—Unit 2
Assessment
4
Unit 3: Matrices and Systems of Equations and Inequalities Summary This unit introduces the concept of matrices. It begins by exploring methods of solving both linear and nonlinear systems of equations. The unit then leads into the introduction of matrices as a way to solve for systems of equations. The unit examines the various terms used to describe matrices and their properties, as well as how to add, subtract, and multiply matrices. The unit covers inverse matrices and their real-life applications. It also explores the methods for graphing and solving systems of linear inequalities and concludes with solving linear programming problems. Day
Activity/Objective
Type
2 days: 30–31
Solving Systems of Linear Equations Algebraically Solve systems of linear equations algebraically.
Tutorial
2 days: 32–33
Solving Systems of Nonlinear Equations Algebraically Solve systems of nonlinear equations algebraically.
Tutorial
2 days: 34–35
Gauss-Jordan Elimination Solve systems of linear equations using Gauss-Jordan elimination, with two or three variables.
Tutorial
2 days: 36–37
Matrices and Matrix Operations Perform calculations on matrices using matrix operations.
Tutorial
2 days: 38-39
Matrix Multiplication Perform matrix multiplication.
Tutorial
2 days: 40–41
Solving Systems of Linear Equations with an Inverse Matrix Solve systems of linear equations using an inverse matrix.
Tutorial
2 days: 42–43
Solving Systems of Inequalities with Graphs Find the graphical solutions to a system of inequalities.
Tutorial
2 days: 44–45
Solving Linear Programming Problems Geometrically Use the geometric approach to find the solution to a linear programming problem.
Tutorial
2 days: 46–47
Unit Activity and Discussion—Unit 3
Unit Activity Discussion
1 day: 48
Posttest—Unit 3
Assessment
5
Unit 4: Analytic Geometry: Conic Section Summary This unit introduces and explores the conic sections. The unit starts with the introduction of each conic—the parabola, the circle, the ellipse, and the hyperbola. The unit examines the conic sections, their standard forms, graphs, and real-life applications. Then the unit covers the conics in more depth by explaining the translation and rotation of the coordinate axes. The unit concludes with solving the parametric form of equations and also covers their real-life applications. Day
Activity/Objective
Type
2 days: 49–50
Conic Sections and Parabolas Understand the conic section while exploring parabolas and their graphs.
Tutorial
2 days: 51–52
Ellipses and Circles and Their Graphs Explore ellipses and circles and their graphs.
Tutorial
2 days: 53–54
Hyperbolas and Their Graphs Explore hyperbolas and their graphs.
Tutorial
2 days: 55–56
Translation and Rotation of Axes Examine translation and rotation of axes.
Tutorial
2 days: 57–58
Parametric Equations of Conics Examine parametric equations of conics.
Tutorial
2 days: 59–60
Unit Activity and Discussion—Unit 4
Unit Activity Discussion
1 day: 61
Posttest—Unit 4
Assessment
6
Unit 5: Sequences, Induction, and Probability Summary This unit brings together the topics of sequences, induction, and probability. The unit begins by introducing sequences and series and explores the various types of sequences. The unit then transfers to mathematical induction as a method of finding proofs. Lastly, the unit investigates probability, including permutations and combinations, and basic statistics involved with probability. This unit is important in precalculus because many real-world situations involve sequences and probabilities. Day
Activity/Objective
Type
1 day: 62
Sequences and Series Gain an understanding of sequences and series.
Tutorial
1 day: 63
Arithmetic and Geometric Sequences Explore arithmetic and geometric sequences.
Tutorial
2 days: 64–65
Mathematical Induction Demonstrate an understanding of mathematical induction.
Tutorial
1 day: 66
Counting Techniques Use counting techniques to determine the number of outcomes.
Tutorial
2 days: 67–68
Sample Spaces and Probability Explore sample spaces and probability.
Tutorial
2 days: 69–70
Random Variables Investigate discrete and continuous random variables.
Tutorial
2 days: 71–72
The Normal Curve Use the normal curve to compute probabilities.
Tutorial
2 days: 73–74
The Binomial Formula and Probability Examine the binomial formula and its use in probability.
Tutorial
2 days: 75–76
Unit Activity and Discussion—Unit 5
Unit Activity Discussion
1 day: 77
Posttest—Unit 5
Assessment
7
Unit 6: Limits: Introduction to Calculus Summary This unit serves as a bridge from precalculus to calculus by introducing the fundamental concepts used in calculus, specifically limits, derivatives, and integrals. The unit begins by introducing the concept of a limit, and describes an iterative process to estimate limits using a table of values. The unit then explores limits in more depth, describing some basic arithmetic properties, as well as how to use limits to determine the continuity of a function. Lastly, the unit introduces derivatives by determining the slope of a tangent line at a point, and integrals by finding the area under a curve. Day
Activity/Objective
Type
2 days: 78–79
Limits of Functions Explore the definition of a limit.
Tutorial
2 days: 80–81
Finding the Limit of a Function Examine techniques used to find the limit of a function.
Tutorial
2 days: 82–83
The Tangent Line Problem Explore the tangent line problem.
Tutorial
2 days: 84–85
The Area Under a Curve Explore the area under a curve.
Tutorial
2 days: 86–87
Unit Activity and Discussion—Unit 6
Unit Activity Discussion
1 day: 88
Posttest—Unit 6
Assessment
1 day: 89
Semester Review
1 day: 90
End-of-Semester Test
Assessment
8