Precalculus A - Greenways Academy
Syllabus
Precalculus A 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. For example, this is the first time you’ll learn about numbers outside the real number system, and you’ll begin to treat functions and polynomials like numbers—plugging them into equations as you would a number.
Course Goals By the end of this course, you will be able to do the following: Work with linear and quadratic equations and inequalities. Define and work with complex numbers. Graph linear equations. Transform functions. Graph special types of functions, such as piecewise functions. Work with linear, quadratic, inverse, exponential, logarithmic, rational, and polynomial functions. Work with trigonometric functions and the unit circle. Graph trigonometric functions and their inverses.
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.
1 © 2013 EDMENTUM, INC.
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 A 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: Equations and Inequalities Summary This unit provides the conceptual understanding of equations and inequalities. It begins by identifying different types of equations and inequalities and methods of solving them. It then explores linear equations in one and two variables and how to solve for a given variable. This unit also includes inequalities and how to solve for and represent their answers. The unit concludes by exploring absolute value equations and complex numbers and the correct steps to solve for them, as well as the three equation types, radical, rational, and power equations. This unit is essential in precalculus because it is a building block for later units.
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
Linear Equations Simplify and solve linear equations.
Tutorial
1 day: 3
Problem Solving Examine and apply problem-solving techniques.
Tutorial
1 day: 4
Linear Inequalities Solve linear inequalities and write their solutions using various notations.
Tutorial
2 days: 5–6
Equations and Inequalities with Absolute Value Solve equations and inequalities that involve absolute value
Tutorial
2 days: 7–8
Quadratic Equations and Inequalities Solve quadratic equations and inequalities.
Tutorial
2 days: 9–10
Complex Numbers Define and work with complex numbers.
Tutorial
2 days: 11–12
Other Types of Equations Solve other types of equations, including those involving radicals and power functions.
Tutorial
2 days: 13–14
Unit Activity and Discussion—Unit 1
Unit Activity Discussion
1 day: 15
Posttest—Unit 1
Assessment
3
Course Orientation
Unit 2: Graphs Summary This unit provides a basic understanding and application of graphs. The unit starts with the coordinate system and reviews how to graph equations. It shows how, given two graphed points, you can find the distance, midpoint, and equations of lines. It goes on to explain linear equations and their applications in-depth. It concludes by explaining linear regression, which allows for multiple points of data to be included when finding a linear equation that represents a set of data. Day
Activity/Objective
Type
1 day: 16
Graphing Points and Equations Use the Cartesian coordinate system to graph points and equations.
Tutorial
2 days: 17–18
Distances and Circles in the Coordinate Plane Calculate distances, compute midpoints, and graph circles in the coordinate plane.
Tutorial
2 days: 19–20
Slope and Intercept in Linear Equations Find and compare the slope and intercept of linear equations.
Tutorial
2 days: 21–22
Graphing Linear Equations Graph linear equations based on information about points on the line, other parallel lines, or slope and y-intercept and use graphs to solve real-world problems.
Tutorial
2 days: 23–24
Using Linear Models to Solve Problems Use linear models to solve problems.
Tutorial
2 days: 25–26
Regression Lines and Correlation Coefficients Find linear regression lines and correlation coefficients.
Tutorial
2 days: 27–28
Unit Activity and Discussion—Unit 2
Unit Activity Discussion
1 day: 29
Posttest—Unit 2
Assessment
4
Unit 3: Functions and Their Graphs Summary This unit introduces the concept of functions, a core concept in precalculus. The rest of the course will be based on functions and function notation. The unit begins by describing functions in terms of their graphs and how transformations produce changes in the graphs. The unit shows how to combine functions and find inverse functions. It also presents many real-world applications of functions, showing how to represent them as functions and solve them both mathematically and graphically.
Day
Activity/Objective
Type
2 days: 30–31
Introduction to Functions and Function Notation Define functions, identify types of functions, and use function notation.
Tutorial
2 days: 32–33
Graphs of Functions Analyze the graphs of functions.
Tutorial
2 days: 34–35
Piecewise Functions Analyze the graphs of piecewise functions.
Tutorial
2 days: 36–37
Transformation of Functions Investigate transformations of functions.
Tutorial
2 days: 38–39
Combining Functions Combine functions to form new functions.
Tutorial
2 days: 40–41
Quadratic Functions Graph, model, and solve quadratic functions.
Tutorial
2 days: 42–43
Inverse Functions Define and use inverse functions.
Tutorial
2 days: 44–45
Unit Activity and Discussion—Unit 3
Unit Activity Discussion
1 day: 46
Posttest—Unit 3
Assessment
5
Unit 4: Polynomial and Rational Functions Summary This unit is important in precalculus because it extends your previous knowledge of quadratic functions into the realm of polynomial and rational functions, whose solutions require a variety of methods of factoring and finding roots. It then explains how to graph these functions, both by hand and with the use of a calculator. This unit concludes with how to solve polynomial and rational inequalities. This unit will allow you to work through many real-world examples based on polynomial and rational functions. Day
Activity/Objective
Type
2 days: 47–48
Polynomial Functions Examine polynomial functions.
Tutorial
2 days: 49–50
Graphs of Polynomial Functions Examine graphs of polynomial functions.
Tutorial
2 days: 51–52
Graphing Polynomial Functions Graph polynomial functions.
Tutorial
2 days: 53–54
Rational Functions Investigate rational functions.
Tutorial
2 days: 55–56
Polynomial and Rational Inequalities Examine polynomial and rational inequalities.
Tutorial
2 days: 57–58
Further Analysis of Rational and Polynomial Functions Use partial fraction decomposition and examine real and complex solutions of rational and polynomial functions.
2 days: 59–60
Unit Activity and Discussion—Unit 4
Unit Activity Discussion
1 day: 61
Posttest—Unit 4
Assessment
6
Unit 5: Exponential and Logarithmic Functions Summary This unit is important in precalculus because many real-world problems can be explained and solved using exponential and logarithmic functions. The unit begins by introducing the basic forms of these functions and explaining how to solve for and graph them. It then discusses how to apply the functions to real-world situations and solve them. Overall, this unit covers how to recognize, write, and graph various exponential and logarithmic functions and use their properties to manipulate expressions and solve equations. Day
Activity/Objective
Type
2 days: 62–63
Exponential Functions Examine exponential functions.
Tutorial
2 days: 64–65
Exponential Models Investigate exponential models.
Tutorial
2 days: 66–67
Logarithmic Functions Examine logarithmic functions.
Tutorial
2 days: 68–69
Logarithmic Models Investigate logarithmic models
Tutorial
1 day: 70
Solving Exponential and Logarithmic Functions Solve exponential and logarithmic functions.
Tutorial
2 days: 71–72
Unit Activity and Discussion—Unit 5
Unit Activity Discussion
1 day: 73
Posttest—Unit 5
Assessment
7
Unit 6: Trigonometric Functions Summary In this unit, you will learn about one of the key concepts in precalculus, trigonometry. This unit introduces the trigonometric functions by describing the interrelationships between the unit circle, angles, and their relationship to the Cartesian plane. This unit begins by reviewing some basic concepts relating to angles and expands to include the unit circle and trigonometric functions. It then covers the definition, description, and illustration of the key concepts of the trigonometric functions, including their relationship to angles, the unit circle, right triangles, graphs, periodic functions, and inverse trigonometric functions. Day
Activity/Objective
Type
2 days: 74–75
Angles and Their Measures Examine angles and their measures.
Tutorial
2 days: 77–77
Trigonometric Functions and the Unit Circle Examine trigonometric functions using a unit circle.
Tutorial
2 days: 78–79
Problems Involving Right Triangles Find unknown measurements in right triangles.
Tutorial
2 days: 80–81
Trigonometric Functions Examine trigonometric functions.
Tutorial
2 days: 82–83
Trigonometric Graphs Examine trigonometric graphs.
Tutorial
2 days: 84–85
Inverse Trigonometric Functions Examine inverse trigonometric functions.
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
Precalculus A 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. For example, this is the first time you’ll learn about numbers outside the real number system, and you’ll begin to treat functions and polynomials like numbers—plugging them into equations as you would a number.
Course Goals By the end of this course, you will be able to do the following: Work with linear and quadratic equations and inequalities. Define and work with complex numbers. Graph linear equations. Transform functions. Graph special types of functions, such as piecewise functions. Work with linear, quadratic, inverse, exponential, logarithmic, rational, and polynomial functions. Work with trigonometric functions and the unit circle. Graph trigonometric functions and their inverses.
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.
1 © 2013 EDMENTUM, INC.
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 A 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: Equations and Inequalities Summary This unit provides the conceptual understanding of equations and inequalities. It begins by identifying different types of equations and inequalities and methods of solving them. It then explores linear equations in one and two variables and how to solve for a given variable. This unit also includes inequalities and how to solve for and represent their answers. The unit concludes by exploring absolute value equations and complex numbers and the correct steps to solve for them, as well as the three equation types, radical, rational, and power equations. This unit is essential in precalculus because it is a building block for later units.
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
Linear Equations Simplify and solve linear equations.
Tutorial
1 day: 3
Problem Solving Examine and apply problem-solving techniques.
Tutorial
1 day: 4
Linear Inequalities Solve linear inequalities and write their solutions using various notations.
Tutorial
2 days: 5–6
Equations and Inequalities with Absolute Value Solve equations and inequalities that involve absolute value
Tutorial
2 days: 7–8
Quadratic Equations and Inequalities Solve quadratic equations and inequalities.
Tutorial
2 days: 9–10
Complex Numbers Define and work with complex numbers.
Tutorial
2 days: 11–12
Other Types of Equations Solve other types of equations, including those involving radicals and power functions.
Tutorial
2 days: 13–14
Unit Activity and Discussion—Unit 1
Unit Activity Discussion
1 day: 15
Posttest—Unit 1
Assessment
3
Course Orientation
Unit 2: Graphs Summary This unit provides a basic understanding and application of graphs. The unit starts with the coordinate system and reviews how to graph equations. It shows how, given two graphed points, you can find the distance, midpoint, and equations of lines. It goes on to explain linear equations and their applications in-depth. It concludes by explaining linear regression, which allows for multiple points of data to be included when finding a linear equation that represents a set of data. Day
Activity/Objective
Type
1 day: 16
Graphing Points and Equations Use the Cartesian coordinate system to graph points and equations.
Tutorial
2 days: 17–18
Distances and Circles in the Coordinate Plane Calculate distances, compute midpoints, and graph circles in the coordinate plane.
Tutorial
2 days: 19–20
Slope and Intercept in Linear Equations Find and compare the slope and intercept of linear equations.
Tutorial
2 days: 21–22
Graphing Linear Equations Graph linear equations based on information about points on the line, other parallel lines, or slope and y-intercept and use graphs to solve real-world problems.
Tutorial
2 days: 23–24
Using Linear Models to Solve Problems Use linear models to solve problems.
Tutorial
2 days: 25–26
Regression Lines and Correlation Coefficients Find linear regression lines and correlation coefficients.
Tutorial
2 days: 27–28
Unit Activity and Discussion—Unit 2
Unit Activity Discussion
1 day: 29
Posttest—Unit 2
Assessment
4
Unit 3: Functions and Their Graphs Summary This unit introduces the concept of functions, a core concept in precalculus. The rest of the course will be based on functions and function notation. The unit begins by describing functions in terms of their graphs and how transformations produce changes in the graphs. The unit shows how to combine functions and find inverse functions. It also presents many real-world applications of functions, showing how to represent them as functions and solve them both mathematically and graphically.
Day
Activity/Objective
Type
2 days: 30–31
Introduction to Functions and Function Notation Define functions, identify types of functions, and use function notation.
Tutorial
2 days: 32–33
Graphs of Functions Analyze the graphs of functions.
Tutorial
2 days: 34–35
Piecewise Functions Analyze the graphs of piecewise functions.
Tutorial
2 days: 36–37
Transformation of Functions Investigate transformations of functions.
Tutorial
2 days: 38–39
Combining Functions Combine functions to form new functions.
Tutorial
2 days: 40–41
Quadratic Functions Graph, model, and solve quadratic functions.
Tutorial
2 days: 42–43
Inverse Functions Define and use inverse functions.
Tutorial
2 days: 44–45
Unit Activity and Discussion—Unit 3
Unit Activity Discussion
1 day: 46
Posttest—Unit 3
Assessment
5
Unit 4: Polynomial and Rational Functions Summary This unit is important in precalculus because it extends your previous knowledge of quadratic functions into the realm of polynomial and rational functions, whose solutions require a variety of methods of factoring and finding roots. It then explains how to graph these functions, both by hand and with the use of a calculator. This unit concludes with how to solve polynomial and rational inequalities. This unit will allow you to work through many real-world examples based on polynomial and rational functions. Day
Activity/Objective
Type
2 days: 47–48
Polynomial Functions Examine polynomial functions.
Tutorial
2 days: 49–50
Graphs of Polynomial Functions Examine graphs of polynomial functions.
Tutorial
2 days: 51–52
Graphing Polynomial Functions Graph polynomial functions.
Tutorial
2 days: 53–54
Rational Functions Investigate rational functions.
Tutorial
2 days: 55–56
Polynomial and Rational Inequalities Examine polynomial and rational inequalities.
Tutorial
2 days: 57–58
Further Analysis of Rational and Polynomial Functions Use partial fraction decomposition and examine real and complex solutions of rational and polynomial functions.
2 days: 59–60
Unit Activity and Discussion—Unit 4
Unit Activity Discussion
1 day: 61
Posttest—Unit 4
Assessment
6
Unit 5: Exponential and Logarithmic Functions Summary This unit is important in precalculus because many real-world problems can be explained and solved using exponential and logarithmic functions. The unit begins by introducing the basic forms of these functions and explaining how to solve for and graph them. It then discusses how to apply the functions to real-world situations and solve them. Overall, this unit covers how to recognize, write, and graph various exponential and logarithmic functions and use their properties to manipulate expressions and solve equations. Day
Activity/Objective
Type
2 days: 62–63
Exponential Functions Examine exponential functions.
Tutorial
2 days: 64–65
Exponential Models Investigate exponential models.
Tutorial
2 days: 66–67
Logarithmic Functions Examine logarithmic functions.
Tutorial
2 days: 68–69
Logarithmic Models Investigate logarithmic models
Tutorial
1 day: 70
Solving Exponential and Logarithmic Functions Solve exponential and logarithmic functions.
Tutorial
2 days: 71–72
Unit Activity and Discussion—Unit 5
Unit Activity Discussion
1 day: 73
Posttest—Unit 5
Assessment
7
Unit 6: Trigonometric Functions Summary In this unit, you will learn about one of the key concepts in precalculus, trigonometry. This unit introduces the trigonometric functions by describing the interrelationships between the unit circle, angles, and their relationship to the Cartesian plane. This unit begins by reviewing some basic concepts relating to angles and expands to include the unit circle and trigonometric functions. It then covers the definition, description, and illustration of the key concepts of the trigonometric functions, including their relationship to angles, the unit circle, right triangles, graphs, periodic functions, and inverse trigonometric functions. Day
Activity/Objective
Type
2 days: 74–75
Angles and Their Measures Examine angles and their measures.
Tutorial
2 days: 77–77
Trigonometric Functions and the Unit Circle Examine trigonometric functions using a unit circle.
Tutorial
2 days: 78–79
Problems Involving Right Triangles Find unknown measurements in right triangles.
Tutorial
2 days: 80–81
Trigonometric Functions Examine trigonometric functions.
Tutorial
2 days: 82–83
Trigonometric Graphs Examine trigonometric graphs.
Tutorial
2 days: 84–85
Inverse Trigonometric Functions Examine inverse trigonometric functions.
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
