RWM102: Algebra « Saylor.org – Free Online Courses Built by Professors

Algebra

Purpose of Course show close

This introductory mathematics course is for you if you have a solid foundation in arithmetic (that is, you know how to perform operations with real numbers, including negative numbers, fractions, and decimals). Numbers and basic arithmetic are used often in everyday life in both simple situations, like estimating how much change you will get when making a purchase in a store, as well as in more complicated ones, like figuring out how much time it would take to pay off a loan under interest. The subject of algebra focuses on generalizing these procedures. For example, algebra will enable you to describe how to calculate change without specifying how much money is to be spent on a purchase--it will teach you the basic formulas and steps you need to take no matter what the specific details of the situation are. Likewise, accountants use algebraic formulas to calculate the monthly loan payments for a loan of any size under any interest rate. In this course, you will learn how to work with formulas that are already known from science or business to calculate a given quantity, and you will also learn how to set up your own formulas to describe various situations by translating verbal descriptions to mathematical language. In the later units of this course, you will discover another tool used in mathematics to describe numbers and analyze relationships: graphing. You will learn that any pair of numbers can be represented by a point on a coordinate plane and that a relationship between two quantities can be represented by a line or a curve. Units 6, 7, and 8 may seem more abstract than the earlier ones, as you will deal with expressions that contain mostly variables and not too many numbers. While the procedures you will master in these units might seem to have little practical application, you have to keep in mind that they result in formulas that describe very real situations in business, accounting, and science. Knowing how to perform various operations with algebraic expressions will eventually enable you to solve quadratic and even more complex equations. You will explore a variety of real-world scenarios that can be described by these kinds of equations. For example, if a ball is thrown up in the air, solving a quadratic equation will help you find out when it will hit the ground. As another example, if you know the area of a rectangular garden, then you can use a quadratic equation to find the length of each side.

Course Information show close

Welcome to RWM102. General information on this course and its requirements can be found below.

Course Designer: Inna Shpiro

Primary Resources: This course comprises a range of different free, online materials. However, the course makes primary use of the following materials:

CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra and Basic Algebra Concepts
CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts
Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra
Khan Academy’s “Equation Basics”
Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers”
Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications II: Solving Problems”
Dr. James Sousa’s MathIsPower4U Algebra Videos

Please note that the CK-12 resources require you to sign in with Google, Facebook, Twitter, or a CK-12 account. You may set up a CK-12 account for free by clicking on “Create an Account” and following the steps provided.

Also, please note that some webpages might take some time to load, as many of the webpages contain a lot of mathematical symbols.

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will need to complete the activities in each unit as well as the Final Exam. You will also need to complete:

Sub-subunit 2.2.5 Assessment
Sub-subunit 2.3.2 Assessment
Subunit 4.4 Assessment
Sub-subunit 5.5.2 Assessment
Sub-subunit 6.4.2 Assessment
Subunit 7.3 Assessment
Sub-subunit 8.3.2.3 Assessment
Sub-subunit 8.4.2 Assessment
Sub-subunit 9.5.2 Assessment

Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through the resources, activities, and assessments in each unit.

In order to pass this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.

Time Commitment: This course should take you a total of 131.25 hours to complete. Each unit includes a time advisory that lists the amount of time you are expected to spend on each subunit. These should help you plan your time accordingly. It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take you 6.75 hours. Perhaps you can sit down with your calendar and decide to complete subunits 1.1 and 1.2 (a total of 3.5 hours) on Monday night; subunit 1.3 (a total of 3.25 hours) on Tuesday night; etc.

Tips/Suggestions: As noted in the “Course Requirements” section, there is a prerequisite for this course. It may be helpful to review RWM101: Foundations before you begin this course.

Please make sure to take comprehensive notes as you work through each resource. Mark down any important equations, formulas, and definitions that stand out to you. These notes will serve as a useful review as you study for your Final Exam.

Common Core Standards: In addition to listing course-wide and unit-specific learning outcomes, this course also makes note of all Common Core Standards that are covered by the learning material. Notations of these standards appear next to the learning outcomes with which they are aligned (i.e., A-SSE.1).

Educational standards comprise the set of skills and concepts the students are to master at any given level. They provide goals for the educators and expectations for the students.

Information on Common Core Standards can be found on Common Core Standards Initiative website: http://www.corestandards.org/the-standards.

High School Algebra Standards (with the description of each standard) can be found at http://www.corestandards.org/Math/Content/HSA/.

This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.

Learning Outcomes show close

NOTE: Each of the learning outcomes listed below has been aligned with one or more of the Common Core standards in mathematics. This alignment is reflected in the numbered notation listed alongside each outcome below. For more information on Common Core standards, please read here.

Upon successful completion of this course, you will be able to:

evaluate and simplify algebraic expressions; (HSA-SSE.A.1)
solve linear equations and inequalities in one variable; (HSA-REI.B.3)
solve systems of linear equations and inequalities; (HSA-REI.C.5, HSA-REI.C.6, HSA-REI.D.12)
solve literal equations for a given variable; (HSA-CED.A.4, HSA-REI.B.3)
translate verbal phrases to algebraic (variable) expressions;
define percent, and solve basic percent problems;
apply simple interest formula to problems involving loans and saving accounts;
apply uniform motion formula to problems involving motion of one or two objects;
solve word problems by identifying a variable and creating an equation or an inequality; (HSA-CED.A.1)
solve word problems by identifying two or more variables and creating a system of equations or inequalities; (HSA-CED.A.3)
plot points on the coordinate plane, and determine the coordinates of any point on the coordinate plane;
graph linear equations and inequalities in two variables on the coordinate plane; (HSA-CED.A.2, HSA-REI.D.10, HSA-REI.D.12)
calculate a slope of a line passing through two given points;
write an equation of a straight line in point-slope or slope-intercept form;
solve word problems by creating a graph of a straight line and interpreting the meaning of the slope and intercepts of the line in the context of a problem; (HSA-CED.A.2)
perform operations with algebraic exponential expressions using the rules of exponents;
perform operations with polynomials; (HSA-APR.A.1)
identify polynomials that can be factored and determine appropriate factoring strategy; and (HSA-SSE.A.2)
solve quadratic equations by factoring. (HSA-SSE.B.3, HSA-REI.B.4)

Course Requirements show close

In order to take this course, you must:

√ have access to a computer;

√ have continuous broadband Internet access;

√ have the ability/permission to install plug-ins or software (e.g. Adobe Reader or Flash);

√ have the ability to download and save files and documents to a computer;

√ have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.);

√ have competency in the English language;

√ have read the Saylor Student Handbook; and

√ have completed RWM101: Real World Math I: Foundations.

Unit Outline show close

1 Variables and Variable Expressions
1.1 Definition and Meaning of Variable Expressions
1.1.1 Variables and Variable Expressions
1.1.2 Constant Terms and Coefficients
1.2 Evaluating Variable Expressions
1.2.1 Replacing Variables with Their Values
1.2.2 Using Order of Operations Agreement to Simplify the Resulting Numerical Expression
1.3 Simplifying Variable Expressions
1.3.1 Properties of Real Numbers
1.3.2 Combining Like Terms
1.3.3 Removing the Parentheses
1.3.4 Simplifying General Variable Expressions

2 Linear Equations
2.1 Introduction to Equations
2.1.1 Definition of an Equation and a Solution of an Equation
2.1.2 Addition/Subtraction Property of Equations
2.1.3 Multiplication/Division Property of Equations
2.2 Solving Equations
2.2.1 Basic One-Step Equations
2.2.2 Equations with Variables on One Side
2.2.3 Equations of the Form ax + b = cx + d
2.2.4 Equations Containing Parentheses
2.2.5 Classifying Equations According to the Number of Solutions: Identities and Contradictions
2.3 Literal Equations
2.3.1 Solving Literal Equation for One of the Variables
2.3.2 Formulas

3 Word Problems
3.1 Translating English to Math
3.1.1 Mathematical Symbols and Expressions for Common Words and Phrases
3.1.2 Translating Verbal Expression into Mathematical
3.2 Translating Sentences into Equations
3.2.1 Number Problems
3.2.2 Consecutive Integer Problems
3.2.3 General Statement Problems
3.3 Basic Uniform Motion Problems
3.3.1 Applying Uniform Motion Equation
3.3.2 Problems Involving Objects Moving in Opposite Directions
3.3.3 Uniform Motion Problems
3.4 Creating Equations to Solve Problems
3.4.1 Value Mixture Problems
3.4.2 Percent Mixture Problems
3.4.3 More Uniform Motion Problems

4 Inequalities
4.1 Definition and Notation of an Inequality
4.2 Graphing Inequalities on a Number Line
4.3 Solving Linear Inequalities
4.3.1 Solving One-Step Inequalities
4.3.2 Solving Multi-Step Inequalities
4.4 Application Problem

5 Graphs of Linear Equations and Inequalities
5.1 Graphing Points in the Rectangular Coordinate Plane
5.2 Linear Equations in Two Variables
5.2.1 Ordered Pairs as Solutions of an Equation in Two Variables
5.2.2 Graphing Equations in Two Variables of the Form y = mx + b
5.2.3 Graphing Equations in Two Variables of the Form Ax + By = C
5.2.4 Graphs of the Equations in the Forms x = C and y = C
5.3 Slopes and Intercepts of a Straight Line
5.3.1 Intercepts of a Straight Line
5.3.2 Slope of a Straight Line
5.3.3 Applications: Real-World Interpretations of Slope and Intercepts
5.4 Equations of Straight Line
5.4.1 Slope-Intercept Form
5.4.2 Point-Slope Form
5.4.3 Standard Form
5.4.4 Writing the Equation of a Line Passing through a Given Point, Parallel or Perpendicular to a Given Line
5.4.5 Application Problems
5.5 Linear Inequalities in Two Variables
5.5.1 Graphing Linear Inequality of Two Variables on the Coordinate Plane
5.5.2 Application Problems

6 Systems of Linear Equations and Inequalities
6.1 Solving Systems of Linear Equations
6.1.1 Solution of a System of Linear Equations
6.1.2 Solving Systems of Linear Equations by Graphing
6.1.3 Solving Systems of Linear Equations Using the Substitution Method
6.1.4 Solving Systems of Linear Equations Using the Elimination Method
6.1.5 Strategy for Solving Systems of Linear Equations: Choosing a Method
6.1.6 Classifying Systems by the Number of Solutions
6.2 Solving Word Problems by Using Systems of Equations
6.3 Systems of Linear Inequalities
6.3.1 Graphing Systems of Linear Inequalities
6.3.2 Applications of Systems of Linear Inequalities

7 Operations with Monomials
7.1 Algebraic Exponential Expressions
7.2 Rules of Exponents
7.2.1 Product of Exponents, Power of Exponent, and Power of a Product
7.2.2 Quotient of Exponents and Power of a Quotient
7.2.3 Negative Exponents
7.3 Simplifying Power Expressions
7.4 Multiplying and Dividing Monomials
7.4.1 Multiplying Monomials
7.4.2 Monomial Division

8 Operations with Polynomials
8.1 Classification of Polynomials
8.2 Addition and Subtraction of Polynomials
8.2.1 Horizontal Format
8.2.2 Vertical Format
8.3 Multiplying Polynomials
8.3.1 Multiplying Polynomial by a Monomial
8.3.2 Multiplying a Polynomial by a Polynomial
8.4 Dividing Polynomials
8.4.1 Dividing a Polynomial by a Monomial
8.4.2 Dividing a Polynomial by a Binomial

9 Factoring Polynomials
9.1 Factoring Monomial from a Polynomial
9.1.1 Identifying Greatest Common Factor (GCF) of Two or More Monomials and of Other Expressions
9.1.2 Rewriting Polynomial as a Product of a Monomial and a Different Polynomial
9.2 Factoring Polynomials by Grouping
9.2.1 Factoring out a Common Binomial Factor
9.2.2 Factoring Four-Term Polynomials by Grouping
9.3 Factoring Quadratic Trinomials
9.3.1 Factoring Trinomials of the Form x^2 + bx + c
9.3.2 Factoring Trinomials of the Form ax^2 + bx + c
9.3.3 Factoring Trinomials of the Form x^2 + bxy + cy^2 and ax^2 + bxy + cy^2
9.3.4 Prime Trinomials
9.4 Special Factoring
9.4.1 Identifying and Factoring Complete Square Trinomial
9.4.2 Identifying and Factoring Difference of Two Squares
9.4.3 Identifying and Factoring Sum and Difference of Two Cubes
9.5 Factoring General Polynomials
9.5.1 Choosing a Strategy for Factoring a Polynomial
9.6 Solving Quadratic Equations by Factoring
9.6.1 Principle of Zero Products and Identifying Solutions
9.6.2 Factoring before Solving
9.6.3 Solving Application Problems

10 Final Exam

Expand All Resources Collapse All Resources

Unit 1: Variables and Variable Expressions

When most people think of math, they think of numbers. However, with this math course, you will be working with letters. In algebra, letters are used to represent numbers. These letters are called variables, because the numbers they represent may vary. For example, if you are paid $10 per hour, your salary can be expressed as 10×h, where h is the number of hours you have worked. You can change the number that letter h stands for according to your particular case in order to calculate your salary.

In this unit, you will learn that the properties of numbers apply to letters as well, and you can use them to work with expressions containing variables. One of the main skills you have to master in this unit is recognizing like terms, because you can add and subtract them as if they were numbers.

Unit 1 Time Advisory show close

Unit 1 Learning Outcomes show close

1.1 Definition and Meaning of Variable Expressions

1.1.1 Variables and Variable Expressions

Lecture: Khan Academy’s “What Is a Variable?”
Link: Khan Academy’s “What Is a Variable?” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan uses an everyday situation to explain how variables are used to represent numbers.

Watching this video and pausing to take notes should take approximately 15 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Variable Expressions”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Variable Expressions” (HTML)

Instructions: Please click on the link above, read the article, and complete practice problems 1–10. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” This article further explains the meaning and usage of variables. It also provides many examples of using variables to represent quantities that you often encounter in the real world as well as exercises to practice choosing appropriate variables and writing variable expressions to describe situations on your own. After you solve the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Variable Expressions’ Problems 1–10” (PDF).

Reading and solving these practice problems should take approximately 30 minutes.

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1.1.2 Constant Terms and Coefficients

Reading: All Math Words Encyclopedia: David McAdams’s “Term”
Link: All Math Words Encyclopedia: David McAdams’s “Term” (HTML)

Instructions: Please click on the link above, read the brief encyclopedia entry, and complete the “Understanding Check” problems. You will receive immediate feedback on whether your response is correct or incorrect when you choose an answer choice. Note that this reading will also cover the topic outlined for sub-subunit 1.3.2.1. You will be using the terminology introduced here throughout the rest of this course.

Reading this section and solving the practice problems should take approximately 15 minutes.

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Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Identifying Variable Parts and Coefficients of Terms”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Identifying Variable Parts and Coefficients of Terms” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This article points out various conventions used in algebra when writing variable expressions, such as always putting a numerical coefficient first and omitting 1 in front of a variable. Use the examples provided as a guide to practice identifying different parts of algebraic expressions. Click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 30 minutes.

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1.2 Evaluating Variable Expressions

1.2.1 Replacing Variables with Their Values

Lecture: Khan Academy’s “Why Aren’t We Using the Multiplication Sign?”
Link: Khan Academy’s “Why Aren’t We Using the Multiplication Sign?” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video gives a very detailed explanation of how to substitute the values of variables in the algebraic expressions. Note how the concept of order of operations is used in some examples. You should be familiar with this concept from your study of arithmetic. You will be reviewing order of operations thoroughly in sub-subunit 1.2.2.1.

Watching this video and pausing to take notes should take approximately 15 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Activity: Khan Academy’s “Evaluating Expressions in One Variable”
Link: Khan Academy’s “Evaluating Expressions in One Variable” (HTML)

Instructions: Please click on the link above and complete the exercise set. It contains easy problems on evaluating simple algebraic expressions containing only one variable. You will encounter more complex problems later in this sub-subunit. Substitute the given value of x into the given algebraic expression and perform the resultant set of numerical operations. Fill in your answer in the appropriate tab on the right side of the page, and click “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Activity: Khan Academy’s “Evaluating Expressions in 2 Variables”
Link: Khan Academy’s “Evaluating Expressions in 2 Variables” (HTML)

Instructions: Please click on the link above, and complete the exercise set. The algebraic expressions here contain more than one variable and you might also see exponents. You should remember how to handle exponents from your study of arithmetic, but you will also be reviewing this in sub-subunit 1.2.2.1.Substitute the given values of the variables into the given algebraic expression and perform the resultant set of numerical operations. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 1.2: Expressions with One or More Variables”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 1.2: Expressions with One or More Variables” (HTML)

Instructions: Please click on the link above, and read the entire article, which explores various real-world situations where evaluating expressions with one or more variables is required. Then, complete practice problems 1–9 and 26–30. Watch the first 2 minutes of the “Variable Expressions” video embedded in the article, if you need help with problems 1–9. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” The rest of the practice problems in this chapter will be assigned later in sub-subunit 1.2.2.2. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 1.2: Expressions with One or More Variables’ Problems 1–9 and 26–30” (PDF).

Reading and solving the practice problems should take approximately 30 minutes.

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1.2.2 Using Order of Operations Agreement to Simplify the Resulting Numerical Expression

1.2.2.1 Order of Operations Review

Note: This sub-subunit reviews the material covered in RWM101: Foundations or any equivalent arithmetic course. Use this section as needed to prepare for sub-subunit 1.2.2.2

Activity: Khan Academy’s “Order of Operations”
Link: Khan Academy’s “Order of Operations” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It contains basic order of operation problems and will help you assess how well you remember this concept from your study of arithmetic. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “PEMDAS”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “PEMDAS” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.”
Order of operations agreement is a convention used by mathematicians to ensure that expressions with many operations are always evaluated the same way, which is consistent with the properties of operations. Because we are used to reading from left to right, it is natural to add and multiply in the same direction, instead of thinking which operations should be performed first. With practice, you will get used to the correct order and will read the expressions accordingly.

Work through examples A and B and Guided Practice problem 1. Complete practice problems 1–7. Make sure to watch the “Order of Operations” video embedded in the text. The rest of the examples and exercises in this text will be assigned later in sub-subunit 1.2.2.2. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘PEMDAS’ Problems 1–7” (PDF).

Reading and solving the practice problems should take approximately 30 minutes.

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1.2.2.2 Evaluating Expressions

Note: This topic is covered by the readings assigned in sub-subunits 1.2.1 and 1.2.2.1.

Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “PEMDAS”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “PEMDAS” (HTML)

Instructions: Please click on the link above, and scroll down to Example C. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.”

In this activity, the examples and exercises require an extra step: substituting the values of the variables in the algebraic expressions. Then, they simply become numerical expressions, which can be evaluated by using order of operations agreement.

Work through Example C and Guided Practice problem 2, and then complete practice problems 8–10. Use the “Order of Operations Example” video embedded in the text for guidance. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘PEMDAS’ Problems 8–10 ” (PDF).

Reading and solving the practice problems should take approximately 15 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 1.2: Expressions with One or More Variables”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 1.2: Expressions with One or More Variables” (HTML)

Instructions: Please click on the link above, scroll down to the Practice Problems section, and complete practice problems 10–25. Watch the “Variable Expressions” video embedded in the text, if you need help. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Algebra Concepts: ‘Expressions with One or More Variables’ Problems 10–25” (PDF).

Reading and solving the practice problems should take approximately 30 minutes.

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1.3 Simplifying Variable Expressions

1.3.1 Properties of Real Numbers

1.3.1.1 Commutative Property of Addition and Multiplication

Activity: Khan Academy’s “Properties of Numbers 1”
Link: Khan Academy’s “Properties of Numbers 1” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It will help you review the basic properties of the operations with numbers. You will study the algebraic applications of these properties in the next assignment. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” tomove to the next problem.

Completing this exercise set should take approximately 10 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Reading: Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers”
Link: Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to the section titled “The Commutative Properties.” Read this section through “Sample Set A,” and work through the exercises in Practice Set A. The solutions to the practice problems are shown directly below each problem.

You know that changing the order in which two numbers are added does not change the result, and, likewise, changing the order in which two numbers are multiplied does not change the result. Because variables represent numbers, this is true for variables as well. With the help of this reading, you will learn to apply this property to any expression.

Reading this section and solving the practice problems should take approximately 15 minutes.

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1.3.1.2 Associative Property of Addition and Multiplication

Reading: Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers”
Link: Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to the section titled “The Associative Properties.” Read this section through “Sample Set B,” and work through the exercises in Practice Set B and Practice Set C. The solutions to the practice problems are shown directly below each problem.

The associative property states that several numbers can be added or multiplied in any order. Later in the course, you will be using this property often when you will have to switch terms around in order to simplify algebraic expressions.

Reading this section and solving the practice problems should take approximately 15 minutes.

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1.3.1.3 Distributive Property of Multiplication over Addition/Subtraction

Reading: Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers”
Link: Connexions: Wade Ellis and Denny Burzynski’s Basic Properties of Real Numbers: “Properties of the Real Numbers” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to the section titled “The Distributive Properties.” Read this section through “Sample Set D,” and work through the exercises in Practice Set D. The solutions to the practice problems are shown directly below each problem.

The distributive property is another property that will be used extensively in simplifying algebraic expressions.

Reading this section and solving the practice problems should take approximately 15 minutes.

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Activity: Khan Academy’s “Number Properties Terminology 1”
Link: Khan Academy’s “Number Properties Terminology 1” (HTML)

Instructions: Please click on the link above, and complete the exercise set. This is a quick review of the definitions of the properties of the operations with real numbers. Select your answer from the choices on the right side of the page, and click “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next question.

Completing this exercise set should take approximately 10 minutes.

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1.3.2 Combining Like Terms

1.3.2.1 Definition and Examples of Like Terms

Note: The definition of a term was already introduced in the reading assigned in sub-subunit 1.1.2. Here, you will learn about like terms.

Reading: All Math Words Encyclopedia: David McAdams’s “Like Terms”
Link: All Math Words Encyclopedia: David McAdams’s “Like Terms” (HTML)

Instructions: Please click on the link above, and read the encyclopedia entry. Note that the like terms have the same variables in the same exponents but might have different numerical coefficients. You will need to recognize like terms in order to add and subtract them, which you will do in sub-subunit 1.3.2.2.

Reading this entry should take approximately 5 minutes.

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1.3.2.2 Simplifying Expressions by Combining Like Terms

Lecture: Khan Academy’s “Combining Like Terms”
Link: Khan Academy’s “Combining Like Terms” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video explains in detail why and how the terms defined as like can be added and subtracted. This is a basic algebraic tool which you will be using throughout the course. It is important to understand that terms that are not like terms cannot be combined together.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Combining Like Terms”
Link: Khan Academy’s “Combining Like Terms” (HTML)

Instructions: Please click on the link above, and complete the exercise set, which provides practice combining like terms. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Combining Like Terms”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Combining Like Terms” (HTML)

Instructions: Please click on the link above, and read the entire webpage. After you have reviewed the examples, you are ready to practice simplifying expressions. Click on the “new problem” button at the end of the page to try a practice problem, and then select the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 30 minutes.

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1.3.3 Removing the Parentheses

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Simplifying Expressions Like –a(3b– 2c– d)”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Simplifying Expressions Like –a (3b– 2c– d)” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Removing, or opening, the parenthesis is another algebraic tool that allows you to work with expressions that contain parenthesis even when the variables inside them cannot be combined. Click on the “new problem” button at the end of the article to try a practice problem, and then select the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 30 minutes.

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1.3.4 Simplifying General Variable Expressions

Lecture: Khan Academy’s “Combining Like Terms and the Distributive Property”
Link: Khan Academy’s “Combining Like Terms and the Distributive Property” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In general algebraic expressions, you will need to open the parenthesis and then combine the like terms. Due to commutative and associative properties, you can move the like terms around in order to combine them.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Combining Like Terms With Distribution”
Link: Khan Academy’s “Combining Like Terms With Distribution” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It will provide practice using the number properties, particularly commutative and distributive, to simplify simple algebraic expressions. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Simplifying Algebraic Expressions”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Simplifying Algebraic Expressions” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Dr. Sousa explains a few slightly more complicated examples of simplifying algebraic expressions. Note that the last example in the video contains parenthesis within brackets. According to the order of operations, you need to first remove the inner grouping symbol of parenthesis and then simplify the expression within the brackets.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Exercise Supplement”
Link: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Exercise Supplement” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Complete the odd-numbered exercises for 47–55. Click on the “Show Solution” link next to each problem to see the correct solution.

Solving the practice problems should take approximately 30 minutes.

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Unit 2: Linear Equations

You, like other people, probably use equations in everyday life without even realizing it. Calculating a unit price to figure out which brand is cheaper when making a purchase, converting inches into feet, estimating how much time it would take you to drive to your destination at a certain speed all involve solving equations mentally. In this unit, you will learn formal procedures for solving equations. You can probably recall the basic rules from the math courses you have taken in the past. In this unit, you will review these rules while focusing on the formal logical definition of equation as a statement and its solution as a number that makes this statement true. You will apply the skills from Unit 1 to simplify the sides of an equation before attempting to solve it. You will also work with equations that contain more than one variable, and you will learn that because variables always represent numbers, you can use the same rules to find the specific variable you are looking for.

Unit 2 Time Advisory show close

Unit 2 Learning Outcomes show close

2.1 Introduction to Equations

2.1.1 Definition of an Equation and a Solution of an Equation

Reading: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable”
Link: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable” (HTML)

Instructions: Please click on the link above. You may read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Read the following sections: “Define Linear Equations in One Variable” and “Solutions to Linear Equations in One Variable.” Then, scroll down to “Exercises,” and complete exercises 1–5. Click on the “Show Solution” link for each problem to check your answer.

An important point to take away from this reading is that equation is defined as a statement (containing a variable), which may or may not be true, depending on the value of the variable. To solve an equation means to find all the values of the variable for which the statement is true. In the following sub-subunits, you will focus on finding these values.

Reading this section and solving the practice problems should take approximately 30 minutes.

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2.1.2 Addition/Subtraction Property of Equations

Lecture: Khan Academy’s “Why Do We Do the Same Thing to Both Sides: Simple Equations”
Link: Khan Academy’s “Why Do We Do the Same Thing to Both Sides: Simple Equations” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, you will find an ingenious virtual demonstration of a balanced scale representing an equation. This video provides a detailed explanation of why an equation does not change if the same thing is added to (or subtracted from) both sides.

Watching this video and pausing to take notes should take approximately 10 minutes.

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Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “The Addition Property of Equality”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “The Addition Property of Equality” (HTML)

Instructions: Please click on the link above, and read the entire webpage. After you review the examples, you can use addition property to determine whether two equations are equivalent. Click on the “new problem” button at the end of the article to try a practice problem and check your answer. Continue this process by clicking on “new problem,” and try to solve at least 10 practice problems or more, if necessary.

Reading this section and solving the practice problems should take approximately 30 minutes.

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2.1.3 Multiplication/Division Property of Equations

Lecture: Khan Academy’s “Intuition Why We Divide Both Sides”
Link: Khan Academy’s “Intuition Why We Divide Both Sides” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, the analogy between an equation and a balanced scale is again used to explain why an equation remains the same when both sides are multiplied or divided by the same number or expression.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “The Multiplication Property of Equality”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “The Multiplication Property of Equality” (HTML)

Instructions: Please click on the link above, and read the entire webpage. After you review the examples, you can use multiplication property to determine whether two equations are equivalent. Click on the “new problem” button at the end of the article to try a practice problem and check your answer. Continue this process by clicking on “new problem,” and try to solve at least 10 practice problems or more, if necessary.

Reading this section and solving the practice problems should take approximately 30 minutes.

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2.2 Solving Equations

2.2.1 Basic One-Step Equations

2.2.1.1 Equations of the Form x + a = b and x – a = b

Lecture: Khan Academy’s “One Step Equation Intuition Exercise Intro”
Link: Khan Academy’s “One Step Equation Intuition Exercise Intro” (YouTube)

Instructions: Please click on the link, and watch the brief video. This is an introduction to a hands-on (well, almost) exercise of finding the weight of Spice Man by keeping the scale balanced. To locate the link to the exercise, note that there is a list of videos (marked by a camera symbol) and exercises (marked by a star) on the left-hand side of the webpage. The link to the current video (“One Step Equation Intuition Exercise Intro”) is highlighted in gray, and below this appears the link to the exercise (“One Step Equation Intuition”). Please click on this link, and complete the exercise.

Watching this video and completing the exercise should take approximately 15 minutes.

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Lecture: Khan Academy’s “Adding and Subtracting the Same Thing to Both Sides”
Link: Khan Academy’s “Adding and Subtracting the Same Thing to Both Sides” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video explains how to solve the basic one-step equations you will encounter in algebra and other math courses.

Watching this video and pausing to take notes should take approximately 10 minutes.

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Activity: Khan Academy’s “One Step Equations”
Link: Khan Academy’s “One Step Equations” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It consists of equations that could be solved by either adding or subtracting a number from both sides in order to isolate the variable. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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2.2.1.2 Equation of the Form ax = b and x/a = b

Lecture: Khan Academy’s “Simple Equations”
Link: Khan Academy’s “Simple Equations” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video provides examples of another type of basic equation that can be solved in one step. Note that the example in the last three minutes of this video, which involves combining like terms on one side of the equation, covers the topic outlined in sub-subunit 2.2.2.3.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Solving One-Step Equations 2”
Link: Khan Academy’s “Solving One-Step Equations 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is one example of a one-step equation written in a form x/a =b. Sal Khan uses this example to highlight that division is really the same operation as multiplication.

Watching this video and pausing to take notes should take approximately 10 minutes.

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Activity: Khan Academy’s “One Step Equations with Multiplication”
Link: Khan Academy’s “One Step Equations with Multiplication” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Multiply or divide both sides of the equations in order to isolate the variable. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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Lecture: Khan Academy’s “Algebra: Linear Equations 1”
Link: Khan Academy’s “Algebra: Linear Equations 1” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Note that this video contains examples of the equations of the form ax = b, but in some of them a and b are fractions. Instead of dividing by a fraction (a), you can multiply both sides of the equation by its reciprocal.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “One-Step Equations and Inverse Operations”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’sAlgebra Concepts: “One-Step Equations and Inverse Operations” (HTML)

Instructions: Please click on the link above and read the entire webpage, which provides a review and summary of what you have already learned as well as examples of some real-world situationsmodeled by one-step equations. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.”

After reading, complete practice problems 1–16. Watch the “One Step Equations” video embedded in the text, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘One-Step Equations and Inverse Operations’ Problems 1–16 ” (PDF).

Reading and solving the practice problems should take approximately 30 minutes.

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2.2.2 Equations with Variables on One Side

2.2.2.1 Equations of the Form ax + b = c

Lecture: Khan Academy’s “Why Do We Do the Same Thing to Both Sides: Two Step Equations”
Link: Khan Academy’s “Why Do We Do the Same Thing to Both Sides: Two Step Equations” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Now that you are familiar with basic equations, this video introduces an equation that requires more than one step to solve. Note that x is found by performing the operations in the inverse order: first subtraction and then division.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Equations 2”
Link: Khan Academy’s “Equations 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video contains a very detailed explanation of how to solve two more equations of the form ax + b = c.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Activity: Khan Academy’s “2-Step Equations”
Link: Khan Academy’s “2 -Step Equations” (HTML)

Instructions: Please click on the link above, and complete the exercise set to practice solving simple two-step equations. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 15 minutes.

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Lecture: Khan Academy’s “Algebra: Linear Equations 2”
Link: Khan Academy’s “Algebra: Linear Equations 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Note that one of the examples discussed contains fractions, but it still can be solved as any other equation of the form ax + b = c. There is another way to approach this kind of equation, which will be shown in the sub-subunit 2.2.2.2.

Watching this video and pausing to take notes should take approximately 15 minutes.

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2.2.2.2 Equations of the Form ax + b = c Containing Fractions

Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving Linear Equations Involving Fractions”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving Linear Equations Involving Fractions” (HTML)

Instructions: Please click on the link above, read the introduction to the examples, and work through the examples carefully. Note that these equations are solved here by a method different than the one that was used in the “Algebra: Linear Equations 2” video in sub-subunit 2.2.2.1. Instead of performing operations with fractions, Dr. Burns eliminates the fractions from the equation by multiplying both sides by their common denominator. Click on the “new problem” button at the end of the article to try a practice problem and check your answer. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 30 minutes.

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2.2.2.3 Combining Like Terms on the Same Side before Solving

Note: The example of an equation containing like terms on the same side is explained in the last 3 minutes of the “Simple Equations” video assigned in sub-subunit 2.2.1.2. If necessary, take about 15 minutes to review the video.

2.2.3 Equations of the Form ax + b = cx + d

Lecture: Khan Academy’s “Equations 3”
Link: Khan Academy’s “Equations 3” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Now, you are moving on to solving slightly more complicated equations, such as ones that contain variables on both sides as seen in this video. You can transform this equation into an already familiar form by subtracting a variable expression from both sides.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Solving Equations 2”
Link: Khan Academy’s “Solving Equations 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another example of an equation with variables on both sides.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Equations with Variables on Both Sides”
Link: Khan Academy’s “Equations with Variables on Both SIdes” (HTML)

Instructions: Please click on the link above, and complete the exercise set to practice solving equations with variables on both sides. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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Lecture: Khan Academy’s “Algebra: Linear Equations 3”
Link: Khan Academy’s “Algebra: Linear Equations 3” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. The examples shown in this video require combining like terms on one side as well as moving variable expressions from one side to another.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving Linear Equations, All Mixed Up”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving Linear Equations, All Mixed Up” (HTML)

Instructions: The webpage linked above provides practice solving various equation types that have been introduced so far. Please click on the link, and work through the examples. Then, click on the “new problem” button at the end of the page to try a practice problem and check your answer. Continue this process by clicking on “new problem,” and solve10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Working through the examples and solving the practice problems should take approximately 30 minutes.

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2.2.4 Equations Containing Parentheses

Lecture: Khan Academy’s “Solving Equations with the Distributive Property”
Link: Khan Academy’s “Solving Equations with the Distributive Property” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video will help you to move on to even more complex equations. In the example shown, you have to remove the parenthesis from the both sides of the equation before attempting to solve it.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Solving Equations with the Distributive Property 2”
Link: Khan Academy’s “Solving Equations with the Distributive Property 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is an interesting example of an equation with variables on both sides involving fractions. Sal Khan chooses to multiply both sides of the equation by the common denominator and uses distributive property in order to get rid of the fractions.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Multistep Equations with Distribution”
Link: Khan Academy’s “Multistep Equations with Distribution” (HTML)

Instructions: Please click on the link above, and complete the exercise set to practice solving equations containing parentheses. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 40 minutes.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Distributive Property for Multi-Step Equations”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Distributive Property for Multi-Step Equations” (HTML)

Instructions: Please click on the link above, read the webpage to review some more examples of equations with parenthesis, and complete practice problems 1–22. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Remember that you need to simplify both sides of equation before solving. Watch the “Multi-Step Equations” video embedded in the text, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Distributive Property for Multi-Step Equations” Problems 1–22 ” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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2.2.5 Classifying Equations According to the Number of Solutions: Identities and Contradictions

Until now, all linear equations you have solved always have only one solution. From the following article, you will learn that this is not always the case.

Reading: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable”
Link: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable” (HTML)

Instructions: Please click on the link above. You may read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Read the section titled “Conditional Equations, Identities, and Contradictions.”

Reading these definitions and examples should take approximately 15 minutes.

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Assessment: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable”
Link: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable” (HTML)

Instructions: Please click on the link above. You may also download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to “Exercises,” and complete exercises 26–35. Keep in mind that these equations could have only one solution, infinite solutions, or no solution. Click on the “Show Solution” link next to each problem to check your answer.

This assessment should take approximately 45 minutes.

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2.3 Literal Equations

2.3.1 Solving Literal Equation for One of the Variables

Literal equations contain two or more variables. The trick is to focus on isolating the variable you need to find and to treat the rest of the variables as if they are known.

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Solving a Formula for a Variable”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Solving a Formula for a Variable” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Note that literal equations, just like any others, can be solved by doing the same thing to both sides.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: Flat World Knowledge: John Redden’s Linear Equations and Inequalities: “Solve for a Variable – Algebra”
Link: YouTube: Flat World Knowledge: John Redden’s Linear Equations and Inequalities: “Solve for a Variable – Algebra” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video provides another example of solving a literal equation. Note that the steps taken to isolate a variable are very similar to the one in the second example in Dr. Sousa’s video.

Watching this video and pausing to take notes should take approximately 10 minutes.

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Activity: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable”
Link: Connexions: John Redden’s Elementary Algebra: “Solving Linear Equations in One Variable” (HTML)

Instructions: Please click on the link above. You may also download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to the section titled “Linear Literal Equations,” and work through Example 16. Then, scroll down to “Exercises,” and complete exercises 36–40. Click on the “Show Solution” link next to each problem to check your answer.

Reading this section and solving the practice problems should take approximately 20 minutes.

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Activity: Khan Academy’s “Solving Equations in Term of a Variable”
Link: Khan Academy’s “Solving Equations in Term of a Variable” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Solve the problem, and select your answer from the choices given on the right side of the page. Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, it will prompt you to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 30 minutes.

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2.3.2 Formulas

Lecture: Khan Academy’s “Example of Solving for a Variable”
Link: Khan Academy’s “Example of Solving for a Variable” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. A formula is an equation that expresses a relationship between two or more quantities. When one of these quantities needs to be rewritten in terms of others, the formula becomes a literal equation.

In this video, the formula for the perimeter of a rectangle (equation used to find a perimeter when its length and width are known) is solved for the width. That is, the result is an equation used to find the width of a rectangle, when its perimeter and length are known. Note that Sal Khan shows two different ways to arrive at the answer.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Solving for a Variable 2”
Link: Khan Academy’s “Solving for a Variable 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. The formula shown in this video converts Fahrenheit temperature into Celsius. Solving for Fahrenheit temperature results in a formula that converts Celsius temperature into Fahrenheit. Note that there are two possible approaches to do this, but only one is shown in the video. You might want to try the second approach (distribute 5/9 over the parenthesis) to see which one is more convenient.

Watching this video, pausing to take notes, and trying the second approach on your own should take approximately 15 minutes.

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Assessment: Connexions: Wade Ellis and Denny Burzynski’s Solving Linear Equations and Inequalities: “Proficiency Exam”
Link: Connexions: Wade Ellis and Denny Burzynski’s Solving Linear Equations and Inequalities: “Proficiency Exam” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Complete exercises 1–11. This exercise set will allow you to assess your mastery of the concepts from Unit 2. Click on the “Show Solution” link next to each problem to check your answer.

Completing this assessment should take approximately 45 minutes.

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Unit 3: Word Problems

In this unit, you will apply the skill of solving equations you mastered in Unit 2 to solve various types of word problems. When you encounter a word problem, you have to remember to read it carefully and think critically about what quantity you are asked to find, what quantities are known, and what the relationship is between them. This will help you set up an equation that will give you an answer to the problem. For example, if you know the discounted price of an item and need to find the original price, remember that the percent of the discount is taken from this original price, which is something you do not know!

You probably already know how to solve some of the problems covered in this unit, such as percent problems or uniform motion problems that can be solved by one arithmetic operation. However, you will now approach these types of problems from the algebraic point of view, which will enable you to move on to more complex problems. In these problems, you cannot just add/subtract and multiply/divide the quantities given and arrive at the answer. These problems can only be solved by setting up an equation.

Unit 3 Time Advisory show close

Unit 3 Learning Outcomes show close

3.1 Translating English to Math

3.1.1 Mathematical Symbols and Expressions for Common Words and Phrases

Reading: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications I: Translating Words to Mathematical Symbols”
Link: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications I: Translating Words to Mathematical Symbols” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Read the section titled “Translating Word to Symbols” through “Sample Set A.” Also, copy Table 1 into your notes for future reference. You will use this table in the next assignment and in sub-subunit 3.1.2.

Reading this section and taking notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Writing Expressions”
Link: Khan Academy’s “Writing Expressions” (HTML)

Please click on the link above, and complete the exercise set. Use the article in the previous assignment to help you translate these verbal expressions into algebraic expressions. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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3.1.2 Translating Verbal Expression into Mathematical

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Write Algebraic Expressions” and “Ex 2: Writing Basic Algebraic Expressions”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Write Algebraic Expressions” (YouTube) and “Ex 2: Writing Basic Algebraic Expressions” (YouTube)

Instructions: Please click on the links above, watch these videos, and take notes. In these videos, Dr. Sousa explains two examples of translating a given real-life situation to algebraic language. As you watch, pay attention to how the keywords such as more and less translate into mathematical operations.

Watching these videos and pausing to take notes should take approximately 15 minutes.

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Activity: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications I: Translating Words to Mathematical Symbols”
Link: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications I: Translating Words to Mathematical Symbols” (HTML)

Instructions: Please click on the link above, and scroll down to Practice Set A. Using Sample Set A as a guide, complete exercises 1–7 and check your solutions (located below each problem). Developing the skill of translating verbal expressions to mathematical is the first step to solving word problems, which you will be doing in subunits 3.2 through 3.4. Note that you may also download this module as a PDF by clicking on the appropriate link at the bottom of the webpage.

Completing these exercises should take approximately 15 minutes.

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3.2 Translating Sentences into Equations

3.2.1 Number Problems

Reading: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications II: Solving Problems”
Link: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications II: Solving Problems” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Read the section titled “Five Step Method.” This method will guide you through solving problems in this and subsequent subunits of Unit 3. Read Examples 1 and 2 in Sample Set A, and try Exercises 1 and 2 from Practice Set A on your own. The solutions to the practice problems are shown directly below each problem.

Reading this section and solving the practice problems should take approximately 30 minutes.

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Activity: Mathematics and Multimedia: Guillermo Bautista’s The Mathematics Word Problem Solving Series: “Part 1”, “Part 2”, “Part 3”, and “Part 4”
Link: Mathematics and Multimedia: Guillermo Bautista’s The Mathematics Word Problem Solving Series: “Part 1” (HTML), “Part 2” (HTML), “Part 3” (HTML), and “Part 4” (HTML)

Instructions: On the webpages linked above, the author explains how to solve problems involving numbers by translating verbal description of the relationship between the numbers into an equation. First, click on the link to Part 1. Read the webpage and work through the solutions to problems 1 and 2. Then, click on the link to Part 2, and attempt to solve problems 4 and 5 on your own. Then, compare your solutions to the ones given by the author. Next, click on the link Part 3, and try solving problems 7, 8, and 9. Then, compare your solutions to the ones given. Finally, click on the link to Part 4, and solve problem 12. After you attempt this problem, check your answer.

Working through the solutions to all the problems should take approximately 45 minutes.

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3.2.2 Consecutive Integer Problems

Lecture: Khan Academy’s “Integer Sums”
Link: Khan Academy’s “Integer Sums” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video will introduce you to the word problems involving consecutive integers, or integers that follow one another. Pay attention to the explanation of how to express one consecutive integer in terms of another. Note the difference between the problems: the first one is about consecutive integers, whereas the second one is about consecutive odd integers.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Sum of Consecutive Odd Integers”
Link: Khan Academy’s “Sum of Consecutive Odd Integers” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another problem involving consecutive odd integers.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Mathematics and Multimedia: Guillermo Bautista’s The Mathematics Word Problem Solving Series: “Part 1”, “Part 2”, and “Part 4”
Link: Mathematics and Multimedia: Guillermo Bautista’s The Mathematics Word Problem Solving Series: “Part 1” (HTML), “Part 2” (HTML), and “Part 4” (HTML)

Instructions: The webpages linked above contain information on solving number problems. You will focus on the problems involving consecutive integers. First, click on the link to Part 1. Scroll down to problem 3, and work through its solution. Again, note how the fact that integers follow one another is expressed algebraically. Then, click on the link to Part 2, and try solving problem 6 on your own. Then, compare your solution to the one given by the author. Finally, click on the link to Part 4, solve problem 10, and check your answer.

Working through these solutions should take approximately 30 minutes.

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Activity: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications II: Solving Problems”
Link: Connexions: Wade Ellis and Denny Burzynski’s Algebraic Expressions and Equations: “Applications II: Solving Problems” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to Example 3 in Sample Set A. Review the algebraic method of solving problems involving consecutive, odd, or even integers. Then, complete exercises 4, 34, 36, and 38. Click on the “Show Solutions” link next to each problem to check your answer.

Reading this section and solving the practice problems should take approximately 30 minutes.

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Activity: Khan Academy’s “Integer Sums”
Link: Khan Academy’s “Integer Sums” (HTML)

Please click on the link above, and complete the exercise set. It provides more practice solving problems involving consecutive integers. Keep in mind that after you have solved your equation, you still have to answer the question in the problem. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 45 minutes.

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3.2.3 General Statement Problems

Lecture: Khan Academy’s “Algebraic Word Problem”
Link: Khan Academy’s “Algebraic Word Problem” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is an example of using algebraic solution to a real-life situation. Watch how each piece of information given about the shelves is translated into algebraic expression and then used to create an equation.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Problem Solving with Equations”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Problem Solving with Equations” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. You may skip the first four minutes of the video (problems 1 and 2) and focus on problems 3 and 4. Note that while these problems (and the problem discussed in Khan Academy’s “Algebraic Word Problem”) are seemingly about different situations, yet the same basic approach is used to solve these problems.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Linear Equation Word Problems”
Link: Khan Academy’s “Linear Equation Word Problems” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Solving these word problems will help you apply your algebraic and critical thinking skills to various real-life situations. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Linear Equation Application - Car Repair with Labor and Parts”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Linear Equation Application - Car Repair with Labor and Parts” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. This is another example of solving a word problem using a linear equation.

Watching this video and pausing to take notes should take approximately 15 minutes.

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3.3 Basic Uniform Motion Problems

3.3.1 Applying Uniform Motion Equation

A motion is uniform when the speed, or rate, of the motion is constant. For example, if a car moves with the speed of 30 mph without accelerating or slowing down, then it is in uniform motion. The distance that the car travels during a given time can be found according to the uniform motion equation: distance = rate × time.

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Distance = Rate x Time Application Problem”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Distance = Rate x Time Application Problem” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this video, the uniform motion equation is used to find speed and distance.

Watching this video and pausing to take notes should take approximately 15 minutes.

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3.3.2 Problems Involving Objects Moving in Opposite Directions

Activity: Mathematics and Multimedia: Guillermo Bautista’s The Mathematics Word Problem Solving Series: “Math Word Problems: Solving Motion Problems Part 2”
Link: Mathematics and Multimedia: Guillermo Bautista’s The Mathematics Word Problem Solving Series: “Math Word Problems: Solving Motion Problems Part 2” (HTML)

Instructions: Please click on the link above, work through the problems on this webpage on your own, and then review the solutions. This webpage contains examples of two common types of motion problems in which the object travel in the opposite directions, either towards each other or away from each other.

Working through these solutions should take approximately 15 minutes.

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3.3.3 Uniform Motion Problems

Reading: The Saylor Foundation’s “Uniform Motion Problems Involving Current, Headwind, or Moving Sidewalk”
Link: The Saylor Foundation’s “Uniform Motion Problems Involving Current, Headwind, or Moving Sidewalk” (PDF)

Instructions: Read this brief example of uniform motion problems, and then attempt the practice problems on the second page. When you have finished, you may check your answers against the Saylor Foundation’s “Answer Key to Uniform Motion Problems Involving Current, Headwind, or Moving Sidewalk” (PDF).
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3.4 Creating Equations to Solve Problems

3.4.1 Value Mixture Problems

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Simple Word Problems Resulting in Linear Equations”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Simple Word Problems Resulting in Linear Equations” (HTML)

Instructions: Please click on the link above, and read the entire webpage. The problems you are going to solve here all have to do with mixtures of two different types of a product, costing a different price. Click on the “new problem” button at the end of the webpage to try a practice problem, and check your answer. Continue this process by clicking on “new problem,” and solve five problems. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 1 hour.

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Lecture: Khan Academy’s “Mixture Problems 1”
Link: Khan Academy’s “Mixture Problems 1” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another example of a mixture problem, where the unit price of the mixture of two products is unknown.

Watching this video and pausing to take notes should take approximately 15 minutes.

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3.4.2 Percent Mixture Problems

3.4.2.1 Definition of a Percent, Basic Percent Equation and Basic Percent Problems

Note: This sub-subunit reviews the material covered in RWM101: Foundations or any equivalent arithmetic course. Use this section as needed to prepare for sub-subunit 3.4.2.2.

Activity: Khan Academy’s “Percentage Word Problems 1”
Link: Khan Academy’s “Percentage Word Problems 1” (HTML)

Instructions: Please click on the link above, and complete the exercise set. This will help you assess your proficiency in solving problems involving percents, which you have studied in Arithmetic. You will review the concept and application of percents again in the next assignment. Fill in your answer in the appropriate tab on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 40 minutes.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 3.7: Percent Problems”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 3.7: Percent Problems” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Watch the “How to Solve Percent Equations” video embedded in the text to review how to solve basic percent problems by setting up equations, if you need help. Then, complete practice problems 16–30. Use the embedded “Percent Problems” video (skipping the first five minutes) as a guide, if you need help with these problems. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’‘Chapter 3.7: Percent Problems’ Problems 16–30 ” (PDF).

Reading and working on the practice problems should take approximately 45 minutes.

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3.4.2.2 Solving Percent Mixture Problems

Note: The problems in this section involve mixing two solutions with different concentrations of a certain substance.

Lecture: Khan Academy’s “Mixture Problems 2”
Link: Khan Academy’s “Mixture Problems 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. While different in context, algebraically this problem is similar to mixture problems in sub-subunit 3.4.1. Note the steps Sal Khan takes to choose a variable, translate all the information given into algebraic expressions, and set up an equation.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Mixture Problems 3”
Link: Khan Academy’s “Mixture Problems 3” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another example of a percent mixture problem. This time, the percent concentration in one part of the mixture is unknown.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Larry Green’s “Problems that Involve Mixing 2 Beakers of Acid”
Link: Larry Green’s “Problems that Involve Mixing 2 Beakers of Acid” (HTML)

Instructions: Please click on the link above. This interactive applet will guide you one step at a time through setting up the table you need to solve a typical percent mixture problem. This website focuses only on the type of problems in which amounts of both solutions being mixed are unknown. Try to go through at least three problems to achieve proficiency.

Solving these practice problems should take approximately 45 minutes.

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Activity: Larry Green’s “Money and Mixing Word Problems”
Link: Larry Green’s “Money and Mixing Word Problems” (HTML)

Instructions: Please click on the link above. This website will help you solve various mixture problems (including the ones involving percents) one step at a time. Try to solve at least five problems or more, if necessary. You can also click on “More information on Money and Mixing Problems” at the bottom of the page to see more examples of mixture and percent mixture problems.

Solving the practice problems should take approximately 45 minutes.

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3.4.3 More Uniform Motion Problems

Note: The problems in this sub-subunit still require the use of the uniform motion equation that you used in subunit 3.3; however, these problems cannot be solved by just applying the formula. You will have to express all given information algebraically as well as set up and solve an equation.

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Find the Rate of Two Cyclists Traveling toward Each Other”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Find the Rate of Two Cyclists Traveling toward Each Other” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this problem, two objects (cyclists, in this case) are traveling in opposite directions toward each other. Note how Dr. Sousa uses the table to organize all the information given in the problem.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Find the Distance between Two Cities Using the Distance = Rate * Time Formula”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Find the Distance between Two Cities Using the Distance = Rate * Time Formula” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this problem, there is only one traveler, but she travels to another city and then back. Again, a table is useful to organize all the information given. Note that the task here is to find the distance, but the variable chosen to be denoted, x, is the time it takes to complete a one-way trip. You will find that this is a convenient approach in most problems where time is not given directly.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Find When Two People Leaving the Same Place at Different Rates and Time Will Meet”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Find When Two People Leaving the Same Place at Different Rates and Time Will Meet” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this example, two people travel in the same direction and one has to catch up to the other. Note that the key information you need to set up the equation is not given in the problem explicitly: both people will have traveled the same distance by the time one overtakes the other.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Larry Green’s “Distance = Rate times Time”
Link: Larry Green’s “Distance = Rate times Time” (HTML)

Instructions: Please click on the link above. This website will guide you through solving various uniform motion problems one step at a time. Try to solve at least five problems or more, if needed. You can also click on “Information on Distance-Rate-Time Problems” at the bottom of the page and scroll down to the section titled “Motion Problems” to see one more solved example of a uniform motion problem.

Solving these practice problems should take approximately 45 minutes.

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Unit 4: Inequalities

You probably use inequalities, just like equations, in everyday life without thinking about it. Every time you go to the store, you need to decide whether you have enough money to pay for the items you need to purchase. The inequality you need to solve is the Amount of Money has to be GREATER than the Total Cost of Items. In this unit, very similar to Unit 2: Equations, you will generalize the procedure for solving inequalities. You will learn that an inequalityis a statement and its solution is a set of numbers that makes this statement true. You will explore which properties of inequalities are the same and which are different from the properties of equations.

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Unit 4 Learning Outcomes show close

4.1 Definition and Notation of an Inequality

Reading: Connexions: Wade Ellis’s and Denny Burzynski’s Solving Linear Equations and Inequalities: “Linear Inequalities in One Variable”
Link: Connexions: Wade Ellis’s and Denny Burzynski’s Solving Linear Equations and Inequalities: “Linear Inequalities in One Variable” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Read the text until the section titled “The Algebra of Linear Inequalities.” This part of the text is an introduction to linear equalities. As you can see from the given examples, inequalities are very similar to equations, except that they state that one expression is smaller (or greater) than another, rather than equal.

Reading this text should take approximately 15 minutes.

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4.2 Graphing Inequalities on a Number Line

Reading: All Math Words Encyclopedia: David McAdams’ “Inequality”
Link: All Math Words Encyclopedia: David McAdams’ “Inequality” (HTML)

Instructions: Please click on the link above, read the brief encyclopedia entry, and focus on the table under the title “Graphing One Variable Inequalities.” In your notes, make sketches of how four different types of inequalities (in the first four rows of the table) are represented on the number line. Use these notes as a reference throughout the rest of the Unit 4.

Reading this section and taking notes should take approximately 20 minutes.

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Lecture: Khan Academy’s “Inequalities on a Number Line”
Link: Khan Academy’s “Inequalities on a Number Line” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan explains the connection between algebraic inequality statement and its representation on the number line.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Inequalities on a Number Line”
Link: Khan Academy’s “Inequalities on a Number Line” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Solve the problem, and select your answer from the choices given on the right side of the page (or, in some problems, fill in the answer tab). Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, you will be prompted to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 20 minutes.

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4.3 Solving Linear Inequalities

4.3.1 Solving One-Step Inequalities

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Inequalities”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Inequalities” (HTML)

Instructions: Please click on the link above, read the webpage, and watch the videos embedded in the text. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” After reading the article and working through guided practice examples, complete practice problems 1–12. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Linear Inequalities’ Problems 1–12 ” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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Activity: Khan Academy’s “One Step Inequalities”
Link: Khan Academy’s “One Step Inequalities” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Solve the problem, and fill in the answer on the right side of the page. Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, you will be prompted to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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4.3.2 Solving Multi-Step Inequalities

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Multi-Step Inequalities”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Multi-Step Inequalities” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Note that solving inequalities involves the same procedure as solving equations: simplifying both sides, bringing the variable terms to the same side, and isolating the variable. After reading the article and working through guided practice examples, complete the practice problems 1–15. Make sure to watch the embedded “Multi-Step Inequalities” video, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Multi-Step Inequalities’ Problems 1–15 ” (PDF).

Reading and solving the practice problems should take approximately 1 hour.

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Activity: Khan Academy’s “Multi-Step Linear Inequalities”
Link: Khan Academy’s “Multi-Step Linear Inequalities” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Solve the problem, and fill in the answer on the right side of the page. Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, you will be prompted to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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4.4 Application Problem

Reading: CK-12’s Algebra II-Second Edition: “Chapter 6.1: Solving Inequalities”
Link: CK-12’s Algebra II-Second Edition: “Chapter 6.1: Solving Inequalities” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the Introduction, and then scroll down to Example 3. Study the solutions for Example 3 (parts a–d), which explain how to translate the verbal statements of inequality into algebraic. Pay attention to the words such as at least or at most, which indicate that greater or equal or less or equal notation should be used.

Reading the introduction and studying the solutions to Example 3 should take approximately 15 minutes.

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Lecture: Khan Academy’s “Writing and Using Inequalities 3”
Link: Khan Academy’s “Writing and Using Inequalities 3” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is an example of a word problem that can be solved by using an inequality.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Inequalities”
Link: Khan Academy’s “Inequalities” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is an example of a real-world situation that can be described with an inequality.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Writing and Using Inequalities 2”
Link: Khan Academy’s “Writing and Using Inequalities 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another, slightly more complicated example of an application of inequalities.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Assessment: Connexions: Wade Ellis and Denny Burzynski’s Solving Linear Equations and Inequalities: “Proficiency Exam”
Link: Connexions: Wade Ellis and Denny Burzynski’s Solving Linear Equations and Inequalities: “Proficiency Exam” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Complete exercises 13–16 and 24. This exercise set will allow you to assess your mastery of concepts from Unit 4. Click on the “Show Solution” link next to each problem to check your answer.

Completing this assessment should take approximately 30 minutes.

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Unit 5: Graphs of Linear Equations and Inequalities

This unit is an introduction to graphing relationships between the two quantities on the coordinate plane. A graph helps visualize how one quantity depends on another. In this course, you will only graph relationships that can be described by a linear equation, and their graphs are always straight lines. Graphing is an important tool that aids in analyzing relationships, both in abstract and in applied math. The material in this unit will help you become comfortable with graphing pairs of numbers on the coordinate plane and understand how the equations and relationships can be represented by lines. As an example, you might want to graph how the location of a train depends on the time since the train departed from the station. If the train is moving with constant speed, this graph would be a straight line. The slant of this line (which, as you will learn in this unit, is called the slope) will depend on the speed: the greater the speed, the steeper the line. If the line is going up (looking from left to right), it tells you that the distance is growing with time, which means that the train is moving away from the station. Otherwise, if the line is going down, it tells you that the distance is decreasing, which means that the train is approaching the station. You will see that a lot of information about the train’s journey can be gathered from just one graph.

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Unit 5 Learning Outcomes show close

5.1 Graphing Points in the Rectangular Coordinate Plane

Note: In this subunit, and throughout the rest of Unit 5, please use graph paper and a ruler to create accurate graphs of points on a coordinate plane and straight lines.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Points in the Coordinate Plane”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Points in the Coordinate Plane” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article, watch the videos embedded in the text, and try to play the interactive “Coordinate Plane Game,” linked under the “Try This” heading. Then,complete practice problems 1–10. Once you complete the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12’s Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Points in the Coordinate Plane’ Problems 1–10 ” (PDF).

Reading and solving the practice problems should take approximately 1 hour.

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Activity: Khan Academy’s “Graphing Points and Naming Quadrants”
Link: Khan Academy’s “Graphing Points and Naming Quadrants” (HTML)

Instructions: Please click on the link above, and complete the interactive exercise set. Drag the orange dot to the given point on the coordinate plane, then select the quadrant that contains this point from the choices on the right side of the page. Once you are done, select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, you will be prompted to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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5.2 Linear Equations in Two Variables

Note: Recall that in Unit 2 a solutionof an equation was defined as a number that make this equation a true statement. A solution of linear equations in 2 variables can be defined the same way, except now it is a pair of numbers (one for each variable) that makes the equation a true statement. In this subunit, you will learn how to find these solutions and represent them on a graph.

5.2.1 Ordered Pairs as Solutions of an Equation in Two Variables

Lecture: Khan Academy’s “Descartes and Cartesian Coordinates”
Link: Khan Academy’s “Descartes and Cartesian Coordinates” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video is an introduction to a branch of mathematics known as Coordinate Geometry, which studies the connection between algebraic equations and properties of lines and curves representing them on the coordinate plane. While this lecture mostly covers the material related to sub-subunit 5.2.2, it will help you understand Dr. Burns’ article assigned in this sub-subunit (5.2.1).

Watching this video and pausing to take notes should take approximately 30 minutes.

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Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Introduction to Equations and Inequalities in Two Variables”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Introduction to Equations and Inequalities in Two Variables” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Note that this reading also covers the material that will be discussed is subunits 5.3 and 5.6. These are the two main concepts you need to understand after reading this article:
- how to identify whether a pair of numbers is a solution of a given equation or inequality in two variables; and
- if you graph all solutions (ordered pairs of numbers) of a linear equation in two variables on a coordinate plane, they will create a straight line.
Practice applying these concepts by clicking on the “new problem” button at the end of the article, trying a problem, and checking your answer. Attempt to solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 1 hour.

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Lecture: Khan Academy’s “Ordered Pair Solutions of Equations”
Link: Khan Academy’s “Ordered Pair Solutions of Equations” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, you will see an example of how to determine whether a pair of numbers is a solution of a given equation in two variables.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Ordered Pair Solutions to Linear Equations”
Khan Academy’s “Ordered Pair Solutions to Linear Equations“ (HTML)

Instructions: Please click on the link above, and complete the exercise set. Determine which one of the given ordered pairs is a solution to the given (algebraically or graphically) linear equation; then select your answer from the choices on the right side of the page. Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, you will be prompted to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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5.2.2 Graphing Equations in Two Variables of the Form y = mx + b

Lecture: Khan Academy’s “Algebra: Graphing Lines 1”
Link: Khan Academy’s “Algebra: Graphing Lines 1” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the lines shown on graph paper. This video shows how to create a graph of a linear equation by generating a few solutions (ordered pairs) and plotting them on a coordinate plane. All linear equations in this video are such that y is expressed in terms of x: y = mx + b.

Watching the video, pausing to take notes, and graphing the lines should take approximately 30 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex 2: Graph a Linear Equation Containing Fractions Using a Table of Values”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex 2: Graph a Linear Equation Containing Fractions Using a Table of Values” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the line shown on graph paper. This is also an example of graphing a linear equation written in the form y = mx + b, but the focus is on the fact that the coefficient in front of x is a fraction. Because we can pick any values of x to generate the ordered pairs for plotting, it is convenient to choose the ones that will produce integer values of y, which are easier to graph.

Watching the video, pausing to take notes, and graphing the line should take approximately 15 minutes.

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5.2.3 Graphing Equations in Two Variables of the Form Ax + By = C

Lecture: Khan Academy’s “Plotting (x, y) Relationships”
Link: Khan Academy’s “Plotting (x, y) Relationships” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the line shown on graph paper. In this video, Sal Khan again graphs a linear equation by generating its solutions. However, this time an extra step is required: the original equation needs to be rewritten with the variable y on one side and everything else on the other side, so values of x can be plugged in and the values of y can be easily calculated.

Watching the video, pausing to take notes, and graphing the line should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex 2: Graph a Linear Equation in Standard Form Using a Table of Values”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex 2: Graph a Linear Equation in Standard Form Using a Table of Values” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the line shown on graph paper. This is another example of graphing a linear equation written in the form Ax + By = C. Note how Dr. Sousa chooses the values of x to create a table of ordered pairs convenient for plotting.

Watching the video, pausing to take notes, and graphing the line should take approximately 15 minutes.

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Lecture: Khan Academy’s “Ordered Pair Solutions of Equations 2”
Link: Khan Academy’s “Ordered Pair Solutions of Equations 2” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the line shown on graph paper. In addition to graphing a linear equation, this video highlights the connection between the solutions of an equation and all points on the straight line, or the graph of the equation.

Watching the video, pausing to take notes, and graphing the line should take approximately 15 minutes.

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5.2.4 Graphs of the Equations in the Forms x = C and y = C

Note: In this section and the following sub-subunit 5.2.5, you will explore how to graph two special cases of linear equations.

5.2.4.1 Graph of a Vertical Line

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Graph a Vertical Line Using a Table of Values”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Graph a Vertical Line Using a Table of Values” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the line shown on graph paper.

Watching the video, pausing to take notes, and graphing the line should take approximately 15 minutes.

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5.2.4.2 Graph of a Horizontal Line

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Graph a Horizontal Line Using a Table of Values”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Graph a Horizontal Line Using a Table of Values” (YouTube)

Instructions: Please click on the link above, watch the video, take notes, and graph the line shown on graph paper.

Watching the video, pausing to take notes, and graphing the line should take approximately 15 minutes.

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Activity: Khan Academy’s “Graphing Linear Equations”
Link: Khan Academy’s “Graphing Linear Equations” (HTML)

Instructions: Please click on the link above, and complete this interactive exercise set. Make a table of ordered pairs of numbers (x, y), and then drag the two points on the given line to the corresponding places on the coordinate plane. Once you are done, select “Check Answer” to see if your graph is correct or incorrect. If it is incorrect, you will be prompted to try again. If your graph is correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Activity: Khan Academy’s “Identifying Linear Relationships”
Link: Khan Academy’s “Identifying Linear Relationships” (HTML)

Instructions: Please click on the link above, and complete the interactive exercise set, in which you have to get information about a linear relationship from its graph. Enter your answer into the appropriate tab on right side of the page, or select one from the given choices. Then, select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If your graph is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 15 minutes.

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5.3 Slopes and Intercepts of a Straight Line

5.3.1 Intercepts of a Straight Line

Note: Intercepts are the points where a line (or any other graph) intersects x- and y-axis. Most graphing procedures involve identifying and graphing at least one of the intercepts.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Intercepts by Substitution”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Intercepts by Substitution” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” After reading and working through the examples, complete practice problems 1–10. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12’s Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Intercepts by Substitution’ Problems 1–10 ” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Intercepts and the Cover-up Method”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Intercepts and the Cover-up Method” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Once on the webpage, please make sure you are viewing the “Basic Level” material.This reading explains another method for identifying intercepts of the line from its equation. It also provides some examples of real-world applications of intercepts. After reading and working through the examples, complete practice problems 3–22. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12’s Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Intercepts and the Cover-up Method’ Problems 3–22 ” (PDF).

Reading and solving the practice problems should take approximately 1 hour and 30 minutes to complete.

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Activity: Khan Academy’s “Solving for the x-intercept”
Link: Khan Academy’s “Solving for the x-intercept” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Find the value of the x-intercept using one of the methods you learned in the previous assignments, and enter it into the answer tab on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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Activity: Khan Academy’s “Solving for the y-intercept”
Link: Khan Academy’s “Solving for the y-intercept” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Find the value of the y-intercept using one of the methods you learned in the previous assignments, and enter it into the answer tab on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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5.3.2 Slope of a Straight Line

Note: Slope is a major characteristic of a straight line. Slope is a number that indicates how steep the line is (relatively to the horizontal) and whether the line is going up or down.

5.3.2.1 Definition of Slope and Slope Formula

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK-12 Editors’ “Chapter 4.4: Slope”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK-12 Editors’ “Chapter 4.4: Slope” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” After reading and working through the examples, complete practice problems 4–23. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: ‘Chapter 4.4: Slope’ Problems 4–23 ” (PDF).

Reading and solving the practice problems should take approximately 1 hour and 30 minutes to complete.

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Activity: Khan Academy’s “Identifying Slope of a Line”
Link: Khan Academy’s “Identifying Slope of a Line” (HTML)

Instructions: Please click on the link above, and complete the exercise set. You will be required to find the slope using the slope formula, or to identify a line with a given slope. Enter your answer into the tab on the right side of the page, or select one of the given choices. Then, select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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5.3.2.2 Slopes of Parallel and Perpendicular Lines

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Parallel and Perpendicular Lines”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Parallel and Perpendicular Lines” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This reading defines parallel and perpendicular lines and illustrates the relationship between their slopes.

Reading this section should take approximately 15 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Comparing Equations of Parallel and Perpendicular Lines”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Comparing Equations of Parallel and Perpendicular Lines” (HTML)

Instructions: Please click on the link above. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” In the following exercises, you will apply the concepts and formulas from Dr. Burns’ article. Work through Examples A, B, and C, and watch the video embedded in the text. Then, complete practice problems 1–4 and 11. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12’s Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Comparing Equations of Parallel and Perpendicular Lines’ Problems 1–4 and 11 ” (PDF).

Working through the examples and solving the practice problems should take approximately 45 minutes.

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5.3.3 Applications: Real-World Interpretations of Slope and Intercepts

Note: In this sub-subunit, you will learn that slope represents the rate with which one quantity changes with respect to another quantity and the intercepts usually represent the starting and ending points of a process.

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Rate and Slope”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Rate and Slope” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video introduces the meaning of the slope as a rate of change.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Slope Application Involving Population Growth”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Slope Application Involving Population Growth” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, the slope formula is used to calculate rate (in this case, rate of population growth per year).

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Slope Application Involving Production Costs”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Slope Application Involving Production Costs” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Here, the slope formula is again used to calculate rate (in this case, production cost per item).

Watching this video and pausing to take notes should take approximately 15 minutes.

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5.4 Equations of Straight Line

Note: While working on subunits 5.2 and 5.3, you have probably noticed that linear equations are usually written in one of these forms:

y = mx +b, known as the slope-intercept form. As you will learn in this subunit, m in this equation is the slope and bis the y-intercept.

Ax + By = C, known as the standard form.

There is also another possible form, used to write an equation of the line with a given slope mpassing through a given point (x₁, y₁). It is known as the point-slope form: y – y₁ = m(x – x₁).

In this subunit, you will be identifying the line’s properties from these equations and converting equations from one form to another.

5.4.1 Slope-Intercept Form

5.4.1.1 Graphing Equation of a Line in a Slope-Intercept Form

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Slope-Intercept Form”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Slope-Intercept Form” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Then, graph the linear equations given in practice problems 1–5 and 7, and complete practice problems 16–21. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Slope-Intercept Form’ Problems 1–5, 7, and 16–21” (PDF).

Reading and solving the practice problems should take approximately 1 hour and 30 minutes.

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Activity: Khan Academy’s “Line Graph Intuition”
Link: Khan Academy’s “Line Graph Intuition” (HTML)

Instructions: Please click on the link above, and complete this interactive exercise set, which tests your understanding of the concept of slope. Follow the directions on the right side of the panel (you will be asked to graph a line either by dragging points or adjusting values of its slope and intercept). Once you are done, select “Check Answer” to see if your graph is correct or incorrect. If it is incorrect, you will be prompted to try again. If your graph is correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 15 minutes.

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5.4.1.2 Writing Equation of a Line in a Slope-Intercept Form

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Finding Equations of Lines”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Finding Equations of Lines” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This article explains how to find an equation of a line in slope-intercept point when the slope and one point on the line are given, or when two points on the line are given. Practice writing equations by clicking on the “new problem” button at the end of the article, trying a problem, and checking your answer. Attempt to solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Activity: Khan Academy’s “Slope Intercept Form”
Khan Academy’s “Slope Intercept Form” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Here, you have to again write the slope-intercept equation of a line passing through the given points. Enter your values of the slope and the y-intercept into the appropriate tabs on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If your graph is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 45 minutes.

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5.4.2 Point-Slope Form

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Linear Equations in Point-Slope Form”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Linear Equations in Point-Slope Form” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Then, complete practice problems 3–17. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Linear Equations in Point-Slope Form’ Problems 3–17 ” (PDF).

Reading and solving the practice problems will take approximately 1 hour.

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Activity: Khan Academy’s “Point Slope Form”
Link: Khan Academy’s “Point Slope Form” (HTML)

Instructions: Please click on the link above, and complete the exercise set. You will have to write the point-slope equations of a line passing through the given points. Note the following before attempting this exercise set:
·         Notations f(x₁) and f(x₂) are being used here instead of y₁ and y₂.
·         More than one correct answer is possible for each problem.

Enter your values of y₁ and x₁ into the appropriate tabs on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 45 minutes.

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Activity: Khan Academy’s “Converting Between Point-Slope and Slope-Intercept”
Link: Khan Academy’s “Converting Between Point-Slope and Slope-Intercept” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Write the equations given in point-slope form as equations in slope-intercept form. Enter your values of the slope and the y-intercept into the appropriate tabs on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again.If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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5.4.3 Standard Form

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Forms of Linear Equations”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Forms of Linear Equations” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Then,complete practice problems 4–27. Use the “Linear Equations in Standard Form” video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Forms of Linear Equations’ Problems 4–27 ” (PDF).

Reading and solving the practice problems will take approximately 1 hour.

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Activity: Khan Academy’s “Converting Between Slope-Intercept and Standard Form”
Link: Khan Academy’s “Converting Between Slope-Intercept and Standard Form” (HTML)

Instructions: Please click on the link above, and complete the exercise set. You will have to convert the equations in standard form into equations in slope-intercept form and vice versa. Enter your values of A, B, and C or your values for m and b into the appropriate tabs on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 45 minutes.

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Activity: Khan Academy’s “Finding the Equation of a Line”
Link: Khan Academy’s “Finding the Equation of a Line” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It provides more practice writing an equation of a line in any form. For some problems in this set, note the following:
·         Notations f(x₁) and f(x₂) are sometimes used instead of y₁ and y₂.
·         More than one correct answer is possible for some problems.

Enter your answers in the appropriate tabs on the right side of the page. Select “Check answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 40 minutes.

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5.4.4 Writing the Equation of a Line Passing through a Given Point, Parallel or Perpendicular to a Given Line

Note: You have learned about the special relationship between the slopes of parallel lines and perpendicular lines in sub-subunit 5.3.2. Now that you know how write the equations of lines, you can write the equations of the lines parallel or perpendicular to the given line.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Equations of Parallel Lines”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Equations of Parallel Lines” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Then, complete practice problems 2–10. Use the “Linear Equations in Standard Form” video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Equations of Parallel Lines’ Problems 2–10” (PDF).

Reading and solving the practice problems will take approximately 45 minutes.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Equations of Perpendicular Lines”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Equations of Perpendicular Lines” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Then, complete practice problems 3–21. Please note that for problem 13, there is a mistake in the text, and the equation for Line 1 should read as 5y = 3x + 1. Use the “Linear Equations in Standard Form” video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Equations of Parallel Lines’ Problems 2–10 ” (PDF).

Reading and solving the practice problems will take approximately 45 minutes.

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Activity: Khan Academy’s “Equations of Parallel and Perpendicular Lines”
Link: Khan Academy’s “Equations of Parallel and Perpendicular Lines” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Use the given graph and the scratch pad to draw the line parallel or perpendicular to the given. This will help you to eye the values of its slope and y-intercept. Enter your values of m and b into the appropriate tabs on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 45 minutes.

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5.4.5 Application Problems

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Linear Equation Application (Write a Cost Equation)”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Linear Equation Application (Write a Cost Equation)” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is an example of writing a linear equation in slope-intercept form to describe a real-world situation (in this case, a cost of a variable number of items).

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Linear Equation Application (Write a Profit Equation)”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Linear Equation Application (Write a Profit Equation)” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another example of a linear equation in slope-intercept form that describes a real-world situation.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Given a Linear Model, Interpret the Meaning of Slope and Make Predictions”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Given a Linear Model, Interpret the Meaning of Slope and Make Predictions” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In the example discussed in this video, the linear equation of a process (motion of the plane) is given. You will learn how to obtain the information about the motion from this equation.

Watching this video and pausing to take notes should take approximately 15 minutes.

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5.5 Linear Inequalities in Two Variables

Note: Before starting this subunit, spend about 15 minutes briefly reviewing Dr. Burns’ article “Introduction to Equations and Inequalities in Two Variables” assigned in sub-subunit 5.2.1.

5.5.1 Graphing Linear Inequality of Two Variables on the Coordinate Plane

Note: Recall that the solutions of a linear equation or an inequality in two variables are ordered pairs of numbers that make the equation or the inequality a true statement. Both equations and inequalities in two variables have an infinite number of solutions. However, while all solutions of a linear equation belong to one straight line, the solutions of a linear inequality will take up an entire region on a coordinate plane. The following reading explains how to identify this region and show it on the graph.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Inequalities in Two Variables”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Inequalities in Two Variables” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article, watch the videos embedded in the text, and work through Examples A and B. Then, complete practice problems 1–9. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Linear Inequalities in Two Variables’ Problems 1–9

Reading and solving the practice problems should take approximately 1 hour to complete.

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Activity: Khan Academy’s “Graphs of Inequalities”
Link: Khan Academy’s “Graphs of Inequalities” (HTML)

Instructions: Please click on the link above, and complete this interactive exercise set. You will have to write a linear inequality represented by a given graph. Fill in your answer in the appropriate tab on the left side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Activity: Khan Academy’s “Graphing and Solving Linear Inequalities”
Link: Khan Academy’s “Graphing and Solving Linear Inequalities” (HTML)

Instructions: Please click on the link above, and complete the interactive exercise set, which will help you obtain mastery in graphing and interpreting the solutions of the linear inequalities in two variables. Drag the blue dots to move the line to the correct position and select the appropriate shading and line option (solid or dashed) to represent the given inequality. Then, determine whether the given ordered pairs of numbers are the solutions to this inequality. Once you are done, select “Check Answer.” If your graph and answers are incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

This interactive exercise set should take approximately 30 minutes.

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5.5.2 Application Problems

Note: In this sub-subunit, you will learn how graphing linear inequalities can be applied to real-world situations.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Inequalities in Two Variables”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Inequalities in Two Variables” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Scroll down and work through Example C and Guided Practice. Then, complete practice problems 14 and 16. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Linear Inequalities in Two Variables’ Problems 14 and 16 ” (PDF).

Working through the examples and solving the practice problems should take approximately 30 minutes.

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Assessment: Connexions: Wade Ellis and Denny Burzynski’s Graphing Linear Equations and Inequalities in One and Two Variables: “Proficiency Exam”
Link: Connexions: Wade Ellis and Denny Burzynski’s Graphing Linear Equations and Inequalities in One and Two Variables: “Proficiency Exam” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Complete exercises 3–25. This exercise set will allow you to assess your mastery of most of the concepts covered in Unit 5. Click on the “Show Solution” button next to each problem to check your answer.

Completing this assessment should take approximately 1.25 hours.

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Unit 6: Systems of Linear Equations and Inequalities

You have seen in Units 2 and 5 that linear equations in one variable usually have 1 solution and linear equations in two variables have infinitely many solutions. What would happen if two linear equations in two variables had to be solved together? That would mean that a pair of numbers would have to satisfy both equations at the same time. This pair of numbers would be the solution of a system of linear equations. Some of the mixture problems from Unit 3 could be solved by setting up a system of linear equations, because they involve two given relationships between two variables. In this unit, you will learn three different methods of solving systems of linear equations and use them to solve a variety of world problems. You will also find solutions of systems of linear inequalities in two variables and to apply them to real-life situations. For example, if you want to determine the price for and amount of two types of candy for a party, you have the constraints of the total amount of candy needed (GREATER than a given amount) and the amount of money you can spend (LESS than the amount you have). The quantities of two types of candy you buy have to satisfy both constraints. In this unit, you will learn how to identify all the possibilities for similar problems.

Unit 6 Time Advisory show close

Unit 6 Learning Outcomes show close

6.1 Solving Systems of Linear Equations

6.1.1 Solution of a System of Linear Equations

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Introduction to Systems of Equations”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Introduction to Systems of Equations” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You can skim through the beginning of the article, as it reviews the concepts you already know. Focus on the definition of a system of equations and its solution as well as the discussion of how many solutions a system of equations can have. This discussion will also be relevant to the concepts covered in sub-subunit 6.1.6.

Studying this reading should take approximately 20 minutes.

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Lecture: Khan Academy’s “Testing a Solution for a System of Equations”
Link: Khan Academy’s “Testing a Solution for a System of Equations” (YouTube)

Instructions: Please click on the link above, watch the video and take notes. This video demonstrates how to find out whether an ordered pair of numbers is a solution of a given system of equations.

Watching this video and pausing to take notes should take approximately 10 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Graphs of Linear Systems”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Graphs of Linear Systems” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” This article contains practice problems for the concepts you have just learned. Work through Example A, and complete practice problems 9–12. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “
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6.1.2 Solving Systems of Linear Equations by Graphing

Note: In this sub-subunit and throughout the rest of Unit 6, please use graph paper and a ruler to create accurate graphs of equations and inequalities in two variables.

Lecture: Khan Academy’s “Graphing System of Equations”
Link: Khan Academy’s “Graphing System of Equations” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video demonstrates the method of solving systems of linear equations in two variables by graphing.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Graphing System of Equations”
Link: Khan Academy’s “Graphing System of Equations” (HTML)

Instructions: Please click on the link above, and complete this interactive exercise set. Graph the equations in the system by using intercepts or slope-intercept method, whichever is more appropriate. Then, determine the coordinates of the point of intersection, fill in the appropriate tabs on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Graphs of Linear Systems”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Graphs of Linear Systems” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” This article contains more practice problems for solving systems of equations by graphing. Work through Example B and the example in the Guided Practice section, then complete practice problems 13–22. Watch the “Solving Linear Systems by Graphing” video embedded in the text if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “
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6.1.3 Solving Systems of Linear Equations Using the Substitution Method

Lecture: Khan Academy’s “Practice Using Substitution for Systems”
Link: Khan Academy’s “Practice Using Substitution for Systems” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, a system of equation is solved using substitution.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Reading: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Solving Systems Using Substitution”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Solving Systems Using Substitution” (HTML)

Instructions: Please click on the link above, read the introduction and section titled “Example (a system with a unique solution).” This reading provides a step-by-step method of solving any system of linear equations in two variables by substitution. However, as you will see in the rest of the subunit 6.1, some systems are better suited for solving by substitution than others.

Reading these sections should take approximately 15 minutes.

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Activity: Khan Academy’s “Systems of Equations with Substitution”
Link: Khan Academy’s “Systems of Equations with Substitution” (HTML)

Instructions: Please click on the link above, and complete this interactive exercise set. Solve the given system of equations by substitution, then fill in your values of x and y in the appropriate tabs on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move on to the next problem.

Completing this exercise set should take approximately 45 minutes.

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6.1.4 Solving Systems of Linear Equations Using the Elimination Method

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Solving Systems Using Elimination”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Solving Systems Using Elimination” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This reading explains the concepts behind the elimination method and provides different examples of systems that are convenient to solve by elimination. You will see more solved examples and practice problems in the rest of this sub-subunit.

Reading this webpage should take approximately 30 minutes.

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Lecture: Khan Academy’s “Simple Elimination Practice”
Link: Khan Academy’s “Simple Elimination Practice” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, you will see an example of a system that is ideal for solving by elimination.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Systems of Equations with Simple Elimination”
Link: Khan Academy’s “Systems of Equations with Simple Elimination” (HTML)

Please click on the link above, and complete the exercise set. It gives you an opportunity to practice the examples similar to the one you saw in the Khan Academy’s video. Fill in your values for x and y in the appropriate tabs on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 45 minutes.

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Lecture: Khan Academy’s “Example 2: Solving Systems by Elimination”
Link: Khan Academy’s “Example 2: Solving Systems by Elimination” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. The system discussed in this video can also be solved by elimination, but an extra step is required before one of the variables can be eliminated.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Systems with Elimination Practice”
Link: Khan Academy’s “Systems with Elimination Practice” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this example, both equations need to be replaced by equivalent in order to eliminate one of the variables.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Systems of Equations with Elimination”
Link: Khan Academy’s “Systems of Equations with Elimination” (HTML)

Instructions: Please click on the link above, and complete this interactive exercise set. Solve the given system of equations by elimination, then fill in your values of x and y in the appropriate tabs on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move on to the next problem. Refer back to the examples in Dr. Burns’ “Solving Systems Using Elimination” in this sub-subunit as a guide, if you need help.

Completing this exercise set should take approximately 45 minutes.

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6.1.5 Strategy for Solving Systems of Linear Equations: Choosing a Method

Note: You have seen three different methods for solving systems of linear equations in two variables. In this sub-subunit, you will focus on choosing the most efficient method of solving a particular system.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Comparing Methods for Solving Linear Systems”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Comparing Methods for Solving Linear Systems” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the entire article and watch the videos embedded in the text. Focus on the summary of the three methods in the “Guidance” section, which highlights when each method is most appropriate. After reading, complete practice problems 1–6. Try to choose the most efficient method for solving each system.

Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Comparing Methods for Solving Linear Systems’ Problems 1–6 ” (PDF).

Reading, watching videos, and solving the practice problems should take approximately 1 hour to complete.

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Activity: Khan Academy’s “Systems of Equations”
Link: Khan Academy’s “Systems of Equations” (HTML)

Please click on the link above, and complete the exercise set. It provides more practice solving various systems of equations. While Khan Academy suggests a method for solving each system, you might want to think whether this method is most appropriate and possibly choose another one. Fill in your values for x and y in the appropriate tabs on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 1 hour.

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6.1.6 Classifying Systems by the Number of Solutions

Note: Recall from the Dr. Burns’ text in sub-subunit 6.1.1 that the systems of linear equations in two variables can have one, none, or infinitely many solutions. In this sub-subunit, you will explore the last two cases in greater detail.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 7.5: Special Types of Linear Systems”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 7.5: Special Types of Linear Systems” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Also, make sure to watch the “Special Types of Linear Systems” video embedded in the text. Then, complete practice problems 7–24. Keep in mind that you do not have to actually solve a system to classify it as inconsistent or dependent; it is enough to show that slopes of both lines are the same and the y- intercepts are different (in case of inconsistent system) or that the equations in the system are equivalent (in case of dependent system). Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 7.5: Special Types of Linear Systems’ Problems 7–24 ” (PDF).

Reading, watching the video, and solving the practice problems should take approximately 1 hour and 30 minutes.

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Activity: Khan Academy’s “Solutions to Systems of Equations”
Link: Khan Academy’s “Solutions to Systems of Equations” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Use your skills acquired in the previous assignment to determine the number of solutions of each system. Select your answer from the choices given on the right side of the page. Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, it will prompt you to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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6.2 Solving Word Problems by Using Systems of Equations

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Simple Word Problems Resulting in a System Equations”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! Topics in Algebra II: “Simple Word Problems Resulting in a System Equations” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This reading reviews the step-by-step process for solving word problems: finding information in the text of the problem, translating verbal statements into mathematical statements, and solving the resultant system of equations. After reading and working through the example, click on “new problem” at the bottom of the webpage, and solve seven problems. Alternatively, you may generate a worksheet of seven problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The solutions will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 1 hour.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Applications of Linear Systems”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Applications of Linear Systems” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the entire text, and watch the videos embedded in the text. (You may skip Example B, as you have already seen it in sub-subunit 6.1.6.) This reading discusses a variety of real-world situations which can be described by systems of linear equations. Note that some of these systems can be inconsistent or dependent, and you will learn to interpret what this means in the context of the given situation. After reading, complete practice problems 1–10. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Applications of Linear Systems’ Problems 1–10 ” (PDF).

Reading, watching the videos, and solving the practice problems should take approximately 1 hour.

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Activity: Khan Academy’s “Systems of Equations Word Problems”
Link: Khan Academy’s “Systems of Equations Word Problems” (HTML)

Please click on the link above, and complete the exercise set. It contains more word problems that can be solved by setting up and solving a system of equations. Fill in your values for x and y in the appropriate tabs on the right side of the page, and select “Check Answer.” If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 1 hour.

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6.3 Systems of Linear Inequalities

6.3.1 Graphing Systems of Linear Inequalities

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 7.6: Systems of Linear Inequalities”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 7.6: Systems of Linear Inequalities” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article until the section “Linear Programming – Real-World Systems of Linear Inequalities.” Scroll down to the section “Practice Problems,” and watch “Systems of Linear Inequalities” video embedded in the text. Then, complete practice problems 6–16. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 7.6: Systems of Linear Inequalities’ Problems 6–16 ” (PDF).

Reading, watching the video, and solving the practice problems should take approximately 1 hour and 30 minutes.

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Activity: Khan Academy’s “Graphing Systems of Inequalities”
Link: Khan Academy’s “Graphing Systems of Inequalities” (HTML)

Instructions: Please click on the link above, and complete the interactive exercise set. Drag the points to move each line to correct position and select appropriate shading and line option (solid or dashed) to represent each inequality. Once you are done, select “Check Answer.” If your graph is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

This interactive exercise set should take approximately 30 minutes.

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Activity: Khan Academy’s “Graphing and Solving Systems of Inequalities”
Link: Khan Academy’s “Graphing and Solving Systems of Inequalities” (HTML)

Instructions: Please click on the link above, and complete the interactive exercise set, which is similar to the one in the previous assignment and which will help you obtain mastery in graphing and interpreting the solutions of the systems of inequalities. Besides graphing the systems as described in the instructions to the previous assignment, you will also need to determine whether the given ordered pairs of numbers are the solutions to each system. Once you are done, select “Check Answer.” If your graph and answers are incorrect, you will be prompted to try again. If they are correct, you can click on “Correct! Next Question” to move to the next problem.

This interactive exercise set should take approximately 30 minutes.

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6.3.2 Applications of Systems of Linear Inequalities

Note: Linear programming is a process of analyzing a real-world situation by using a system of several linear inequalities.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Programming”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Linear Programming” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the entire article, and watch the videos embedded in the text. Then, complete practice problems 1–11. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Linear Programming’ Problems 1–11 ” (PDF).

Reading, watching the videos, and solving the practice problems should take approximately 1 hour and 30 minutes.

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Assessment: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 7: Systems of Equations and Inequalities”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: “Chapter 7: Systems of Equations and Inequalities” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Scroll down to the section “System of Equations and Inequalities; Counting Methods: Review,” and complete the odd-numbered problems for 3–41, but do not complete problem 37. This set of practice problems will allow you to assess your mastery of the concepts in Unit 6. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra Concepts: ‘Chapter 7: Systems of Equations and Inequalities’Problems 3–41 Odd” (PDF).

This assessment should take approximately 1 hour to complete.

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Unit 7: Operations with Monomials

Starting with this unit, you will work with expressions that consist mostly of letters (variables) and do not seem to have very much to do with numbers. You will see how the rules governing operations with these expressions arise from the properties of operations with numbers, particularly distributive property and order of operations. This unit focuses on expressions called monomials. These are expressions that contain only one term (recall from Unit 1 what “term” means). The fact that monomial contains the Greek word mono, which means one, can help you remember this definition. For example, expression ab is a monomial, but a + b is not, because it contains two terms.

Unit 7 Time Advisory show close

Unit 7 Learning Outcomes show close

7.1 Algebraic Exponential Expressions

Note: You are already familiar with the concepts of exponents and powers from arithmetic. In this subunit, you will review it, emphasizing the odd and even powers of signed (positive or negative) numbers. Then, the concept of exponents will be applied to variables and variable expressions.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 8.1: Exponential Properties Involving Products”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 8.1: Exponential Properties Involving Products” (HTML)

Instructions: Please click on the link above to open the webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article until the section titled “Product of Powers Property.” You will read the remainder of the chapter in sub-subunit 7.2.1. For this subunit on algebraic exponential expressions, complete practice problems 1–13. The definition of a power will be used in the following subunit to derive all of the rules you need to work with exponential expressions. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 8.1: Exponential Properties Involving Products’ Problems 1–13 ” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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Lecture: YouTube: Flat World Knowledge: John Redden’s Real Numbers and Their Operations: “Simplify Even Exponents – Algebra”
Link: YouTube: Flat World Knowledge: John Redden’s Real Numbers and Their Operations: “Simplify Even Exponents – Algebra” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video points out the difference in exponential expressions such as (-2)⁴ and -2⁴ and explains why the first one equals a positive number while the second one equals a negative.

Watching this video and pausing to take notes should take approximately 15 minutes.

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7.2 Rules of Exponents

7.2.1 Product of Exponents, Power of Exponent, and Power of a Product

Note: In this sub-subunit, you will learn about these three rules of exponential expressions:

to multiply powers of the same base, ADD the exponents;

to raise a power to another power, MULTIPLY the exponents; and

to raise a product to a power, EACH factor of the product has to be raised to this power.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ “Chapter 8.1: Exponential Properties Involving Products”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ “Chapter 8.1: Exponential Properties Involving Products” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article, starting with the section titled “Product of Powers Property.” Then, complete practice problems 28–43. Watch the “Exponent Properties Involving Products” video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 8.1: Exponential Properties Involving Products’ Problems 28–43 ” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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7.2.2 Quotient of Exponents and Power of a Quotient

Note: In this sub-subunit, you will learn about the following two rules of exponential expressions:

to divide powers of the same base, SUBTRACT the exponents; and

to raise a quotient to a power, BOTH the numerator and denominator have to be raised to this power.

Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 8.2: Exponential Properties Involving Quotients”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 8.2: Exponential Properties Involving Quotients” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” After reading the article, complete the practice problems 1–26. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 8.2: Exponential Properties Involving Quotients’ Problems 1–26 ” (PDF).

Reading and solving the practice problems should take approximately 1 hour to complete.

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7.2.3 Negative Exponents

Note: In this sub-subunit, the definition of a power will be extended to include negative exponents. While this might seem counterintuitive (one cannot multiply a number by itself a negative number of times), there is a way to raise a number to a negative power by taking a reciprocal of the same number raised to a positive power. You will learn why this definition makes sense and learn to apply it to simplify exponential expressions.

Lecture: Khan Academy’s “Negative Exponent Intuition”
Link: Khan Academy’s “Negative Exponent Intuition” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan explains why the reasoning behind the definition of a negative power as a reciprocal of a positive power. This lecture will also help you understand why the rules of exponents that you have learned so far apply to the negative exponents as well.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Reading: Connexions: Wade Ellis and Denny Burzynski’s Basic Operations with Real Numbers: “Negative Exponents”
Link: Connexions: Wade Ellis and Denny Burzynski’s Basic Operations with Real Numbers: “Negative Exponents” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. This article provides a lot of practice applying the definition of negative exponents and simplifying expressions containing negative exponents. Using Sample Sets A, B, and C for guidance, complete the exercises in Practice Sets A, B, and C. Then, complete exercises 33–48. You may access the solutions to the problems by clicking on the “Show Solution” link next to the problem.

Reading and completing the exercises should take approximately 2 hours.

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7.3 Simplifying Power Expressions

Note: In this subunit, you will use all of the rules that you have learned so far to simplify all kinds of expressions, containing both positive and negative exponents. The goal of simplifying is to rewrite an expression without any parentheses or negative exponents.

Lecture: Khan Academy’s “Exponent Properties 5”
Link: Khan Academy’s “Exponent Properties 5” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video provides a detailed explanation of an example of taking a power of a product.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Exponent Properties 6”
Link: Khan Academy’s “Exponent Properties 6” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another example of simplifying exponential expressions, but this time the example involves a quotient.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Exponent Properties 7”
Link: Khan Academy’s “Exponent Properties 7” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This example is more complicated, as it involves raising a product to a negative power.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Larry Green’s “Practice with Exponents”
Link: Larry Green’s “Practice with Exponents” (HTML)

Instructions: Please click on the link above, and read the instructions on how to use this interactive applet. Solve five problems correctly at the “Middle” level, and then move up to “Difficult” and correctly solve 5 more problems. Use the “Hint” button if you are not sure where to start. The link “Information about Exponent Rules” at the bottom of the page provides a review and examples of all the relevant rules of exponents.

Solving the practice problems should take approximately 30 minutes.

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Activity: Khan Academy’s “Simplifying Expressions with Exponents”
Link: Khan Academy’s “Simplifying Expressions with Exponents” (HTML)

Instructions: Please click on the link above, and complete the exercise set. This is another opportunity to review and practice simplifying exponential expressions. Once you determine the resultant exponents of each variable in the given expression, enter them in the appropriate tabs on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Assessment: Connexions: Wade Ellis and Denny Burzynski’s Basic Operations with Real Numbers: “Negative Exponents”
Link: Connexions: Wade Ellis and Denny Burzynski’s Basic Operations with Real Numbers: “Negative Exponents” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to the Exercises section, and complete exercises 133–143. Click on the “Show Solution” button for each problem to check your answer.

Completing this assessment should take approximately 30 minutes.

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7.4 Multiplying and Dividing Monomials

Note: Monomials, as you will learn in this subunit, are expressions composed of a product of a number and variables in positive exponents. Multiplying and dividing monomials are the first steps in performing operations with polynomials, which will be discussed in Unit 8.

7.4.1 Multiplying Monomials

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Multiplying Monomials”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Multiplying Monomials” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this video, Dr. Sousa explains how to multiply monomials using the rules of exponents.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Multiplication of Monomials by Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Multiplication of Monomials by Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the section titled “Guidance,” and work through Example A. Then, complete practice problems 1–5. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Multiplication of Monomials by Polynomials’ Problems 1–5 ” (PDF).

Reading and solving the practice problems should take approximately 15 minutes.

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7.4.2 Monomial Division

Reading: The Saylor Foundation’s “Monomial Division”
Link: The Saylor Foundation’s “Monomial Division” (PPT)

Instructions: Read these examples and practice problems on monomial division. The answers to the practice problems are provided at the end of the presentation.
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Unit 8: Operations with Polynomials

In this unit, you will become familiar with a special type of algebraic expressions, called a polynomial. A polynomial, as opposed to a monomial, is an expression that contains two or more terms. (The word poly means many in Greek.) Usually the polynomials you will work with will look like x⁵ + 2x³ + x + 2, or similar. Polynomials have various special properties that you will be analyzing in the future math courses. In this course, you will learn how to recognize, classify, add, subtract, multiply, and divide polynomials. You will apply the skills of combining like terms and using distributive property in order to perform these operations. These skills are helpful when you are dealing with the motion of two or more objects: for example, when you need to calculate when and where one runner will overtake another runner. For another example, these skills are useful if you need to know how much interest you are earning from two or more savings accounts.

Unit 8 Time Advisory show close

Unit 8 Learning Outcomes show close

8.1 Classification of Polynomials

Note: In this subunit, you will be introduced to a certain kind of algebraic expression called a polynomial expression. You will learn how to identify polynomials and classify them according to their degree and number of terms.

Reading: Connexions: Wade Ellis and Denny Burzynski’s Elementary Algebra: “Algebraic Expressions and Equations: Classification of Expressions and Equations”
Link: Connexions: Wade Ellis and Denny Burzynski’s Elementary Algebra: “Algebraic Expressions and Equations: Classification of Expressions and Equations” (HTML)

Instructions: Please click on the link above. You may also download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Read the webpage until the section titled “Classification of Polynomial Equations.” Note that while it is not necessary to memorize all the new vocabulary words you will encounter, you should know their meaning, as they will be used often in all the materials in this course from now on. After reading and working through the examples, scroll down to the “Exercises” section and complete exercises 10–34. The solutions to the exercises are shown directly below each problem.

Reading, taking notes, and completing the exercises should take approximately 1 hour.

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Lecture: Khan Academy’s “Polynomials 2”
Link: Khan Academy’s “Polynomials 2” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This short video shows that the total value of a variable number of $20, $10, and $5 bills is a polynomial expression. One can substitute the number of each type of bills into this polynomial to calculate the total amount of money.

Watching this video and pausing to take notes should take approximately 15 minutes.

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8.2 Addition and Subtraction of Polynomials

8.2.1 Horizontal Format

Lecture: Khan Academy’s “Addition and Subtraction of Polynomials”
Link: Khan Academy’s “Addition and Subtraction of Polynomials” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, you will see examples of polynomial expressions of various degrees and number of terms. Then, you will work through two examples of adding and subtracting two polynomials. You will notice that all you have to do to add or subtract polynomials horizontally is open the parentheses (in case of subtraction) and combine like terms. This video also contains an example of applications of polynomials in geometry.

Watching this video and pausing to take notes should take approximately 30 minutes.

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8.2.2 Vertical Format

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Adding and Subtracting Polynomials”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Adding and Subtracting Polynomials” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this video, adding and subtracting polynomials is performed in both horizontal and vertical formats. Note that in the vertical format, the terms of the same degree (or like terms) are aligned one under another, much like the digits of the same place value are aligned in addition or subtraction of decimals and large numbers.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Adding Polynomials with Multiple Variables”
Link: Khan Academy’s “Adding Polynomials with Multiple Variables” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, two polynomials, each containing two variables, are added vertically. Note how like terms are aligned under each other and the rest of the terms are simply rewritten as they cannot be combined.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Addition and Subtraction of Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Addition and Subtraction of Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Scroll down to the section titled “Practice Problems,” and complete problems 1–10. Use either horizontal or vertical format of adding/subtracting. Make sure to write the result as a polynomial in standard form. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Addition and Subtraction of Polynomials’ Problems 1–10 ” (PDF).

Solving the practice problems should take approximately 15 minutes.

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Activity: Connexions: Wade Ellis and Denny Burzynski’s Elementary Algebra: “Algebraic Expressions and Equations: Combining Polynomials Using Multiplication”
Link: Connexions: Wade Ellis and Denny Burzynski’s Elementary Algebra: “Algebraic Expressions and Equations: Combining Polynomials Using Multiplication” (HTML)

Instructions: Please click on the link above. You may also download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Scroll down to Sample Set D, and work through examples 18–21. Then, complete exercises 25–28 from Practice Set D. To check your answers, click on “Show Solution” link next to each problem.

Completing the examples and exercises should take approximately 15 minutes.

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8.3 Multiplying Polynomials

8.3.1 Multiplying Polynomial by a Monomial

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Multiplication of Monomials by Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Multiplication of Monomials by Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.”Scroll down to “Example B,” and work through Example B, Example C, and Guided Practice. Watch the videos embedded in the text, if you need help with these examples. Then, complete practice problems 6–11. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Multiplication of Monomials by Polynomials’ Problems 6–11 ” (PDF).

Working through the examples and solving the practice problems should take approximately 30 minutes.

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8.3.2 Multiplying a Polynomial by a Polynomial

Note: When you multiply polynomials, you have to distribute each term of the first polynomial over the second polynomial, which means you have to multiply each term of the first polynomial by each term of the second polynomial. The resources in this sub-subunit will show you various techniques for keeping track of all the resultant terms. The multiplication result will usually contain like terms, which can be combined.

8.3.2.1 Multiplying Binomials (FOIL)

Note: When you multiply a binomial by a binomial (two terms by two terms), your result will have to have four terms. Sometimes two of these terms will be alike, and you will have to combine them to simplify the result. Depending on the binomials you are multiplying, the final result might be a binomial, a trinomial, or a four-term polynomial.

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Basic FOIL”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Basic FOIL” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This reading explains a mnemonic for multiplying binomials, known as FOIL, which stands for First, Outer, Inner, Last, the order in which the terms are multiplied. Practice the basic binomial multiplication problems on this webpage in order to gain mastery and to be able to do these types of problems quickly.

Click on the “new problem” button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on “new problem” five times. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 30 minutes.

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Lecture: Khan Academy’s “Multiplying Binomials”
Link: Khan Academy’s “Multiplying Binomials” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan uses the distributive property to multiply two binomials and then shows that if the order of the binomials in the multiplication problem is switched, then the result remains the same.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “More Complicated FOIL”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “More Complicated FOIL” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Here, you will find slightly more complicated binomial multiplication examples. Click on the “new problem” button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on “new problem” five times. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 30 minutes.

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8.3.2.2 Special Products of Binomials

Note: Special products of binomial are the results of the following multiplication:

Multiplying a binomial by itself (in other words, squaring it):

(a ± b)² = a² ±2ab +b²

A trinomial in the form of a trinomial on the right-hand side of the above formula is known as a complete square trinomial.

Multiplying the binomials that are the sum and difference of the same terms:

(a – b)(a + b) = a² – b²

A binomial in the form of a binomial on the right-hand side of the above formula is known as a difference of two squares.

These formulas, sometimes also called classic quadraticformulas, are important to remember. Recognizing when they can be applied is a necessary skill for this and many other higher-level math courses.

8.3.2.2.1 Complete Square

Lecture: Khan Academy’s “Square a Binomial”
Link: Khan Academy’s “Square a Binomial” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan performs the multiplication of a binomial by itself twice: once using FOIL and once using a classic quadratic formula (which he also derives). This will help you understand why this formula always works.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Simplifying (a+ b)2 and (a– b)2”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Simplifying (a+ b)² and (a– b)²” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This reading provides practice with squaring simple binomials. Click on the “new problem” button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on “new problem” five times. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 15 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.3: Special Product of Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.3: Special Product of Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read this entire webpage, which contains examples of squaring all kind of binomials. Scroll down to the Practice Set, and complete exercises 1–13. Use the “Special Products of Binomials” video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 9.3: Special Product of Polynomials’ Problems 1–13” (PDF).

Solving the practice problems should take approximately 30 minutes.

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8.3.2.2.2 Difference of Two Squares

Lecture: Khan Academy’s “Special Polynomial Products 1”
Link: Khan Academy’s “Special Polynomial Products 1” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan performs the multiplication of a binomial by the binomial with same terms but the opposite sign between them twice: once using FOIL and once using a classic quadratic formula (which he also derives). This video will help you understand why this formula always works.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.3: Special Product of Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.3: Special Product of Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Note that you have already come across this reading in sub-subunit 8.3.2.2.1. Briefly review this webpage, which contains examples of applying the difference of two squares formula to all kind of binomials. Scroll down to the Practice Set, and complete exercises 14–23. Use the “Special Products of Binomials” video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 9.3: Special Product of Polynomials’ Problems 14–23 ” (PDF).

Solving the practice problems should take approximately 30 minutes.

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8.3.2.3 Multiplying Polynomials with Any Number of Terms

Lecture: Khan Academy’s “Multiplying Polynomials”
Link: Khan Academy’s “Multiplying Polynomials” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video contains an example of multiplication of a binomial of a trinomial. Note the method used to make sure each term of one polynomial is multiplied by each term of another.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Multiplication of Polynomials”
Link: Khan Academy’s “Multiplication of Polynomials” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video serves as a review of the subunit on polynomial multiplication as it contains a variety of examples done in different ways.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.2: Multiplication of Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.2: Multiplication of Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Scroll down to the Practice Set, and solve problems 7–21. This set of problems contains exercises involving multiplying polynomials with any number of terms. Some of the problems involve multiplying three polynomials. In this case, multiply any two of the three polynomials first and then multiply the third polynomial by the simplified result. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 9.2: Multiplication of Polynomials’ Problems 7–21” (PDF).

Solving the practice problems should take approximately 45 minutes.

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Assessment: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.9: Chapter 9 Review”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.9: Chapter 9 Review” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Complete exercises 13–28. This exercise set will allow you to assess your mastery of classification and operations with polynomials (except for division). Once you have completed the practice problems, check your answers against the Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 9.9: Chapter 9 Review’ Problems 13–28” (PDF).

This assessment should take approximately 1 hour to complete.

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8.4 Dividing Polynomials

Note: Because polynomials can be multiplied, it follows that they can also be divided. The quotient, or the result of the division, of two polynomials, can be defined the same way as a quotient of two real numbers: it is a polynomial such that if it is multiplied by the divisor, then the result is the original polynomial. To find such a polynomial by guessing and checking, however, is not practical. In this subunit, you will learn the technique for dividing two polynomials. You will find that it is somewhat similar to long division of real numbers.

8.4.1 Dividing a Polynomial by a Monomial

Reading: Connexions: Wade Ellis and Denny Burzynski’s Elementary Algebra: “Rational Expressions: Dividing Polynomials”
Link: Connexions: Wade Ellis and Denny Burzynski’s Elementary Algebra: “Rational Expressions: Dividing Polynomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF version of this text by clicking on the appropriate link at the bottom of the page. Read the section titled “Dividing a Polynomial by a Monomial,” work through the examples in Sample Set A, and complete the exercises in Practice Set A. The solutions to the exercises are shown directly below each problem.

Reading this section and solving the practice problems should take approximately 30 minutes.

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Lecture: Khan Academy’s “Polynomial Divided by Monomial”
Link: Khan Academy’s “Polynomial Divided by Monomial” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows an example of dividing a polynomial by a monomial.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Dividing Multivariable Polynomial with Monomial”
Link: Khan Academy’s “Dividing Multivariable Polynomial with Monomial” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is another example of division of a polynomial by a monomial, but two variables are involved.

Watching this video and pausing to take notes should take approximately 15 minutes.

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8.4.2 Dividing a Polynomial by a Binomial

Lecture: Khan Academy’s “Dividing Polynomials 1”
Link: Khan Academy’s “Dividing Polynomials 1” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows an example of dividing a quadratic trinomial by a binomial. Note that the result can be checked by multiplication, just like as the result of the division of real numbers.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Polynomial Division”
Link: Khan Academy’s “Polynomial Division” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video introduces the method of dividing polynomials known as long division. Sal Khan first tries out this method by dividing a simple binomial by a monomial, which you already know how to do. Then, he generalizes the procedure in order to divide a trinomial by a binomial. Note that he checks his result later by factoring, a procedure you will learn about in Unit 9. You will see more examples of long division in the next two Khan Academy videos as well.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Lecture: Khan Academy’s “Algebraic Long Division”
Link: Khan Academy’s “Algebraic Long Division” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows an example of dividing a cubic four-term polynomial by a binomial. This time, the long division method produces a remainder.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Activity: Connexions: Wade Ellis and Denny Burzynski’s Rational Expressions: “Dividing Polynomials”
Link: Connexions: Wade Ellis and Denny Burzynski’s Rational Expressions: “Dividing Polynomials” (HTML)

Instructions: Please click on the link above. You may read the HTML page, or you can download a PDF version of this text by clicking on the appropriate link at the bottom of the page. Scroll down to Sample Set B, work through examples 4 and 5, and then work through example 6 in Sample Set C. (Note that example 6 focuses on dividing a polynomial where one of the coefficients is zero.) Then, complete the exercises 6–9 in Practice Set B and exercises 10–13 in Practice Set C. The solutions to the exercises are shown when you click on the “Show Solution” link.

Solving the practice problems should take approximately 1 hour.

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Assessment: Connexions: Kenny M. Felder’s Advanced Algebra II: “Rational Expressions Homework – Dividing Polynomials”
Link: Connexions: Kenny M. Felder’s Advanced Algebra II: “Rational Expressions Homework – Dividing Polynomials” (HTML)

Instructions: Please click on the link above to open the webpage. You may work from the HTML page, or you can download a PDF version of this exercise set by clicking on the appropriate link at the bottom of the page. Complete exercises 1–8. Pay careful attention to the directions in each exercise, as they are not all the same. Once you have completed the exercises, check your answers against The Saylor Foundation’s “Answer Key to Connexions: Kenny M. Felder’s Advanced Algebra II: ‘Rational Expressions Homework – Dividing Polynomials’ Problems 1–8” (PDF).

This assessment should take approximately 45 minutes.

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Unit 9: Factoring Polynomials

Factoring, the procedure you will learn to perform in this unit, is multiplication in reverse: instead of multiplying two polynomials, you will need to write a given polynomial as a product of two or more different expressions. Factoring is an important tool that you will use in solving quadratic, and, later, higher degree polynomial equations. Quadratic equations occur a lot in problems that involve rectangular objects and their areas: planning gardens, framing photographs, carpeting the floors, and so on.

Unit 9 Time Advisory show close

Unit 9 Learning Outcomes show close

9.1 Factoring Monomial from a Polynomial

9.1.1 Identifying Greatest Common Factor (GCF) of Two or More Monomials and of Other Expressions

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Recognizing Products and Sums; Identifying Factor and Terms”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Recognizing Products and Sums; Identifying Factor and Terms” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Reading and understanding algebraic expressions, much like translating sentences from a foreign language, is a skill that takes time to develop. Prior to learning how to factor an algebraic expression (that is, to write the expression as a product), it is essential to have fluency in distinguishing which parts of the expressions are multiplied and which are added/subtracted. This reading provides a useful review of how to identify which algebraic expression is a product of several factors and which one is a sum of several terms.

Click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Identifying Common Factors”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Identifying Common Factors” (HTML)

Instructions: Please click on the link above, and read the entire webpage. This set of examples provides practice with identifying common factors, both monomial and binomial, in algebraic expressions. Click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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9.1.2 Rewriting Polynomial as a Product of a Monomial and a Different Polynomial

Note: Now that you have practice identifying common factors, you can begin to factor polynomials by rewriting them as a product of a common factor and another polynomial. This process, as Dr. Burns points out below in “Factoring Simple Expressions,” is the result of the distributive property of real numbers applied in the opposite direction: instead of multiplying a number by a parenthesis, you will be writing the given result as a product of a number and a parenthesis.

Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Simple Expressions”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Simple Expressions” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Note that some examples involve factoring out a common binomial factor. This reading also covers the topics outlined for subunit 9.2, including sub-subunits 9.2.1 and 9.2.2.

Try a few simple factoring problems yourself: click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Activity: Khan Academy’s “Factoring Linear Binomials”
Link: Khan Academy’s “Factoring Linear Binomials” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Determine if the terms of the given expression have a common factor. If they do, write the factored expression in the tab on the right side of the page. If not, your answer will be the same as the original expression. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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Reading: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Monomial Factors of Polynomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Monomial Factors of Polynomial” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the section titled “Finding the Greatest Common Binomial Factor.” Work through the examples and Guided Practice, and complete practice problems 1–10. Watch the “Polynomial Equations in Factored Form” video embedded in the text, if you need help. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Monomial Factors of Polynomials’ Problems 1–10” (PDF).

Note that all terms in the polynomials that you have encountered in this subunit have a common factor. If there is no factor (other than 1) common to all of the terms, then the polynomial cannot be written as a product of a monomial and a different polynomial.

Reading and solving the practice problems should take approximately 30 minutes.

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9.2 Factoring Polynomials by Grouping

Note: This subunit focuses on the factoring of four-term polynomials that can be factored by grouping (that is, grouping the terms into 2 pairs). Not all polynomials can be factored this way. This technique will be applied to factoring quadratic trinomials later in sub-subunit 9.3.2.2.

9.2.1 Factoring out a Common Binomial Factor

Note: Review Dr. Burns’ “Factoring Simple Expressions” reading assigned in sub-subunit 9.1.2, focusing on the problems that require factoring out a common binomial factor. You should spend approximately 15 minutes reviewing this material.

9.2.2 Factoring Four-Term Polynomials by Grouping

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Factor by Grouping”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Factor by Grouping” (YouTube)

Instructions: Please click on the link above, watch the first 6 minutes of the video, and take notes. Note the steps taken to factor a four-term polynomial.
- Separate the polynomial into two groups of two terms.
- Identify a common factor in each of the group and factor it out.
- Check that the resultant expressions contain a common binomial factor. If they do not, then the polynomial cannot be factored, at least not when the terms are grouped this way.
- Factor out a common binomial factor and rewrite the polynomial as a product of two binomials.
Watching this video and taking notes should take approximately 30 minutes.

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Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.7: Factoring Polynomials Completely”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: “Chapter 9.7: Factoring Polynomials Completely” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Scroll down to the section titled “Factoring by Grouping,” and work through Example 3. Then, scroll down to the Practice Set and solve problems 11–15, 17, 18, and 23–27. All of these problems are four-term polynomials that can be factored by grouping. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Basic Algebra: CK12 Editors’ ‘Chapter 9.7: Factoring Polynomials Completely’ Problems 11–15, 17, 18, 23–27” (PDF).

Working through the example and solving the practice problems should take approximately 45 minutes.

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9.3 Factoring Quadratic Trinomials

Note: Factoring quadratic trinomials is an important algebraic tool. In this course, you will use this tool to solve quadratic equations in subunit 9.6.

9.3.1 Factoring Trinomials of the Form x^2 + bx + c

9.3.1.1 Revisiting FOIL: Working Backwards

Note: You have seen that a polynomial can be written as a product of a monomial and a different polynomial due to distributive property. In this sub-subunit, you will review the FOIL procedure for multiplying two binomials in order to analyze how it can be reversed in order to write the given result (a trinomial) as a product of two binomials.

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Basic Concepts Involved in Factoring Trinomials”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Basic Concepts Involved in Factoring Trinomials” (HTML)

Instructions: Please click on the link above, and read the entire webpage. You may skip the section titled “Key Ideas for Finding Numbers that Work.”

Reading this section and taking notes should take approximately 15 minutes.

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9.3.1.2 Factoring Trinomials of the Form x^2 + bx + c When c Is Positive

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form x^2+ bx+ c, Where c > 0”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form x²+ bx+ c , Where c > 0” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Note that in one of the examples the trinomial cannot be factored. You will learn how to identify such trinomials in sub-subunit 9.3.4. Then, try factoring trinomials yourself: click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Lecture: Khan Academy’s “Factoring Trinomials with a Leading 1 Coefficient”
Link: Khan Academy’s “Factoring Trinomials with a Leading 1 Coefficient” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. Sal Khan uses both inverse FOIL reasoning and factoring by grouping to factor a quadratic trinomial of the form x² + bx + c with positive c.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Factoring Polynomials 1”
Link: Khan Academy’s “Factoring Polynomials 1” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Factor each trinomial and enter the result in the tab on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to moveto the next problem.

Completing this exercise set should take approximately 30 minutes.

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9.3.1.3 Factoring Trinomials of the Form x^2 + bx + c When c Is Negative

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form x^2 + bx+ c, Where c < 0”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form x²+ bx+ c, Where c < 0” (HTML)

Instructions: Please click on the link above, and read the entire webpage. Note that in one of the examples the trinomial cannot be factored. You will learn how to identify such trinomials in sub-subunit 9.3.4. Then, try factoring trinomials yourself: click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Factoring a Trinomial with Leading Coefficient of 1 – the Basics”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Factoring a Trinomial with Leading Coefficient of 1 – the Basics” (YouTube)

Instructions: Please click on the link above, watch the first 7 minutes of video (introduction and four examples), and take notes. This video reviews the procedure for factoring quadratic trinomials of the form x² + bx + c.

Watching this video and taking notes should take approximately 30 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Factor Trinomials When A Equals 1”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Factor Trinomials When A Equals 1” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows four examples of factoring polynomials of the form ax² + bx + c.

Watching this video and taking notes should take approximately 15 minutes.

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9.3.2 Factoring Trinomials of the Form ax^2 + bx + c

Note: There are two methods traditionally used to factor quadratic trinomials of the form ax² + bx + c: using trial factors (also known as guess and check) and by grouping (also known as ac, or bridge method). You will try out both methods and can use a preferred method in the future. You will also develop a sense of when a trinomial can be factored instantly by guessing and when this would take a long time and factoring by grouping is a better choice.

9.3.2.1 Using Trial Factors (“Guess and Check” Method)

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form ax^2 + bx+ c”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form ax²+ bx+ c” (HTML)

Instructions: Please click on the link above, and read the webpage until the section titled “Example: ‘Factor by Grouping’ Method.” Try factoring trinomials using this method: click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve five problems. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

While Dr. Burns points out in this reading that listing all trial factors and checking their products can be tedious, some trinomials can be factored fairly quickly using this method. For example, if either a or c or both are prime, their only factors are 1 and itself, and this limits the number of trial factors. Also, if c is positive, both binomial factors will have to contain the same sign, and this limits the number of trial factors as well.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Factoring Trinomials: Trial and Error and Grouping”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Factoring Trinomials: Trial and Error and Grouping” (YouTube)

Instructions: Please click on the link above, watch the first five minutes of the video (introduction and two examples), and take notes. This video explains how to factor the trinomials of the form ax² + bx + c by guessing and checking.

Watching this video and taking notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Factor Trinomials When A Is NOT Equal to 1 - Trial and Error Method”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex: Factor Trinomials When A Is NOT Equal to 1 - Trial and Error Method” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows two more examples of factoring the trinomials of the form ax² + bx + c by guessing and checking.

Watching this video and taking notes should take approximately 15 minutes.

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9.3.2.2 Factoring by Grouping (Bridge or ac Method)

Reading: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form ax^2 + bx+ c”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Factoring Trinomials of the Form ax²+ bx+ c” (HTML)

Instructions: Note that you have already come across this resource in sub-subunit 9.3.2.1, but you should revisit this resource again for review and additional practice. Please click on the link above, and read the section titled “Example: ‘Factor by Grouping’ Method.” After working through the example, try factoring trinomials using this method yourself: click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem” and solve five problems. You can also create a worksheet of five problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Activity: Larry Green’s “The AC Method”
Link: Larry Green’s “The AC Method” (HTML)

Instructions: Please click on the link above. This interactive applet will guide you through factoring trinomials by grouping one step at a time. Try to factor at least five trinomials or more. You can also click on “Information on the AC method” at the bottom of the page and scroll down to the section titled “The AC Method” to review the procedure for the method, examples, and more exercises.

Studying this applet and solving the problems should take approximately 30 minutes.

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Lecture: Khan Academy’s “Factoring Trinomials with a non-1 Leading Coefficient by Grouping”
Link: Khan Academy’s “Factoring Trinomials with a non-1 Leading Coefficient by Grouping” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows an example of factoring a trinomial by grouping.

Watching this video and pausing to take notes should take approximately 15 minutes.

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9.3.2.3 When a Is Negative

Activity: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Factorization of Quadratic Expressions with Negative Coefficients”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Factorization of Quadratic Expressions with Negative Coefficients” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Work through Example E. In this example, the issue of factoring a trinomial with a = -1 is avoided by factoring (-1) out. Then, the resultant trinomial has the form of x2 + bx + c and can be factored as such. Then, work on practice problems 13–16 and 18. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Factorization of Quadratic Expressions with Negative Coefficients’ Problems 13–16, and 18” (PDF).

Reading and solving the practice problems should take approximately 30 minutes.

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Lecture: CK-12’s Algebra Concepts: “Factoring ax2 + bx + c Trinomials (a Is Negative) - Overview”
Link: CK-12’s Algebra Concepts: “Factoring ax² + bx + c Trinomials (a Is Negative) - Overview” (HTML)

Instructions: Please click on the link above, watch the video, and take notes. In this video, a quadratic trinomial with negative a is factored by grouping. If it seems a bit confusing to do it this way, you can avoid it by factoring (-1) out, as was done in the previous example. Then, the resultant trinomial can be factored either using trial factors or by grouping, and a negative sign will remain in front of the final product.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Factoring Polynomials by Grouping”
Link: Khan Academy’s “Factoring Polynomials by Grouping” (HTML)

Instructions: Please click on the link above, and complete the exercise set to practice factoring various trinomials by grouping. Enter the factored expression in the tab on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 40 minutes.

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9.3.3 Factoring Trinomials of the Form x^2 + bxy + cy^2 and ax^2 + bxy + cy^2

Lecture: YouTube: Flat World Knowledge: John Redden’s Factoring and Solving by Factoring: “Factor a Trinomial – Algebra”
Link: YouTube: Flat World Knowledge: John Redden’s Factoring and Solving by Factoring: “Factor a Trinomial – Algebra” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. You will see an example of a trinomial containing two variables being factored. Note that the inverse FOIL reasoning can still be applied to determine how the binomial factors are going to look like.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: Flat World Knowledge: John Redden’s Factoring and Solving by Factoring: “Factor a Trinomial – Algebra”
Link: YouTube: Flat World Knowledge: John Redden’s Factoring and Solving by Factoring: “Factor a Trinomial – Algebra” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. You will see another example of a trinomial containing two variables being factored. Inverse FOIL reasoning, together with the method of trial factors, is again applied to determine how the binomial factors are going to look like.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Factoring Polynomials with Two Variables”
Link: Khan Academy’s “Factoring Polynomials with Two Variables” (HTML)

Instructions: Please click on the link above, and complete the exercise set. Solve the problem and type your answer in the answer tab. Select “Check Answer” to see if you got the answer correct or incorrect. If you get the answer incorrect, it will prompt you to try again. Once you get the answer correct, you can click on “Correct! Next Question” to move on to the next problem.

This activity should take approximately 30 minutes.

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9.3.4 Prime Trinomials

Reading: The Saylor Foundation’s “Prime Trinomials”
Link: The Saylor Foundation’s “Prime Trinomials” (PDF)

Instructions: Read this brief example of prime trinomials, and then attempt the practice problems on the third page. When you have finished, you may check your answers against the Saylor Foundation’s “Answer Key to Prime Trinomials Practice Problems” (PDF).
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9.4 Special Factoring

Note: Recall from sub-subunit 9.3.2.2 that the products of some binomial result in special kinds of polynomials, called complete square trinomial and difference of two squares. Because you are working on factoring, which is a procedure inverse to multiplying, and you are given a polynomial that looks like a complete squareor difference of two squares, the polynomial can be factored immediately with the help of classic quadraticformulas (used in the reverse direction):

a²± 2ab +b² = (a ± b)²

a²– b² = (a – b)(a + b)

There also will be two new special factoring formulas introduced in this subunit, known as sum and difference of two cubes:

a³+ b³ = (a + b)(a² – ab + b²)

a³– b³ = (a – b)(a² + ab + b²)

9.4.1 Identifying and Factoring Complete Square Trinomial

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra I Concepts: “Factorization Using Perfect Square Trinomials”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra I Concepts: “Factorization Using Perfect Square Trinomials” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article until the section titled “Solving Quadratic Polynomial Equations by Factoring.” Watch the videos embedded in the text, and work through Guided Practice Examples. Then, complete practice problems 1–8. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra I Concepts: ‘Factorization Using Perfect Square Trinomials’ Problems 1–8” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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Lecture: Khan Academy’s “Factoring Perfect Square Trinomials”
Link: Khan Academy’s “Factoring Perfect Square Trinomials” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows how to check whether the given trinomial is a complete square and factor it if it is. (If it is not, factoring by grouping should be attempted.)

Watching this video and pausing to take notes should take approximately 15 minutes.

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9.4.2 Identifying and Factoring Difference of Two Squares

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Factorization Using Difference of Squares”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Factorization Using Difference of Squares” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article, watch the videos embedded in the text and work through Guided Practice Examples. Then, complete practice problems 1–10. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Factorization Using Difference of Squares’ Problems 1–10” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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Lecture: Khan Academy’s “Factoring Difference of Squares”
Link: Khan Academy’s “Factoring Difference of Squares” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows how to check whether the given binomial is a difference of two squares and factor it if it is.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Activity: Khan Academy’s “Factoring Difference of Squares 1”
Link: Khan Academy’s “Factoring Difference of Squares 1” (HTML)

Instructions: Please click on the link above, and complete the exercise set to practice factoring simple Difference of Two Squares binomials. Enter the factored expression in the tab on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 20 minutes.

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Activity: Khan Academy’s “Factoring Difference of Squares 2”
Link: Khan Academy’s “Factoring Difference of Squares 2” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It contains slightly more complicated Difference of Two Squares binomials. Enter the factored expression in the tab on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to moveto the next problem.

Completing this exercise set should take approximately 20 minutes.

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9.4.3 Identifying and Factoring Sum and Difference of Two Cubes

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Factoring a Sum or Difference of Cubes”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Factoring a Sum or Difference of Cubes” (YouTube)

Instructions: Please click on link above, watch the first five minutes of the video, and take notes. In this video, the sum of two cubes formula is proven by using polynomial division and there is an example of factoring a binomial that is a difference of two cubes.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Factoring Sum of Cubes”
Link: Khan Academy’s “Factoring Sum of Cubes” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this video, Sal Khan shows why the sum of two cubes formula is true (in a different way than in Dr. Sousa’s video) and uses it to factor a binomial.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Ex1: Factor a Sum or Difference of Cubes”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Ex1: Factor a Sum or Difference of Cubes” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. You will see two more examples of factoring binomials using the sum and difference of two cubes formulas.

Watching this video and pausing to take notes should take approximately 15 minutes.

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9.5 Factoring General Polynomials

Activity: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Factoring Completely”
Link: CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: “Factoring Completely” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the section titled “Guidance,” and work through Example A.Scroll down to the Guided Practice, and work through the examples. Then, complete practice problems 1–10. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Melissa Kramer’s Algebra Concepts: ‘Factoring Completely’ Problems 1–10” (PDF).

Reading and solving the practice problems should take approximately 45 minutes.

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Activity: Khan Academy’s “Factoring Polynomials 2”
Link: Khan Academy’s “Factoring Polynomials 2” (HTML)

Instructions: Please click on the link above, and complete the exercise set. These trinomials require two steps to be factored completely – factoring out the greatest common factor and then factoring the remaining trinomial. Enter the final result in the tab on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Activity: Khan Academy’s “Factoring Difference of Squares 3”
Link: Khan Academy’s “Factoring Difference of Squares 3” (HTML)

Instructions: Please click on the link above, and complete the exercise set. It contains binomials that require two steps to be factored – factoring out the greatest common factor and using the Difference of Two Squares formula. Enter the factored expression in the tab on the right side of the page. Select “Check Answer” to see if your answer is correct. If it is incorrect, you will be prompted to try again. If it is correct, you can click on “Correct! Next Question” to move to the next problem.

Completing this exercise set should take approximately 30 minutes.

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Lecture: Khan Academy’s “Factoring Trinomials by Grouping 6”
Link: Khan Academy’s “Factoring Trinomials by Grouping 6” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows an example of a trinomial that requires two factoring steps in order to be factored completely: factoring out a common monomial factor and factoring by grouping.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: Khan Academy’s “Difference of Cubes Factoring”
Link: Khan Academy’s “Difference of Cubes Factoring” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This video shows an example of a binomial that requires two factoring steps in order to be factored completely: factoring out a common monomial factor and using the difference of two cubes formula.

Watching this video and pausing to take notes should take approximately 15 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to the Khan Academy.
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Lecture: YouTube: Flat World Knowledge: John Redden’s Factoring and Solving by Factoring: “Factor Difference of Squares – Algebra”
Link: YouTube: Flat World Knowledge: John Redden’s Factoring and Solving by Factoring: “Factor Difference of Squares – Algebra” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. In this example, the difference of two squares formula is applied twice in order to factor the expression completely: once to the original polynomial, and then to the new binomial factor.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Lecture: YouTube: Flat World Knowledge: John Redden’s John Redden’s Factoring and Solving by Factoring: “Factoring Special Binomials – Algebra”
Link: YouTube: Flat World Knowledge: John Redden’s John Redden’s Factoring and Solving by Factoring: “Factoring Special Binomials – Algebra” (YouTube)

Instructions: Please click on the link above, watch the video, and take notes. This is an example of an interesting polynomial that can be attempted to be factored either by using the difference of two squares or difference of two cubes formulas. Whichever one you choose, the resultant factors can still be factored further using another special binomial factoring formula. In this video, the difference of two squares formula is used first. As an exercise, try an alternative method (e.g. applying difference of two cubes first) and try to show that the results will in fact be the same.

Watching this video and pausing to take notes should take approximately 15 minutes.

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Assessment: Connexions: Wade Ellis and Denny Burzynski’s Factoring Polynomials: “Proficiency Exam”
Link: Connexions: Wade Ellis and Denny Burzynski’s Factoring Polynomials: “Proficiency Exam” (HTML)

Instructions: Please click on the link above. You can read the HTML webpage or download a PDF version of the text by clicking on the appropriate link at the bottom of the page. Complete exercises 3–13. This exercise set will allow you to assess your mastery of factoring polynomials. Click on the “Show Solution” link next to each problem to check your answer.

Completing this assessment should take approximately 45 minutes.

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9.5.1 Choosing a Strategy for Factoring a Polynomial

Reading: The Saylor Foundation’s “Choosing a Strategy for Factoring a Polynomial”
Link: The Saylor Foundation’s “Choosing a Strategy for Factoring a Polynomial” (PPT)

Instructions: Read these examples and practice problems on factoring a polynomial. The answers to the practice problems are provided at the end of the presentation.
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9.6 Solving Quadratic Equations by Factoring

Note: In Unit 2, you learned how to solve all kinds of linear equations, or equations where the highest degree of the unknown variable is 1. In this subunit, you will be introduced to one of the methods of solving quadratic equations, or equations where the highest degree of the unknown variable is 2 – factoring. Factoring breaks up a quadratic equation into two linear equations. Thus, a quadratic equation that can be solved by factoring will generally have two solutions.

9.6.1 Principle of Zero Products and Identifying Solutions

Reading: YouTube: MathIsPower4U: Dr. Sousa’s “The Zero-Product Property”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “The Zero-Product Property” (YouTube)

Instructions: Please click on link above, watch the first five minutes of the video, and take notes. In this video, the sum of two cubes formula is proven by using polynomial division. Also, there is an example of factoring a binomial that is a difference of two cubes.

Watching this video and pausing to take notes should take approximately 15 minutes.

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9.6.2 Factoring before Solving

Lecture: YouTube: MathIsPower4U: Dr. Sousa’s “Solving Quadratic Equations by Factoring”
Link: YouTube: MathIsPower4U: Dr. Sousa’s “Solving Quadratic Equations by Factoring” (YouTube)

Instructions: Please click on link above, watch the video, and take notes. In this video, you will see a few examples solved by various methods of factoring.

Watching this video and pausing to take notes should take approximately 30 minutes.

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Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving Simple Quadratic Equations by Factoring”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving Simple Quadratic Equations by Factoring” (HTML)

Instructions: Please click on the link above, and work through the examples. Then, click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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Activity: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving More Complicated Quadratic Equations by Factoring”
Link: Dr. Carol Burns’ One Mathematical Cat, Please! A First Course in Algebra: “Solving More Complicated Quadratic Equations by Factoring” (HTML)

Instructions: Please click on the link above, and work through solving the examples. Then, click on the “new problem” button at the end of the page to try a practice problem. After answering the problem, click on the “check your answer” button. Continue this process by clicking on “new problem,” and solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled “Click Here for a Randomly Generated Worksheet and Answers.” The answers will be provided at the end of the worksheet.

Reading this section and solving the practice problems should take approximately 45 minutes.

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9.6.3 Solving Application Problems

Reading: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Solving Problems by Factoring”
Link: CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: “Solving Problems by Factoring” (HTML)

Instructions: Please click on the link above. You may also download a PDF, mobi, or ePub version of this text by clicking on the appropriate link under “Download.” Read the article, and watch the videos embedded in the text. Then, complete practice problems 1–9. Once you have completed the practice problems, check your answers against The Saylor Foundation’s “Answer Key to CK-12: Andrew Gloag, Anne Gloag, and Eve Rawley’s Algebra Concepts: ‘Solving Problems by Factoring’ Problems 1–9” (PDF).

Reading and solving the practice problems should take approximately 1 hour to complete.

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Final Exam

Final Exam: The Saylor Foundation’s “RWM102 Final Exam”
Link: The Saylor Foundation’s “RWM102 Final Exam” (HTML)

Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
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