Unit 2: Order of Operations
In life, we often have procedures that everybody uses to avoid problems. When driving a car, for example: if you want to change lanes, you have to first look to make sure the lane is clear, activate your turn signal, check the lane again, move into the lane, and deactivate your turn signal. You do not move into the lane, activate your signal, make sure the lane is clear, and deactivate your signal. That can, and eventually will, cause a serious accident. In order to avoid costly errors, mathematicians had to agree on the series of steps that are needed to simplify expressions involving the four basic operations, grouping symbols, and exponents. This series of steps is known as the "order of operations" and is more commonly known as either PEMDAS or "Please Excuse My Dear Aunt Sally, she Left to Right." This tells us in which order to simplify the expression. (Tip: it is multiply OR divide and add OR subtract - whichever you see first.)
Mathematicians also needed a way to quickly write out a repeated multiplication problem, like 2 x 2 x 2 x 2 x 2, so they invented the use of exponents. This unit will introduce you to the process of working with basic exponents. As you go higher, you will learn more about exponents.
Another topic you will learn about in this unit is the concept of "greatest common factor." Mathematically, the greatest common factor (GCF) is the largest number you can divide two or more numbers by. In real life, it also makes appearances, both mathematical and non-mathematical. A detective trying to make connections between an arrested criminal and a suspected accomplice is going to be less interested in the facts that they have both eaten at McDonald's and both like strawberry milkshakes than in the fact that the suspected accomplice has been the criminal's best friend for twenty years. That fact is far greater to the investigation.
The last topic you will cover is related to greatest common factor but is different. It is known as "least common multiple." Here, you are trying to determine the smallest number that two numbers can both divide into. Again, it appears in life. Let's say your favorite radio station is running a promotion: every fifth caller receives free concert tickets, and every twelfth caller receives a free gas card. How long will it take before they have a caller who receives both prizes on the same phone call? This is an example of using the least common multiple. (In case you are wondering, it would be the 60th caller who won both prizes.)
Completing this unit should take you approximately 22 hours.
Upon successful completion of this unit, the student will be able to:
- find the greatest common factor and least common multiple of whole numbers;
- calculate basic exponents;
- calculate problems using negative numbers;
- apply the order of operations to simplify expressions; and
- solve real-world mathematical problems involving the four operations.
2.1: Greatest Common Divisor/Factor
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to solve problems finding the greatest common divisor/factor.
Read the section on "The Greatest Common Factor (GCF)" through "Sample Set A." Look closely at the section titled "A Method for Determining the Greatest Common Factor." Then, complete "Practice Set A," exercises 1-4 and the even-numbered problems for 6-20. The solutions to the problems are shown below each problem. These problems will allow you to practice finding the greatest common factor between numbers.
Complete the "Greatest Common Divisor" assignment. This assignment incorporates concepts from the greatest common divisor. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2. For this assignment, you will compute your answer and type it into the answer box. You may then click on "Check Answer" to see if you were correct or if you need to try again.
2.2: Least Common Multiple
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to solve problems finding the least common multiple.
Read the "Least Common Multiple" section. Start with the "Multiples" section, and then complete "Practice Set A," exercises 1-3. Continue with the "Common Multiples" section, and then complete "Practice Set B," exercises 6-8. Finish by reading "The Least Common Multiple (LCM)" section, and then complete "Practice Set C," exercises 11-13. Then, complete the even-numbered problems for 16-44. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice finding the least common multiple between numbers.
Complete the "Least Common Multiple" assignment. This assignment incorporates concepts from the least common multiple. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2. For this assignment, you will compute your answer and type it into the answer box. You may then click on "Check Answer" to see if you were correct or if you need to try again.
Watch this video, which consists of two least common multiple and greatest common factor word problems. Pause the video after each problem has been given, and try to work out the answer on your own before coming back to the video to check your answer. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2.
2.3: Negative Numbers
2.3.1: Negative Numbers Introduction
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to understand how negative numbers work.
2.3.2: Adding Negative Numbers
Study the section titled "Adding Integers with Like Signs" on pages 115-117 of the textbook. You may stop when you reach the section titled "Adding Integers with Unlike Signs" on page 117. The material can also be located through the bookmark on the left side (2 The Integers, 2.2 Adding Integers, "Adding Integers with Like Signs"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1 and 2 on page 115 and Example 3 on page 116. This material provides examples of adding two positive and two negative integers.
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to add negative numbers.
2.3.3: Adding Integers with Different Signs
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to add integers with different signs.
Study the section titled "Adding Integers with Unlike Signs" on pages 117-119 of the textbook. You may stop when you reach the section titled "Properties of Addition of Integers" on page 119. The material can also be located through the bookmark on the left side (2 The Integers, 2.2 Adding Integers, "Adding Integers with Like Signs"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Example 4 on page 117 and Example 5 on page 118. This material provides examples of adding one positive and one negative integer.
Complete the odd-numbered exercises for 13-35 and 65-83 on pages 124 and 125 of the textbook. The exercises can also be located through the bookmark on the left side (2 The Integers, 2.2 Adding Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to recognize addition properties as well as determine the profit and loss for a company. The solutions to these problems are located in the "Answers" section on page 126 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.2 Adding Integers, "Answers").
2.3.4: Adding/Subtracting Negative Numbers
Complete the odd-numbered exercises for 1-23 and 51-59 on pages 133 and 134 of the textbook. The exercises can also be located through the bookmark on the left side (2 The Integers, 2.3 Subtracting Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to recognize subtraction and properties and apply the concepts when dealing with a temperature change and comparing highest and lowest points. The solutions to these problems are located in the "Answers" section on page 135 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.3 Subtracting Integers, "Answers").
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to subtract integers.
Study the section titled "Subtracting Integers" on pages 128 through 132. The material can also be located through the bookmark on the left side (2 The Integers, 2.3 Subtracting Integers, "Subtracting Integers"), which will take you directly to the reading. After reading and studying the "Subtracting Integers" section, complete the "You Try It" problems beside Example 1 on page 129 and Example 4 on page 131. This material provides examples of subtracting integers.
Complete the "Adding Negative Numbers" assessment. This quiz incorporates concepts from negative numbers introduction, adding negative numbers, adding integers with different signs, and adding/subtracting negative numbers. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2. For this quiz, you will compute your answer and type it into the answer box. You may then click on "Check Answer" to see if you were correct or if you need to try again.
Complete the "Negative Number Word Problems" assignment. This assignment incorporates concepts from negative numbers introduction, adding negative numbers, adding integers with different signs, and adding/subtracting negative numbers. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2. For this assignment, you will compute your answer and type it into the answer box. You may then click on "Check Answer" to see if you were correct or if you need to try again.
2.3.5: Multiplying Positive and Negative Numbers
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to multiply integers with different signs.
Study the sections titled "Multiplication and Division of Integers" on pages 137 and 138 as well as "Multiplying by Minus One" and "The Product of Two Integers" on pages 140-142 of the textbook. The material can also be located through the bookmark on the left side (2 The Integers, 2.4 "Multiplication and Division of Integers") and (2 The Integers, 2.4 Multiplication and Division of Integers, "Multiplying by Minus One"), which will take you directly to the readings. After reading and studying these sections, complete the "You Try It" problems beside Examples 1 and 2 on pages 141 and 142. This material provides examples of multiplication of integers and rules associated with each problem.
Complete the odd-numbered exercises for 17-47 as well as problem 85 on pages 145 through 147 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to recognize multiplication properties and apply these concepts. The solutions to these problems are located in the "Answers" section on page 147 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Answers").
2.3.6: Dividing Positive and Negative Numbers
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to explain how to divide integers with different signs.
Study the section titled "Division of Integers" on pages 143 and 144 of the textbook. The material can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Division of Integers"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Example 4 on page 144. This reading provides examples of division of integers and rules associated with each problem.
Complete the odd-numbered exercises for 61-83 on page 146 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to recognize division properties and apply these concepts. The solutions to these problems are located in the "Answers" section on page 147 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Answers").
Complete the "Multiplying and Dividing Negative Numbers" assignment. This assignment incorporates concepts from multiplying and dividing negative numbers. It may help to review the concepts associated with the questions by revisiting the Khan videos in Unit 2. For this assignment, you will compute your answer and type it into the answer box. You may then click on "Check Answer" to see if you were correct or if you need to try again.
2.4: Exponent Basics
2.4.1: Understanding Exponents
Take notes as you watch these videos. Watch the first presentation carefully two or three times until you are able to explain how to write a problem in exponential notation. Watch the second presentation carefully two or three times until you are able to explain how to simplify a problem in exponential notation.
Read the section titled "Exponential Notation." Read the introductory text on exponential notation through "Sample Set A." Then, complete "Practice Set A," exercises 1-6. Continue with the "Reading Exponential Notation" subsection. Then, complete the odd-numbered exercises for 15-29 and 31-57. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to understand the basics of exponential notation.
2.4.2: Level One Exponents
Watch this video and take notes. Watch the presentation carefully two or three times until you are able to explain how to simplify exponents with a positive and negative bases as well as exponents to the zero power.
Study Example 3 located on page 142 of the textbook. After reading and studying this example, complete the "You Try It" problems beside Example 3. This material provides an example of exponential notation with negative signs and bases.
Complete the odd-numbered problems for 49-59 on page 146 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to simplify exponent problems with negative bases. The solutions to these problems are located in the "Answers" section on page 147 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.4 Multiplication and Division of Integers, "Answers").
Complete the assessment that tests your knowledge on positive and zero exponents. This incorporates concepts from positive exponents with a positive or negative base and zero exponents. You can review the concepts associated with the questions with the Khan videos in Unit 2. Answer each question by inputting your calculation into the answer box. You may click on "Check Answer" to check if your answers are correct or if you need to try again.
2.5: Order of Operations
2.5.1: Introduction to Order of Operations
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to apply the process of the order of operations.
Read the section titled "Order of Operations with Integers" on pages 148-150 of the textbook, stopping at "Evaluating Fractions." The material can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 2 and 4 on page 149. This material provides examples of order of operations with integers.
Complete the odd-numbered problems for 1-39 on page 152 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to apply the process of the order of operations. The solutions to these problems are located in the "Answers" section on page 154 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Answers").
2.5.2: Order of Operations
Take notes as you watch these videos. Watch the presentations carefully two or three times until you are able to apply the process of the order of operations to more complicated problems.
Complete the "You Try It" problem beside Example 3 on page 149. Then, complete the odd-numbered problems for 45-53 as well as 59, 61, 67, 69, 73, 75, and 77 on page 153 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Exercises"), which will take you directly to the material. These exercises will provide you with the opportunity to apply the process of the order of operations to more complicated problems. The solutions to these problems are located in the "Answers" section on page 154 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Answers").
Review the section titled "Evaluating Fractions," including Example 6, on page 150 of the textbook. The material can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Evaluating Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problem beside Example 6 on page 150. This material provides examples of order of operations while evaluating fractions.
Complete the odd-numbered problems for 81-103 on pages 153 and 154 of the textbook. These exercises can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Exercises"), which will take you directly to the material. These exercises will provide you with the opportunity to apply the process of the order of operations to more complicated problems. The solutions to these problems are located in the "Answers" section on page 154 of the textbook. The solutions page can also be located through the bookmark on the left side (2 The Integers, 2.5 Order of Operations with Integers, "Answers").
Complete the assessment, which tests your knowledge on order of operations. You can review these concepts associated with the questions with the Khan videos in subunit 2.5. Compute the answer to the given problem, and input your response into the answer box. Then, click on "Check Answer" to see if you were correct or if you need to try again.