Unit 3: Fractions
Fractions are very easy to work with if you learn the rules. After all, fractions are everywhere.
Have you ever eaten a Hershey's chocolate bar? It is conveniently broken up into little pieces, allowing you the option to devour in big bites or to savor tiny little morsels. Let's say you have a Hershey's bar sitting on your dining room table. Your oldest child cheerfully announces that she has eaten half of the bar, and her younger brother has eaten a quarter of the bar. If you know how to work with fractions, you can quickly calculate how much of the bar is left.
Fractions appear in many other situations such as sale prices, measurements, money, gardening; the list of applications is virtually endless. In this unit, you will learn to work with fractions. You will learn how to reduce them, how to add/subtract/multiply/divide them, and how to apply them to real-world situations.
Completing this unit should take you approximately 38 hours.
Upon successful completion of this unit, the student will be able to:
- identify parts of a fraction;
- recognize equivalent fractions and fractions in lowest terms;
- recognize various forms of a fraction;
- convert between mixed numbers and improper fractions;
- determine how to find the least common denominator between fractions;
- use equivalent fractions to add and subtract fractions and mixed numbers with like and unlike denominators;
- determine how to solve multiplication and divisions problems with fractions and mixed numbers; and
- solve real-world mathematical problems involving the four operations with fractions.
3.1: Fraction Basics
3.1.1: Identifying Fraction Parts
Take notes as you watch these videos. Watch the first presentation carefully two or three times until you are able to identify the numerator and denominator of a fraction. Watch the second presentation carefully two or three times until you are able to identify parts of a fraction.
Read "Fractions of Whole Numbers," "The Parts of a Fraction," and "Reading and Writing Fractions" sections. Then, complete "Practice Set A," exercises 1-5 and "Practice Set B," exercises 6-17. The solutions to the problems are shown directly below each problem. These problems will allow you to practice identifying numerators and denominators as well as writing fractions by using words.
Complete every fourth problem for 18-46 and the even-numbered problems for 48-64. The solutions to the problems are shown directly below each problem. These problems will allow you to practice identifying numerators and denominators. This includes determining the numerator and denominator from problems such as "you need 3/4 of a cup of sugar to make a batch of cookies."
3.1.2: Equivalent Fractions
Take notes as you watch these videos. Watch the first presentation carefully two or three times until you are able to identify equivalent fractions. Watch the second presentation carefully two or three times until you are able to compare fractions.
Read the "Equivalent Fractions" section through "Sample Set A." Then, complete "Practice Set A," exercises 1-5. The solutions to the problems are shown directly below each problem. These problems will allow you to practice identifying if pairs of fractions are equivalent.
Complete the assignments, which test your knowledge on equivalent fractions. You can review the concepts associated with the questions the Khan videos in subunit 3.1.2. 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.
Complete the assignments, which test your knowledge on equivalent fractions. You can review the concepts associated with the questions the Khan videos in subunit 3.1.2. 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.
3.1.3: Fraction Types
3.1.3.1: Proper and Improper Fractions
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to identify proper and improper fractions.
Read the "Positive Proper Fractions" and "Positive Improper Fractions" sections. This reading provides examples of proper and improper fractions. Then, complete the odd-numbered problems 11-25. The solutions to the problems are shown directly below each problem. These problems will allow you to determine a proper fraction, improper fraction, or a mixed number.
3.1.3.2: Mixed Numbers and Improper Fractions
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to understand mixed numbers and improper fractions.
3.1.3.3: Converting Fractions
3.1.3.3.1: Changing Improper Fractions to Mixed Numbers
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to understand changing an improper fraction to a mixed number.
- College of the Redwoods: "Pre-Algebra Textbook: Changing Improper Fractions to Mixed Fractions" File
Review the section titled "Changing Improper Fractions to Mixed Fractions" on pages 293-295 of the textbook, stopping at "Multiplying and Dividing Mixed Fractions." The material can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Changing Improper Fractions to Mixed Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 4-6 on pages 294 and 295. This material provides examples of converting an improper fraction to a mixed fraction.
Read the "Converting Improper Fractions to Mixed Numbers" section. Then, complete "Practice Set A," exercises 1-6. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting an improper fraction to a mixed number. Then, complete the odd-number exercises 27-39. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting an improper fraction to a mixed number.
3.1.3.3.2: Changing Mixed Numbers to Improper Fractions
Read the "Converting Mixed Numbers to Improper Fractions" section. Then, complete "Practice Set B," exercises 7-10. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting a mixed number to an improper fraction.
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to understand changing an improper fraction to a mixed number.
Complete the odd-numbered problems 41-55. The solutions to the problems are shown directly below each problem. These problems will allow you to practice converting a mixed number and an improper fraction.
Complete the assignment that tests your knowledge of converting fractions. This assignment incorporates concepts from converting mixed numbers to improper fractions and vice versa. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.1.3.3. 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.
3.1.4: Fractions in Lowest Terms
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to reduce fractions to lowest terms.
Read the "Reducing Fractions to Lowest Terms" section. Then, complete "Practice Set B," exercises 6-11 and "Practice Set C," exercises 12-17. The solutions to the problems are shown directly below each problem. These problems will allow you to practice reducing fractions to their lowest terms. Then, complete the odd-numbered problems for 61-77 as well as the odd-numbered problems for 89-113. The solutions to the problems are shown directly below each problem. These problems will allow you to practice reducing fractions to their lowest terms as well as determine fractional parts of a day.
3.1.5: Fractions in Highest Terms
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to solve an application of raising fractions to highest terms.
Read the "Raising Fractions to Higher Terms" section. Then, complete "Practice Set D," exercises 18-22. The solutions to the problems are shown directly below each problem. These problems will allow you to find the missing numerator and denominator. Complete the odd-numbered problems for 39-53. The solutions to the problems are shown directly below each problem. These problems will allow you to practice reducing fractions to their lowest terms.
3.1.6: Finding Common Denominators
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to rewrite fractions with a least common denominator.
3.2: Fractions Operations
3.2.1: Adding and Subtracting
3.2.1.1: Adding Fractions with Like Denominators
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able add fractions with common denominators.
Read the "Addition of Fractions with Like Denominators" section. Then, complete "Practice Set A," exercises 1-4. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice adding fraction with like denominators.
Complete the assignment that tests your knowledge of adding fractions. This assignment incorporates concepts from adding fractions with like denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.1. 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.
3.2.1.2: Subtracting Fractions with Like Denominators
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able subtract fractions with common denominators.
Read the "Subtraction of Fractions with Like Denominators" section. Then, complete "Practice Set B," exercises 5-9. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice subtracting fractions with like denominators. Then, complete the odd-numbered problems for 11-35 and 39. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice adding and subtracting fractions with like denominators.
Complete the assignment that tests your knowledge of subtracting fractions. This assignment incorporates concepts from subtracting fractions with like denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.1. 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.
3.2.1.3: Adding Fractions with Unlike Denominators
Take notes as you watch these videos. View the presentations carefully two or three times until you are able add fractions with unlike denominators.
Read the "Addition and Subtraction of Fractions" section stopping at "Sample Set A," Example 3. Then, complete "Practice Set A," exercise 1. The solutions to the problem can be revealed by the "Show Solution" link under each problem. This reading and exercise will allow you to practice adding fractions with unlike denominators.
Complete the assignment that tests your knowledge of adding fractions. This assignment incorporates concepts from adding fractions with unlike denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.1. 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.
3.2.1.4: Subtracting Fractions with Unlike Denominators
Read starting at "Sample Set A," Example 3, through the "Addition and Subtraction of Fractions" section. Then, complete "Practice Set A," exercises 2-5. The solutions to the problems are shown directly below each problem. These problems will allow you to practice subtracting fractions with unlike denominators. Then, complete the even-numbered problems 6-36. The solutions to the problems are shown directly below each problem. These problems will allow you to add and subtract fractions with unlike denominators.
Complete the assignment that tests your knowledge of subtracting fractions. This assignment incorporates concepts from subtracting fractions with unlike denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.1. 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.
3.2.1.5: Adding and Subtracting Fractions with Different Signs
Study the "Adding and Subtracting Fractions with Different Denominators" on pages 277-279 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.4 Adding and Subtracting Fractions, "Adding and Subtracting Fractions with Different Denominators"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 4 and 5 on page 279. This material provides examples of subtracting fractions with different signs.
Complete the odd-numbered exercises for 69-103 on pages 287 through 288 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.4 Adding and Subtracting Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to add and subtract fractions with different signs. The solutions to these problems are located in the "Answers" section on page 289 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.4 Adding and Subtracting Fractions, "Answers").
Complete the assessment that tests your knowledge on adding and subtracting fractions. This quiz incorporates concepts from adding and subtracting fractions with like and unlike denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.1. 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.
3.2.2: Adding and Subtracting Mixed Numbers
3.2.2.1: Adding Mixed Numbers with Like Denominators
Take notes as you watch these videos. View each presentation carefully two or three times until you are able add mixed numbers with common denominators.
3.2.2.2: Subtracting Mixed Numbers with Like Denominators
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able subtract mixed numbers with common denominators.
3.2.2.3: Adding Mixed Numbers with Unlike Denominators
Take notes as you watch these videos. View each presentation carefully two or three times until you are able add mixed numbers with unlike denominators.
Study the "Adding Mixed Fractions" section on pages 301-303 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.6 Adding and Subtracting Mixed Fractions, "Adding Mixed Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 1, 2, and 3 on pages 301 and 302. This material provides examples of subtracting fractions with different signs.
3.2.2.4: Subtracting Mixed Numbers with Unlike Denominators
Take notes as you watch these videos. View each presentation carefully two or three times until you are able subtract mixed numbers with unlike denominators.
Study the "Subtracting Mixed Fractions" section 9 on pages 304-307 of the textbook, stopping at Example 9. The material may also be located through the bookmark on the left side (4 Fractions, 4.6 Adding and Subtracting Mixed Fractions, "Subtracting Mixed Fractions"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems next to Examples 5 through 8 on pages 304-306. This material provides examples of subtracting mixed fractions.
Complete the assignments that tests your knowledge of adding and subtracting mixed numbers. These assignments incorporate concepts from adding and subtracting mixed numbers with like and unlike denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.2. 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.
Complete the assignments that tests your knowledge of adding and subtracting mixed numbers. These assignments incorporate concepts from adding and subtracting mixed numbers with like and unlike denominators. You can review the concepts associated with the questions with the Khan videos in sub-subunit 3.2.2. 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.
3.2.3: Applications of Adding and Subtracting Fractions
Take notes as you watch these videos. View each presentation carefully two or three times until you are able understand applications where you add and subtract.
Review the "Adding and Subtracting Mixed Fractions" section through Example 9 on pages 301-307 of the textbook; you read this section in sub-subunits 3.2.1 and 3.2.2. The material may also be located through the bookmark on the left side (4 Fractions, 4.6 Adding and Subtracting Mixed Fractions), which will take you directly to the reading. After reviewing this section, complete the "You Try It" problem next to Example 4 on page 303 and Example 9 on page 307. This material contains applications of adding and subtracting fractions.
Complete the following problems: problem 39 for "Addition and Subtraction of Fractions with Like Denominators"; problems 50, 52, and 54 for "Addition and Subtraction of Fractions with Unlike Denominators"; problems 42, 44, and 46 for "Addition and Subtraction of Mixed Numbers". The solutions to the problems can be revealed by the "Show Solution" link below each problem, or may be found directly below the problem. These problems will allow you to practice adding and subtracting fractions with unlike denominators, which include finding amounts needed for recipes and finding the cost after an increase.
3.2.4: Multiplying
3.2.4.1: Multiplying Fractions
Take notes as you watch this video. Watch the presentation carefully two or three times until you are able to multiply fractions.
Study the "Multiply Fractions" section on pages 249-252, stopping at Example 5 on page 252 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.2 Multiplying Fractions), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems next to Examples 2, 3, and 4 on pages 251 and 252. This material contains examples of multiplying fractions. Then, complete the odd-numbered exercises for 5-27 on page 260 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.2 Multiplying Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to multiply fractions. The solutions to these problems are located in the "Answers" section on page 263 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.2 Multiplying Fractions, "Answers").
3.2.4.2: Multiplying Mixed Numbers
Take notes as you watch these videos. View the presentations carefully two or three times until you are able to multiply mixed numbers.
Read "Multiplication of Mixed Numbers" section stopping at the "Powers and Roots of Fraction" section. Then, complete "Practice Set C," exercises 17-20. The solutions to the problems are shown directly below each problem. These problems will allow you to practice multiplying mixed numbers.
Complete the odd-numbered exercises for 25-47 on pages 297 and 298 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to multiply mixed numbers. The solutions to these problems are located in the "Answers" section on page 299 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Answers").
3.2.5: Dividing
3.2.5.1: Dividing Fractions
Take notes as you watch these videos. View the presentations carefully two or three times until you are able to divide fractions.
Study the "Division" section on pages 267 and 268 of the textbook. The material may also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Division"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems next to Examples 2, 3, and 4 on pages 267 and 268. This material contains examples of dividing fractions. Then, complete the odd-numbered exercises for 33-67 on page 271 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to divide fractions. The solutions to these problems are located in the "Answers" section on page 273 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Answers").
3.2.5.2: Dividing Mixed Numbers
Take notes as you watch these videos. View the presentations carefully two or three times until you are able to divide fractions.
Read Examples 7-9 under "Division of Fractions" section, stopping at the "Powers and Roots of Fraction" section. Then, complete "Practice Set B," exercises 11-13. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice dividing mixed numbers.
Complete the odd-numbered exercises for 49-71 on pages 298 and 299 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to divide mixed numbers. The solutions to these problems are located in the "Answers" section on page 299 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Answers").
3.2.6: Applications of Multiplying and Dividing Fractions
Take notes as you watch these videos. Each link will direct you to a series of videos and practice exercises that address these topics. Watch both presentations carefully until you are able to apply the multiplication and division of fractions to a word problem.
Study the "Parallelograms" and "Triangles" sections on pages 255-258 of the textbook, stopping at "Identifying the Base and Altitude." The material may also be located through the bookmark on the left side (4 Fractions, 4.3 Dividing Fractions, "Parallelograms"), which will take you directly to the reading. After reading and studying this section, complete the "You Try It" problems beside Examples 8 and 9 on pages 256 and 257. This material contains examples of dividing fractions.
Then, complete the odd-numbered exercises for 57-69 on page 262 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.2 Multiplying Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to apply multiplying fractions by finding the areas of parallelograms and triangles. The solutions to these problems are located in the "Answers" section on page 263 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.2 Multiplying Fractions, "Answers").
Then, complete problems 73 and 75 on page 299 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Exercises"), which will take you directly to the material. These exercises will provide you with the opportunity to practice dividing fractions by dividing fields and cutting jewelry into pieces. The solutions to these problems are located in the "Answers" section on page 299 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.5 Multiplying and Dividing Mixed Fractions, "Answers").
Complete the assessment that tests your knowledge on applications of multiplying and dividing fractions. You can review the concepts associated with these questions with the Khan videos in sub-subunits 3.2.4 and 3.2.5. 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.
Complete the assessment that tests your knowledge on applications of multiplying and dividing fractions. You can review the concepts associated with these questions with the Khan videos in sub-subunits 3.2.4 and 3.2.5. 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.
3.3: Order of Operations
3.3.1: Order of Operations with Fractions
Read "The Order of Operations" section, stopping at "Sample Set A," Example 5. Then, complete "Practice Set A," exercises 1-5 and 7. The solutions to the problems are shown directly below each problem. These problems will allow you to practice the order of operations with fractions.
Complete the odd-numbered exercises for 17-35 on pages 321 and 322 of the textbook. These exercises can also be located through the bookmark on the left side (4 Fractions, 4.7 Order of Operations with Fractions, "Exercises"), which will take you directly to the assignment. These exercises will provide you with the opportunity to practice dividing fractions. The solutions to these problems are located in the "Answers" section on page 324 of the textbook. The solutions page can also be located through the bookmark on the left side (4 Fractions, 4.7 Order of Operations with Fractions, "Answers").
3.3.2: Complex Fractions
Read "Simple Fractions and Complex Fractions" and "Converting Complex Fractions to Simple Fractions" sections, stopping at "Sample Set A," Example 5. Then, complete "Practice Set A," exercises 1-6. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to practice the order of operations with complex fractions. Then, complete the odd-number problems for 7-25. The solutions to the problems can be revealed by the "Show Solution" link under each problem. These problems will allow you to use order of operations to simplify complex fractions.