Contents Common Core State Standards Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12

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2 Contents Common Core State Standards Lesson Computing Unit Rates Lesson Identifying the Constant of Proportionality RP. 7.RP..b Lesson Lesson Solving Real-World Problems with Ratios and Percents Using Proportional Relationships to Solve Multi-Step Problems RP. 7.RP. Lesson Adding and Subtracting Rational Numbers..... Lesson 6 Multiplying Rational Numbers Lesson 7 Dividing Rational Numbers Lesson 8 Problem Solving with Rational Numbers Lesson 9 Factoring and Expanding Linear Expressions... 8 Lesson 0 Applying Properties to Solve Problems NS..c, 7.NS..d 7.NS..a 7.NS..b 7.NS. 7.EE., 7.EE. 7.EE., 7.NS..c Lesson Solving Multi-Step Real-World Problems Lesson Solving Word Problems Algebraically EE. 7.EE..a Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC L e ss o n

3 Common Core State Standards Lesson Using Inequalities to Solve Problems Lesson Scale Drawings Lesson Solving problems with Circles Lesson 6 Area, Volume, and Surface Area Lesson 7 Drawing Inferences about a Population EE..b 7.G. 7.G. 7.G.6 7.SP. Lesson 8 Making Predictions with Experimental Probability SP.6 Lesson 9 Probability Models Lesson 0 Tree Diagrams SP.7.a, 7.SP.7.b 7.SP.8.b Glossary Math Tools Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC

4 LESSON Computing Unit Rates Plug In Using Ratios A ratio compares two quantities. It can be expressed in different ways. The ratio of apples to oranges is 8 to. Equivalent ratios can be written as the same fraction in simplest form. A table of equivalent ratios shows how two quantities are related. Write: 8 or 8: or 8 to Ice Cubes Used Glasses of Iced Tea Ok! I can write a ratio as a fraction, with a colon (:), or with the word to I see! All of the ratios ice cubes of glasses are equivalent because they can be written as the simplified fraction. Words to Know ratio a comparison of two quantities 6 or 6: or 6 to equivalent ratios ratios with the same value that can be expressed as the same fraction in simplest form A DISCUSS How can you use the table above to find the number of ice cubes in a given number of glasses of iced tea? You can describe quantities with a ratio. DO Write the ratio of markers to boxes. Compare the quantities. Write the ratios in different ways. 0 markers boxes : to Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Lesson

5 B You can use multiplication and division to find equivalent ratios for 6. Complete the table of equivalent ratios. DO Multiply both quantities to find equivalent ratios. Divide both quantities to find more equivalent ratios. Computing Unit Rates The numbers in the top row are the numerators of the fractions for the ratios. 6 C You can use equivalent ratios to solve problems. DO Janet runs,60 feet in minutes. Dan runs,70 feet in minutes. If they keep the same paces, who would run farther in minutes? Feet Minutes Janet s Running Times, Write the ratio of feet to minutes for each person. Complete the table to write equivalent ratios. Feet Minutes Dan s Running Times, Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC PRACTICE Complete the table of equivalent ratios. Feet Yards Solve. Use the table to compare the ratios. would run farther in minutes. would run DISCUSS 6 9 8,60 feet and would run 7,90 feet. How could you find the number of feet run for a number of minutes that is not in the table? Text Messages Cost ($) Alex reads pages in 0 minutes. Jenna reads pages in minutes. They are both reading a 60-page magazine. Who will finish reading the magazine first?

6 POWER UP Finding Rates from Complex Fractions A rate is a ratio that compares quantities with different units of measure. pounds 6 dollars This ratio compares pounds to dollars, so it is a rate. A complex fraction has a fraction in the numerator, denominator, or both. A complex fraction can be rewritten as a division expression. I can think of a complex fraction as a fraction divided by a fraction. A rate that compares fractions can be written as a complex fraction. A mouse can move about miles in hour. This complex fraction has a fraction only in the denominator. Words to Know rate a ratio that compares quantities with different units of measure 0 miles hours complex fraction a fraction that has a fraction in the numerator, denominator, or both 7 8 DISCUSS Why do you think it will be helpful to write rates as complex fractions in math problems? A You can use complex fractions to write rates that include a fraction. DO Alexandra completed of her homework in hours. Write this rate as a complex fraction. Decide what the rate compares. Write a complex fraction with the given quantities. The rate compares the amount of homework to. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC 6 Lesson

7 Computing Unit Rates B You can use complex fractions to write rates that compare two fractions. DO Jamal walks 7 8 mile in hour. How can you write this rate as a complex fraction? I get it! The units of the rate, miles, tell me hours which values are in the numerator and the denominator of the complex fraction. Understand which quantities are being compared. Write the complex fraction. miles DISCUSS Trevor read of a book in days, and he writes this rate as. What can you tell Trevor about his work? PRACTICE Write the rate as a complex fraction. A recipe calls for cup of flour to make of a batch of cookies. A band marched of the parade route in hour. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC A car used of a tank of gas to travel 8 of the total distance. A sprinkler system uses gallon of water every hour. 7

8 READY TO GO Computing Unit Rates A unit rate is a ratio comparing quantities with different units of measure to a denominator equal to. 7 8 inch year This is a unit rate because it compares 7 8 inch to year. A unit rate always compares a quantity to unit of another kind of quantity. You can express a rate written as a complex fraction as a unit rate. 8 Since any number remains the same when divided by : 8 8 The rate can be written as the unit rate 8. I get it! I just simplify the complex fraction and then divide the fraction by. Words to Know unit rate a ratio that compares the number of units of one quantity to unit of a second quantity. dollars per pound cost unit rate pound $ lb 6 miles per hour unit rate miles hour mi 6 hr DISCUSS Why might it be helpful to find unit rates? lesson link Plug In POWER UP GO! A ratio compares two quantities. A ratio can be written in three ways. : to A ratio that includes a fraction can be written as a complex fraction. 6 I see! I can use what I know about ratios and complex fractions to find unit rates. Then I can use unit rates to solve problems. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC 8 Lesson

9 Computing Unit Rates work together You can use labeled Fraction Strips to help you solve unit rate problems involving complex fractions. The rate of cups to hours is written as a complex fraction. The complex fraction is written as a division problem and simplified. The unit rate is. The Fraction Strips model the problem. Tara will drink cups of water in hour. Tara drinks cups of water in hour. At this rate, how much water will Tara drink in an hour? cups hours 6 Division lets me simplify fractions, even complex fractions! I can use fraction strips to model division. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC A You can use unit rates to solve problems. DO A recipe calls for cup of sugar to make gallon of iced tea. How much sugar is needed to make gallon of iced tea? Write the rate of sugar to iced tea as a complex fraction. Write the complex fraction as a division problem, then rewrite as multiplication. Divide both the numerator and the denominator by the denominator to find the unit rate. DISCUSS sugar iced tea The unit rate is. How can you be sure that you have found the unit rate? cup of sugar is needed to make gallon of iced tea. Fraction Strips can be found on p

10 Ready to Go practice Find the unit rate. Use Fraction Strips to help you. A recipe calls for eggs for every teaspoon of salt. Kathy s heart beats 60 times in minute. Fraction Strips can be found on p.. The unit rate is eggs for teaspoon. The unit rate is times in minute. Find the unit rate. mile in hour 0 words in minute 0 REMEMBER A unit rate compares units of one quantity to unit of a different quantity. The unit rate is miles in hour. The unit rate is words in minute. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC 0 Lesson

11 Computing Unit Rates Compare the unit rates. Alice skates mile in hour. Elizabeth skates mile in 9 hour. Who skates farther in hour? Alice s Rate Elizabeth s Rate Alice skates miles in hour. Elizabeth skates miles in hour. skates farther in hour. Solve. I see! Miles per hour means number of miles in hour. 6 Maurice walked 6 miles on the treadmill in hours. How many miles per hour did Maurice walk? 7 On average, a person who weighs 0 pounds burns 8 calories in 6 hour while playing basketball. At this rate, how many calories are burned in hour? Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC DISCUSS Determine the Rate Rachel wants to find the best unit price for potatoes at a grocery store. She sees a 0-pound bag for $.90, a -pound bag for $.7, and a -pound bag for $0.99. How can Rachel find the price per pound? Which bag has the least unit price? What is the unit price for this bag? I can express a mixed number as an improper fraction and then write it as part of a complex fraction.

12 Ready to Go Problem solving READ PLAN CHANGING THE PACE Gabby has weeks to read a novel. She reads pages of a novel in 7 week. At this rate, will Gabby read the 00-page book in time? What is the problem asking you to find? whether Gabby can read pages in weeks What do you need to know to solve the problem? Gabby reads pages in week. How can you solve the problem? Find the unit rate. Then multiply the unit rate by Compare the number to 00 pages. weeks. SOLVE Write the rate as a complex fraction. 7 CHECK Write the complex fraction as a division problem. Then rewrite as multiplication. 7 Multiply and simplify. pages per week Find the number of pages she can read in weeks. pages Multiply the unit rate you found by 7 week. pages week 7 pages the number of pages should be with this rate. Will Gabby finish the 00-page book in time? Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Lesson

13 Computing Unit Rates practice Use the problem-solving steps to help you. Marita sold cups of lemonade in hour. Jennifer sold 9 cups of fruit punch in hour. Which girl sold more drinks per hour? I remember! I need to find the unit rate first. Then I ll use the unit rate to find the answer. CHECKLIST Read Plan Solve Check A -pound box contained 6 fruit tarts. How many tarts would be in a one-pound box? CHECKLIST Read Plan Solve Check Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC A coach compared the scoring of four players over the season. Lorraine played in 8 games and scored 8 points. Jana played in games and scored 68 points. Maggie played in 9 games and scored points. Nikki played in 7 games and scored 6 points. Which player scored the greatest number of points per game? CHECKLIST Read Plan Solve Check

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15 Contents Student Edition Contents vi Instructional Overview viii Student Edition Overview viii Teacher s Manual: An Annotated Guide xi Understanding the Common Core State Standards xvii Adapted from the Common Core State Standards Initiative Web site: Common Core State Standards Correlation Chart xviii Standards for Mathematical Practice xxiv Common Core State Standards Lesson Computing Unit Rates Using Ratios Finding Rates from Complex Fractions Computing Unit Rates Lesson Identifying the Constant of Proportionality Unit Rates as Ratios Graphing a Proportional Relationship Identifying the Constant of Proportionality Lesson Solving Real-World Problems with Ratios and Percents... 8 Writing Equivalent Forms: Fraction/Decimal/Percent Writing Equations to Solve Ratio Problems Solving Real-World Problems with Ratios and Percents 6.RP., 6.RP..a 7.RP. 7.RP. 6.RP. 7.RP..d 7.RP..b.NF.6, 6.RP..C 7.RP..c, 7.RP. 7.RP. Lesson Using Proportional Relationships to Solve Multi-Step Problems Using Percents Solving One-Step Ratio and Percent Problems Using Proportional Relationships to Solve Multi-Step Problems 6.RP..c 7.RP. 7.RP. Lesson Adding and Subtracting Rational Numbers Representing Opposite Situations Adding a Number and its Opposite Adding and Subtracting Rational Numbers Lesson 6 Multiplying Rational Numbers Multiplying Fractions Using Rules to Multiply Integers Multiplying Rational Numbers Lesson 7 Dividing Rational Numbers Finding Quotients of Fractions Using Rules to Divide Integers Dividing Rational Numbers 6.NS., 7.NS..a 7.NS..b 7.NS..c, 7.NS..d.NF..a 7.NS..a 7.NS..a 6.NS. 7.NS..b 7.NS..b Duplicating this page is prohibited by law. 0 Triumph Learning, LLC contents iii

16 Lesson 8 Problem Solving with Rational Numbers Problem Solving with Whole Numbers Simplifying Complex Fractions Problem Solving with Rational Numbers Lesson 9 Factoring and Expanding Linear Expressions Recognizing and Generating Equivalent Expressions Using the Distributive Property to Factor an Expression Factoring and Expanding Linear Expressions Lesson 0 Applying Properties to Solve Problems Estimation Strategies Using Properties of Operations to Solve Problems Applying Properties to Solve Problems Lesson Solving Multi-Step Real-World Problems Solving Single-Step Mathematical Problems with Decimals Solving Single-Step Mathematical Problems with Fractions Solving Multi-Step Real-World Problems Lesson Solving Word Problems Algebraically Solving One-Step Algebraic Equations Solving Two-Step Algebraic Equations Solving Word Problems Algebraically Lesson Using Inequalities to Solve Problems Solving One-Step Inequalities Writing Two-Step Inequalities Using Inequalities to Solve Problems Lesson Scale Drawings Writing and Solving Proportions Using Proportions to Find Lengths of Sides Scale Drawings Lesson Solving Problems with Circles Radius, Diameter, and Circumference of a Circle Area of a Circle Solving Problems with Circles Lesson 6 Area, Volume, and Surface Area Area of Triangles and Parallelograms Volume of Rectangular Prisms Area, Volume, and Surface Area Common Core State Standards 7.NS. 7.NS. 7.NS. 6.EE., 6.EE. 7.EE. 7.EE., 7.EE. 7.EE. 7.EE., 7.NS..c 7.EE., 7.NS..c 7.EE. 7.EE. 7.EE. 6.EE.7 7.EE..a 7.EE..a 6.EE.8 7.EE..b 7.EE..b 6.RP. 7.G. 7.G. 7.G. 7.G. 7.G. 6.G. 6.G. 7.G.6 iv co n t e n ts Duplicating this page is prohibited by law. 0 Triumph Learning, LLC

17 Common Core State Standards Lesson 7 Drawing Inferences about a Population Ṙecognizing Statistical Questions and Describing Data Identifying a Representative Sample of a Population Drawing Inferences about a Population Lesson 8 Making Predictions with Experimental Probability Probability Finding Experimental Probability Making Predictions with Experimental Probability Lesson 9 Probability Models Exploring Theoretical Probability Exploring Experimental Probability Probability Models Lesson 0 Tree Diagrams Probability of Compound Events Sample Space of Compound Events Tree Diagrams 6.SP., 6.SP. 7.SP. 7.SP. 7.SP. 7.SP.6 7.SP.6 7.SP. 7.SP.6 7.SP.7.a, 7.SP.7.b 7.SP.8.a 7.SP.8.b 7.SP.8.b Appendix: Math Tools A Duplicating this page is prohibited by law. 0 Triumph Learning, LLC contents v

18 Student Edition Contents Contents Common Core State Standards Lesson Computing Unit Rates Lesson Identifying the Constant of Proportionality RP. 7.RP..b Lesson Lesson Solving Real-World Problems with Ratios and Percents Using Proportional Relationships to Solve Multi-Step Problems RP. 7.RP. Lesson Adding and Subtracting Rational Numbers..... Lesson 6 Multiplying Rational Numbers Lesson 7 Dividing Rational Numbers Lesson 8 Problem Solving with Rational Numbers Lesson 9 Factoring and Expanding Linear Expressions... 8 Lesson 0 Applying Properties to Solve Problems NS..c, 7.NS..d 7.NS..a 7.NS..b 7.NS. 7.EE., 7.EE. 7.EE., 7.NS..c Lesson Solving Multi-Step Real-World Problems Lesson Solving Word Problems Algebraically EE. 7.EE..a Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Lesson vi student edition contents 0 Triumph Learning, LLC

19 Common Core State Standards Lesson Using Inequalities to Solve Problems Lesson Scale Drawings Lesson Solving Problems with Circles Lesson 6 Area, Volume, and Surface Area Lesson 7 Drawing Inferences about a Population EE..b 7.G. 7.G. 7.G.6 7.SP. Lesson 8 Making Predictions with Experimental Probability SP.6 Lesson 9 Probability Models Lesson 0 Tree Diagrams SP.7.a, 7.SP.7.b 7.SP.8.b Glossary Math Tools Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC 0 Triumph Learning, LLC student edition contents vii

20 LESSON Computing Unit Rates Plug In Using Ratios FOUNDATIONAL UNDERSTANDING ON-LEVEL TARGET Plug In Using Ratios Student Edition. pp. power up Finding Rates from Complex Fractions ready to go Computing Unit Rates OBJECTIVES CONCEPTS AND SKILLS VOCABULARY Use ratios to describe quantities. Find equivalent ratios. Use equivalent ratios to solve problems. Write a rate as a complex fraction. Find unit rates. Understand the concept of a unit rate. ạ b associated with a ratio a:b with b 0, and use tables to find and compare equivalent ratios. CCSS: 6.RP., 6.RP..a Compute unit rates as complex fractions. CCSS: 7.RP. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. CCSS: 7.RP. ratio equivalent ratios rate complex fraction unit rate MATERIALS Math Tool: Two-Color Counters, p. A9 (Student Edition p. 6) Index cards (suggested) Counters in two different colors (suggested) Build Background Talk to students about reasons to use ratios in real life. For example, Max makes baskets out of attempts. Jackson makes 6 baskets out of 8 attempts. Who will make more baskets in 6 attempts? Explain that using ratios helps answer that question. MP Have students discuss additional examples of real situations that involve using ratios. Tell students they will work on writing ratios to represent quantities, finding equivalent ratios, and using equivalent ratios to solve problems. english language learners MP ELL students may need extra support for writing ratios. Give students two different colored counters or have them use the Math Tool: Two-Color Counters. Have students practice writing ratios for different groupings of the counters. Introduce and Model Introduce Concepts and Vocabulary Guide students through the information about ratios and equivalent ratios. Emphasize that to find equivalent ratios, students must multiply or divide the numerator and denominator by the same value. Use Words to Know to clarify their understanding of vocabulary. MP6 Have students explain the terms ratio and equivalent ratios to a partner. Support Discussion MP Have partners discuss briefly before group discussion. As needed, remind students of how to read a ratio table. Prompt: How do you read a ratio table? Sentence Starter: A ratio table allows Lesson 0 Triumph Learning, LLC

21 LESSON Plug In Computing Unit Rates Using Ratios A ratio compares two quantities. It can be expressed in different ways. The ratio of apples to oranges is 8 to. Equivalent ratios can be written as the same fraction in simplest form. A table of equivalent ratios shows how two quantities are related. B You can use multiplication and division to find equivalent ratios for 6. Complete the table of equivalent ratios. Multiply both quantities to find equivalent ratios. Divide both quantities to find more equivalent ratios. DO Computing Unit Rates The numbers in the top row are the numerators of the fractions for the ratios Lesson Write: 8 or 8: or 8 to Ice Cubes Used Glasses of Iced Tea Ok! I can write a ratio as a fraction, with a colon (:), or with the word to I see! All of the ratios ice cubes of glasses are equivalent because they can be written as the simplified fraction. C You can use equivalent ratios to solve problems. Janet runs,60 feet in minutes. Dan runs,70 feet in minutes. If they keep the Feet same paces, who would run Minutes farther in minutes? DO Janet s Running Times,60,80 7,90 0, Words to ratio Know a comparison of two quantities 6 or 6: or 6 to equivalent ratios ratios with the same value that can be expressed as the same fraction in simplest form Write the ratio of feet to minutes for each person. Complete the table to write equivalent ratios. Feet Minutes Dan s Running Times,70,60,90 8, DISCUSS How can you use the table above to find the number of ice cubes in a given number of glasses of iced tea? Possible response: Find the column with that number of glasses in the bottom row to read the related number of ice cubes in the top row. A You can describe quantities with a ratio. Write the ratio of markers to boxes. Compare the quantities. Write the ratios in different ways. DO PEN MARKER PEN MARKER PEN MARKER 0 markers boxes 0 0 : 0 to Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Use the table to compare the ratios. Dan would run farther in minutes. He would run 8,60 feet and Janet would run 7,90 feet. How could you find the number of feet run for a number of minutes that is not in the table? Possible response: by multiplying or dividing both quantities of an equivalent ratio by the same value to produce the given number of minutes DISCUSS PRACTICE Complete the table of equivalent ratios. Feet 6 9 Yards Text Messages Cost ($) Solve. Alex reads pages in 0 minutes. Jenna reads pages in minutes. They are both reading a 60-page magazine. Who will finish reading the magazine first? Jenna Lesson Model Application DO A Remind students that the order of the quantities in a problem determine the order of the numbers in the ratio. DO B Guide students as they find equivalent ratios. Explain that equivalent ratios have the same value. DO C Monitor to make sure that students are completing the tables correctly. If needed, set up the table for students. Support Discussion MP Have partners discuss briefly before group discussion. As needed, remind students of ways to find equivalent ratios. COMMON ERRORs When writing ratios, students may reverse the values. Remind them that order is important: whichever word comes first, its number comes first in the ratio. Have students circle the first quantity so that they remember that is the first number in the ratio. Prompt: How do you find an equivalent ratio? Sentence Starter: To find an equivalent ratio... Practice and Assess Ask students to complete practice items on page independently or in pairs. Monitor ongoing work. Observe whether students are correctly computing equivalent ratios. Use the chart below as needed to address any difficulties. Observation Students have difficulty completing ratio tables. Action Have students practice simplifying ratios and identifying equivalent ratios. Make sets of index cards for students to practice in pairs. On one set of index cards, write ratios that can be simplified. On the second set of index cards, write the simplified ratios. Have students work together to find the equivalent ratios. 0 Triumph Learning, LLC Computing Unit Rates

22 power up Finding Rates from Complex Fractions FOUNDATIONAL UNDERSTANDING ON-LEVEL TARGET Plug In Using Ratios power up Finding Rates from Complex Fractions Student Edition. pp. 6 7 ready to go Computing Unit Rates OBJECTIVES CONCEPTS AND SKILLS VOCABULARY Use ratios to describe quantities. Find equivalent ratios. Use equivalent ratios to solve problems. Write a rate as a complex fraction. Find unit rates. Understand the concept of a unit rate. ạ b associated with a ratio a:b with b 0, and use tables to find and compare equivalent ratios. CCSS: 6.RP., 6.RP..a Compute rates as complex fractions. CCSS: 7.RP. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. CCSS: 7.RP. ratio equivalent ratios rate complex fraction unit rate MATERIALS index cards (suggested) english language learners MP6 Students may have difficulty with the fraction terms numerator and denominator. Have students make an index card to use as a reference. Students should define the terms and label an example. Build Background Talk to students about reasons to write rates as complex fractions. For example, Leslie rowed her boat a.. mile in hour. What is her average speed in miles per hour? Explain that writing a rate as a complex fraction can be used to answer this question. MP Have students discuss additional examples of real situations in which they would write rates as complex fractions. Tell students they will write rates as complex fractions. Introduce and Model Introduce Concepts and Vocabulary Guide students through the information about writing rates using complex fractions. Emphasize that the numerator of the complex fraction is the first quantity being compared and the denominator of the fraction is the second quantity being compared. Use Words to Know to clarify their understanding of vocabulary. MP6 Have students write the terms rate and complex fraction in their own words. Then have volunteers share their definitions. Support Discussion MP7 Have partners discuss briefly before group discussion. As needed, remind students of how they used ratios. Prompt: How did you use ratios to solve problems? Sentence Starter: It is helpful to write rates as complex fractions because Lesson 0 Triumph Learning, LLC

23 Computing Unit Rates POWER up A rate is a ratio that compares quantities with different units of measure. pounds 6 dollars This ratio compares pounds to dollars, so it is a rate. A complex fraction has a fraction in the numerator, denominator, or both. A complex fraction can be rewritten as a division expression. Words to rate Know a ratio that compares quantities with different units of measure 0 miles hours Finding Rates from Complex Fractions I can think of a complex fraction as a fraction divided by a fraction. A rate that compares fractions can be written as a complex fraction. A mouse can move about miles in hour. This complex fraction has a fraction only in the denominator. complex fraction a fraction that has a fraction in the numerator, denominator, or both 7 8 I get it! The units of the rate, miles, tell me hours B You can use complex fractions to write rates which values are in the numerator and the that compare two fractions. denominator of the complex fraction. Jamal walks 7 8 mile in hour. How can you write this rate as a complex fraction? 7 Understand which quantities are being compared. miles 8 Write the complex fraction. hour DO Trevor read of a book in days, and he writes this rate as. What can you tell Trevor about his work? Possible response: He is incorrect. The numerator and denominator of a complex PRACTICE fraction should be the quantities given, so Trevor s rate is. Write the rate as a complex fraction. DISCUSS A recipe calls for cup of flour to make A band marched of the parade route of a batch of cookies. in hour. Lesson Why do you think it will be helpful to write rates as complex fractions in math problems? Possible response: It will be helpful because many quantities, such as weight and time, are in fractional units. I can also find equivalent ratios. A You can use complex fractions to write rates that include a fraction. Alexandra completed of her homework in hours. Write this rate as a complex fraction. Decide what the rate compares. The rate compares the amount of homework Write a complex fraction with the to hours it took to complete. given quantities. DISCUSS DO Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC A car used of a tank of gas to travel 8 of the total distance. 8 A sprinkler system uses gallon of water every hour. 6 Lesson 7 Model Application DO A Explain that complex fractions are fractions with a numerator, denominator, or both that are also fractions. Guide students through writing the rate as a complex fraction. DO B Monitor that students write the first quantity being compared in the numerator and the second quantity being compared in the denominator. Support Discussion MP Have partners discuss briefly before group discussion. As needed, remind students that it is important to understand which quantities are being compared in the rate. spotlight on mathematical PRACTICES MP..Critique Others Reasoning Help students think about Trevor s reasoning critically by asking probing questions: What does the complex fraction that represents the rate look like? Does Trevor s fraction describe the rate? Prompt: How do you write a rate? Sentence Starter: When using a complex fraction to write a rate Practice and Assess Ask students to complete practice items on page 7 independently or in pairs. Monitor ongoing work. Observe whether students correctly write equivalent rates for the complex fractions. Use the chart below as needed to address any difficulties. Observation Students write incorrect equivalent ratios. Action Have students reread how to find equivalent ratios from the previous section. Then review with students how to multiply fractions by whole numbers. 0 Triumph Learning, LLC Computing Unit Rates

24 ready to go Computing Unit Rates FOUNDATIONAL UNDERSTANDING ON-LEVEL TARGET Plug In Using Ratios power up Finding Rates from Complex Fractions ready to go Computing Unit Rates Student Edition. pp. 8 OBJECTIVES CONCEPTS AND SKILLS VOCABULARY Use ratios to describe quantities. Find equivalent ratios. Use equivalent ratios to solve problems. Write a rate as a complex fraction. Find unit rates. Understand the concept of a unit rate. ạ b associated with a ratio a:b with b 0, and use tables to find and compare equivalent ratios. CCSS: 6.RP., 6.RP..a Compute rates as complex fractions. CCSS: 7.RP. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. CCSS: 7.RP. ratio equivalent ratios rate complex fraction unit rate MATERIALS Lesson Quiz, Assessment Manual. pp. Lesson Quiz Answer Key, Assessment Manual Math Tool: Fraction Strips, pp. A and A (Student Edition pp. 09 and ) Index cards (suggested) english language learners MP7 The vocabulary in this lesson may be confusing because all the terms are related. Have students make a diagram on an index card showing how the terms ratio, rate, and unit rate are related. Next to each term, students should provide an example. Build Background Talk to students about reasons to use unit rates. For example, Alexis walked her dog.. mile in hour. Sophia walked her dog.. mile in hour. Who will walk her dog farther in an hour? Explain that unit rates can be used to answer this question. MP Have students discuss additional examples of real situations that require the unit rates. Tell students they will find unit rates and solve problems involving unit rates. Introduce and Model Introduce Concepts and Vocabulary Guide students through the information about unit rates. Emphasize that when a rate is simplified so that it has a denominator of unit, it is called a unit rate. Use Words to Know to clarify their understanding of vocabulary. MP6 Have students explain the term unit rate to a partner. Support Discussion MP Have partners discuss briefly before group discussion. If needed, recall how to simplify a rate. Prompt: What is true about all unit rates? Sentence Starter: By using a unit rate, I can... lesson link Connect to Foundational Understanding Skills learned in the Plug In and Power Up are referenced in the Lesson Link. Explain that rates and unit rates are special types of ratios and that a unit rate is a special type of rate. Ratios, rates, and unit rates can be used to make comparisons and solve problems. 6 Lesson 0 Triumph Learning, LLC

25 Computing Unit Rates READY TO go A unit rate is a ratio comparing quantities with different units of measure to a denominator equal to. 7 8 inch year This is a unit rate because it compares 7 8 inch to year. A unit rate always compares a quantity to unit of another kind of quantity. Computing Unit Rates You can express a rate written as a complex fraction as a unit rate. 8 Since any number remains the same when divided by : The rate can be written as the unit rate. I get it! I just simplify the complex fraction and then divide the fraction by. WORk TOgEThER You can use labeled Fraction Strips to help you solve unit rate problems involving complex fractions. The rate of cups to hours is written as a complex fraction. The complex fraction is written as a division problem and simplified. The unit rate is. The Fraction Strips model the problem. Tara will drink cups of water in hour. Tara drinks cups of water in hour. At this rate, how much water will Tara drink in an hour? cups hours 6 Division lets me simplify fractions, even complex fractions! I can use fraction strips to model division. Lesson Words to unit rate Know a ratio that compares the number of units of one quantity to unit of a second quantity. dollars per pound cost unit rate pound $ lb 6 miles per hour unit rate miles 6 mi hour hr Why might it be helpful to find unit rates? Possible response: Unit rates are easy to compare to each other. Unit rates make it easy to find equivalent rates. lesson link Plug In POWER up go! DISCUSS A ratio compares two quantities. A ratio can be written in three ways. : to A ratio that includes a fraction can be written as a complex fraction. 6 I see! I can use what I know about ratios and complex fractions to find unit rates. Then I can use unit rates to solve problems. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC A You can use unit rates to solve problems. A recipe calls for cup of sugar to make gallon of iced tea. Fraction Strips can be found How much sugar is needed to make gallon of iced tea? on p. 09. Write the rate of sugar to iced tea sugar as a complex fraction. iced tea Write the complex fraction as a division problem, then rewrite as multiplication. 0 Divide both the numerator and the denominator by the denominator to find the unit rate The unit rate is. 6 cup of sugar is needed to make gallon of iced tea. DO How can you be sure that you have found the unit rate? Possible response: I can check that the denominator is. A unit rate always has in the denominator. DISCUSS 8 Lesson 9 Work Together Explain that students will use fraction strips to model complex fractions. The number of two thirds in represents the simplified fraction for the number of cups of water Tara will drink in. hour. DO A Monitor students as they use unit rates to solve problems. Remind students of how to simplify fractions using division. Support Discussion MP6 Have partners discuss briefly before group discussion. As needed, have students explain how a ratio and a unit rate are related. spotlight on mathematical PRACTICES MP6 Attend to Precision Help students think about unit rates. What makes a unit rate different from a rate? Prompt: What is true about all unit rates? Sentence Starter: A unit rate has 0 Triumph Learning, LLC Computing Unit Rates 7

26 READY TO go Computing Unit Rates PRACTICE Find the unit rate. Use Fraction Strips to help you. A recipe calls for eggs for every teaspoon of salt. Kathy s heart beats 60 times in minute. 60 Fraction Strips can be found on p.. Compare the unit rates. Alice skates mile in hour. Elizabeth skates mile in 9 hour. Who skates farther in hour? Alice s Rate Elizabeth s Rate The unit rate is 0 eggs for teaspoon. The unit rate is 80 times in minute. Alice skates miles in hour. Elizabeth skates miles in hour. Elizabeth skates farther in hour. Solve. I see! Miles per hour means number of miles in hour. Find the unit rate. mile in hour 0 words in minute 0 REMEMBER A unit rate compares units of one quantity to unit of a different quantity. 6 Maurice walked 6 miles on the treadmill in hours. How many miles per hour did Maurice walk? 7 On average, a person who weighs 0 pounds burns 8 calories in 6 hour while playing basketball. At this rate, how many calories are burned in hour? miles 98 calories The unit rate is 9 8 miles in hour The unit rate is words in minute. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC I can express a mixed Determine the Rate number as an improper Rachel wants to find the best unit price for potatoes fraction and then at a grocery store. She sees a 0-pound bag for $.90, write it as part of a a -pound bag for $.7, and a complex fraction. -pound bag for $0.99. Possible response: She can find the unit rate, or price per pound, How can Rachel find the price per pound? for each bag. Which bag has the least unit price? the -pound bag What is the unit price for this bag? $0. per pound DISCUSS 0 Lesson ADDITIONAL PRACTICE Provide students with additional practice to model and solve: Leon bikes 0 miles in hour. What is his average speed in miles per hour? Patti runs miles in hour. What is her average speed in miles per hour? spotlight on MATHEMATICAL LANGUAGE MP6 Support students in using mathematical language as they work: What is the rate? Is the rate a complex fraction? How did you find the unit rate? Support Independent Practice Remind students to read the REMEMBER. If needed, ask: Does each unit rate have a denominator of? 7 Which quantity goes in the numerator? Which quantity goes in the denominator? Support Discussion MP Have partners discuss briefly before group discussion. As needed, have students discuss how they have compared different ratios and rates. Prompt: How can you compare different rates? Sentence Starter: The bag that has the lowest per-pound price is Problem Solving Model the Four-Step Method MP MP Guide students through the four-step method using think-aloud strategies. Point out the comparison clue words at this rate. Think Aloud Gabby can read pages in 7 week. To determine if she will finish the book on time, I must find the unit rate, the number of pages she reads in week. 8 Lesson 0 Triumph Learning, LLC

27 READY TO go Computing Unit Rates PROblEm solving READ PlAn CHANGING THE PACE Gabby has weeks to read a novel. She reads pages of a novel in 7 week. At this rate, will Gabby read the 00-page book in time? What is the problem asking you to find? whether Gabby can read 00 pages in weeks PRACTICE Use the problem-solving steps to help you. Marita sold cups of lemonade in hour. Jennifer sold 9 cups of fruit punch in hour. Which girl sold more drinks per hour? Jennifer I remember! I need to find the unit rate first. Then I ll use the unit rate to find the answer. CHECKLIST READ PLAN SOLVE CHECK Lesson What do you need to know to solve the problem? Gabby reads pages in 7 week. solve How can you solve the problem? Find the unit rate. Then multiply the unit rate by Compare the number to 00 pages. Write the rate as a complex fraction. 7 weeks. A -pound box contained 6 fruit tarts. How many tarts would be in a one-pound box? 8 CHECKLIST READ PLAN SOLVE CHECK ChECk Write the complex fraction as a division 7 problem. Then rewrite as multiplication Multiply and simplify. pages per week Find the number of pages she can read in weeks. 7 pages Multiply the unit rate you found by 7 week. 0 pages week 7 7 pages The number of pages should be with this rate. Will Gabby finish the 00-page book in time? No Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC A coach compared the scoring of four players over the season. Lorraine played in 8 games and scored 8 points. Jana played in games and scored 68 points. Maggie played in 9 games and scored points. Nikki played in 7 games and scored 6 points. Which player scored the greatest number of points per game? Lorraine CHECKLIST READ PLAN SOLVE CHECK Lesson Support Problem-Solving Practice Have students use the Checklist as they complete each step. Prompt: Can you show me the unit rate for each girl? Prompt: How did you simplify the ratio? Prompt: How did you compare the ratios? Explore Student Thinking MP8 Have partners compare their work on a problem and describe their results. COMMON ERRORS When finding the unit rate, students may not simplify the denominator to a single unit. Remind students that the unit rate must have a denominator of. Assess Use the table below to observe whether students accurately calculated unit rates and to address any difficulties as needed before the quiz. When all students are ready, assign the Lesson Quiz. Observation Errors in finding unit rates are frequent; general confusion about solving problems involving unit rates. Observation Makes occasional errors when finding unit rates; some understanding of solving unit rate problems. Observation Finds unit rates and solves unit rate problems accurately. Action Have students draw bar diagrams to model the rates and to find the unit rate. Action Provide additional practice problems for finding unit rates. Encourage students to carefully follow the steps to finding the unit rate. Action Assign the Lesson Quiz. 0 Triumph Learning, LLC Computing Unit Rates 9

28 Target Foundational Mathematics Assessments 7 First Edition

29 Contents Lesson Quizzes Lesson Quiz.... Lesson Quiz... 6 Lesson Quiz... 8 Lesson Quiz... 0 Lesson Quiz... Lesson 6 Quiz... Lesson 7 Quiz... 6 Lesson 8 Quiz... 8 Lesson 9 Quiz... 0 Lesson 0 Quiz... Lesson Quiz... Lesson Quiz... 6 Lesson Quiz... 8 Lesson Quiz... 0 Lesson Quiz... Lesson 6 Quiz... Lesson 7 Quiz... 6 Lesson 8 Quiz... 8 Lesson 9 Quiz... 0 Lesson 0 Quiz.... Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC Performance Tasks Proportions Task A... Proportions Task B... 6 Number System Task A Number System Task B Expressions and Equations Task A... Expressions and Equations Task B... Geometry Task A Geometry Task B Probability Task A Probability Task B... 6 Practice Tests Practice Test... 6 Practice Test

30 LESSON Quiz Choose the best answer.. Which is not a complex fraction? A. B. C D. 8. Which is the ratio to 6 written as a unit rate? A. B. C. 6 8 D. Write each rate as a complex fraction.. A recipe calls for cup of flour to make of a batch of brownies.. A scout troop cleaned up of the park in hour.. A car used 8 tank of gas during 9 0 of the whole distance of a trip of the class did of the homework. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC

31 Lesson Quiz Find the unit rate mile in hour 8. 0 pages in 6 hour The unit rate is 9. 8 mile in minutes miles in hour. The unit rate is 0. gallon in 6 hour pages in hour. Lesson Quizzes The unit rate is mile in minute. The unit rate is gallons in hour. Solve.. Becky ran 6 mile in 8 minutes. Sarah ran 8 mile in minutes. Who ran farther in minute? Becky ran mile in minute. Sarah ran mile in minute. ran farther in minute.. Andrew texted words in minute. Jennifer texted words in minute. Who texted more words in minute? Andrew texted minute. Jennifer texted minute. words in words in texted more words in minute. Duplicating any part of this book is prohibited by law. 0 Triumph Learning, LLC. Justin walked his dog miles in hours. How many miles per hour did Justin and his dog walk?. Ricardo wants to know when he got the best price for gas. He paid $.08 for gallons two weeks ago. Last week he paid $. for 9 gallons, and this week he paid $.70 for gallons of gas. In which week did Ricardo get the best price per gallon? Explain how you know.. Eliora burns 0 calories in 6 hour while dancing. At this rate, how many calories does Eliora burn in hour? 6. Kathy has weeks to finish her report. She wrote 8 pages in 7 week. At this rate, will Kathy finish her 0-page report on time? Explain how you know.