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LESSON 7.9 Algebra • Fractions and Properties of Addition FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards 4.NF.B.3c Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. MATHEMATICAL PRACTICES MP2 Reason abstractly and quantitatively. MP7 Look for and make use of structure. F C R Coherence: Standards Across the Grades Before Grade 4 After 3.NF.A.1 4.NF.B.3c 5.NF.A.1 5.NF.A.2 F C R Rigor: Level 1: Understand Concepts....................Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency.......On Your Own Level 3: Applications..................................Think Smarter and Go Deeper Learning Objective Use the properties of addition to add fractions. Language Objective Students present to a partner a step-by-step demonstration of how you can add fractions with like denominators using the properties of addition. Materials MathBoard FC R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 383J. About the Math Professional Development Why Teach This When students apply the Commutative and Associative Properties of Addition to fractions, they discover they can use mental math to add some fractions. For example, they can use the properties to change the order or grouping of addends so the sum of two fractions is 1. Encourage students to look in each exercise for fractions that have a sum of 1. Then have them decide how to use the properties to change the order or grouping of the fractions so that the mixed numbers whose fractions have a sum of 1 are next to each other or added first. Professional Development Videos 435A Chapter 7 Interactive Student Edition Personal Math Trainer Math on the Spot 1 ENGAGE Daily Routines Common Core Problem of the Day 7.9 Jess is laying down tiles in rows. She will have more than 4 rows, but less than 10 rows. She wants to lay down the same number of tiles in each row. She starts with 63 tiles. What numbers of rows could have the same number of tiles? 7 and 9 rows Vocabulary Interactive Student Edition Multimedia eGlossary with the Interactive Student Edition Essential Question How can you add fractions with like denominators using the properties of addition? Making Connections Ask students to tell what they know about addition properties. • What are some addition properties? Possible answer: Commutative and Associative Properties of Addition • How can the Commutative and Associative Properties help you add 25 + 86 + 75 mentally? Possible answer: The Commutative Property lets me change the order of 86 and 75. The Associative Property lets me group 25 and 75 to get 100. Then I just have to add 100 + 86 to get 186. Learning Activity What is the problem the students are trying to solve? Connect the story to the problem. Ask the following questions. Vocabulary Builder Use this activity to remind students of the Commutative Property of Addition and the Associative Property of Addition. Write each of the following equalities on the board. Have students name the property that is shown. Then ask students to explain how the property makes it easier to find the sum of the fractions. Possible explanation: the two fractions that have a sum of 1 are next to each other or added first. __ + 1 __ + 3 __ = 5 __ + 3 __ + 1 __ Commutative 1. 5 • Suppose you are trying to find the sum of 3 fractions mentally. If possible, what sum do you think would be helpful to make with 2 of the fractions? Possible answer: 1 • What is the relationship between the numerator and denominator of a fraction that is equivalent to 1? They are the same. Literacy and Mathematics View the lesson opener with the students. Then, have students complete the following activity. • Have students work in small groups to create a table that lists the Commutative and Associative Properties of Addition, explains each property, and provides a whole number and a fraction example for each property. 8 8 8 8 8 8 1 11 5 1 5 Associative ___ ___ ___ ___ ___) + ___ 2. + ( + ) = ( + 11 12 12 12 12 12 12 1 + ___ 7 = ___ 3 + ___ 7 + ___ 1 Commutative 3 + ___ 3. ___ 10 10 10 10 10 10 __ + 7 __) + 2 __ = 8 __ + (7 __ + 2 __) Associative 4. (8 9 9 9 9 9 9 How can you add fractions with like denominators using the proper ties of addition? Lesson 7.9 435B LESSON 7.9 2 EXPLORE 4.NF.B.3c Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Add and subtract mixed numbers with like DO NOT EDIT--Changes must beeach made through “Filewith info”an equivalent fraction, and/or by using properties of operations denominators, e.g., by replacing mixed number CorrectionKey=B and the relationship between addition and subtraction. Connect Essential Question How can you add fractions with like denominators using the properties of addition? connect The Associative and Commutative Properties of Addition can help you group and order addends to find sums mentally. You can use mental math to combine fractions that have a sum of 1. ELL Strategy: Develop Meanings • The Commutative Property of Addition states that when the order of two addends is changed, the sum is the same. For example, 4 + 5 = 5 + 4. • The Associative Property of Addition states that when the grouping of addends is changed, the sum is the same. For example, (5 + 8) + 4 = 5 + (8 + 4). The terms Commutative Property of Addition and Associative Property of Addition may be unfamiliar academic words. The map shows four lighthouses in the Florida Keys and their distances apart in miles. The Dry Tortugas Lighthouse is the farthest west, and the Alligator Reef Lighthouse is the farthest east. 5 + 43 ___ 6 + 34 ___ 5 Add. 70 ___ 10 10 5 70__ 5 34__ 6 43__ 10 10 ) + _ 10 = (_ +_ 6 43__ Use the Commutative Property to order the addends so that the fractions with a sum of 1 are together. Use the Associative Property to group the addends that you can add mentally. Add the grouped numbers, and then add the other mixed number. 105 ) + _ 10 = (_ 6 ___ 148 10 =_ Write the sum. So, the distance from the Dry Tortugas Lighthouse to the Alligator Reef Lighthouse, traveling between the four lighthouses, 6 148___ 10 miles. is _ Chapter 7 435 4_MNLESE342255_C07L09.indd 435 10/8/14 4:30 PM Reteach 7.9 Enrich 7.9 3 2 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B 1 Lesson 7.9 Reteach Name Use addition properties to help you solve each problem. 1. 6135316 The Associative Property of Addition states that when the grouping of addends is changed, the sum is the same. (3 1 6) 1 4 5 3 1 (6 1 4) 7 1 65 3 1 4_ _. Use the properties and mental math to add 10 _ 8 8 8 _ 1 47 _ 1 65 _ 10 3 8 8 8 Step 2 Use the Commutative Property to order the addends so that the fractions 3 1 6_ 5 1 47 _ _ 1 47 _ 1 65 _ 5 10 _ with a sum of 1 are together. 10 3 8 8 8 8 8 8 3 1 6_ 7 5 1 4 _ Step 3 Use the Associative Property to group 5 10 __ 8 8 8 the addends that you can add mentally. ( 3. ) _ 5 (17) 1 4 7 8 Step 5 Write the sum. _ 5 21 7 8 Robyn cut a length of ribbon into four pieces to wrap four gifts. The 7 lengths she cut were 16 __ 12 inches, 3 9 2 __ __ 10 __ 12 inches, 4 12 inches, and 10 12 inches. If she used the whole ribbon, how long was her ribbon? Ben’s family likes bananas. On Monday, they ate 1 3_4 pounds of bananas. On Tuesday, they ate 2 2_4 pounds. On Wednesday, they ate 2 1_4 pounds. On Thursday, they ate 1 2_4 pounds. How many pounds of bananas did Ben’s family eat during the four days? ( 1 5 ) 2 5 ( 4 5 ( ) 11 12 5 12 7 12 13___11___ 4. 2___ 11 7 __ 12 Chapter Resources © Houghton Mifflin Harcourt Publishing Company 4_MNLEAN343122_C07R09.indd 21 ) 4 10 7 10 3 10 16___ 2. 5___ 11___ 2 9_ 5 4 13 __ 10 7 8 ( 3 8 ) 1 8 5. 4__1 6__1__ 3 11 _ 8 7-21 Emily enjoys riding her bike. During a five-day biking trip, she rode 8 1_8 miles, 4 3_8 miles, 5 4_8 miles, 2 7_8 miles, and 6 _18 miles. How many miles in all did she ride during the trip? 4. Ms. Cleary runs a catering business. She is buying fruit to make a large order 3 for fruit salad. She buys 5 __ 10 pounds 4 of apples, 3 __ 10 pounds of oranges, 1 3 __ 2 __ 10 pounds of bananas, 4 10 pounds of 4 __ green grapes, and 5 10 pounds of red grapes. How many pounds of fruit did Ms. Cleary buy in all? 27 miles 5 pounds 20__ 10 8 pounds Use the properties and mental math to find the sum. 1. 3__11__ 14__ 2. 9 inches 41__ 12 Step 1 Look for fractions that combine to make 1. Step 4 Add the grouped numbers and then add the other mixed number. Lesson 7.9 Enrich Mixing Properties Properties of addition can help you group and order addends so you can use mental math to find sums. The Commutative Property of Addition states that when the order of two addends is changed, the sum is the same. 3 4 ) ( 1 4 3. 7__1 513__ 5. 16 2 6 ( 1 6 Explain how you used the commutative and associative properties to help you add the mixed numbers. Possible answer: I first looked for fractions that had a sum of 1. I then ordered and ) 4 6 6. 9__1 4__17__ Differentiated Instruction Name Algebra • Fractions and Properties of Addition any two numerators equals the denominator of the fractions. 435 Chapter 7 10 5 5 6 70__ 34__ 43__ 6 + 34 ___ 5 + 43 ___ 5 =_ 10 10 10 70 ___ +_ +_ 10 10 10 10 = 1 5 + 5 = 10 and __ 10 need to use the Commutative Property first because 4 6 __ __ 10 + 10 make a sum of 1, which is easy to add mentally. 34 150 Use the properties to order and group. © Houghton Mifflin Harcourt Publishing Company • Image Credits: (tr) ©Danita Delimont/Alamy MATHEMATICAL PRACTICES MP6 Attend to precision. 5 __ • How would your first step change if 70 10 4 __ was 70 10 instead? Possible answer: I would not 43 160 70 150 What is the distance from the Dry Tortugas Lighthouse to the Alligator Reef Lighthouse, traveling between the four lighthouses? Unlock the Problem MP8 Look for and express regularity in repeated reasoning. • When adding three mixed numbers with like denominators, how do you know if you can reorder and/or regroup the addends so that the fractions with a sum of 1 are together? Possible answer: check if the sum of Number and Operations— Fractions—4.NF.B.3c MATHEMATICAL PRACTICES MP2, MP7, MP8 Unlock Unlock the the Problem Problem • Write both terms on the board. • Have students draw cars driving around. Tell them the cars are commuting. Have them write Commutative Property of Addition under the picture and write an example of it. • Then have students draw people they associate (socialize) with. Tell them the people are associating, or are spending time together in a group. Have them write Associative Property of Addition under the picture and write an example of it. How can you find the distance between lighthouses in the Florida Keys? Read the problem to find out. Encourage students to think about how the properties can help them add mentally. • Which fractions in the expression have a 5 5 __ sum of 1? How do you know? __ 10 + 10 ; Lesson 7.9 Fractions and Properties of Addition Remind students of the definitions of each property they will use in this lesson. It may help to discuss the common meanings of commute and associate. ALGEBRA Name _ 21 1 6 grouped the addends so the fractions with a sum of 1 were together. Reteach Chapter Resources © Houghton Mifflin Harcourt Publishing Company 2/18/14 3:45 PM 4_MNLEAN343122_C07E09.indd 22 7-22 Enrich 11/13/14 12:26 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A Try This! Make sure students understand why the Commutative Property was used first. • Why is the order of 2 and 3 _23 changed? The Try This! Use the properties and mental math to solve. Show each step, and name the property used. order is changed so that the mixed numbers whose fraction parts add to 1 are next to each other. 1 1_3 + (2 + 3 2_3 ) ( ) _ + 32 _+2 11 Commutative Property of Addition ( Associative Property of Addition 3 3 _ + 32 _ +2 11 3 3 ) • Why is the Associative Property used to change the way the numbers are grouped? The grouping was changed so that we add 1 _13 and 3 _23 first to get the sum, 4 + 1. (1 + 4) + 2 5+2=7 Share Share and and Show Show Math Talk Use Math Talk to focus on students’ understanding of how to use the Commutative and Associative Properties to add 3 mixed numbers. MATH BOARD 1. Complete. Name the property used. (3104 + 5102 ) + 106 = (5102 + 3104 ) + _ = 5 2 + (3 4 + _) 10 10 ____ ____ ____ ____ ____ ____ ____ 6 __ 10 6 __ 10 2 + = 5 ____ 4 10 _ Commutative Property of Addition _____ Associative Property of Addition _____ Math Talk 2 or 91 _ 9__ =_ 5 10 Possible description: use the Commutative Property to order the mixed numbers so that the fractions that have a sum of 1 are together and then add 2 5_8 . (1 1_3 + 1 2_3 ) + 2 5_8 = 3 + 2 5_8 = 5 5_8 3 EXPLAIN MATHEMATICAL PRACTICES 2 Reason Abstractly Describe how you could use the properties to find the sum 1 1_3 + 2 5_8 +1 2_3 . Share and Show Use the properties and mental math to find the sum. (278 + 382) + 118 __ __ 3. ( ) __ + 1 + 3 __ 12 5 5 3 4. __ 15 5 __ or 71 __ 72 5. 6 6 6 4 8 6 3 1 + 11 __) + 2 __ (1__ 4 4 4 __ 51 4 6. 5 2) + 3 __ 4 + 1 __ (12 __ 9 9 9 2 17 __ 9 7. 9) 8 + ____ 3 + (1____ ____ 12 12 12 2 8 or 2__ 2 ___ 3 12 436 4_MNLESE342255_C07L09.indd 436 12/02/14 2:54 PM Advanced Learners Logical Individual •For the following sums, have students work backward to write at least two addends for each sum. They should use at least two properties when finding the addends. 1 1. 7__ 5 2. 8__ 3 8 __ 3. 15 4. 42 5 Possible answer for Exercise 1: 1 = 1 7__ 6 + 1__ 3 3 1 = 2 + 3 + 1 + 1__ 3 2 1 __ = 2 + 3 + __ + __ + 11 3 3 2 1 __ __ = 2 + 3 + 3 + + 3 2 1 __ __ 1 = 23 + 3 + 1__ 3 3 3 __ 11 BOARD The first problem connects to the learning model. Use the checked exercises for Quick Check. __ + (5 5 __ + 4 3 __) 53 © Houghton Mifflin Harcourt Publishing Company __ © Houghton Mifflin Harcourt Publishing Company 2. MATH Quick Check Quick Check If If Then 3 2 31 2 1 Rt I Rt I a student misses the checked exercises Differentiate Instruction with • Reteach 7.9 • Personal Math Trainer 4.NF.B.3c • RtI Tier 1 Activity (online) COMMON ERRORS Error Students may not recognize pairs of COMMON fractions that have a sumERRORS of 1. Example In the expression 2 _95 ∙ (1 _91 ∙ 3 9_4 ), the ease of adding _95 and _94 is not recognized. Springboard to Learning Use examples to demonstrate that when the sum of the numerators of two fractions with like denominators is the same as the denominators, the sum of the fractions is 1. 3 Lesson 7.9 436 Name On On Your Your Own Own On Your Own Use the properties and mental math to find the sum. If students complete the checked exercises correctly, they may continue with the remaining exercises. Suggest to students that they circle the fraction pairs in Exercises 8–10 that have a sum of 1. 8. (451_3 + 61_3) + 382_3 ( 9. _1 + 103 1_ + 12 2 2 116 1 90 _ 3 4 ELABORATE 11. DEEPER Pablo is training for a marathon. He ran 5 4_8 miles on Friday, 6 5_8 miles on Saturday, and 7 4_8 miles on Sunday. How many miles did he run on all three days? Problem Solving • Applications ) 12. ( ) 5 + 10 + 11__ 5 10. 3 __ 10 10 25 DEEPER At lunchtime, Dale’s Diner served a total of 2 2_6 pots of vegetable soup, 3 5_6 pots of chicken soup, and 4 3_6 pots of tomato soup. How many pots of soup were served in all? 10 2_3 pots 19 5_8 miles MATHEMATICAL PRACTICES OqnakdlRnkuhmf¤@ookhb`shnmr OqnakdlRnkuhmf¤@ookhb`shnmr SMARTER Use the expressions in the box for 13–14. Exercise 14 is a multistep problem that requires students to find the value of four expressions and then compare those values. Have volunteers describe how they can find the value of expression C, which has fractions with different denominators. 13. Which property of addition would you use to 8 Associative SMARTER 14. Which two expressions have the ( Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. © Houghton Mifflin Harcourt Publishing Company Use this video to help students model and solve this type of Think Smarter problem. SMARTER D 1_ + 4_ + 2_ 3 3 2_5 + (5 4_5 + 2 1_5 ) = 3 2_5 + (2 _51 + 5 4_5 ) (1 _18 + 3 _56 ) + 2 2_6 + _5 ) + 3 3_ 8 8 3 3 Match the equation with the property used. 6 6 3 6 6 3 __ __ __ __ __ __ 12 + ( 12 + 12 ) = ( 1 2 + 12 ) + 12 (4 _16 ) C _37 + _12 + 4_7 same value? 15. ) 8 B _1 + 2 2 A and C Math on the Spot Video Tutor ( A 1_ + _78 + 4_ regroup the addends in Expression A? = (3 _56 = 1 _18 + 4 _1 ) + 2 2_ + ( 8_5 6 + 3 _38 ) 6 • • • • • • Commutative Property Associative Property Chapter 7 • Lesson 9 SMARTER For Exercise 15, students must understand the difference in the application of the Commutative Property and the Associative Property. If students classify all four equations incorrectly, they may have switched the definitions of the two properties. If students classify some of the equations correctly and others incorrectly, they may not understand the meaning of the properties themselves. 437 Chapter 7 437 MATHEMATICAL PRACTICES ANALYZEt-00,'034536$563&t13&$*4*0/ Pose a Problem DEEPER Costumes are being made for the high school musical. The table at the right shows the amount of fabric needed for the costumes of the male and female leads. Alice uses the expression 7 3_8 + 1 5_8 + 2 4_8 to find the total amount of fabric needed for the costume of the female lead. 16. Pose a Problem Costume Fabric (in Yards) Female Lead Costume Male Lead Costume Silk 7 3_8 1 2_8 Felt 1 5_8 2 3_8 Cotton 2 4_8 5 6_8 Material To find the value of the expression using mental math, Alice used the properties of addition. Have students read Exercise 16 and describe the steps Alice followed to solve the problem. Ask a volunteer to record each step on the board. DEEPER __ + 15 __ + 24 __ = 73 __ + 15 __ + 24 __ 73 8 8 8 8 8 8 ( ) Challenge your advanced learners to write addition problems with three addends that are mixed numbers, such that the fraction parts of two of the three mixed numbers have a sum of 1. Their problems should be able to be solved by using the Commutative and Associative Properties. Invite volunteers to share their work with the class. MP7 Look for and make use of structure. • When adding three mixed numbers, how does grouping addends whose fractions have a sum of 1, if possible, make it easier to add the mixed numbers mentally? Alice added 7 + 1 and was able to quickly add 3_ and 5_ to the sum of 8 to get 9. She added 24 _ 8 8 8 to 9, so her answer was 114_. 8 So, the amount of fabric needed for the costume of the female lead actor is 114_ yards. 8 Write a new problem using the information for the costume for the male lead actor. Solve your problem. Check your solution. Pose a Problem Possible problem: Alice used the expression 1 _28 + 2 _38 + 5 _68 to find the total amount of fabric needed for the costume of the male lead. What is the total amount of fabric needed for the Possible solution: Alice rewrote the expression as (1 2_8 + 5 6_8 ) + 2 3_8 and simplified it by adding the wholenumber parts and the fraction parts in the parentheses. Then she added the mixed number: (1 + 5 + 1) + 2 3_8 = 9 3_8 . So, the male lead’s costume needed 9 3_8 yards of fabric. Possible answer: only one fraction is left in the addends, so it can just be written in the sum. Three whole numbers are left to add. • MATHEMATICAL 7 Identify Relationships Explain how using the properties of PRACTICE addition makes both problems easier to solve. Possible explanation: the properties make the problems easier to solve because you can rearrange the mixed numbers so that their fraction parts add to 1. © Houghton Mifflin Harcourt Publishing Company costume? 438 5 EVALUATE Formative Assessment Essential Question Using the Language Objective DIFFERENTIATED INSTRUCTION D INDEPENDENT ACTIVITIES Properties of Addition to change the order and grouping of the addends so the mixed numbers whose fractions have a sum of 1 are next to each other or are added first. Differentiated Centers Kit Activities Fraction Bingo! Literature Sleeping Half the Day Away Reflect Have students present to a partner a step-by-step demonstration to answer the Essential Question. How can you add fractions with like denominators using the properties of addition? I can use the Commutative and Associative Games Fraction Concentration Math Journal WRITE Math Describe how the Commutative and Associative Properties of Addition can make adding mixed numbers easier. (BNFT Students complete purple Activity Card 6 by creating pictorial models of fractions and finding equivalent fractions. Students read about what fraction of the day different kinds of animals sleep. Students practice adding and subtracting fractions and mixed numbers to reveal a hidden picture. Lesson 7.9 438 Practice and Homework Lesson 7.9 Name Fractions and Properties of Addition COMMON CORE STANDARD—4.NF.B.3c Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Use the properties and mental math to find the sum. Practice and Homework 2 1 1 1__ __ 1 2 __ 1. 5 1 3 Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers. ( 3 __ 1 3 5 __ 1 2 7 __ 2. 10 1 ) 3 8 ( 8 ) 8 __ 1 3 2 __ 1 5 4 __ 3. 8 1 5 ( 5 5 ) 1 1 (4) 5_ 3 9 1_3 _ _ 165 8 _ 2 1 5 1__ __ 1 4 __ 4. 6 3 4 ( 4 ) 5. 4 17 2_5 _ (6 3__6 1 10 4__6) 1 9 __62 _ 263 6 _ _ 162 4 _ 6. (6 2__5 1 1 4__5) 1 3 __51 _ 112 5 _ Problem Problem Solving Solving 7. Nate’s classroom has three tables of different lengths. One has a length of 4 1_2 feet, another has a length of 4 feet, and a third has a length of 2 1_2 feet. What is the length of all three tables when pushed end to end? 8. Mr. Warren uses 2 1_4 bags of mulch for his garden and another 4 1_4 bags for his front yard. He also uses 3_4 bag around a fountain. How many total bags of mulch does Mr. Warren use? © Houghton Mifflin Harcourt Publishing Company 11 feet 9. _ bags 71 4 Math Describe how the Commutative and Associative WRITE Properties of Addition can make adding mixed numbers easier. Check students’ work. Chapter 7 439 Chapter 7 439 Lesson Check (4.NF.B.3c) 1. A carpenter cut a board into three pieces. One piece was 2 5_6 feet long. The second piece was 3 1_6 feet long. The third piece was 1 5_6 feet long. How long was the board? 5 feet 7_ 6 2. Harry works at an apple orchard. He picked 45 7_8 pounds of apples on Monday. He picked 42 3_8 pounds of apples on Wednesday. He picked 54 1_8 pounds of apples on Friday. How many pounds of apples did Harry pick those three days? Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. _ pounds 1423 8 Spiral Review (4.OA.B.4, 4.NBT.B.5, 4.NBT.B.6, 4.NF.B.3c) 3. There were 6 oranges in the refrigerator. Joey and his friends ate 3 2_3 oranges. How many oranges were left? 4. Darlene was asked to identify which of the following numbers is prime: 2, 12, 21, 39 Which number should she choose? 5. A teacher has 100 chairs to arrange for an 2 6. Nic bought 28 folding chairs for $16 each. assembly into equal rows. Write one way the chairs could be arranged. Include the number of rows and the number of chairs in each row. How much money did Nic spend on chairs? Possible answer: 10 rows of 10 chairs $448 © Houghton Mifflin Harcourt Publishing Company 1 2_ 3 FOR MORE PRACTICE GO TO THE 440 Personal Math Trainer Lesson 7.9 440