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LESSON 5.4 Factors and Multiples FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards 4.OA.B.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. MATHEMATICAL PRACTICES MP6 Attend to precision. MP7 Look for and make use of structure. F C R Coherence: Standards Across the Grades Before Grade 4 After 3.OA.C.7 4.OA.B.4 5.OA.B.3 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 Understand the relationship between factors and multiples, and determine whether a number is a multiple of a given number. Language Objective Students list three ways in which factors and multiples are related. Materials MathBoard F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 277H. About the Math Professional Development Teaching for Depth Students can use counters to help them determine the factors of a number. They can model the number by forming equal groups with the counters. For example, they can show equal groups of 3 when looking for the factor 3. Finding Whether 3 is a Factor of Each Number 3 5 10 12 no counters left over factor counters left over not a factor counters left over not a factor no counters left over factor Students can look at the modeled number to determine whether the counters form equal groups with none left over. If there are no counters left over, the model shows that the number of counters in each equal group is a factor of the number. For example, the models for 3 and 12 show that 3 is a factor for those numbers. The counter model for determining factors also connects to the concept of ﬁnding multiples. When students establish the factors of a number, they then also know that the number is a multiple of those factors. For example, since 3 and 12 have 3 as a factor, 3 and 12 are both multiples of 3. Professional Development Videos 299A Chapter 5 1 ENGAGE Daily Routines Common Core Problem of the Day 5.4 The table shows the price of a carton of each item at Jack’s Sporting Goods. Each carton has 8 balls. Jack's Sporting Goods with the Interactive Student Edition Essential Question How are factors and multiples related? Making Connections Ask students to tell what they know about factors. • What is a factor? A factor is a number multiplied by another number to ﬁnd a product. Item Price • Name the factors of 20. 1, 2, 4, 5, 10, 20 Carton of Soccer Balls $192 Learning Activity Carton of Basketballs $256 Mr. Neal bought a carton of basketballs. How much did he spend for 2 basketballs? $64 Vocabulary common multiple Interactive Student Edition Multimedia eGlossary Fluency Builder Common Core Fluency Standard 4.NBT.B.6 What is the problem the students are trying to solve? Connect the story to the problem. Ask the following questions. • How can you use division to tell whether a number is a factor of another number? Possible answer: Divide the number by the possible factor. If the number divides evenly, then it is a factor. Literacy and Mathematics View the lesson opener with the students. Then, choose one or more of the following activities. • Have the class make a list of the factors of 36. Then have the class count by each factor and tell whether they say 36 each time. • Have students make one list of the numbers they say when they count by 6s to 72 and another list of the numbers they say when they count by 8s to 72. Then have them identify the numbers that are common to both lists. Division Facts Have students complete these division facts as quickly as possible. 56 ÷ 7 = 8 24 ÷ 8 = 3 27 ÷ 3 = 9 45 ÷ 9 = 5 30 ÷ 5 = 6 42 ÷ 6 = 7 35 ÷ 7 = 5 20 ÷ 5 = 4 36 ÷ 4 = 9 72 ÷ 9 = 8 48 ÷ 8 = 6 36 ÷ 6 = 6 How are factors and multiples related? Lesson 5.4 299B LESSON 5.4 2 EXPLORE 4.OA.B.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. Lesson 5.4 Name Unlock the Problem Factors and Multiples Read and discuss the problem with the class. Make sure students understand that each row of the display should have 3 animals and there should be no animals left over. MP1 Make sense of problems and persevere in solving them. • What must be true about the number of animals in the sets that Mason could buy? Unlock Unlock the the Problem Problem Toy animals are sold in sets of 3, 5, 10, and 12. Mason wants to make a display with 3 animals in each row. Which sets could he buy, if he wants to display all of the animals? ↑ factor factor • How many animals are sold in each set? 3, 5, 10, or 12 ↑ multiple of 3 multiple of 4 One Way Find factors. MP7 Look for and make use of structure. Review factors and divisibility with students. Then have students tell whether 3 is a factor of each number in the problem. Tell whether 3 is a factor of each number. Think: If a number is divisible by 3, then 3 is a factor of the number. Yes, 3 is divisible by 3. Is 3 a factor of 3? _____ No, 5 is not divisible by 3. Is 3 a factor of 5? _____ No, 10 is not divisible by 3. Is 3 a factor of 10? _____ Another Way Yes, 12 is divisible by 3. Is 3 a factor of 12? _____ Before completing the exercise, have a volunteer write the numbers for skip counting by 3 from 3 through 12 on the board. 3, 6, 9, 12 • Why could Mason buy sets of 3 or 12 for his display? Possible answer: those are the numbers that 12 . 3 3 is a factor of _ and _ Another Way Find multiples. Multiply and make a list. 9 ,_ 12 , _ 15 ,… 3 ,_ 6 ,_ _ 1×3 © Houghton Mifflin Harcourt Publishing Company Use Math Talk to help students understand the relationship between factors and multiples. • Three is a factor of 24. Is it also true that 24 is a multiple of 3? Explain. Possible answer: 3 3 × 4 = 12 ↑ One Way Math Talk • How many animals will be in each row? The product of two numbers is a multiple of each number. Factors and multiples are related. Possible answer: the number must have 3 as a factor. are multiples of 3, so they would make 3 animals in each row of the display. Operations and Algebraic Thinking—4.OA.B.4 MATHEMATICAL PRACTICES MP6, MP7 Essential Question How are factors and multiples related? MATHEMATICAL PRACTICES 2×3 3×3 4×3 5×3 3 12 are multiples of 3. _ and _ 3 and _ 12 toy animals. So, Mason could buy sets of _ Math Talk Possible explanation: a whole number is always a multiple of its factors. For example, 3 is a factor of 24, so I know that 24 is a multiple of 3. MATHEMATICAL PRACTICES 6 Explain how you can use what you know about factors to determine whether one number is a multiple of another number. Chapter 5 299 since 3 is a factor of 24, 3 can be multiplied by a whole number to get 24. Any number that is the product of 3 and a counting number is a multiple of 3. ELL Strategy: Model Language Explain the meaning of multiples, and have students repeat the word. • Write the multiples of 2 up to 30 on the board. Say them chorally with students: 2, 4, 6, ... • The multiples of 2 are the same as skip counting by 2. • Student pairs practice counting by naming multiples of 2. Have each explain the meaning of multiples. 3 Reteach 5.4 1 Lesson 5.4 Reteach Name Differentiated Di D iff f erentiated IIn nst stru ruct ctio ion n Instruction Enrich 5.4 2 Lesson 5.4 Enrich Name Factors and Multiples Multiple Dates On January 1, 2011, the Petersons began a new allowance program for their four children: You know that 1 3 10 5 10 and 2 3 5 5 10. So, 1, 2, 5, and 10 are all factors of 10. Every third day, beginning January 3, Adrian will get his allowance. Every fourth day, beginning January 4, Beth will get her allowance. Every fifth day, beginning January 5, Zoe will get her allowance. Every seventh day, beginning January 7, Eddie will get his allowance. You can skip count to find multiples of a number: Count by 1s: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . . Count by 2s: 2, 4, 6, 8, 10, 12, . . . Count by 5s: 5, 10, 15, 20, 25, . . . Count by 10s: 10, 20, 30, 40, . . . 1. What is the first day that Adrian and Beth will get their allowances on the same day? 2. What is the first day that Beth and Zoe will get their allowances on the same day? 3. What is the first day that Adrian and Eddie will get their allowances on the same day? 4. What is the first day that Adrian, Beth, and Zoe will get their allowances on the same day? 5. Stretch Your Thinking How many days will it be until Note that 10 is a multiple of 1, 2, 5, and 10. A number is a multiple of all of its factors. January 12 A common multiple is a multiple of two or more numbers. So, 10 is a common multiple of 1, 2, 5, and 10. 1. 3 2. January 20 Multiply to list the next five multiples of 3. , 6 , 9 , 12 , 15 , 18 January 21 Multiply to list the next five multiples of 7. 7 , 14 , 21 , 28 , 35, 42 Is the number a factor of 8? Write yes or no. 3. 2 4. yes 8 March 1 5. yes 15 6. 20 no no all four children will get their allowances on the same day? Explain. 420 days; Possible explanation: I multiplied 3 3 4 3 Is the number a multiple of 4? Write yes or no. 7. 2 no 299 Chapter 5 Chapter Resources © Houghton Mifflin Harcourt Publishing Company 8. 12 9. yes 16 yes 5-11 10. 5 3 7 to find the first common multiple of 3, 4, 5, and 7. 18 no 3 3 4 3 5 3 7 5 3 3 20 3 7 5 60 3 7, or 420. Reteach Chapter Resources © Houghton Mifflin Harcourt Publishing Company 5-12 Enrich DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A Common Multiples Read the definition for common multiple. Example Common Multiples A common multiple is a multiple of two or more numbers. Ask a volunteer to read the problem aloud. • How can you tell on which dates Tony and Amanda work together? Those dates have both a Example Find common multiples. Tony works every 3 days and Amanda works every 5 days. If Tony works June 3 and Amanda works June 5, on what days in June will they work together? circle and a box around them. They are June 15 and 30. • Why are 15 and 30 common multiples of 3 and 5? because 15 and 30 are multiples of both 3 and 5 Circle multiples of 3. Draw a box around multiples of 5. June Sun Mon Tue Wed Thu Fri Sat 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Math Talk Use Math Talk to help students recognize the relationship between factors and multiples. • What is 5 in relationship to 15? Possible answer: 5 is a factor of 15. • What is 15 in relationship to 5? Possible Think: The common multiples have both a circle and a box. answer: 15 is a multiple of 5. 15 and _ 30 . The common multiples are _ 15 and June _ 30 . So, Tony and Amanda will work together on June _ Math Talk MATH BOARD Identify Relationships Discuss how factors and multiples are related. Give an example. 1. Multiply to list the next five multiples of 4. 4 ,_ 8 ,_ 12 , _ 16 , _ 20 , _ 24 _ 1×4 3. 6 yes _ 4. 16 yes _ 5. 18 no _ no _ Is the number a multiple of 6? Write yes or no. 6. 3 7. 6 no _ Share and Show Possible example: 5 is a factor of 15. 15 is a multiple of 5. Is the number a factor of 6? Write yes or no. 2. 3 3 EXPLAIN MATHEMATICAL PRACTICES 7 8. 16 yes _ 9. 18 no _ yes _ © Houghton Mifflin Harcourt Publishing Company Share Share and and Show Show MATH BOARD The first problem connects to the learning model. Use the checked exercises for Quick Check. Quick Check Quick Check If If 3 2 31 2 1 Rt I Rt I a student misses the checked exercises 300 Then 4_MNLESE342224_C05L04.indd 300 Advanced Learners 27/02/14 3:12 PM Visual Partners •Present this problem: Rosa makes different breads on different days. She makes rolls every third day, loaves every fourth day, donuts every sixth day, and muffins every eighth day. If she baked all of the items today, in how many days will she bake all the items on the same day again? 24 days •Have students work in pairs to solve the problem. •Have students make up a problem similar to the problem above. Have pairs of students trade problems and solve. Differentiate Instruction with • Reteach 5.4 • Personal Math Trainer 4.OA.B.4 • RtI Tier 1 Activity (online) COMMON ERRORS COMMON ERRORS Error Students may confuse factors and multiples. Example 1 8 is a factor of 6. 3 is a multiple of 6. Springboard to Learning Have students use a number line to find multiples of a number. Multiples of a number are equal to or greater than the number. Remind students that factors of a number are always less than or equal to the number. Lesson 5.4 300 Name On Your Own On On Your Your Own Own If students complete the checked exercises correctly, they may continue with the remaining exercises. For Exercise 14, after students list the next nine multiples of each number, have them circle the common multiples in each list before writing the answer on the third line of the exercise. MP8 Look for and express regularity in repeated reasoning. • How are the numbers in each list in Exercises 15 and 16 related? Possible answer: in Is the number a multiple of 3? Write yes or no. 10. 4 11. 8 no _ 12. 24 no _ 13. 38 yes _ no _ 14. List the next nine multiples of each number. Find the common multiples. 4, 6, 8, 10, 12, 14, 16, 18, 20 Multiples of 2: 2, _______ 16, 24, 32, 40, 48, 56, 64, 72, 80 Multiples of 8: 8, _______ 8, 16 Common multiples: ______ MATHEMATICAL PRACTICE 8 Generalize Algebra Find the unknown number. 48 15. 12, 24, 36, _ Exercise 15, the numbers are the ﬁrst 3 multiples of 12. In Exercise 16, they are the ﬁrst 4 multiples of 25. • How does this help you ﬁnd the unknown number in each list? Possible answer: I know that I 125 16. 25, 50, 75, 100, _ Tell whether 20 is a factor or multiple of the number. Write factor, multiple, or neither. 17. 10 need to write the next multiple in each list. 18. 20 multiple ___ With Exercises 17–19, point out how factors and multiples are related using more numerical examples, if students are still having difﬁculty. SMARTER factor, multiple ___ neither ___ Write true or false. Explain. 20. Every whole number is a multiple of 1. DEEPER 19. 30 21. Every whole number is a factor of 1. True; for every number n, n × 1 = n. False; 2 is not a factor of 1 SMARTER Julio wears a blue shirt every 3 days. Larry wears a blue shirt every 4 days. On April 12, both Julio and Larry wore a blue shirt. What is the next date that they will both wear a blue shirt? 12 Sun 19 1 26 8 15 22 Sun 29 • What are ﬁve multiples of 1,000? MP6 Attend to precision. • Is 25 a multiple or a factor of 50? 25 is a factor of 50 because 25 × 2 = 50. SMARTER For Exercises 20 and 21, students need to recall that any number multiplied by 1 is that number. This makes 1 a factor of every number. Watch for students who confuse multiples and factors. For Exercise 22, students may ﬁnd more than one common multiple of 3 and 4. They must recognize that only the least of these multiples will give them the ﬁrst date after April 12 that Julio and Larry will both wear a blue shirt. Math on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com. 301 Chapter 5 © Houghton Mifflin Harcourt Publishing Company Possible answer: 1,000; 2,000; 3,000; 4,000; 5,000 22. April 24 13 14April 15 16 17 Mon 21 Tue Wed Thu 24 Fri 20 22 23 2 3 4 5 6 27 28 29 30 31 9 10 11 12 13 16 17 18 19 20 23September 24 25 26 27 Mon Tue Wed Thu Fri 30 1 2 3 18 Sat 25 7 14 21 28 Sat 4 Chapter 5 • Lesson 4 301 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A MATHEMATICAL PRACTICES MODEL • REASON • MAKE SENSE Problem Problem Solving Solving •• Applications Applications 4 ELABORATE Complete the Venn diagram. Then use it to solve 23–25. Factors of 48 23. What multiples of 4 are not factors of 48? 2 20, 28, 32, 36, 40, 44 48 8 4 3 4, 8, 12, 16, 24, 48 Pose a Problem Look back at Problem 24. Write a similar problem by changing the numbers. Then solve. DEEPER Check students’ work. 40 12 20 MATHEMATICAL PRACTICES 28 MP4 Model with mathematics. Discuss each part of the Venn diagram. • What are the factors of 48? Where are they shown? 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; on 44 16 32 WRITE Math 6 Problem Solving • Applications 36 24 1 24. What factors of 48 are multiples of 4? 25. First Twelve Multiples of 4 Show Your Work the left side and in the overlapping section • List 12 multiples of 4 starting with 4. Where are they shown? 4, 8, 12, 16, 20, 24, 26. Kia paid $10 for two charms. The price of each charm was a multiple of $2. What are the possible prices of the charms? 28, 32, 36, 40, 44, 48; on the right side and in the overlapping section Possible answers: $4 and $6; $8 and $2 27. • How can you describe the numbers in the overlapping section? They are factors of 48 and MATHEMATICAL PRACTICE 7 Look for Structure The answer is 9, 18, 27, 36, 45. What is the question? multiples of 4. Possible question: What are five multiples of 9? 28. MP7 Look for and make use of structure. • In Exercise 27, what is the relationship of the numbers in the list? Possible answer: the Math How do you know whether a number is a WRITE multiple of another number? Possible answer: a number is a multiple of numbers are the first 5 multiples of 9. another number if it is divisible by the number. 29. SMARTER SMARTER For numbers 29a–29e, select True or False for each 29a. The number 45 is a multiple of 9. True False 29b. The number 4 is a multiple of 16. True False 29c. The number 28 is a multiple of 4. True False 29d. The number 4 is a factor of 28. True False 29e. The number 32 is a factor of 8. True False © Houghton Mifflin Harcourt Publishing Company statement. Students who answer True to number 29b do not understand the concept of a multiple. These students may have recognized the pair of numbers in this problem as part of a basic fact and incorrectly assumed that must mean that the number is a multiple of the other. 302 4_MNLESE342224_C05L04.indd 302 27/02/14 2:52 PM DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES 5 EVALUATE Formative Assessment Essential Question Using the Language Objective Reflect Have students make a list to answer the Essential Question. How are factors and multiples related? Differentiated Centers Kit Games Factor Farm Activities First One Out Students complete orange Activity Card 5 by using multiplication to find numbers that match given products. Students practice determining factors of whole numbers. Games Possible answer: a factor times a factor is a product. The product is a multiple of either factor. Math Journal WRITE Math Write a word problem that can be solved by finding the numbers that have 4 as a factor. Lesson 5.4 302 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=A Practice and Homework Name Lesson 5.4 Factors and Multiples COMMON CORE STANDARD—4.OA.B.4 Gain familiarity with factors and multiples. Is the number a multiple of 8? Write yes or no. Practice and Homework 1. 4 Think: Since 4 × 2 = 8, 4 is a factor of 8, not a multiple of 8. 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. 2. 8 3. 20 yes no 4. 40 yes no List the next nine multiples of each number. Find the common multiples. 8, 12, 16, 20, 24, 28, 32, 36, 40 5. Multiples of 4: 4, ___________ 14, 21, 28, 35, 42, 49, 56, 63, 70 Multiples of 7: 7, ___________ 28 Common multiples: __________ 6, 9, 12, 15, 18, 21, 24, 27, 30 6. Multiples of 3: 3, ___________ 18, 27, 36, 45, 54, 63, 72, 81, 90 Multiples of 9: 9, ___________ 9, 18, 27 Common multiples: __________ Tell whether 24 is a factor or multiple of the number. Write factor, multiple, or neither. multiple 7. 6 ___ neither 8. 36 ___ factor 9. 48 ___ Problem Problem Solving Solving © Houghton Mifflin Harcourt Publishing Company 10. Ken paid $12 for two magazines. The cost of PROFESSIONAL DEVELOPMENT 11. Jodie bought some shirts for $6 each. each magazine was a multiple of $3. What are the possible prices of the magazines? Marge bought some shirts for $8 each. The girls spent the same amount of money on shirts. What is the least amount they could have spent? $3 and $9; $6 and $6 12. $24 Math Write a word problem that can be solved by WRITE finding the numbers that have 4 as a factor. Check students’ work. Chapter 5 Mathematical Practices in Your Classroom 4_MNLESE342224_C05P04.indd 303 PROFESSIONAL PROFESSIONAL DEVELOPMENT DEVELOPMENT 2/13/14 5:59 PM Mathematical Practices in Your Classroom CCSS.Math.Practice.MP6 Attend to precision. Depth of understanding is exhibited when students can use precise and appropriate mathematical language to describe mathematical concepts and to answer questions about those concepts. In this lesson, students connect the concepts of factors and multiples, and precision of language is vital. When having class discussions, suggest to students that they think about whether they are using the correct math vocabulary before they respond. Students can reference the vocabulary definitions if needed. 303 Chapter 5 303 Ask questions such as the following to help students differentiate factors and multiples: • Are the factors of a number less than, equal to, or greater than the number? Can you give an example? The factors are less than or equal to the number. For example, 1, 2, 4, 5, 10, and 20 are all factors of 20. • How can we be sure that 10 is a factor of 20? Multiply 10 by another number to see if the product is 20. 2 times 10 is equal to 20. So, 10 is a factor of 20. • How do you find multiples of 20? Multiply 20 by counting numbers starting with 1, 2, and 3. The answers are the multiples: 20, 40, 60. So, 20 is a multiple of 20. Lesson Check (4.OA.B.4) 1. Of the numbers listed below, which are NOT multiples of 4? 2. What number is a common multiple of 5 and 9? 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. 2, 4, 7, 8, 12, 15, 19, 24, 34 2, 7, 15, 19, 34 Possible answer: 45 Spiral Review (4.OA.A.3, 4.NBT.A.2, 4.NBT.B.4, 4.NBT.B.5) tiles into a rectangular array of 4 rows. How many tiles will be left over? 2 tiles 5. There are 18 rows of seats in the auditorium. There are 24 seats in each row. How many seats are in the auditorium? 432 seats 4. Jerome added two numbers. The sum was 83. One of the numbers was 45. What was the other number? 38 6. The population of Riverdale is 6,735. What is the value of the 7 in the number 6,735? © Houghton Mifflin Harcourt Publishing Company 3. Jenny has 50 square tiles. She arranges the 700 FOR MORE PRACTICE GO TO THE 304 Personal Math Trainer Lesson 5.4 304