Multiples and Factors
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- Cecilia Hart
- 6 years ago
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1 Part 1: Introduction Multiples and Factors Develop Skills and Strategies CCSS 4.OA.B.4 In previous lessons, you multiplied and divided numbers. Now you can use multiplication and division to find factors and multiples, and then learn a way to classify a number by how many factors it has. Take a look at this problem. A garden has several rows of pumpkin plants. Each row has 10 plants. How many pumpkin plants could be in the garden? Explore It Use the math you already know to solve the problem. How many pumpkin plants are in each row? How many pumpkin plants would be in 2 rows? In 4 rows? In 5 rows? How do you know there are not a total of 45 pumpkin plants in the garden? How can you describe how many pumpkin plants there can be? 54
2 Part 1: Introduction Find Out More If there were 5 rows of 10 pumpkin plants, there would be 50 plants. The numbers 5 and 10 are a factor pair of 50 because they are two factors whose product is 50. To find other factor pairs of 50, think of other ways 50 pumpkin plants could be planted in equal rows. You could have 2 rows of You could also have one row of and 25 are a factor pair of 50, and so are 1 and 50. When you multiply numbers, the product is a multiple of each factor. 50 is a multiple of all six of its factors: 1, 2, 5, 10, 25, and 50. You can use skip-counting to check if a number is a multiple of another number. If you start at 0 and count by 25s, you will reach 50. So, 50 is a multiple of 25. Numbers like 50 that have more than one factor pair are called composite numbers. Prime numbers have only one factor pair: the number and 1. The number 1 is neither prime nor composite. Is there more than one way to plant 11 carrot seeds in equal rows? Is there more than one way to plant 24 onions in equal rows? Reflect 1 Describe the difference between prime numbers and composite numbers. 55
3 Part 2: Modeled Instruction Read the problem below. Then explore different ways to use multiples to solve it. Leona has 5 cups of oats. She needs 2 cups of oats for one full batch of oatmeal muffins. Can she use all of her oats by making multiple full batches? Picture It You can use a model to help understand the problem. The model shows the oats Leona has, divided into 2-cup measuring cups. 2 cups 1 cup 2 cups 1 cup 2 cups 1 cup Model It You can also use a number line to help understand the problem. The number line shows multiples of 2 circled. To find the multiples of 2, you can start at 0 and skip count by 2s. You can see that the multiples of 2 are always even
4 Part 2: Guided Instruction Connect It Now you will further explore the problem from the previous page. 2 Why does the model use measuring cups that hold 2 cups of oats? 3 How can you tell from the measuring cup model that 2 does not go into 5 evenly? 4 What do the circled numbers on the number line represent? 5 How can you tell from the number line that Leona can t use all 5 cups of oats in 2-cup batches? 6 How many cups of oats would Leona use in 3 batches of muffins? 7 Explain how you can use multiples of 2 to know whether a number, such as 5, is divisible by 2. Try It Use what you just learned to solve these problems. 8 There are 4 bottles of water in a pack. Patrick needs 20 bottles of water for his soccer team. Can he buy exactly 20 bottles in packs of 4? 9 What are the first five multiples of the number 9? 57
5 Part 3: Modeled Instruction Read the problem below. Then explore different ways to use factor pairs to solve it. Marcela is putting 40 stickers in her sticker book and wants to put the same number of stickers in each row. Find all the ways she can arrange the stickers. Model It You can use arrays to help understand the problem. One way Marcela can arrange her stickers is in 8 rows of 5. 8 and 5 are a factor pair. Using the same factors, Marcela could also arrange them in 5 rows of 8. Model It You can also use area models to help understand the problem. Two more ways Marcela can arrange her stickers are 10 rows of 4 or 2 rows of and 10 are a factor pair, and 2 and 20 are a factor pair. Using the same factors, Marcela could also arrange them in 4 rows of 10 or 20 rows of 2. 58
6 Part 3: Guided Instruction Connect It Now you will explore the problem from the previous page further. 10 What are two more ways to arrange the stickers into even rows? 11 List all of the factor pairs of 40: 12 The numbers in a factor pair are the factors. How many factors does 40 have? 13 Why might it be helpful to always start with the number 1 and work up when finding factors? 14 Explain how to use arrays or area models to find factor pairs. Try It Use what you just learned to solve these problems. 15 Brad is playing with pattern blocks. He has 18 blocks and wants to make an array with the same number of blocks in each row. What are the different ways he could arrange the blocks? 16 What are the factors of the number 27? 59
7 Part 4: Modeled Instruction Read the problem below. Then explore different ways to understand it. Janae has 36 pennies. Nate has 23 pennies. Who has a composite number of pennies? Picture It You can use models to help understand the problem. 36 pennies can be divided into 3 equal stacks of pennies can t be divided into more than one equal stack. Janae Nate Model It You can also use area models to help understand the problem. With composite numbers, you can make area models that are more than just one row wide. Janae Nate 60
8 Part 4: Guided Instruction Connect It Now you will explore the problem from the previous page further. 17 What factor pair is shown by Janae s stacks of pennies? 18 Is 36 a prime or composite number? How do you know? 19 Is 23 a prime or composite number? How do you know? 20 Explain how you can use models to decide if a number is prime or composite. Try It Use what you just learned to solve these problems. 21 Mrs. Reynaldo is picking up 17 playground balls after recess and she wants to put the same number of balls into each ball bin. What are the different ways she could group the balls? 22 Is 17 a prime number or a composite number? 61
9 Part 5: Guided Practice Study the model below. Then solve problems Student Model Any number that has 0 or 5 in the ones place is a multiple of 5! School pictures are sold with 9 pictures on a sheet. Hallie needs 45 pictures for her family and classmates. Can she buy exactly 45 pictures in sheets of 9? Look at how you could show your work using a picture. Pair/Share How else could you solve this problem, without using models? Solution: Hallie can buy exactly 45 pictures. She needs 5 sheets. I noticed that 2 goes into every even number! 23 There are 12 levels in Liang s new video game. If he plays the same number of levels each day, what are all the possibilities for the number of days he could spend playing the game? Show your work. Pair/Share Why do you need to find the factors of 12 to solve this problem? Solution: 62
10 Part 5: Guided Practice 24 A basketball team scored 37 points in one quarter. Is the number 37 prime or composite? Show your work. Starting with 1 is a good way to find factors! Solution: Pair/Share Why do you need to find the factors of 37 to solve this problem? 25 Grant walks 2 miles every day. Which could NOT be the number of miles that Grant has walked after some number of days? Circle the letter of the correct answer. What do you know about multiples of 2? A 2 B 3 C 10 D 18 Noelle chose B as the correct answer. How did she get that answer? Pair/Share With your partner, discuss why answer A is incorrect. 63
11 Part 6: Common Core Practice Solve the problems. 1 Simon is organizing his 36 toy cars into equal-sized piles. Which list shows all of the possible numbers of cars that could be in each pile? A 2, 3, 4, 6 B 1, 2, 3, 4, 6 C 2, 3, 4, 6, 9, 12, 18 D 1, 2, 3, 4, 6, 9, 12, 18, 36 2 Reggie ate 31 raisins. How many factor pairs does 31 have? Is it a prime number or a composite number? A B C D It is prime because it has 0 factor pairs. It is prime because it has 1 factor pair. It is composite because it has 1 factor pair. It is composite because it has 2 factor pairs. 3 Sara is playing a memory card game with 24 cards. She wants to lay the cards out in rows so that each row has the same number of cards. Shade in the boxes to show one way that she could lay out the cards. 64
12 Part 6: Common Core Practice 4 Tell whether each sentence is True or False. a. The number 96 is a multiple of 12. That means all of the factors of 12 are also factors of 96. True False b. The number 1 is prime. True False c. The number 1 is composite. True False d. The number 2 is prime. True False e. The number 9 has four factors. True False 5 There are 15 cousins playing a game. They need to divide evenly into teams. Draw a model to show one way they can split into teams. Then decide if 15 is a prime number or a composite number. Show your work. Answer The number of cousins, 15, is a number. 6 A pack of toy cars contains 12 cars. If Sylvia buys only packs of 12, what are two possible numbers of cars that she could buy? Show your work. Answer Sylvia could buy cars or cars. Self Check Go back and see what you can check off on the Self Check on page
13 Develop Skills and Strategies (Student Book pages 54 65) Multiples and Factors Lesson Objectives Use understanding of multiplication facts to list all the factors of a given whole number. Use understanding of multiplication and division facts to determine if a whole number is a multiple of another number. Apply understanding of multiples and factors to solving problems. Prerequisite SkilLs Multiply and divide within 100. Understand that multiplication and division are inverse operations. Know that the numbers being multiplied in a multiplication problem are called factors. Vocabulary factor pair: two numbers that are multiplied together to give a product multiple: the product of the number and any other whole number (0, 4, 8, 12, etc. are multiples of 4) composite number: a number that has more than one pair of factors prime number: a number that has only one pair of factors: itself and 1 Review the following key terms. multiplication: an operation used to find the total number of items in equal-sized groups division: an operation used to separate a number of items into equal-sized groups factor: numbers that are multiplied together to get a product product: the answer to a multiplication problem The Learning Progression In this lesson, students extend the idea of composition and decomposition of numbers to multiplication. They connect their previous understanding of multiplication and division to the language of multiples and factors. Students decompose a number into equal groups and express the decomposition with factor pairs using number lines, arrays, and area models. They also realize that any whole number is a multiple of each of its factors. To find all the factors for a product, students learn to search systematically and check to see if 2 is a factor of a number, then 3, then 4, and so on. They begin to think about all whole numbers as being either prime or composite in terms of having factors and being multiples. This understanding of multiples and factors not only supports grade level problem-solving work but will also support later work with exponential notation in factoring numbers. Ready Lessons Teacher Toolbox Tools for Instruction Interactive Tutorials Prerequisite Skills Teacher-Toolbox.com 4.OA.B.4 4.OA.4 CCSS Focus 4.OA.B.4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. standards FOR MatheMaticaL Practice: SMP 2, 5, 7 (See page A9 for full text.) 60
14 Part 1: Introduction At a glance Students use a multiplication problem and an array to review key ideas about multiplication. Part 1: introduction Multiples and Factors Develop skills and strategies ccss 4.oa.b.4 Step By Step Tell students that this page models a multiplication problem that is similar to ones they ve solved in the past. in previous lessons, you multiplied and divided numbers. now you can use multiplication and division to find factors and multiples, and then learn a way to classify a number by how many factors it has. take a look at this problem. A garden has several rows of pumpkin plants. Each row has 10 plants. How many pumpkin plants could be in the garden? Have students read the problem at the top of the page. Work through the Explore It questions as a class. For the last two questions, ask student pairs to explain their answers. Remind students of their previous experiences with relating skip-counting to multiplication. Mathematical Discourse If the problem said there were 5 pumpkins in each row instead of 10, how many pumpkins could there be? 5, 10, 15, 20, 25, and so forth Why does counting by 5 tell you all the possible amounts? Students may say that counting by 5 adds 5 each time and then relate it to repeated addition or multiplication. Would counting by 10 tell you possible numbers of pumpkins for rows of 5 pumpkins? Students may see that counting by 10 would give some of the possibilities but not all. (It would give every other possibility.) 54 explore it use the math you already know to solve the problem. How many pumpkin plants are in each row? 10 How many pumpkin plants would be in 2 rows? 20 In 4 rows? 40 In 5 rows? 50 How do you know there are not a total of 45 pumpkin plants in the garden? because all of the rows have 10 pumpkin plants. to have a total of 45, you would have to have 4 rows of 10 and 1 row of 5. How can you describe how many pumpkin plants there can be? there can be any number you get to by counting by tens or multiplying by ten. 61
15 At a glance Part 1: Introduction Students learn about factor pairs, multiples, and prime and composite numbers. Part 1: introduction Find out More Step By Step Read Find Out More as a class. Introduce the term factor pair. Ask students to work with a partner to find all the factor pairs that give the product 50. Explain that you only need to list the factor pairs once ( is the same as ). List the 3 factor pairs on the board. Ask students to count how many different factors there are [6] and then write them in order on the board. Ask students to think of some other numbers that only have two factors and list them on the board. [2, 3, 5, 7, 11, 13, etc.] Introduce the terms prime and composite. Make sure students understand that all numbers that have more than two factors are called composite numbers. Look at the carrot picture and the onion picture as a class and discuss why the number 11 is prime and the number 24 is composite. Ask students to think about the number one and apply what they learned about primes and composites. Point out that one is a special number that doesn t fit into either category. The number one has only a single factor. There is no factor pair because there is only one number. The number one has only one factor (instead of the two factors that prime numbers have). Have students discuss the Reflect question in pairs and share with the class. Use the Mathematical Discourse questions to assess students understanding. If there were 5 rows of 10 pumpkin plants, there would be 50 plants. The numbers 5 and 10 are a factor pair of 50 because they are two factors whose product is 50. To find other factor pairs of 50, think of other ways 50 pumpkin plants could be planted in equal rows. You could have 2 rows of You could also have one row of and 25 are a factor pair of 50, and so are 1 and 50. When you multiply numbers, the product is a multiple of each factor. 50 is a multiple of all six of its factors: 1, 2, 5, 10, 25, and 50. You can use skip-counting to check if a number is a multiple of another number. If you start at 0 and count by 25s, you will reach 50. So, 50 is a multiple of 25. Numbers like 50 that have more than one factor pair are called composite numbers. Prime numbers have only one factor pair: the number and 1. The number 1 is neither prime nor composite. reflect Is there more than one way to plant 11 carrot seeds in equal rows? Mathematical Discourse Is there more than one way to plant 24 onions in equal rows? 1 Describe the difference between prime numbers and composite numbers. the only factor pair prime numbers have is the number and 1. composite numbers have other factor pairs, too. Are the numbers 2, 4, 6, 8, and 10 all composite numbers? Why or why not? The number 2 is prime, but all other even numbers are composite because they are multiples of 2. Are these odd numbers also prime numbers: 1, 3, 5, 7, 9, 15? Why or why not? Some odd numbers are prime, like 3 and 5, but other numbers, like 9 and 15, have more than two factors
16 Part 2: Modeled Instruction At a glance Students explore a problem in which a number cannot be divided evenly into groups. They use models, number lines, and multiples to help understand the problem. Step By Step Read the problem at the top of the page as a class. Read Picture It. Remind students that picturing a problem or acting it out can help them to understand the problem. Ask students for the total number of cups of oats [5] and the number of cups it takes for each batch. Ask students to underline what the problem is asking. [Can she use all of her oats by making multiple full batches?] Part 2: Modeled instruction read the problem below. then explore different ways to use multiples to solve it. Leona has 5 cups of oats. She needs 2 cups of oats for one full batch of oatmeal muffins. Can she use all of her oats by making multiple full batches? Picture it you can use a model to help understand the problem. The model shows the oats Leona has, divided into 2-cup measuring cups. 2 cups 1 cup 2 cups 1 cup 2 cups 1 cup Model it you can also use a number line to help understand the problem. The number line shows multiples of 2 circled. To find the multiples of 2, you can start at 0 and skip count by 2s. You can see that the multiples of 2 are always even Draw the 5 cups on the board and ask for a volunteer to circle cups that will make full batches. Ask students to answer the problem s question and have a volunteer share and explain why Leona can or cannot use all of her oats. Read Model It. Remind students that they can use multiples or counting by a certain number as a strategy to see if the number is a factor of the product. Ask, Can you skip count by 2s to get to 5? [no] Is 2 a factor of 5? [no] Point out that students can use all these ways to answer the problem: picturing it, using a number line, and asking if 2 is a multiple of 5. Make it clear to students that this problem shows a situation in which you divide a number in equal groups and have something left over. 56 Real-World Connection Ask students to think of and share with a partner a time when they wanted to divide something up into equal groups and there was something left over. Examples: pairing socks, splitting a deck of cards between 5 people, sharing food, and so forth. 63
17 At a glance Part 2: Guided Instruction Students revisit the problem on page 56. They reflect upon the different ways to think about and solve the problem and the method they prefer. Step By Step Ask students to read the problem again and then read and discuss the Connect It questions together. Ask students to think about the problem in terms of division and finding how many cups of 2 there are in 5 cups of oats. Help them make the important connection that they can use what they know about factors and multiples to solve the problem. Remind students that they can just count by multiples of 2 to see that 2 does not divide evenly into 5. Let students know that when dividing into a number, they can think whether the divisor is a factor of the number they are dividing. This tells them if the factor divides evenly into the number or if there will be something left over. Ask students to work with a partner on the Try It problems and be ready to share their answers. Part 2: guided instruction connect it now you will further explore the problem from the previous page. 2 Why does the model use measuring cups that hold 2 cups of oats? each batch of muffins uses 2 cups of oats. 3 How can you tell from the measuring cup model that 2 does not go into 5 evenly? 4 What do the circled numbers on the number line represent? the multiples of 2 5 How can you tell from the number line that Leona can t use all 5 cups of oats in 2-cup batches? the 5 isn t circled, so it isn t a multiple of 2. 6 How many cups of oats would Leona use in 3 batches of muffins? 7 Explain how you can use multiples of 2 to know whether a number, such as 5, is divisible by 2. try it the third measuring cup is only half full. Possible explanation: When you skip count by 2s, you don t land on 5: 2, 4, 6. this means that 5 is not an even number, and therefore, 5 is not divisible by 2. use what you just learned to solve these problems. 8 There are 4 bottles of water in a pack. Patrick needs 20 bottles of water for his soccer team. Can he buy exactly 20 bottles in packs of 4? yes 9 What are the first five multiples of the number 9? 6 9, 18, 27, 36, Visual Model Using a hundreds chart to find multiples. Many students benefit from using a concrete hundreds chart to count out multiples. Write the problem on the board. Instruct students to count by multiples of 6 on the chart to see if the number divides evenly and if 42 is a multiple of 6. Ask students to do the same for the problem Point out that counting past 41 (to 44) shows that 41 is not a multiple of 4. Review with students whether the two products are prime or composite and why. TRY it solutions 8 Solution: Yes; Students may draw 20 objects and divide them into groups of 4, use a number line to show skips of 4, or write multiples of 4 to Solution: 9, 18, 27, 36, 45; Students may repeatedly add 9 to find the next term. ERROR ALERT: Students may confuse multiples with factors and write 1, 3, 9. Have the students write the number sentences corresponding to the first 5 multiples of 9. [ , , , , ] Have students identify the factors and products, and then connect the products with a factor of 9 to multiples of 9. 64
18 Part 3: Modeled Instruction At a glance Students explore using area models to understand and solve problems that involve finding all the factor pairs of a product. Step By Step Read the problem at the top of the page as a class. Focus students attention on the array in the first Model It and ask students what factor pair the model represents. [5 3 8] Point out that the array can be turned to represent and still show 40 stickers. Introduce the area models in the second Model It. Explain that area models show one whole area partitioned into rows and columns with no gaps between the squares or tiles. Create a 2-by-20 area model on the board and outline the area. Ask students for the factor pairs for the models shown on the page and write them on the board. Review with students whether 40 is a prime or composite number and how they know. Part 3: Modeled instruction read the problem below. then explore different ways to use factor pairs to solve it. Marcela is putting 40 stickers in her sticker book and wants to put the same number of stickers in each row. Find all the ways she can arrange the stickers. Model it you can use arrays to help understand the problem. One way Marcela can arrange her stickers is in 8 rows of 5. 8 and 5 are a factor pair. Using the same factors, Marcela could also arrange them in 5 rows of 8. Model it you can also use area models to help understand the problem. Two more ways Marcela can arrange her stickers are 10 rows of 4 or 2 rows of and 10 are a factor pair, and 2 and 20 are a factor pair. Using the same factors, Marcela could also arrange them in 4 rows of 10 or 20 rows of Concept Extension Practice using area models to understand and find factors. Materials: graph paper Instruct students to work in pairs and use graph paper to draw out area models that show all the ways to get a product of 39. Have them write the factor pairs next to each model. Ask students to describe the models they drew and state the factor pairs. Do the same for the product 43. Discuss whether the products are composite or prime numbers and how they knew they found all the ways to model the product. Mathematical Discourse Which of these numbers has the most factors: 30 or 36? 36 Why does 36 have so many more factors? The number 36 is a multiple of 1, 2, 3, 4, 6, 9, 12, 18, and 36. A lot more numbers divide evenly into 36 than into 30. How can you be sure you have found all the ways to show or model a product? Go in order, starting with 2, and ask if you can make a rectangle with an area equal to the product and one side length of 2. Then repeat with each number 2, 3, 4, and so on, until you begin to find factor pairs you already found. 65
19 At a glance Part 3: Guided Instruction Students revisit the problem on page 58 and solve it by using what they know about multiples and factors. Step By Step Tell students that the questions in Connect It refer to the problem about the stickers on page 58. Work through problems as a class. Write student responses on the board. Instruct students to individually answer problems 13 and 14. Then, have students turn to a partner and tell each other why models can be helpful for finding factor pairs and have a few students share with the class. Ask students to work in pairs again to complete the Try It questions. Walk around to each pair, listen to, and join in on discussions at different points. Part 3: guided instruction connect it now you will explore the problem from the previous page further. 10 What are two more ways to arrange the stickers into even rows? 1 row of 40 and 40 rows of 1 11 List all of the factor pairs of 40: 1 and 40, 2 and 20, 4 and 10, 5 and 8 12 The numbers in a factor pair are the factors. How many factors does 40 have? 8 13 Why might it be helpful to always start with the number 1 and work up when finding factors? Possible answer: starting at the number 1 and checking each number as you count up makes sure you don t miss any factors. 14 Explain how to use arrays or area models to find factor pairs. Possible explanation: the number of rows and number of columns are a factor pair. if you find all the different ways you can arrange the total number of objects into equal rows, you will find all the factor pairs. try it use what you just learned to solve these problems. 15 Brad is playing with pattern blocks. He has 18 blocks and wants to make an array with the same number of blocks in each row. What are the different ways he could arrange the blocks? , , 2 3 9, 9 3 2, 3 3 6, What are the factors of the number 27? 1, 3, 9, Concept Extension Use patterns to find all the pairs for a product. Ask students to find and write all the ways they can make the product 30. As they share, write the facts (1 3 30, etc.) on the board. Then rewrite the list in order. Point out the patterns in the list (the first factor becomes the second factor later on in the list and you can switch the order of factors to find the other pair). Explain that they can use the patterns to help them make sure they ve found all the factor pairs for a product. Have them try using the patterns to find all the ways to make 36. TRY it solutions 15 Solution: , 2 3 9, 3 3 6, 6 3 3, 9 3 2, ; Students may list facts in order or not. ERROR ALERT: Students may miss finding some of the factor pairs. Encourage them to start with the factor 1 and go in order, checking to see if the next number is a factor and then listing the factor pair. 16 Solution: 1, 3, 9, 27; If students list all the facts (i.e., and ), tell them that they only need to list each factor once
20 Part 4: Modeled Instruction At a glance Students explore ways to understand prime and composite numbers. Part 4: Modeled instruction read the problem below. then explore different ways to understand it. Step By Step Read the problem at the top of the page as a class. Explain that the Picture It shows a new way to visualize a product by stacking pennies into equal groups. Ask students what factor pair for 36 is shown in the model [3 3 12] and write it on the board. Be sure students recognize that the two stacks showing 23 pennies are not equal. Ask students to explain why they can t be equal. [23 can t be divided into equal groups, except for groups of 1 or 23.] Janae has 36 pennies. Nate has 23 pennies. Who has a composite number of pennies? Picture it you can use models to help understand the problem. 36 pennies can be divided into 3 equal stacks of pennies can t be divided into more than one equal stack. Model it Janae Nate you can also use area models to help understand the problem. With composite numbers, you can make area models that are more than just one row wide. Use the Model It area models to remind students that they can use area models to show a product. Write the factor pairs shown by Janae s models on the board. [4 3 9, ] Ask students for the other facts they could write using these same factors. [9 3 4, ] Write the factor pair shown by Nate s model on the board. [1 3 23] Ask students for the other fact they could write using these same factors. [23 3 1] 60 Janae Nate Ask students to describe the differences between the picture and the model for 36 and 23. Be sure that students note that 23 cannot be divided evenly by any factors besides 1 and 23. Visual Model Scaffold modeling and listing of factors. Since 36 is a larger number with many factors, some students may need more scaffolded modeling of how to find and list factors. List the factors they ve found so far on the board (1, 3, , 12, 18, 36) and ask students to think of what other factors might be related to the ones shown (like 2 and 4 and 6 and 3). Mathematical Discourse Why are there so many ways to make the product 36? It is a composite number and there are many factors that divide evenly into it. Which method do you like the best for finding all the ways to make the product 36? Why? Possible answer: I like using a rectangular model to systematically check each number starting with 1 so I don t miss any pairs. To find the remaining factors, you may pass out graph paper and have them try to create a model with 2 equal rows and then 6 equal rows. Add the new factors that students find to the list on the board. 67
21 Part 4: Guided Instruction At a glance Students revisit the problem on page 60 and solve it by using what they know about multiples and factors. Step By Step Work through the Connect It questions together. Part 4: guided instruction connect it now you will explore the problem from the previous page further. 17 What factor pair is shown by Janae s stacks of pennies? Is 36 a prime or composite number? composite How do you know? it has factor pairs other than 1 and 36. Emphasize that a prime number has only two factors: the number itself and the number 1. Explain that this means a prime number can only be equally shared in 1 group (or groups equal to the number) or modeled with 1 row in an array (or rows equal to the number). Give students an opportunity to reflect upon their experiences of finding all the ways to show a number as a product. Ask students how these ways are related to determining if a number is prime or composite. 19 Is 23 a prime or composite number? prime How do you know? it only has one factor pair: 1 and Explain how you can use models to decide if a number is prime or composite. if you can divide a number into more than 1 equal part, the number is composite. try it use what you just learned to solve these problems. 21 Mrs. Reynaldo is picking up 17 playground balls after recess and she wants to put the same number of balls into each ball bin. What are the different ways she could group the balls? 1 bin for all 17 balls or 17 bins, each with 1 ball. 22 Is 17 a prime number or a composite number? prime Ask students to work in pairs again to complete the Try It questions. Walk around to each pair, listen to, and join in on discussions at different points. SMP Tip: Again, students are given opportunities to try out different ways to model and think about a problem. They think about which method or tools were most helpful for them. (SMP 5) TRY it solutions 21 Solution: 1 group of 17 or 17 groups of one ball; Students may use words or write factor pairs to show the number of ways to make Solution: Prime. ERROR ALERT: Students who wrote composite may have forgotten that and represent the same factor pair of 17. Remind students that a prime number has 2 factors: itself and 1, not 2 factor pairs. 68
22 Part 5: Guided Practice Part 5: guided Practice Part 5: guided Practice Any number that has 0 or 5 in the ones place is a multiple of 5! study the model below. then solve problems Student Model School pictures are sold with 9 pictures on a sheet. Hallie needs 45 pictures for her family and classmates. Can she buy exactly 45 pictures in sheets of 9? 24 A basketball team scored 37 points in one quarter. Is the number 37 prime or composite? Show your work. Possible answer: ; there are no other factors of 37. Starting with 1 is a good way to find factors! Look at how you could show your work using a picture. Solution: prime Pair/share Why do you need to find the factors of 37 to solve this problem? Pair/share How else could you solve this problem, without using models? Solution: hallie can buy exactly 45 pictures. she needs 5 sheets. 25 Grant walks 2 miles every day. Which could NOT be the number of miles that Grant has walked after some number of days? Circle the letter of the correct answer. a 2 b 3 What do you know about multiples of 2? c 10 I noticed that 2 goes into every even number! Pair/share Why do you need to find the factors of 12 to solve this problem? 23 There are 12 levels in Liang s new video game. If he plays the same number of levels each day, what are all the possibilities for the number of days he could spend playing the game? Show your work. Possible answer: ; ; Solution: 1, 2, 3, 4, 6, or 12 days D 18 Noelle chose B as the correct answer. How did she get that answer? Possible answer: noelle chose b because 3 is an odd number and multiples of 2 are even. Pair/share With your partner, discuss why answer A is incorrect At a glance Students apply what they know about multiples and factors to understand and solve problems. Step By Step Ask students to solve the problems individually. Remind students that some problems require them to show their work. As students work, circulate around the room and give support when needed. When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. solutions Ex Solution: Hallie could buy exactly 45 pictures; She needs 5 sheets; The student sees 5 sheets of 9 pictures or knows Solution: 1, 2, 3, 4, 6, or 12 days; Students may list factors or factor pairs, or use arrays or other diagrams to find the factors of 12. (DOK 2) 24 Solution: Prime; Students may use area models, try to group 37 objects, or list factors or factor pairs of 37. (DOK 1) 25 Solution: B; Noelle could have listed the multiples of 2 or said that 3 is an odd number so it isn t a multiple of 2. Explain to students why the other three answer choices are not correct: A is not correct because 2 miles miles. C is not correct because 2 miles miles. D is not correct because 2 miles miles. (DOK 3) 69
23 Part 5: Common Core Practice Part 6: common core Practice Part 6: common core Practice Solve the problems. 1 Simon is organizing his 36 toy cars into equal-sized piles. Which list shows all of the possible numbers of cars that could be in each pile? A 2, 3, 4, 6 B 1, 2, 3, 4, 6 C 2, 3, 4, 6, 9, 12, 18 D 1, 2, 3, 4, 6, 9, 12, 18, 36 2 Reggie ate 31 raisins. How many factor pairs does 31 have? Is it a prime number or a composite number? A It is prime because it has 0 factor pairs. B It is prime because it has 1 factor pair. C It is composite because it has 1 factor pair. D It is composite because it has 2 factor pairs. 4 Tell whether each sentence is True or False. a. The number 96 is a multiple of 12. That means all of the factors of 12 are also factors of True False b. The number 1 is prime. True 3 False c. The number 1 is composite. True 3 False d. The number 2 is prime. 3 True False e. The number 9 has four factors. True 3 False 5 There are 15 cousins playing a game. They need to divide evenly into teams. Draw a model to show one way they can split into teams. Then decide if 15 is a prime number or a composite number. Show your work. Possible answer: they can split into 3 teams of 5 or 5 teams of 3, or 1 team of 15 or 15 teams of 1. 3 Sara is playing a memory card game with 24 cards. She wants to lay the cards out in rows so that each row has the same number of cards. Shade in the boxes to show one way that she could lay out the cards. Possible answer: Answer The number of cousins, 15, is a composite number. 6 A pack of toy cars contains 12 cars. If Sylvia buys only packs of 12, what are two possible numbers of cars that she could buy? Show your work. Possible answers: ; Answer Sylvia could buy 36 cars or 60 cars. self check Go back and see what you can check off on the Self Check on page At a glance Students individually solve problems involving factors and multiples that might appear on a mathematics test. solutions 1 Solution: D; The piles must each have a factor of 36 cars. (DOK 1) 2 Solution: B; The only factors of 31 are 1 and 31. (DOK 2) 3 Solution: See student book page above for possible student answer. Students could also shade 3 rows of 8 or 2 rows of 12. (DOK 1) 4 Solution: a. True; b. False; c. False; d. True; e. False (DOK 2) 5 Solution: 3 teams of 5, 5 teams of 3, 1 team of 15 and 15 teams of 1 (students may also correctly reason that one team of 15 doesn t work for most games and that 1 person doesn t make a team); 15 is a composite number. (DOK 2) 6 Solution: any two multiples of 12, including 12, 24, 36, 48, 60, and so forth. (DOK 1) 70
24 Differentiated Instruction Assessment and Remediation Ask students to solve this problem: Ella has 1 strip of ribbon that is 26 inches long. What are all the ways she could make equal strips that measure a whole number of inches out of the ribbon? [1 3 26, , , and ] For students who are struggling, use the chart below to guide remediation. After providing remediation, check students understanding. Ask students to name all the ways that a 17-inch-long ribbon can be made into equal strips measuring a whole number of inches. [ or ] If a student is still having difficulty, use Ready Instruction, Level 3, Lesson 6. If the error is... Students may... To remediate... listing only and listing only and not have switched the factors of and to get the factor pairs and not realize that or are also ways to get 26. Help them relate factor pairs to the problem (Ella could cut the ribbon into 2 strips of 13 inches because 2 divides evenly into 26). Use drawings to show that and are different from and , respectively (2 strips of 13 inches is different from 13 strips of 2 inches). Give students a hundreds chart and ask them to go in order, starting with one, and try counting by ones, twos, threes, fours, fives, etc. Explain that if they can skip count by a number and land on 26, then the number is a factor of 26 and the number of times they skip-counted is the other factor in the pair. Help students make an organized list of the factor pairs they find. Hands-On Activity Find patterns with multiples. Materials: Hundreds charts, crayons or markers Student groups or pairs explore visual patterns they create when finding multiples of a number on a hundreds chart. Give each group a number from 2 through 12 for which they will find and color in multiples on a hundreds chart. Light colors should be used so the numbers on the chart are still visible. Instruct each group to label their chart Multiples of and describe the pattern they see on their chart in words. Have each group share and describe the pattern they found. Post the colored charts on the wall and ask the class questions, such as: Are there any charts that look similar in some way? How? Why do you think they are similar? How are the charts for the multiples 3 and 6 the same? Different? How are the charts for 2 and 4 the same? Different? Challenge Activity Find all the prime numbers from Materials: A hundreds chart per pair, crayons or markers Students work in pairs to find all the prime numbers on their hundreds chart. You may mention that a Greek mathematician named Eratosthenes (276 BC) used a similar strategy to explore prime numbers. Have students guess the number of primes on their chart and write the guess on the back of their paper. Instruct pairs to begin by finding all the multiples of two and highlight them on the chart. Have them go in order to mark all the multiples of 3, then 4, etc. Circulate among the pairs and ask questions, such as: Can you show me any prime numbers you ve found? How do you know they are prime? Which numbers are composites? How do you know? How will you know when you ve found all the prime numbers on your chart? Have students share the results of their exploration. 71
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