Lesson 17

Common Multiples

17.1: Notice and Wonder: Multiples (5 minutes)

Warm-up

The purpose of this warm-up is to review factors and multiples while eliciting ideas on common factors and common multiples that will be useful in the activities of this lesson. While students may notice and wonder many things about the numbers they have circled, it is important for students to notice the multiples 4 and 6 have in common and wonder what other multiples they would have in common if the counting sequence continued. 

Launch

Arrange students in groups of 2. Tell students 10 is a multiple of 5 because \(10 = 5 \boldcdot 2\). One way you can find multiples of a number is by skip counting. For example, the multiples of 5 are 5, 10, 15, 20. . . and so on. Give students 1 minute of quiet work time to circle the multiples of 4 and 6. Give students 1 minute to discuss the things they notice and wonder with a partner, followed by a whole-class discussion. 

Student Facing

Circle all the multiples of 4 in this list.

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

Circle all the multiples of 6 in this list.

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

What do you notice? What do you wonder?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If there is time, ask students what other multiples 4 and 6 would have in common if the counting sequence continued. Ask them to explain their reasoning. Record and display their responses for all to see.

17.2: The Florist’s Order (10 minutes)

Activity

Students begin to think about common multiples and the least common multiple when finding ways to place two types of flowers into groups that contain the same number of each flower. Students find all multiples up to 100 for two different numbers. They compare these multiples to determine which ones are the same, and then they determine the least common multiple.

Look for different strategies and representations students use to describe the situation. Some students may draw pictures of groups of flowers, other students may use tables or lists, and other students may do a combination of these.

Launch

Arrange students into groups of 2. Give students 5–7 minutes of quiet work time, then 2 minutes of partner discussion. Follow with whole-class discussion.

Student Facing

A florist can order roses in bunches of 12 and lilies in bunches of 8. Last month she ordered the same number of roses and lilies.

  1. If she ordered no more than 100 of each kind of flower, how many bunches of each could she have ordered? Find all the possible combinations.
  2. What is the smallest number of bunches of roses that she could have ordered? What about the smallest number of bunches of lilies? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share how they organized the different combinations of flowers, and highlight the different strategies they used. Strategies to highlight include tables, lists, snap cubes, and other pictorial representations. Confirm that there are 4 different order combinations, and each time there are 24 more of each flower added. Discuss why the smallest number of flowers of each type is 24, and that it takes 2 bunches of roses to equal 24, and 3 bunches of lilies to equal 24. If there was a group that used snap cubes, ask them to share what this arrangement looks like when represented with two different colors. If possible, use student responses to create a visual display of the concept of least common multiple and display it for all to see throughout the unit.

Speaking: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to communicate the multiple strategies used by different students. Provide sentence frames for students to use when they are comparing and contrasting different representations such as: “All ____ have ____ except ____.”, “What makes ____ different from the others is ____.”
Design Principle(s): Support sense-making; Cultivate conversation

17.3: Least Common Multiple (10 minutes)

Activity

In this activity, students are introduced to the terms common multiple and least common multiple

Launch

Arrange students in groups of 2. Ask students to discuss what they think a common multiple of two whole numbers is with their partner. Give 5 minutes of quiet work time followed by 2 minutes of partner discussion. Follow with whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge. Allow students to use calculators for finding greatest common factors and least common multiples to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Conversing, Representing: MLR2 Collect and Display. While pairs are working, circulate and listen to student talk about the meaning of and reason about least common multiple. Write down phrases (e.g., “the same,” “the smallest,”, “multiply”) and representations (e.g., number lines, lists of multiples) you observe students using. Write these on a visual display, as this will help students use mathematical language as they represent least common multiples.
Design Principle(s): Support sense-making

Student Facing

The least common multiple of 6 and 8 is 24.

  1. What do you think the term “least common multiple” means?
  2. Find all of the multiples of 10 and 8 that are less than 100. Find the least common multiple of 10 and 8.

  3. Find all of the multiples of 7 and 9 that are less than 100. Find the least common multiple of 7 and 9.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. What is the least common multiple of 10 and 20?
  2. What is the least common multiple of 4 and 24?
  3. In the previous two questions, one number is a multiple of the other. What do you notice about their least common multiple? Do you think this will always happen when one number is a multiple of the other? Explain your reasoning. 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of discussion is to clarify the process of finding common multiples and identifying the least common multiple. Ask students to discuss a way to find the least common multiple of any two numbers with a partner. Then in whole-class discussion, invite students to describe their process for finding least common multiples. Display several pairs of numbers and ask students to describe their process for finding common multiples and the least common multiple. Encourage students to use the terms “common multiple” and “least common multiple” in their responses.

17.4: Prizes on Grand Opening Day (15 minutes)

Activity

In this activity, students continue to explore common multiples in context. Prizes are being given away to every 5th, 9th, and 15th customer. Students list the multiples of each number when determining which customers get prizes and when customers get more than one prize. Customers who get more than one prize represent pairwise least common multiples. It is also true that the first customer who gets all 3 prizes represents the least common multiple of all three numbers, but this idea goes beyond the standards being addressed, and there aren't enough customers for this to happen. Students reason abstractly about common multiples and least common multiple to solve problems in context (MP2).

Monitor for students using these strategies:

  • List numbers from 1 to 50 and skip count to identify common multiples.
  • Analyze common multiples of pairs of numbers, rather than all three numbers at once.
  • Denote multiples of different numbers with different shapes, colors, or other notations. Identify common multiples as numbers that have multiple designations.

Launch

Arrange students in groups of 2. Encourage students to discuss their reasoning with their partner as they work. Give students 10 minutes work time followed by a whole-class discussion.

Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem, without solving it for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., The bakery is giving away free prizes on opening day.). In the second read, ask students to describe quantities in the story (e.g., total of first 50 people get prizes, every fifth person gets a bagel, every 9th person a muffin, and every 12th person a slice of cake). In the third read, ask students to draw representations of the situation, and then brainstorm possible mathematical solution strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem, keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​

Student Facing

Lin’s uncle is opening a bakery. On the bakery’s grand opening day, he plans to give away prizes to the first 50 customers that enter the shop. Every fifth customer will get a free bagel. Every ninth customer will get a free blueberry muffin. Every 12th customer will get a free slice of carrot cake.

  1. Diego is waiting in line and is the 23rd customer. He thinks that he should get farther back in line in order to get a prize. Is he right? If so, how far back should he go to get at least one prize? Explain your reasoning.
  2. Jada is the 36th customer.

    1. Will she get a prize? If so, what prize will she get?
    2. Is it possible for her to get more than one prize? How do you know? Explain your reasoning.
  3. How many prizes total will Lin’s uncle give away? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

For each unique strategy, select one group to display their work for all to see and explain their reasoning. Sequence their presentations in the order they are presented in the Activity Narrative. If one or more of these strategies isn’t brought up by students, demonstrate them. Encourage students to use the terms “common multiple” and “least common multiple.”

Lesson Synthesis

Lesson Synthesis

In this lesson, students learned what a common multiplefor two whole numbers is as well as theleast common multiple. They also developed strategies for how to find the least common multiple of two whole numbers. Discuss:

  • “What are some situations when finding least common multiple is helpful?” (When forming the smallest amount of equal groups, or when events both first happen at the same time.)
  • “Explain what least common multiple means.” (It is the smallest multiple that numbers share.)
  • “How can we can determine least common multiple?” (List multiples of each number until you find the first one that is common to both lists.)

17.5: Cool-down - In Your Own Words Again (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

A multiple of a whole number is a product of that number with another whole number. For example, 20 is a multiple of 4 because \(20 = 5\boldcdot 4\).

A common multiple for two whole numbers is a number that is a multiple of both numbers. For example, 20 is a multiple of 2 and a multiple of 5, so 20 is a common multiple of 2 and 5. 

The least common multiple (sometimes written as LCM) of two whole numbers is the smallest multiple they have in common. For example, 30 is the least common multiple of 6 and 10.

One way to find the least common multiple of two numbers is to list multiples of each in order until we find the smallest multiple they have in common. Let's find the least common multiple for 4 and 10. First, we list some multiples of each number. 

  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44 . . .
  • Multiples of 10: 10, 20, 30, 40, 50, . . .

20 and 40 are both common multiples of 4 and 10 (as are 60, 80, . . . ), but 20 is the smallest number that is on both lists, so 20 is the least common multiple.