Lesson 17

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. 

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. 

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.

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.

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.

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.

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

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.

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.

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.

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.

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.”