In this activity, students begin to think about common multiples and the least common multiple when finding ways to order two types of flowers that come in bunches of different sizes. Students find all multiples up to 100 for two different numbers. They compare these multiples to determine which ones are the same, representing an order with the same numbers of both flowers, and then they determine the least common multiple.

Monitor 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.

The purpose of this activity is for students to share how they organized information when finding different combinations of flowers. Invite students to share their strategies for the first questions. Display or record their responses for all to see. Here are some strategies students may have used:

• Draw a picture to represent bunches of roses and bunches of lilies.
• Make an organized list or table.

If not brought up in students’ explanations, 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 explain that it takes 2 bunches of roses to equal 24, and 3 bunches of lilies to equal 24. 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. 

The purpose of discussion is to formally introduce the least common multiple of two numbers, and to clarify the process of finding common multiples and identifying the least common multiple. Begin by inviting students to share their responses to the first question. If not brought up in students’ explanations, tell students that the least common multiple of two numbers is the smallest product that results from multiplying each of the two numbers by some whole number. 

Next, ask students to discuss a way to find the least common multiple of any two numbers with a partner. Invite groups to share their responses, and record them for all to see. If time allows, display pairs of numbers, and ask students to find the least common multiple.

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 three 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 the 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.

The purpose of this discussion is for students to make connections between multiples and a real-world context. Invite students to share their strategies for determining if Jada would get any prizes as the 36th customer. Then ask students how many prizes in total Lin’s uncle will give away, and discuss the following questions:

• “How can multiples help us solve this problem?” (We can list out the multiples of 5, 9, and 12 up to 50 and add up how many prizes will be given away.)
• “What are the multiples of 5, 9, and 12 up to 50?” (The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. The multiples of 9 are 9, 18, 27, 36, and 45. The multiples of 12 are 12, 24, 36, and 48.)
• “How many total prizes will be given away?” (19 prizes)

If time allows, continue discussing the following questions:

• “How many customers will win 2 prizes?” (Two. Customer 36 will win a muffin and a slice of carrot cake. Customer 45 will win a bagel and a muffin.)
• “How many customers will win all 3 prizes?” (Zero. None of the multiples are common to all three numbers.)