In this activity, students reason abstractly about common multiples and least common multiple to solve problems in context (MP2.) Students find all the common factors of two whole numbers, one representing the number of fiction books and another representing nonfiction books. Then they compare the factors of each whole number to determine the greatest common factor.

Monitor for strategies and representations students use to make sure they account for all possible combinations. Some students may organize their work by number of boxes, checking each time if the total number can be divided into those boxes evenly, without a remainder. Other students may notice that combinations come in pairs. For example, 4 boxes of 12 fiction books can be paired with 12 boxes of 4 fiction books.

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

• Draw a picture to represent boxes and books.
• Make an organized list or table.
• Recognize that factors come in pairs and can always be reversed.

Ask students how they know they have found all possible combinations of boxes, and if necessary, confirm that there are 10 different possibilities for the boxes of fiction books, 7 different possibilities for the boxes of nonfiction books, and 5 different possibilities for the combination boxes.

The purpose of this discussion is to formally introduce the term greatest common factor as the largest factor that two numbers share and to help students extend the connection between factors and the area model to include common factors and greatest common factor. Begin by inviting students to share their thinking on the last question. Record and display their responses for all to see. Here are some questions to discuss:

• “Consider a new bulletin board that is 18 inches tall and 63 inches wide. What are the side lengths of some squares that could completely cover the area with no gaps or overlaps?” (1-inch squares, 3-inch squares, 9-inch squares)
• “If we wanted to use the largest size paper, which size squares should we use?” (the 9-inch squares)

Explain how the 1-, 3-, and 9-inch squares represent the common factors of 18 and 63. The largest square that can tile the area represents the greatest common factor because it is the largest of the common factors and divides evenly into both 18 and 63.