Lesson 16

The purpose of this warm-up is for students to notice factors and common factors of 6 and 10 based on rectangles that can be made with the corresponding number of squares. Students may notice that the height of each pair of images changes, but they might not connect this to factors. The whole-class discussion should focus on how the images reflect factors of 6 and 10. 

Tell students you will show them four pairs of images and their job is to find something that is similar and different about the pairs of images. Tell them to give a signal when they have at least one thing that is similar and one thing that is different. Give students 1 minute of quiet think time followed by a whole-class discussion. 

Ask students to share the things that are alike and different among the pairs of images. Record and display their responses for all to see. If possible, reference the images as the students share and record their responses on the images where appropriate.

If the following two ideas do not come up in the conversation, ask students these questions:

• “2 and 3 are both factors of 6. How is this reflected in the diagram?”
• “2 is a factor of both 6 and 10. How is this reflected in the diagram?”
• “4 is not a factor of either 6 or 10. How is this reflected in the diagram?”

End the discussion defining the term factor as one of two or more numbers, that when multiplied together result in a given product. In this particular case, a factor is the height that will make a rectangle have a given area. 

Students begin to think about common factors and the greatest common factor in the context of finding ways to group equal amounts of baked goods into bags. Students find all common factors of 2 whole numbers, one representing the number of brownies and another representing cookies. They then compare these factors 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 bags, checking each time if the total number can be divided into those bags evenly without remainder. Other students may notice that combinations come in pairs. For example, 4 bags of 12 brownies can be paired with 12 bags of 4 brownies.

Arrange students in groups of 2. Give students 10 minutes work time followed by whole-class discussion. Encourage students to check in with their partner after each question to make sure they get every possible combination of bags.

Students might not find all combinations of factor pairs for each number. If this is the case, ask them to use snap cubes and prompt them to find more combinations. For example, “Is there a way to place 64 snap cubes into 4 groups with no snap cubes left over? How many are in each group?”

For questions 1 and 2, invite students to share how they organized the different combinations of bags, and highlight their different strategies. Sequence responses by first selecting students who found all combinations using pictorial representations, then students who made an organized list or table, and finally students who were able to highlight the fact that factors come in pairs (i.e. the number of bags and the number of brownies can always be reversed). During this discussion, ask students how they know that they have found all possible bag combinations for each number. Confirm that there should be 10 different bag arrangements for the brownies and 7 different bag arrangements for the chocolate chip cookies.

For questions 3 and 4, ask students to compare answers with a partner. Did they find all the combinations? Confirm that there are 5 different bag arrangements, and the greatest number of bags that can be made is 16. Select a group that used snap cubes to share what this arrangement looks like when represented with the two different colors. If possible, create a visual representation of this arrangement that can be displayed for all to see throughout the rest of the unit.

In the last activity, students worked with the concept of common factors in the context of distributing two kinds of baked goods equally. In this activity, students explore common factors of numbers more generally and are introduced to the term greatest common factor. The final question connects the concept of greatest common factor to geometry. They describe what the greatest common factor is and how it applies to a geometric context.

Arrange students in groups of 2. Ask students to discuss what they think a common factor of two numbers is with a partner and select for 1--2 groups to share their thinking. Give 10 minutes of quiet work time followed by whole-class discussion.

Some students may not list all of the factors of a number. Prompt these students to try to find more factors. If additional support is needed, provide graph paper and ask the student if it's possible to make a rectangle area equal to the number with a height of 1, 2, 3, 4, etc. until they are convinced they have all the factors.

Ask students to share their thinking to question 4. Record and display their responses for all to see. Consider asking how their responses would change if the bulletin board was 18 inches tall and 63 inches wide instead. Encourage students to use the terms “common factor” and “greatest common factor” in their explanations.