In this activity, students interpret multiplication expressions and diagrams as the number of groups and the amount in each group, and match representations of the same quantity. This sorting task gives students opportunities to analyze expressions and diagrams closely and make connections (MP7). They then use their insight from the matching activity to generate diagrams for expressions without a match and find their values.

• “What was missing from Han’s diagram? How do you know?” (4 more groups of were missing, because means 7 groups of and there were only 3 groups in Han’s diagram.)
• “If the expression was for 7 groups of instead of , how would Han’s diagram change?” (Each rectangle representing 1 would have 3 equal parts with 1 shaded.)
• Invite students to share the diagrams they drew for the expressions without a match. Ask them to point out the number of groups and the size of the group in each diagram.

This activity prompts students to use their earlier observations to generate a diagram or an expression that represents equal groups of unit fractions when one or the other is given. In one problem, only the total quantity () is given, so students need to reason about the number of groups and the size of each group that could lead to such a value. Finally they analyze two different ways of representing with a diagram, which further illustrates that the value of the expression is .

• See the Lesson Synthesis.