The purpose of this activity is for students to draw two diagrams that represent a unit fraction of a unit fraction. Students work with the same unit fractions in both diagrams. The directions were intentionally written to encourage students to partition a unit square in different ways. Students initially partition the square into thirds in the first problem and into fourths in the second problem. Students may complete the diagrams in a way that makes sense to them. During the Activity Synthesis, students will connect both of the diagrams to the expressions and .

• Ask previously selected students to share their responses in the order given.
• “How are the diagrams the same?” (They both have and in them. They both have of the whole square shaded.)
• “How are the diagrams different?” (In one diagram, I started by showing of the square and then shaded in of . In the other diagram, I started by showing of the square and then shaded in of .)
• Display:, ,
• “How do the diagrams represent these expressions?”
• Adapt diagrams as needed in order to show all three expressions.
• “How do we know that both squares have the same amount shaded?” (They both have shaded.)

The purpose of this activity is for students to deepen their understanding of the relationship between diagrams and multiplication expressions. The expressions are products of unit fractions. Students start with a diagram and first explain how an expression represents the diagram. Then they write their own expression representing a different diagram (MP7).

This activity uses MLR2 Collect and Display. Advances: Reading, Writing.

• “What do you know about the area of the shaded piece?”
• “How can you shade the diagram to show of ? How about of ?”

• “Are there any other words or phrases that are important to include on our display?”
• Ask students to clarify the meaning of a word or phrase.
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display Priya’s diagram and these expressions:
• “How does Priya’s diagram show each expression?” (The rows show halves and of one of the rows is shaded. The columns show fifths and of one of the columns is shaded.)
• Display the second diagram in the activity.
• “How does the shaded piece in the diagram represent ?” (The square is divided into three columns and each of those columns is divided into 5 rows. The shaded part is of one of the columns.)
• “How does the shaded piece represent ?” (The square is divided into 5 rows and each row is divided into 3 columns. The shaded piece is of one of the rows.)
• “How much of the whole rectangle is shaded?” ()