Lesson 2
Represent Unit Fraction Multiplication
Warmup: Which One Doesn’t Belong: Diagrams (10 minutes)
Narrative
Launch
 Groups of 2
 Display the image.
 “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 2–3 minutes: partner discussion
 Share and record responses.
Student Facing
Student Response
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Activity Synthesis
 “How does diagram A represent the expression \(\frac{1}{3} \times \frac{1}{2}\)?” (There is a half shaded and then a third of the half is shaded darker.)
Activity 1: Interpret Diagrams (20 minutes)
Narrative
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 synthesis, students will connect both of the diagrams to the expressions \(\frac {1}{4} \times \frac {1}{3}\) and \(\frac {1}{3} \times \frac {1}{4}\).
Launch
 Groups of 2
Activity
 2 minutes: Independent work time
 10 minutes: partner work time
 Monitor for students who:
 draw diagrams like those in the student responses
 recognize that both diagrams have the same amount shaded
 can explain the different ways they partitioned each diagram
Student Facing

Show \(\frac{1}{3}\) of the square.
Shade \(\frac{1}{4}\) of \(\frac{1}{3}\) of the square.
How much of the whole square is shaded? 
Show \(\frac{1}{4}\) of the square.
Shade \(\frac{1}{3}\) of \(\frac{1}{4}\) of the square.
How much of the whole square is shaded?  How are the diagrams the same and how are they different?
Student Response
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Advancing Student Thinking
If students do not identify how much of the whole square is shaded, ask: “How can you adapt your diagram to show how many pieces would be in the whole square?”
Activity Synthesis
 Ask previously selected students to share their responses in the order given.
 “How are the diagrams the same?” (They both have \(\frac {1}{4}\) and \(\frac {1}{3}\) in them. They both have \(\frac {1}{12}\) of the whole square shaded.)
 “How are the diagrams different?” (In one diagram, I started by showing \(\frac {1}{3}\) of the square and then shaded in \(\frac {1}{4}\) of \(\frac {1}{3}\). In the other diagram, I started by showing \(\frac {1}{4}\) of the square and then shaded in \(\frac {1}{3}\) of \(\frac {1}{4}\).)
 Display:\(\frac{1}{12}\), \(\frac{1}{4}\times\frac{1}{3}\), \(\frac{1}{3}\times\frac{1}{4}\)
 “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 \(\frac {1}{12}\) shaded.)
Activity 2: Write an Expression (15 minutes)
Narrative
The purpose of this activity is for students to deepen their understanding of the relationship between diagrams and multiplication expressions. The expressions here 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.
Supports accessibility for: Conceptual Processing, Memory
Launch
 Groups of 2
Activity
 2 minutes: independent work time
 5–7 minutes: partner work time
MLR2 Collect and Display
 Circulate, listen for, and collect the language students use to describe where they see how Priya’s diagram represents the expressions. Listen for: columns, rows, fifths, halves, tenths, number of pieces, size of the piece. Record students’ words and phrases on a visual display and update it throughout the lesson.
Student Facing
 Explain or show how the expression \(\frac{1}{5} \times \frac{1}{2}\) represents the area of the shaded piece.
 Explain or show how the expression \(\frac{1}{2} \times \frac{1}{5}\) represents the area of the shaded piece.
 Write a multiplication expression to represent the area of the shaded piece. Be prepared to explain your reasoning.
 How much of the whole square is shaded?
Student Response
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Advancing Student Thinking
Activity Synthesis
 “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:
 \(\frac{1}{5} \times \frac{1}{2}\)
 \(\frac{1}{2} \times \frac{1}{5}\)
 “How does Priya’s diagram show each expression?” (The rows show halves and \(\frac{1}{5}\) of one of the rows is shaded. The columns show fifths and \(\frac{1}{2}\) of one of the columns is shaded.)
 Display second diagram in the activity.
 “How does the shaded piece in the diagram represent \(\frac{1}{5} \times \frac{1}{3}\)?” (The square is divided into three columns and each of those columns is divided into 5 rows. The shaded part is \(\frac{1}{5}\) of one of the columns.)
 “How does the shaded piece represent \(\frac{1}{3} \times \frac{1}{5}\)?” (The square is divided into 5 rows and each row is divided into 3 columns. The shaded piece is \(\frac{1}{3}\) of one of the rows.)
 “How much of the whole rectangle is shaded?” (\(\frac{1}{15}\))
Lesson Synthesis
Lesson Synthesis
“Today we wrote multiplication expressions to represent shaded rectangles. We also wrote fractions to represent the size of the shaded piece.”
Display the second image from the second activity.
Display the equations:
\(\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}\)
\(\frac{1}{5} \times \frac{1}{3} = \frac{1}{15}\)
“How do you know these equations are true?” (We can see that the shaded part of the diagram is both \(\frac{1}{3} \times \frac{1}{5}\) of the whole and \(\frac{1}{5} \times \frac{1}{3}\) of the whole. We can also see \(\frac{1}{15}\) because the whole square is divided into 15 equal pieces and one of the equal pieces is shaded.)
Cooldown: How Much is Shaded? (5 minutes)
CoolDown
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