Lesson 2

This warm-up prompts students to carefully analyze and compare different diagrams that represent products of fractions. In making comparisons, students have a reason to use language precisely (MP6). The warm-up also enables the teacher to listen to students as they share their interpretations of the various representations of fraction multiplication, and use their developing vocabulary to describe the characteristics of fractional products.

• 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

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “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.)

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 \(\frac {1}{3} \times \frac {1}{4}\).

• Groups of 2

• 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

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?”

• 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.)

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.

• Groups of 2

• 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.

If students write an expression that does not represent the second diagram, write a correct expression and ask, “How does the expression represent the diagram?”

• “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 the 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?” (\(\frac{1}{15}\))