Lesson 2

Represent Unit Fraction Multiplication

Warm-up: Which One Doesn’t Belong: Diagrams (10 minutes)

Narrative

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.

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

Which one doesn’t belong?
ASquare, length and width, 1. Partitioned into 3 rows of 2 of the same size rectangles. 2 rectangles shaded light blue. 1 rectangle shaded dark blue.
BDiagram. Square, length and width, 1. Partitioned into 5 rows of 2 of the same size rectangles. 1 rectangle shaded.
CDiagram. Square. Length and width, 1. Partitioned horizontally into 6 of the same size rectangles. 1 rectangle shaded.
DDiagram. Square, length and width, 1. Partitioned vertically in half. Left half partitioned horizontally into 3 equal rectangles. 1 rectangle shaded.

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

  1. 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?

    Square, length and width, 1. 
  2. 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?

    Diagram. Square. Length and width, 1. 
  3. 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.

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most important to represent the shaded piece with a multiplication expression. Display the sentence frame: “The next time I write a multiplication expression to represent the shaded part of a square, I will look for . . . .”
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

Priya shaded part of a square.
Diagram. Square, length and width, 1. Partitioned into 2 rows of 5 of the same size rectangles. 1 rectangle shaded.
  1. Explain or show how the expression \(\frac{1}{5} \times \frac{1}{2}\) represents the area of the shaded piece.
  2. Explain or show how the expression \(\frac{1}{2} \times \frac{1}{5}\) represents the area of the shaded piece.
  3. Write a multiplication expression to represent the area of the shaded piece. Be prepared to explain your reasoning.

    Square, length and width, 1. Partitioned into 5 rows of 3 of the same size rectangles. 1 rectangle shaded. 
  4. How much of the whole square is shaded?

Student Response

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Advancing Student Thinking

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?”

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.)

Cool-down: How Much is Shaded? (5 minutes)

Cool-Down

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