Lesson 4

Situations about Multiplying Fractions

Warm-up: Number Talk: More Halving (10 minutes)

Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying unit fractions. These understandings help students develop fluency and will be helpful later in this lesson when students make sense of a unit fraction multiplied by a non-unit fraction.

Launch

  • Display one expression.
  • “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time

Activity

  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Facing

Find the value each expression mentally.

  • \(\frac{1}{2} \times \frac{1}{2}\)
  • \(\frac{1}{3} \times \frac{1}{2}\)
  • \(\frac{1}{4} \times \frac{1}{2}\)
  • \(\frac{1}{5} \times \frac{1}{2}\)

Student Response

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Activity Synthesis

  • “What patterns do you notice?” (The numerators are all 1. The denominators are all even numbers. The fractions are getting smaller. Each time, we find a smaller fraction of \(\frac {1}{2}\).)

Activity 1: The Park (20 minutes)

Narrative

The purpose of this activity is for students to draw a diagram representing the product of a unit fraction and a non-unit fraction. Then students use the diagram to represent the product with an expression and find its value. Students may draw many different diagrams that represent the situation. The context of sports fields was chosen to encourage students to divide the square in thirds, vertically or horizontally and, subsequently, to divide the two thirds that represents the sports in half, horizontally or vertically, to represent the part of the sports section that will be used for soccer fields. The activity synthesis focuses on the expressions and equations that represent the area for the soccer fields. Students reason abstractly and quantitatively throughout as they relate their diagram and the expression representing it to the park (MP2). 

MLR8 Discussion Supports. Synthesis: At the appropriate time, give groups 2–3 minutes to plan what they will say when they present to the class. “Practice what you will say when you share your drawing with the class. Talk about what is important to say, and decide who will share each part.”
Advances: Speaking, Conversing, Representing

Launch

  • Groups of 2
  • “What kinds of things do you see and do in the park?”
    • play frisbee or other games
    • watch the ducks
    • use the swings
    • have a picnic
    • play with my friends
  • 1–2 minutes: independent think time
  • 1–2 minutes: partner discussion

Activity

  • 5 minutes: partner work time
  • Monitor for students who label the diagram to show the different parts of the park.

Student Facing

A city is designing a park on a rectangular piece of land. \(\frac{2}{3}\) of the park will be used for different sports. \(\frac{1}{2}\) of the land set aside for sports will be soccer fields.
  1. Draw a diagram of the situation.

    Diagram, square.

  2. Write a multiplication expression to represent the fraction of the sports section that will be soccer fields.
  3. What fraction of the whole park will be soccer fields? Explain or show your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students do not draw a diagram that represents \(\frac{1}{2} \times \frac{2}{3}\), suggest they draw a diagram to represent the \(\frac{2}{3}\) of the park that will be used for different sports. Ask: “How can you adapt your diagram to show that \(\frac{1}{2}\) of the section used for sports will be soccer fields?”

Activity Synthesis

  • Invite students to share their drawings of the soccer fields.
  • Display the image from the student solution or use a student-generated image.
  • “How does the diagram represent \(\frac{2}{3}\) ?” (There are two vertical slices of the park on the left that are \(\frac{2}{3}\) of the park.)
  • Display expression: \(\frac{1}{2} \times \frac{2}{3}\)
  • “How does the diagram represent \(\frac{1}{2} \times \frac{2}{3}\) ?” (The top half of the left \(\frac{2}{3}\) of the square is shaded darkly.)
  • “How does the diagram represent how much of the whole park will be used for soccer fields, which is also the value of \(\frac{1}{2} \times \frac{2}{3}\)?” (It shows 2 pieces that are each \(\frac{1}{6}\) of the whole.)

Activity 2: A Different Park (15 minutes)

Narrative

The purpose of this activity is for students to relate expressions to a diagram in a situation where they represent the product of a unit fraction and a non-unit fraction. Students work with a diagram that represents a different park. Students write expressions, trade with a partner, and interpret their partner’s expressions and match them to a diagram. As students work together, listen for how they explain why the expressions represent the corresponding areas. While the activity focuses on relating expressions and parts of the diagram, in the synthesis students find the value of products and analyze equations in terms of the park (MP2). As students discuss and justify their decisions while looking through each others’ work, they share mathematical claims and the thinking behind them (MP3).

Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory, Organization

Launch

  • Groups of 2

Activity

  • 5–6 minutes: independent think time
  • 2–3 minutes: partner work time
  • Monitor for students who:
    • can explain why \(\frac{1}{5} \times \frac{1}{2}\) represents the part of the park for the swings.
    • can explain why \(\frac{3}{5} \times \frac{1}{2}\) represents the part of the park for the grass.
    • write \(\frac{4}{5} \times \frac{1}{2}\) to represents the part of the park for the pond.

Student Facing

Here is a diagram for a different park that Elena drew.

Diagram. Rectangle partitioned into 5 rows of 2 of the same size, but different colored rectangles. 

  1. Which part of the park can be represented with the expression \(\frac{3}{5} \times \frac{1}{2}\) ? Explain or show your reasoning.
  2. Pick one of the other parts of the park and write a multiplication expression for the fraction of the park it represents.
  3. Trade expressions with your partner and figure out which part of the park their expression represents. Be prepared to explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students do not identify which section is \(\frac{3}{5}\) of \(\frac{1}{2}\) of the park, suggest they draw a separate diagram to represent each section. Ask: “How would you describe this section in relation to the whole park? How can you represent what you described with a multiplication expression?”

Activity Synthesis

  • Ask previously selected students to share their thinking.
  • Display: \(\frac{1}{5} \times \frac{1}{2} = \frac {1}{10}\)
  • “What part of the diagram does this equation represent?” (It represents the section of the park that is swings. We can see that the swings take up \(\frac{1}{5}\) of \(\frac{1}{2}\) of the park.)
  • Display: \(\frac{3}{5} \times \frac{1}{2} = \frac {3}{10}\)
  • “What part of the diagram does this equation represent?” (It represents the section of the park for grass, which is \(\frac{3}{5}\) of \(\frac{1}{2}\) of the park and it is also \(\frac{3}{10}\) of the whole park.)
  • Display: \(\frac{4}{5} \times \frac{1}{2} = \frac {4}{10}\)
  • “What part of the diagram does this equation represent?” (It represents the section of the park for the pond, which is \(\frac{4}{5}\) of \(\frac{1}{2}\) of the park and it is also \(\frac{4}{10}\) of the whole park.)

Lesson Synthesis

Lesson Synthesis

“Today we represented multiplication of a unit fraction and a non-unit fraction with diagrams and expressions.”

Display the park diagram from the last activity. Display equations:

\(\frac{1}{5}\times \frac{1}{2} = \frac{1}{10}\)

\( \)\(\frac{2}{2} \times \frac{1}{5} = \frac{2}{10}\)

\(\frac{3}{5}\times \frac{1}{2} = \frac{3}{10}\)

\(\frac{4}{5} \times \frac{1}{2} = \frac{4}{10}\)

“Describe to your partner how each equation represents the diagram of the park.”

“What patterns do you notice in the equations?” (Each part of the park is a certain amount of tenths. If we multiply the numerators, we get the numerator in the product. If we multiply the denominators, we get the denominator in the product.)

Cool-down: Area of the Park (5 minutes)

Cool-Down

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