Lesson 8

Divide to Multiply Non-unit Fractions

Warm-up: True or False: A Fraction by a Whole Number (10 minutes)

Narrative

The purpose of this True or False is to elicit the strategies and insights students have for multiplying fractions by whole numbers. Students do not need to find the value of any of the expressions but rather can reason about properties of operations and the relationship between multiplication and division. In this lesson, they will see some ways to find the value of an expression like \(\frac{2}{3} \times 6\).

Launch

  • Display one statement.
  • “Give me a signal when you know whether the statement is true and can explain how you know.”
  • 1 minute: quiet think time

Activity

  • Share and record answers and strategy.
  • Repeat with each statement.

Student Facing

Decide if each statement true or false. Be prepared to explain your reasoning.

  • \(2 \times \left(\frac{1}{3} \times 6\right) = \frac{2}{3} \times 6\)
  • \(2 \times \left(\frac{1}{3} \times 6\right) = 2 \times (6 \div 3)\)
  • \( \frac{2}{3} \times 6 = 2 \times \left(\frac{1}{4} \times6 \right)\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “How can you explain why  \(\frac{2}{3} \times 6 = 2 \times (\frac{1}{4} \times6)\) is false without finding the value of both sides?” (It can't be true because \(\frac{2}{3} \times 6=2\times\frac{1}{3}\times6\).)

Activity 1: Multiply a Whole Number by a Fraction (15 minutes)

Narrative

The purpose of this activity is for students to relate multiplying a non-unit fraction by a whole number to multiplying a unit fraction by the same whole number. After finding the value of \(\frac{1}{5} \times 3\) in a way that makes sense to them, they then consider the value of the products \(\frac{2}{5} \times 3\) and \(\frac{3}{5} \times 3\). In the synthesis students address how they can use the value of \(\frac{1}{5} \times 3\) to find the value other expressions.

Launch

  • Groups of 2

Activity

  • 8 minutes: independent work time
  • Monitor for students who:
    • draw a diagram
    • use division to solve
    • recognize a relationship between \(\frac{1}{5}\times3\)\(\frac{2}{5}\times3\), and \(\frac{3}{5}\times3\) .

Student Facing

Find the value of each expression. Explain or show your reasoning. Draw a diagram if it is helpful. 
  1. \(\frac{1}{5} \times 3\)
  2. \(\frac{2}{5} \times 3\)
  3. \(\frac{3}{5} \times 3\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Ask previously selected students to share their solutions.
  • Display: \(\frac{1}{5} \times 3\), \(\frac{2}{5} \times 3\), \(\frac{3}{5} \times 3\)
  • “How are the expressions the same?” (They all have a 3. They all have some fifths and there is a product.)
  • “How are the expressions different?” (The number of fifths is different. There is 1 and then 2 and then 3.)
  • “How can you use the value of \(\frac{1}{5} \times 3\) to help find the value of \(\frac{2}{5} \times 3\)?” (I can just double the result because it’s \(\frac{2}{5}\) instead of \(\frac{1}{5}\).)
  • “What about \(\frac{3}{5} \times 3\)?” (That’s just another \(\frac{1}{5} \times 3\).)
  • Display diagram from student solution or a student generated diagram like it.
  • “How does the diagram show \(\frac{1}{5} \times 3\)?” (There is 3 total and \(\frac{1}{5}\) of it is shaded.)
  • Display: \(\frac{2}{5} \times 3\).
  • “How could you adapt the diagram to show \(\frac{2}{5} \times 3\)?” (I could fill in 2 of the fifths in each whole instead of 1.)
  • “In the next activity we will study a diagram for \(\frac{2}{5} \times 3\) more.”

Activity 2: Match Expressions to Diagrams (20 minutes)

Narrative

The purpose of this activity is to interpret diagrams in multiple ways, focusing on different multiplication and division expressions. The repeating structure in the diagrams allows for many different ways to find the value and interpret the meaning of the expressions. Encourage students to use words, diagrams, or expressions to explain how the diagram represents each of the expressions.

Monitor for students who:

  • can explain that the diagram represents the multiplication expression \(3 \times \frac{2}{5}\) because it shows 3 groups of \(\frac{2}{5}\)
  • can explain that the diagram represents \(2 \times (3 \div 5)\) because there are 3 wholes divided into 5 equal pieces and 2 of the pieces in each whole are shaded
  • can explain how the diagram represents the relationship between \(\frac{6}{5}\) and \(2 \times (3 \div 5)\)

This activity gives students an opportunity to generalize their learning about fractions, division and multiplication. Students see shaded diagrams in different ways, representing different operations, and begin to see the operations as a convenient way to represent complex calculations (MP8).

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

Engagement: Provide Access by Recruiting Interest. Provide choice. Invite students to decide which expression to start with.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Attention

Launch

  • Groups of 2

Activity

  • 5–10 minutes: partner work time

MLR2 Collect and Display

  • Circulate, listen for, and collect the language students use to describe how each part of the expression represents each part of the diagram.
  • Listen for language described in the narrative.
  • Look for notes, labels, and markings on the diagrams that connect the parts of the diagram to the parts of the expressions.
  • Record students’ words and phrases on a visual display and update it throughout the lesson.

Student Facing

Explain how each expression represents the shaded region.

  1. \(2 \times (3 \div 5)\)

  2. \(\frac{6}{5}\)

  3. \(3 \times \frac{2}{5}\)

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

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students do not choose any expressions that represent the diagram, ask them to describe the diagram. Write down the words and phrases they use and ask, “Which expressions represent the words you used to describe the diagram?”

Activity Synthesis

  • Display the expression: \(3 \times \frac{2}{5}\)
  • “How does the diagram represent the expression?” (It shows 3 groups of \(\frac{2}{5}\).)
  • Display the expression: \(2 \times (3 \div 5)\)
  • “How does the diagram represent the expression?”
  • Display: \(2 \times (3 \div 5)=\frac{6}{5}\)
  • “How do we know this is true?” (We can see both of them in the diagram. \(3\div5\) is the same as \(\frac{3}{5}\) and \(2\times\frac{3}{5}=\frac{6}{5}\)
  • “Are there any other words, phrases, or diagrams that are important to include on our display?”
  • 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.

Lesson Synthesis

Lesson Synthesis

Revisit the chart about the relationship between multiplication and division created in an earlier lesson.

“What would you add to or revise about the relationship between multiplication and division?”

Revise chart as necessary.

Cool-down: Two Thirds (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Section Summary

Student Facing

In this section, we explored the relationship between multiplication and division. We learned that 1 diagram can represent different multiplication and division expressions. For example, we can interpret this diagram with 4 different expressions:

3 diagrams of equal length. 4 equal parts. 1 part shaded. Total length, 1.
  •  \(\frac{3}{4}\) because each rectangle is divided into 4 equal parts and three of them are shaded.
  •  \(3 \times \frac{1}{4}\) because there are 3 parts shaded and each one is \(\frac{1}{4}\) of the rectangle.
  •  \(3 \div 4\) because there are 3 rectangles and each one is divided into 4 equal parts.
  •  \(\frac14 \times 3\) because there are 3 rectangles and \(\frac{1}{4}\) of each one is shaded.

We know that all of these expressions are equal because they all represent the same diagram. We can use any of these expressions to represent and solve this problem:

  • Mai ate \(\frac{1}{4}\) of a 3 pound bag of blueberries. How many pounds of blueberries did Mai eat?