Lesson 18

Represent Situations with Multiplication and Division

Warm-up: Number talk: Three and a Tenth (10 minutes)

Narrative

This Number Talk encourages students to think about multiplication and division involving a whole number and unit fraction. While the order of the factors does not matter for multiplication, as seen in the first two expressions, it does matter for division, as seen in the second pair of expressions. Monitor for students who find the value of the two division expressions using multiplication since the relationship between multiplication and division is the focus of this lesson.

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 of each expression mentally.

  • \(3 \times \frac{1}{10}\)
  • \(\frac{1}{10} \times 3\)
  • \(\frac{1}{10} \div 3\)
  • \(3 \div \frac{1}{10}\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

“How did you find the value of \(3 \div \frac{1}{10}\)?” (I know that there are ten \(\frac{1}{10}\)s in 1 so there are thirty \(\frac{1}{10}\)s in 3.)

Activity 1: Putting it All Together: Multiplication and Division (15 minutes)

Narrative

The purpose of this activity is for students to articulate the relationship between multiplication and division explaining how to solve two different problems using multiplication or division. Students have observed that dividing a whole number by a unit fraction gives the same result as multiplying the whole number by the denominator. They have also observed that dividing a unit fraction by a whole number gives the same result as multiplying the fraction by the unit fraction that has the whole number as a denominator. They also know from prior units and courses that the operations of multiplication and division are closely related. This activity brings these two ideas together, making explicit how one situation and one diagram modeling the situation can be interpreted using either multiplication or division (MP2).

MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to the problems about the neighborhood barbeque dinner. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening

Launch

  • “We are going to solve some problems about a neighborhood barbecue dinner. What do you like to eat for dinner in the summertime?”

Activity

  • 1–2 minutes: independent think time
  • 4–5 minutes: partner work time

Student Facing

  1. Diego’s dad is making hamburgers for the picnic. There are 2 pounds of beef in the package. Each burger uses \(\frac{1}{4}\) pound. How many burgers can be made with the beef in the package?
    1. Draw a diagram to represent the situation.
    2. Write a division equation to represent the situation.
    3. Write a multiplication equation to represent the situation.
  2. Diego and Clare are going to equally share \(\frac{1}{4}\) pound of potato salad. How many pounds of potato salad will each person get?
    1. Draw a diagram to represent the situation.
    2. Write a division equation to represent the situation.
    3. Write a multiplication equation to represent the situation.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Invite students to share their diagrams and equations for the first problem.
  • “How does the diagram help you solve the problem?” (The number of burgers is the number of small rectangles or \(\frac{1}{4}\)s.)
  • “How does the diagram let you interpret the solution using multiplication?” (There are 8 burgers because \(8 \times \frac{1}{4} = 2\).)
  • “How does the diagram let you interpret the solution using division?” (It shows how many \(\frac{1}{4}\)s there are in 2, 8.)

Activity 2: Multiplication or Division? (20 minutes)

Narrative

The purpose of this activity is to for students to make sense of situations, make an appropriate representation in the form of multiplication or division equations, and use that representation to answer questions about the situation (MP2). Students who have a strong understanding of the relationship between division and multiplication can reason about the situations using either operation, though in some cases a representation as division goes beyond grade level which only addresses division of a whole number and a unit fraction. When partners share their work and discuss any disagreements they critique each other's reasoning (MP3).

Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Attention, Organization

Launch

  • Groups of 2

Activity

  • 6 minutes: independent work time
  • 4 minutes: partner work time
  • Monitor for students who:
    • Write multiple equations involving division and multiplication for the same problem.
    • Use diagrams to help them write expressions or equations.

Student Facing

For your set of problems:

  • Write a multiplication or division expression for each situation.
  • Answer the question and write an equation. Make sure to include appropriate units. Draw a diagram, if needed.
  • Trade papers with your partner, and check your partner’s equations. If you disagree, work to reach an agreement.

Partner A:

  1. The distance from Han’s house to Priya’s house is \(\frac{4}{5}\) kilometer. Han has walked \(\frac{3}{4}\) of the way already. How many kilometers has he walked?
  2. Clare’s science class will test water samples in class. If there is a total of \(\frac{1}{2}\) gallon of water and 10 groups, how much water will each group get if they split the water equally?
  3. A container with 3 kilograms of strawberries is \(\frac{1}{5}\) full. How many kilograms can the container hold?

Partner B:

  1. It takes Han 4 minutes to walk \(\frac{1}{3}\) kilometer. How many minutes will it take him to walk 1 kilometer?
  2. Clare’s goal was to collect 4 kilograms of soil sample for her science project. She collected \(2 \frac{2}{3}\) times her goal. How many kilograms of soil did Clare collect?
  3. A container that can hold a \(\frac{1}{2}\) pound of strawberries is \(\frac{3}{5}\) full. How many pounds of strawberries are in the container?

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students are not familiar with the measurement contexts in the situations, incorporate a launch that recalls the sizes of measurements such as kilograms, pounds, gallons, and kilometers. For example, describe a landmark that is 1 kilometer from the school or show students a container that has about \(\frac{1}{2}\) pound of strawberries in it.

Activity Synthesis

  • “How did you know what operation you needed to perform to find the answer?” (I drew a diagram to help me visualize the situation.)
  • “For which problems was it difficult to tell what operation to use?” (I wasn’t sure what to do with the strawberry problem.)
  • For students who used a diagram, “How did drawing the diagram help you write an equation?” (It helped me see what operation could be used to solve the problem and see the result.)
  • Point out equations that correctly represent the same problem (and are thus equivalent) but are expressed differently by displaying the problem about water and the equation: \(\frac{1}{2} \div 10 = \frac{1}{20}\).
  • “Many of you wrote a division equation to represent this problem. What multiplication equation can represent this problem?” (\(10 \times \frac{1}{20} = \frac{1}{2}\))

Lesson Synthesis

Lesson Synthesis

“What do we know about the relationship between multiplication and division?” (I can often use multiplication to solve a division problem. To find \(56 \div 4\) I need to find how many 4s there are in 56 and I can do that with multiplication, first taking 10 of them and then 4 more. Or to find \(3 \div \frac{1}{8}\) I can say there are 8 \(\frac{1}{8}\)s in each whole and then multiply that by 3.)

Create an anchor chart to record student thinking including examples and diagrams.

Cool-down: Diagrams and Equations (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.