Lesson 18

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.

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

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

• “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.)

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).

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

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

• 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.)

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).

• Groups of 2

• 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.

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.

• “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?” (\(10 \times \frac{1}{20} = \frac{1}{2}\))