Lesson 17

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying and dividing fractions. These understandings help students develop fluency and will be helpful later in this lesson when students solve problems about multiplying and dividing fractions.

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “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.

• “¿Cómo se relacionan los dos últimos problemas?” // “How are the last two problems related?” (Dividing by 6 and multiplying by \(\frac{1}{6}\) give the same result.)

This Info Gap activity gives students an opportunity to determine and request the information needed to solve multi-step problems involving multiplication and division of unit fractions. In both cases, the student with the problem card needs to find out the side lengths of the area being covered and the size of the tiles and from there they can figure out how many tiles are needed. The numbers in the problems are chosen so that students can draw diagrams or perform arithmetic directly with the numbers.

In this Info Gap activity, the first problem encourages students to think about multiplying the given fractions. The second problem involves a given area and a missing side length, which may encourage students to represent and solve the problem with a missing factor equation.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2

• Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
• 15 minutes: partner work time
• After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
• Monitor for students who:
  • Draw pictures to arrange the tiles and see how they can fit the tiles together to cover the area.
  • Use division and multiplication in their calculations.

• Invite students to share their responses to the problem about tiling the floor.
• “¿Cómo supieron si usar la multiplicación o la división para resolver el problema?” // “How did you know whether to use multiplication or division to solve the problem?” (I needed to use multiplication to find the area of the floor. I also used multiplication to figure out how many tiles I needed.)
• “¿Cómo resolvieron el problema?” // “How did you solve the problem?” (I found out how many tiles I needed for one square foot by drawing a picture and then multiplied that by the number of square feet in the floor.)
• Invite students to share their responses to the problem about the tiles for the wall.
• “¿Cómo supieron si usar la multiplicación o la división?” // “How did you know whether to use multiplication or division?” (I used multiplication to figure out how many square feet of the wall I could cover with each box and how many square feet were in the wall.)
• “¿Cómo resolvieron el problema?” // “How did you solve the problem?” (I used division to estimate how many boxes of tiles I would need.)

The goal of this activity is to solve a variety of different problems, all of which can be solved by multiplying two fractions though the first problem may also be represented as division of a unit fraction by a whole number. The numbers are complex, making drawings an unlikely strategy to solve the problems. This encourages students to use their understanding of how to multiply fractions or divide with a whole number and a unit fraction. The synthesis focuses on why students chose multiplication or division to solve the problems. 

• Groups of 2

• 6 minutes: independent work time
• 4 minutes: partner discussion
• Monitor for students who:
  • represent the situations with equations
  • explain how their equations represent each situation

If students are not familiar with any of the contexts in the situations consider showing them pictures of rice or tennis balls or asking them to describe a landmark that is approximately a mile from the school.

• Ask previously selected students to share their solutions.
• Display:
  \(\frac{1}{3} \div 11 = \underline{\hspace{1 cm}}\)
• “¿De qué manera la ecuación representa la primera situación?” // "How does the equation represent the first situation?" (The 11 grains of rice together weigh \(\frac{1}{3}\) gram so I need to divide \(\frac{1}{3}\) by 11 to find out how many grams each grain weighs.)
• Dislay: \(\frac {3}{4} \times \frac {9}{10} = \underline{\hspace{1 cm}}\)
• “¿De qué manera la ecuación representa lo que corrió Mai?” // “How does the equation represent Mai's run?” (The \(\frac{9}{10}\) is the full length of the road but she only ran \(\frac{3}{4}\) of it so I multiply \(\frac{9}{10}\) by \(\frac{3}{4}\).)