Lesson 21

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for fraction multiplication. The whole number 60 is intentionally selected to represent the minutes in an hour.  These understandings help students develop fluency and will be helpful in this lesson when students need to be able to interpret the number of minutes in fractional parts of an hour. 

• 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.

• “What is \(\frac{1}{8}\times60?\) How do you know?” (\(7\frac{1}{2}\) because that is half of 15. \(\frac{1}{8}\) is half of \(\frac{1}{4}\).)

The purpose of this activity is for students to think about different activities they might participate in during free time, to define a smaller set of categories, and to collect data. A significant part of this activity is the definition of the categories, which is an important aspect of mathematical modeling (MP4). Encourage students to convert the time for their activities from minutes to hours. Students may notice that some fractions are more meaningful in the context of time such as half, third, and quarter of an hour. Students will further explore the relationship between time and fractions in the activity synthesis.

If a student sees that a specific activity is not on the poster, have them consider if there is an umbrella activity. For example, “What category, that is on a poster, might talking to your friends go under?”

• Groups of 4
• “What are some things you do or like to do if you have free time on the weekend?” (sleep, play video games, play sports, spend time with family, social media, read, go to the movies)

• “For this activity, you’ll think about how you might spend 2 hours of free time and then we’ll record the time we spend for some of those activities.”
• “As you work, I will write down the different categories of your responses on the posters hanging around the room. You will record your times on the posters.”
• As students work, monitor for 4–6 popular activities and write the titles of these activities on the large pieces of poster paper. If necessary, list some examples of the umbrella category.
• 8–10 minutes: independent work time
• Monitor for students who record using multiples of \(\frac{1}{2}\), \(\frac{1}{4}\), and \(\frac{1}{8}\) of an hour.

• “Look at the data of the different categories around the room. What do you notice?” (Most people recorded using halves and fourths. More students play games than read books. There is more data on one poster than on another. There are more small numbers on one poster, it’s hard to compare because there are so many numbers and they are not in any order.)
• If it doesn’t come up, ask, “Why did most people record time as multiples of \(\frac{1}{2}\) and \(\frac{1}{4}\)?” (Because half of 60 minutes is 30 and half of that is 15. I can easily see that on the clock.)
• “Does it make sense to talk about time using \(\frac{1}{8}\) or\(\frac{1}{7}\) of an hour?” (Yes but it’s not a whole number of minutes like \(\frac{1}{2}\) hour or \(\frac{1}{4}\)  hour.)
• “What are some ways we can organize the data?” (Bar graph, line plot, chart, list the numbers in order.)

The purpose of this activity is for students to make and analyze line plots. In this activity, students analyze the free time data collected in the previous activity. They make observations and comparisons to tell the story of their data set.

• Groups of 4
• Give each group one of the data sets from the previous activity.

• 10 minutes: small-group work time
• Monitor for groups who:
  • label the line plot with fractions such as fourths
  • use operations with fractions
  • come up with their own question to analyze the data

• Invite previously selected groups to tell their line plot and data stories.
• “What other information can we tell about our classmates from the line plots?” (More students like this than that. People would like to spend more time on _____, and less time on _____.)