Lesson 14

This warm-up prompts students to make sense of data and quantities before using them to solve problems, by familiarizing themselves with a context and the mathematics that might be involved. Students may be familiar with shoe sizes but may not recognize that each size is associated with a particular measurement.

• Display the shoe-size chart and diagram of insoles.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• Explain that an “insole” is the inside part of a shoe, underneath the foot. Its length is approximately the length of the foot.
• Check students’ interpretation of the data and diagram:
  • “What information do the table and diagram show?”
  • “If someone's shoe size is 5, what’s the length of the insole?”
  • “What shoe size do you wear? What's the length of the insole?”
• “What do you think the missing lengths might be for sizes 3.5 and 5.5?” (For 3.5, the missing length would be \(8\frac{5}{8}\), as there are no other fractions in eighths between \(8\frac{4}{8}\) and \(8\frac{6}{8}\). For size 5.5, it could be \(9\frac{2}{8}\) or \(9\frac{3}{8}\).)
• “Today we'll look at some data and solve some problems related to shoe lengths. The sizing chart here gives us an idea of where the numbers come from and what they mean.”

This activity allows students to integrate several concepts and skills on data analysis and fraction operations. Students plot fractional measurements on a line plot, interpret the data, and find sums or differences of fractions to solve problems in context (MP2). 

To find the difference between the longest and shortest shoe lengths, students can reason in a number of ways, using visual representations or more abstract reasoning. For example, they may:

• use the tick marks on the line plot and count up by eighths from \(7 \frac{6}{8}\) to \(9\frac{5}{8}\)
• count up from \(7\frac{6}{8}\) to 8, then from 8 to 9, and from 9 to \(9\frac{5}{8}\)
• subtract \(7\frac{6}{8}\) from \(9\frac{5}{8}\) in parts: first subtract \(\frac{5}{8}\), then 7, and then another \(\frac{1}{8}\)
• find \(9\frac{5}{8} - 7\frac{6}{8}\) by first decomposing the 9 into \(8 + \frac{8}{8}\) and then subtracting \(7\frac{6}{8}\) from it

To complete the activity, students will need to make sense of the data and the questions, identify relevant numbers or quantities, and persevere in solving problems (MP1).

• Groups of 2
• “Take a minute to read the opening paragraphs and the first problem of the activity. Afterwards, explain to a partner the directions to the first problem.”
• 1 minute: quiet think time
• 1 minute: partner discussion

• “Take a few quiet minutes to work on the activity. Then, discuss your responses with your partner.”
• 5–6 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for the different ways students find the difference between \(9\frac{5}{8}\) and \(7\frac{6}{8}\) as described in the Activity Narrative.
• Identify students who use different strategies to share during synthesis.

• Invite students to share their completed line plot and their responses.
• Focus the discussion on the last two questions about the differences in the longest and shortest shoe lengths. Select previously identified students to share their reasoning strategies, in the order shown in the Activity Narrative.
• As students explain, record and display their reasoning for all to see.
• “Did you use the same strategy to solve the last problem? How did you find out the shoe length of the student who made an error?” (Sample responses: I started at 9 on the number line and moved by eighths 7 times to the left to land at \(8\frac{1}{8}\). I subtracted \(\frac{7}{8}\) from 9.)

In this activity, students analyze a line plot that is incomplete. They relate the list of given fractions to the data on the line plot and use their understanding of equivalence to determine the missing data points. Students also continue to interpret the data and add and subtract fractions to solve problems in context (MP2). 

• Groups of 2
• “How much do you think your feet have grown in the past year? Have you changed to a larger shoe size since third grade?”
• Invite students to share responses.
• “Let’s look at some problems about the change in shoe lengths from third grade to fourth grade.”

• “Take a few quiet minutes to work on the activity. Afterwards, share your thinking with your partner.”
• 5–7 minutes: independent work time
• 4–5 minutes: partner work
• “Be sure to discuss how you know whose data points are missing from the line plot.”

• Select students to display their completed line plot and to share how they decided which data points didn’t get plotted.
• Highlight that one point that is missing could be Jada, Diego, or Han's, as the fractions that represent their change in shoe length, \(1\frac{1}{4}\), are equivalent.
• Select other students to share their responses to questions about Han and Priya's shoe lengths.
• Then, focus the discussion on the last question about Tyler's error, his actual change in shoe length, and how the corrected value might affect the line plot.