Lesson 14

Problems about Fractional Measurement Data

Warm-up: Notice and Wonder: Shoe Sizes (10 minutes)


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.

Student Facing

What do you notice? What do you wonder?

US youth shoe size insole length in inches
1 \(7\frac{6}{8}\)
1.5 8
2 \(8\frac{1}{8}\)
2.5 \(8\frac{2}{8}\)
3 \(8\frac{4}{8}\)
4 \(8\frac{6}{8}\)
4.5 9
5 \(9\frac{1}{8}\)
6 \(9\frac{4}{8}\)
6.5 \(9\frac{5}{8}\)
7 \(9\frac{6}{8}\)

image of shoe size chart, 7 insole lengths. smallest insole length, 7 and 5 eighths.

Student Response

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Activity Synthesis

  • 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.”

Activity 1: Shoe Lengths (20 minutes)


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).

Representation: Access for Perception. Read directions and the first problem aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Attention, Language


  • 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.

Student Facing

Students in a fourth-grade class collected data on their shoe sizes and lengths. They plotted the shoe lengths on a line plot.

Image of two students' shoes.

Dot plot titled Fourth-grade Shoe Lengths from 7 to 10 by 1’s. Hash marks by eighths. Horizontal axis, length, in inches. Beginning at 7 and 6 eighths, the number of X’s above each eighth increment is 1, 0, 3, 1, 3, 0, 5, 3, 0, 0, 0, 1, 0, 0, 0, 1.

The line plot is missing the shoe lengths of six students:

  • 9
  • \(9\frac{1}{8}\)
  • \(8\frac{6}{8}\)
  • \(7\frac{6}{8}\)
  • \(9 \frac{2}{8}\)
  • \(8\frac{1}{8}\)
  1. Complete the line plot with the missing data.
  2. Use the completed line plot to answer the following questions:

    1. What is the largest shoe length?
    2. What is the smallest shoe length?
    3. What is the difference between the largest and smallest shoe lengths? Explain or show your reasoning.

    4. The student who recorded 9 inches for her shoe length made a mistake when reading the shoe chart. Her actual shoe length is \(\frac{7}{8}\) inches shorter.

      What’s her shoe length? Plot her corrected data point on the line plot.

Student Response

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Activity 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.)

Activity 2: Larger Shoes, Anyone? (15 minutes)


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). 

MLR8 Discussion Supports.Display sentence frames to support small-group discussion: “First, I _____ because . . .”, and “I noticed _____ so I . . . .”
Advances: Conversing, Representing


  • 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.”

Student Facing

Ten students recorded their shoe lengths in third grade and then again in fourth grade.

They found how much their feet have grown over a year and organized the data in a table and on a line plot.

 student   change in shoe length (inches) 
Jada \(\frac{5}{4}\)
Priya \(\frac{7}{8}\)
Andre \(\frac{3}{4}\)
Elena \(\frac{1}{2}\)
Han \(1\frac{2}{8}\)
 student  change in shoe length (inches)
Clare \(1\)
Tyler \(1\frac{1}{8}\)
Kiran \(\frac{6}{8}\)
Diego \(1\frac{1}{4}\)
Lin \(\frac{5}{8}\)
Dot plot titled, How Much Have Our Feet Grown? from 0 to 2 by 1’s. Hash marks by eighths. Horizontal axis, change in shoe length, in inches. Beginning at 4 eighths, the number of X’s above each eighth increment is 1, 0, 2, 1, 1, 0, 2.
  1. The line plot shows only seven points. Whose information is missing? Add the three missing points to the line plot.
  2. If Han’s shoe length now is \(9\frac{1}{8}\) inches, what was his shoe length in third grade?
  3. If Priya’s shoe length was \(7\frac{6}{8}\) inches last year, what’s her shoe length this year?
  4. Tyler made a calculation error. What he recorded, \(1\frac{1}{8}\) inches, was \(\frac{3}{8}\) inches off from the actual change in shoe length.

    1. What could be the actual change in his shoe length? Explain or show your reasoning.
    2. How does his error affect the line plot? Explain your reasoning.

Student Response

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Activity Synthesis

  • 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.

Lesson Synthesis

Lesson Synthesis

“Today we used our understanding of fractions to plot and analyze data on line plots. We also added and subtracted fractions to answer questions about measurement data.”

“How was plotting fractional data in halves, fourths, and eighths on a line plot different from plotting whole numbers?” (With whole-number data, we could just count up or down from the labeled tick marks to know where to put a number. With fractions, sometimes it's necessary to think about equivalent fractions first to know where to put a data point. For example, the number line might be partitioned into fourths, but the data might be in eighths or halves.)

“The problems we saw today involved finding differences of two fractions. Did you find the line plots helpful for subtracting two fractions? Why or why not?” (Sample responses: Yes, because I could use the number line and tick marks to help us count up or down, or to know roughly what the difference would be. No, because I could reason about the difference mentally or figure it out on paper.)

Cool-down: Fourth-grade Height Data (5 minutes)


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Student Section Summary

Student Facing

In this section, we added and subtracted fractions with the same denominator, using number lines to help with our reasoning.

First, we learned that a fraction can be decomposed into a sum of smaller fractions. For example, here are a few ways to write \(\frac{6}{10}\):

\(\frac{6}{10}=\frac{4}{10} + \frac{2}{10}\)

\(\frac{6}{10}=\frac{5}{10} + \frac{1}{10}\)

\(\frac{6}{10}=\frac{2}{10} + \frac{2}{10} + \frac{2}{10}\)

If the fraction is greater than 1, it can be decomposed into a whole number and a fraction less than 1. For instance, we can decompose \(\frac{17}{10}\) and rewrite it as \(1\frac{7}{10}\). A number such as \(1\frac{7}{10}\) is called a mixed number.

\(\frac{10}{10} + \frac{7}{10}\)
\(1 + \frac{7}{10} \)

Later, we decomposed fractions into sums and wrote equivalent fractions to help us add and subtract fractions. For example, to find the value of \(1\frac{2}{5} - \frac{3}{5}\), we can:

  • Decompose \(1\frac{2}{5}\) into \(1 + \frac{2}{5} \) or \( \frac{5}{5} + \frac{2}{5}\), which is \(\frac{7}{5}\).
  • Find the value of \(\frac{7}{5} - \frac{3}{5}\), which is \(\frac{4}{5}\).

Finally, we organized and analyzed measurement data on line plots. The data were lengths measured to the nearest inch, \(\frac{1}{2}\) inch, \(\frac{1}{4}\) inch, and \(\frac{1}{8}\) inch.

Dot plot titled Colored Pencil Data from 0 to 6 by 1’s.

Because the measurements have different denominators, we used equivalent fractions to plot them. Then, we used the line plots and what we know about addition and subtraction of fractions to solve problems about the data.