Lesson 7

Multi-step Conversion Problems: Customary Length

Warm-up: Number Talk: Multiples of 12 (10 minutes)

Narrative

This warm-up helps students develop strategies to find multiples of 12 mentally using place value strategies and the distributive property. This prepares students for converting measurements in feet to measurements in inches.

Launch

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

Activity

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

Student Facing

Find the value of each expression mentally.

  • \(45 \times 10\)
  • \(45 \times 2\)
  • \(45 \times 12\)
  • \(46 \times 12\)

Student Response

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Activity Synthesis

  • “How are the values of the products \(45 \times 12\) and \(46 \times 12\) related?” (There is one more 12 in \(46 \times 12\).)

Activity 1: Card Sort: Customary Measurements (15 minutes)

Narrative

The goal of this activity is to compare measurements in the customary length units of inches, feet, and yards. Students first sort the measurements in a way that makes sense to them. Monitor for students who:
  • sort by the unit of measure
  • sort by the way the quantity is written (whole number, mixed number, fraction)

Then, students find the equivalent lengths and list the sets of equivalent lengths in increasing order. Four different lengths have been chosen and each one is presented in inches, feet, and yards. The activity synthesis highlights why expressing all the measurements using one unit is a convenient way to identify common measures and list them in increasing order. Give students access to yard sticks. 

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing, Representing
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual Processing; Memory

Required Materials

Materials to Gather

Materials to Copy

  • Customary Measurement Card Sort

Required Preparation

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

Launch

  • Groups of 2
  • Give each group of students one set of pre-cut cards.
  • Display a yardstick.
  • “What do you notice? What do you wonder?” (It shows feet and inches. It shows 36 inches. I wonder if it’s the same length as a meterstick.)

Activity

  • “In this activity, you will sort some cards into categories of your choosing. When you sort the measurements, you should work with your partner to come up with categories.”
  • 4 minutes: partner work time
  • Select groups to share their categories and how they sorted their cards.
  • Choose as many different types of categories as time allows, but ensure that one set of categories identifies the way the quantity is written (whole number, mixed number, fraction).
  • “Now work with your partner to match the cards with equal measurements. Then, list the groups of matching measurements in increasing order.
  • 3 minutes: partner work time

Student Facing

  1. Your teacher will give you a set of cards that show different measurements. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories.

    (Pause for teacher directions.)

  2. Match the cards with equal measurements. Then, list the groups of matching measurements in increasing order.

Student Response

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Activity Synthesis

  • Invite students to share the matches they made and how they know those cards go together.
  • Attend to the language that students use to describe their matches, giving them opportunities to describe how they know the measurements are equal.
  • Highlight the use of phrases, such as:
    • There are 12 inches in one foot.
    • There are 3 feet in one yard.
    • To find (a fraction) of a foot, I multiplied 12 by the fraction.
  • “How did you compare the sets of measurements? Why?” (I chose one of the units, feet, and compared all of the measurements in that unit.)
  • “Why was it important for all of the measurements to be in the same unit in order to compare?” (That way I can just compare the numbers because they all have the same unit of measure.)

Activity 2: Run a Mile or Two (20 minutes)

Narrative

The goal of this activity is to solve multi-step conversion problems using customary length units. The given information for both problems is in fraction form. Students need to find the perimeter of the different fields and then investigate how many times around each field is a given number of miles. The first problem involves 3 steps:
  • find the perimeter of the rectangular field
  • find the total distance of 6 laps around the field
  • convert from yards to feet or feet to yards

The second problem has only two steps, but the number of laps is unknown and students need to find how many laps make at least 2 miles.

When students critically analyze Priya's claim that six laps of the soccer field is more than a mile, they critique the reasoning of others (MP3).

Required Preparation

  • Before the lesson, figure out a location that students would be familiar with that is about 1 mile away from the school. You will share this location in the launch to help students understand how far 1 mile is.

Launch

  • “About how far is a mile?”
  • 1–2 minutes: quiet think time
  • Record responses for all to see.
  • Describe to students a location that is about a mile away from the school.
  • “About how many feet are in a mile?”
  • “What is an estimate that is too low? Too high? About right?”
  • Record student responses in a table like this one:
    too low about right too high
  • Display: There are 5,280 feet in one mile.
  • Leave the display up throughout the lesson.
  • “We are going to solve some problems about miles.”

Activity

  • 7-10 minutes: partner work time
  • Monitor for students who:
    • find the number of yards in a mile
    • convert yards to feet for the first problem

Student Facing

  1. A rectangular field is 90 yards long and \(42\frac{1}{4}\) yards wide. Priya says that 6 laps around the field is more than a mile. Do you agree with Priya?

    Explain or show your reasoning.

    image of a field.
  2. A different rectangular field is \(408\frac{1}{2}\) feet long and \(240\frac{1}{4}\) feet wide. How many laps around this field would Priya need to run if she wants to run at least 2 miles?

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students don’t have a strategy to solve the first problem, consider asking:

  • “Can you represent a person running one lap around the field.”
  • “How can you figure out the distance of one lap around the field?”

Activity Synthesis

  • Invite students to share their solutions to the first problem.
  • “How did you find how far 1 lap around the field is?” (I multiplied the length and width by 2 and added them.)
  • “How far is one lap?” (\(264\frac{1}{2}\) yards)
  • “How far is 6 laps? How do you know?” (1,587 yards, I multiplied \(264\frac{1}{2}\) by 6.)
  • “How did you find the product \(6 \times 264\frac{1}{2}\)?” (I multiplied 264 by 6. I know that \(6 \times \frac{1}{2}\) is 3 so I added that.)
  • “Is 6 laps more or less than a mile?” (Less, because it’s less than 5,000 feet. Less, because a mile is more than 1,700 yards.)

Lesson Synthesis

Lesson Synthesis

“Today we converted between distances in customary units.”

Display: 10 feet

“How many inches are in 10 feet? How many yards?” (120 inches, \(3 \frac{1}{3}\) yards)

Display: 10 meters

“How many centimeters are in 10 meters? How many kilometers?” (1,000 centimeters, 0.01 kilometer)

“How is converting between metric length units the same as converting between customary length units?” (In each case I multiply by a number when going from a bigger unit to a smaller unit and I divide by a number when going from a bigger unit to a smaller unit.)

“How is converting between metric length units different than converting between customary length units?” (When we convert in metric units we multiply or divide by a power of 10 so the digits in the measurement stay the same. In customary units the division or multiplication is not by a power of 10 so it takes more work, the digits change, and I may need to use fractions instead of decimals.)

Cool-down: Whiteboard Width (5 minutes)

Cool-Down

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Student Section Summary

Student Facing

In this section we studied powers of 10 and conversions between units. We learned that we can write a product of 10s like \(\displaystyle 10 \times 10 \times 10 \times 10\) as \(10^4\). The number 4 is an exponent and it means that there are 4 factors of 10.

We also converted between different measurement units, mostly metric lengths. For example, there are 1,000 millimeters in a meter and 1,000 meters in a kilometer. This means that there are \(1,\!000 \times 1,\!000\) or \(1,\!000,\!000\) millimeters in a kilometer. We could also say that there are \(10^6\) millimeters in a kilometer. We also used our understanding of decimals to make conversions. For example, since there are 1,000 meters in a kilometer that means that each meter is \(\frac{1}{1,000}\) or 0.001 kilometers. So 853 meters can also be written as 0.853 kilometers.