# Lesson 15

Length Measurements

## Warm-up: Which One Doesn’t Belong: Measurements (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare length measurements given in different units, reminding about the relationships between yards, feet, and inches. In making comparisons, students need to attend to both the meaning of each unit and its relationships to other units.

### Launch

• Groups of 2
• Display the four measurements.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn’t belong?

1. 3 feet

2. $$(3 \times 1)$$ yards

3. $$\left(2 \times 18 \right)$$ inches

4. $$\left(\frac{1}{3}+ \frac{1}{3}+ \frac{1}{3}\right)$$ yard

### Activity Synthesis

• “All four quantities measure lengths. What do you notice about these lengths?” (They are in different units. Three of them are equivalent to 1 yard, and one of them is 3 yards.)
• “If we convert the quantities that are equivalent to 1 yard into feet, what will they all be?” (3 feet)
• “What if we convert them into inches?” (36 inches)

## Activity 1: Frisbee Throws (15 minutes)

### Narrative

In this activity, students analyze length measurements, perform multiplication, and convert distances from yards to feet in order to compare and order them. The quantities in yards involve only whole numbers while those feet involve fractional amounts, to encourage students to convert from the larger unit to the smaller one.

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the strategies have in common?”, “How were they different?”, “Why did the different approaches lead to the same (or different) outcome(s)?”

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Consider asking students if they have played frisbee or another game that involves tossing an object. Alternatively, display a video clip of a frisbee game or a frisbee being tossed, or display a frisbee.
• Display the table. “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Read the opening paragraph and the bulleted information in the activity statement.
• “Let’s find out who the top frisbee throwers are in this group of friends.”
• Display a yardstick and a foot-long ruler, if available.

### Activity

• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who convert all the given distances into feet before finding the missing distances and ordering the measurements.

### Student Facing

Six students were throwing frisbees on field day. Here is some information about each person’s first throw.

student  distance
Han  17 yards
Lin  $$51\frac{1}{2}$$ feet
Clare  $$21 \frac{1}{3}$$ feet
Andre 22 yards 2 feet
Elena
Tyler
• Elena’s frisbee went 3 times as far as Clare’s did.
• Andre’s frisbee went 4 times as far as Tyler’s did.

1. Complete the table with Elena and Tyler’s distances. Explain or show your reasoning.
2. Who are the top 3 throwers for that round?

Find out by listing the students and their distances in feet and in order, from longest to shortest.

rank    student    distance (feet)
1
2
3
4
5
6

### Student Response

Students may be unsure how to find Elena’s throwing distance because it requires multiplying a mixed number by a whole number ($$3 \times21 \frac{1}{3}$$). Ask them if it’d help to consider the whole number (21) and the fraction ($$\frac{1}{3}$$) separately (and multiply them separately).

Students may rewrite $$21\frac{1}{3}$$ as $$\frac{64}{3}$$ and multiply it by 3 to get $$\frac{192}{3}$$, but may not know how this number compares to other distances because of its fraction form. Encourage them to think about what whole number this fraction approximates, or reason using smaller fractions with denominator 3. For instance, ask, “How much is $$\frac{30}{3}$$? $$\frac{60}{3}$$? $$\frac{150}{3}$$?”

### Activity Synthesis

• Display the tables from the activity.
• Select students to complete the first table and share their reasoning.
• Then, select students to share their strategies for putting the distances in order. For each strategy, ask if others in the class reasoned the same way.
• “Tyler’s throw was 17 feet and Han’s was 17 yards. Can we tell who threw the frisbee farther without converting one unit to the other? If so, how? If not, why not?” (Yes. We know that 1 yard is greater than 1 foot, so 17 yards must be greater than 17 feet.)
• “How many times as far as Tyler’s distance was Han’s distance?” (3 times as far) “How do you know?” (A yard is 3 times as long as a foot, so 17 yards is 3 times as long as 17 feet.)

## Activity 2: Stone Towers (20 minutes)

### Narrative

In this activity, students apply their knowledge of multiplicative comparison and ability to convert feet and inches to solve a logic puzzle. They use several given clues to determine the heights of four objects. As they use the clues to reason about the heights of the towers and who built them, students reason abstractly and quantitatively (MP2).

Action and Expression: Internalize Executive Functions. Provide students with access to sticky notes (of three different colors if available). Invite students to organize these sticky notes into a table, using the first color to record each person’s name, the second color to record clues about the height of each person’s tower, and the third color to record the actual height of each person’s tower. For students who need additional support, ask “Whose stone tower is the shortest? How tall is that tower?” and “Whose tower is 39 inches tall?” Invite students to explore these questions using the process of elimination. Offer the sentence frame, “It can’t be ____, because ____.”
Supports accessibility for: Conceptual Processing, Memory, Organization

### Launch

• Groups of 4

MLR5 Co-Craft Questions

• Display only the opening paragraph.
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “What do these questions have in common? How are they different?”

### Activity

• “Work with your group to complete the first problem. Then, work on the last problem on your own before discussing it with your group.”
• 8–10 minutes: group work time
• 3–5 minutes: independent work time

### Student Facing

While on an outing, a group of friends had a stone-stacking contest to see who could build the tallest stone tower.

• Andre’s stone tower is 3 times as tall as Diego’s, but Diego didn’t build the shortest tower.
• The tallest tower is 4 feet and 2 inches tall and belongs to Tyler.
• One person built a tower that is 39 inches tall.
• Tyler’s tower is 5 times as tall as the shortest tower.
1. How tall is each person’s stone tower? Be prepared to explain or show your reasoning.

person   tower height (inches)
Andre
Tyler
Clare
Diego
2. Elena came along and built a tower that is 5 times as tall as Diego’s tower. Is Elena’s tower more than 6 feet? Show your reasoning.

### Student Response

Students may conclude that Andre’s tower is 117 inches (or $$39 \times 3$$), not realizing that this contradicts the second clue about Tyler having the tallest tower. Encourage students to check that their responses satisfy all the clues.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Invite students to share how they reasoned about the height of each stone tower. Ask others if they reached the same conclusions but reasoned differently, or if they reached different conclusions.

“One clue says that Tyler’s tower is 5 times as tall as the shortest tower. We know that Tyler’s tower is 4 feet 2 inches. Is it easier to find the height of the shortest tower using 4 feet 2 inches or using 50 inches? Why?” (The first uses two different units, so we’d have to divide 4 feet by 5 and 2 inches by 5, or think about what number multiplied by 5 gives 4 and 2. If we use inches, we’re dealing with one number that is clearly a multiple of 5.)

“How did you decide whether Elena’s tower is greater than 6 feet? Did you convert the 6 feet into inches, convert Diego’s tower into feet, or another way?”

Record strategies. Highlight that while it is often helpful to express a larger unit in terms of a small unit, some problems can be reasoned without doing so.