# Lesson 5

Multi-step Conversion Problems: Metric Length

## Warm-up: True or False: Powers of 10 (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate strategies and understandings they have for multiplying and dividing by powers of 10. They will use these operations when they convert measurements between different metric length units.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$5,\!423 \times 10 = 50,\!423$$
• $$5,\!423 \div 10 = 542.3$$
• $$5,\!423 \div 100 = 54.23$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How did you decide if the equation $$5,\!423 \div 100 = 54.23$$ is true?” (It's like dividing by 10 twice. $$5,\!423 \div 10 = 542.3$$ and $$542.3 \div 10 = 54.23$$. Each number has digits 5, 4, 2, 3 in the same order. The values of the digits' places in 54.23 are $$\frac{1}{100}$$ the values of the corresponding digits' places in 5,423.)

## Activity 1: Walk All Day (15 minutes)

### Narrative

The purpose of this activity is for students to solve multi-step distance problems using centimeters, meters, and kilometers. This gives students an opportunity to think about which units are most helpful for communicating a distance (MP6). When the distance is short, like the length of a single footstep, centimeters or meters both work well. For a longer distance, like the distance a person walks in a day, it is reasonable to use meters or kilometers, but the number of centimeters is very large and more difficult to visualize.

To add movement or make this activity interactive, consider providing groups of 2 or 4 with a centimeter ruler or meter stick to measure their or a classmate’s step before working on the task. While students could use the measurements of their own steps to complete the table, the arithmetic may be more complex and, as a result, it may be harder to observe patterns.

Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Invite students to share the places they could walk in the classroom, at home, in the community, and so on, that would be equivalent to the distances presented in the chart.
Supports accessibility for: Conceptual Processing; Visual-Spatial Processing

### Required Materials

Materials to Gather

### Launch

• Groups of 2 or 4
• “If someone wants to measure the length of one of their footsteps, which unit do you think they should use? Millimeters, centimeters, meters, or kilometers?”
• Highlight that millimeters are too small and kilometers are too large. Both centimeters and meters make sense and the first part of this activity will use those two units of measure.
• “There are 100 centimeters in a meter. About how many steps are in 1 meter?” (2 or 3)
• “There are 1,000 meters in a kilometer. About how many steps are in a kilometer?” (2,000 or 3,000)

### Activity

• 5 minutes: independent work time
• 5 minutes: small-group work time
• Monitor for students who use the following strategies when determining how many kilometers Lin walks during the day:
• multiply 50 centimeters by 8,500 steps to determine the distance in centimeters that Lin walked and divide 425,000 by 100,000
• multiply 0.5 meters by 8,500 steps to determine the distance in meters that Lin walked and then divide 4,250 by 1,000

### Student Facing

Lin has a watch that counts the number of steps she takes during the day and displays those steps in centimeters, meters, or kilometers.

1. Here is a list of activities Lin did on Monday. Next to each activity, write whether it would make sense to display the distance in cm, m, or km.

• walked to her friend’s desk
• walked to the front of the classroom
• walked from her classroom to the bus
• ran twice around the playground
2. The table shows the amount of steps Lin’s watch displayed for each activity. If each of her steps is 50 centimeters, how many centimeters and meters did Lin walk for each activity?
activity number of steps distance (cm) distance (m)
walked to her friend’s desk 5
walked to the front of the classroom 12
walked from her classroom to the bus 250
ran twice around the playground 1,000
3. At the end of the day, Lin’s watch displayed 8,500 steps. Would it make sense for her watch to record the distance in centimeters, meters, or kilometers? Why?
4. How many kilometers did Lin walk that day?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How did you determine how many kilometers Lin walked during the day?”
• Ask previously selected students to share their solutions.
• Display student work or write these equations for all to see:
• $$8,\!500 \times 50= 425,\!000$$
• $$8,\!500 \times 0.5 = 4,\!250$$
• “How does each of these equations represent the situation?” ($$8,\!500 \times 50= 425,\!000$$ represents 8,500 steps that are each 50 centimeters long so that would be a total of 425,000 centimeters.$$8,\!500 \times 0.5 = 4,\!250$$ represents 8,500 steps that are each 0.5 meter long so that would be a total of 4,250 meters.)
• “What is the same about these equations?” (They have the same number of steps. They are both multiplication equations.)
• “What is different about these equations?” (One multiplies the number of steps by 50 centimeters and one multiplies the number of steps by 0.5 meter. The products are different. 4,250 is 100 times smaller than 425,000 because it represents meters instead of centimeters.)
• Display:
• $$425,\!000 \div 100,\!000= 4.25$$
• $$4,\!250 \div 1,\!000 = 4.25$$
• “How does each of these equations represent the situation?” (They both represent the number of kilometers that Lin walked. 425,000 is the distance she walked in centimeters so if you divide it by 100,000, you get the number of kilometers she walked. 4,250 is the distance in meters that she walked so we only have to divide by 1,000 to figure out how many kilometers she walked.)
• Display: 4.25 km, 4,250 m, 425,000 cm
• “Which of these do you think best communicates how far Lin walked this day?” (4.25 kilometers because I can picture how long a kilometer is.)

## Activity 2: Who Ran Farther? (20 minutes)

### Narrative

The purpose of this activity is for students to convert between meters and kilometers to decide which of two measurements is larger. Monitor for students who convert from kilometers to meters, which will give two large whole-number measurements, and for students who convert from meters to kilometers, which will give two decimal numbers. The goal of the activity synthesis is to connect these two different solution strategies.

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to who ran farther, Tyler or Clare. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: partner discussion
• For the third problem, monitor for students who:
• convert from kilometers to meters to find the difference between Clare and Tyler’s runs
• convert from meters to kilometers to find the difference between Clare and Tyler’s runs

### Student Facing

1. Use the table to find the total distance Tyler ran during the week. Explain or show your reasoning.

day distance (km)
Monday 8.5
Tuesday 6.25
Wednesday 10.3
Thursday 5.75
Friday 9.25

2. Use the table to find the total distance Clare ran during the week. Show your reasoning.
day distance (m)
Monday 5,400
Tuesday 7,500
Wednesday 8,250
Thursday 6,750
Friday 7,250

3. Who ran farther, Clare or Tyler? How much farther? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

If a student does not realize that they have to convert the units to compare them, refer to the letters next to the word “distance” in the table headers and ask the student to explain what they mean.

### Activity Synthesis

• Invite students to share strategies for how they added the two sets of numbers (that is, looking for “friendly” pairs of numbers to combine first or adding by place value).
• Invite a student who converted from kilometers to meters to share their solution to the last problem.
• “What worked well in this solution?” (All of the numbers are whole numbers.)
• “What was difficult?” (The numbers were all big.)
• Invite a student who converted from meters to kilometers to share.
• “What worked well?” (The numbers were a good size to visualize.)
• “What was difficult?” (I needed to add and subtract decimals to find out how much farther Tyler ran than Clare.)

## Lesson Synthesis

### Lesson Synthesis

“Today we solved problems and converted length measurements in metric units. We solved problems about how far students walked or ran.”

“How far do you think you walk in a day?” (2 or 3 kilometers because I walk to and from school each day and I think that’s a kilometer and then I run around on the playground a lot during recess. 5 kilometers because I walk to and from school every day and I also usually take my dog out for a walk once or twice a day.)

Consider giving students time to respond in their journals.

Collect some responses and ask students to explain how they know their answers are reasonable.

## Cool-down: Compare Lengths (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.