Lesson 5

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.

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

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

• “¿Cómo decidieron si la ecuación \(5,\!423 \div 100 = 54.23\) es verdadera?” // “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.)

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.

• Groups of 2 or 4
• “Si alguien quisiera medir la longitud de uno de sus pasos, ¿qué unidad creen que debería usar: milímetros, centímetros, metros o kilómetros?” // “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.
• “Hay 100 centímetros en un metro. ¿Aproximadamente cuántos pasos hay en 1 metro?” // “There are 100 centimeters in a meter. About how many steps are in 1 meter?” (2 or 3)
• “Hay 1,000 metros en un kilómetro. ¿Aproximadamente cuántos pasos hay en 1 kilómetro?” // “There are 1,000 meters in a kilometer. About how many steps are in a kilometer?” (2,000 or 3,000)
• Give students access to a meter stick.

• 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

• “¿Cómo decidieron cuántos kilómetros caminó Lin durante el día?” // “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:
  • \(8,\!500 \times 50= 425,\!000\)
  • \(8,\!500 \times 0.5 = 4,\!250\)
• “¿Cómo está representada la situación en cada ecuación?” // “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.)
• “¿En qué se parecen estas ecuaciones?” // “What is the same about these equations?” (They have the same number of steps. They are both multiplication equations.)
• “¿En qué son diferentes estas ecuaciones?” // “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:
  • \(425,\!000 \div 100,\!000= 4.25\)
  • \(4,\!250 \div 1,\!000 = 4.25\)
• “¿Cómo está representada la situación en cada ecuación?” // “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
• “¿Cuál de estas creen que comunica mejor cuánta distancia caminó Lin este día?” // “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.)

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.

• Groups of 2

• 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

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.

• 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.
• “¿Qué funcionó bien en esta solución?” // “What worked well in this solution?” (All of the numbers are whole numbers.)
• “¿Qué fue difícil?” // “What was difficult?” (The numbers were all big.)
• Invite a student who converted from meters to kilometers to share.
• “¿Qué funcionó bien?” // “What worked well?” (The numbers were a good size to visualize.)
• “¿Qué fue difícil?” // “What was difficult?” (I needed to add and subtract decimals to find out how much farther Tyler ran than Clare.)