Lesson 17

This warm-up prompts students to make sense of a problem before solving it, by familiarizing themselves with a context and the mathematics that might be involved. This warm-up gives students a chance to analyze and ask questions about the set of data they will use in a later activity.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• Explain that altitude is the distance of an object from sea level. Most commercial planes that carry passengers fly at an altitude between 33,000 and 41,000 feet. An altitude of 35,000 feet (7 miles) is typical. Lighter airplanes tend to fly at lower altitudes, around 10,000 feet.
• “¿Cuáles de estos aviones podrían ser aeronaves pequeñas? ¿Cuáles podrían ser aviones más grandes de pasajeros?” // “Which of these airplanes might be smaller aircrafts? Which might be larger passenger planes?”

In this activity, students make sense of a situation and decide how to round the quantities in it. They see that their interpretation of the problems and their rounding decisions affect their solutions to the problems. When students describe how they see their rounded quantities in relation to the context, they are thinking abstractly and quantitatively (MP2).

For instance, when answering the first question, students may say that the altitudes of several planes (SK51, AB25, and WN90) are not “about 30,000 feet” because when rounded to the nearest thousand, they round to different numbers. They may consider them differently when they are rounded to the nearest ten-thousand.

The second question prompts students to start considering the implications of using rounded values to solve problems. At this point, it is not necessary for students to clearly articulate why Mai’s suggestion of using rounded altitudes is not reliable for keeping a safe distance between planes. In the next activity, students will look more closely at the implications of rounding in the same context.

• Groups of 2–4
• “¿Se han preguntado cuántos aviones están en el aire en un momento dado? ¿Cuál sería su estimación?” // “Have you wondered how many planes are in the air at any given time? What would be your estimate?”
• 30 seconds: Share estimate with a partner.
• Share and record responses.
• “Una fuente de datos reportó que en 2017 el número de aviones que estuvieron en el cielo al mismo tiempo varió desde aproximadamente 3,300 (en el momento menos transitado) ¡hasta más de 12,000 (en horas pico)!” // “One data source reported that, in 2017, the number of planes that are in the sky at the same time ranged from about 3,300 (when it is the least busy) to over 12,000 (at peak times)!”
• “Con esa cantidad de aviones volando al tiempo, es extremadamente importante que se mantengan a distancias seguras, especialmente cuando están cerca de aeropuertos concurridos” // “With that many planes in flight at once, it is extremely important for planes to keep a safe distance from one another, especially around busy airports.”

• “Completen la actividad con su grupo” // “Work with your group to complete the activity.”
• 8–10 minutes: group work time
• Monitor for the different ways students decide whether a number is “about 30,000” and test the validity of Mai’s strategy.

Students may rely on intuition or may not use a consistent strategy to decide if the planes’ altitudes are “about 30,000 feet.” For instance, they might say, “All the numbers in the 30 or 31 thousands look close enough to 30,000.” Consider asking: “¿Qué cuenta como ‘lo suficientemente cerca’?” // “What counts as ‘close enough’?” and “¿33,975 o 28,999 están lo suficientemente cerca?” // “Is 33,975 or 28,999 close enough?” Encourage them to think of a more consistent way to decide.

• Invite students to share their responses to the first question. Discuss reasons for any disparity in students’ lists.
• “¿Cómo decidieron cuáles números incluir en su lista de ‘aproximadamente 30,000 pies’ y cuáles excluir?” // “How did you decide which numbers to include in your list of ‘about 30,000 feet’ and which to exclude?” (Sample responses:
  • Rounded all numbers to the nearest 10,000.
  • Excluded numbers less than 20,000 and more than 35,000, and then rounded numbers in the rest to the nearest 1,000.)
• Point out that students may answer the question differently depending on how they round the numbers.
• “¿Alguien puede dar un ejemplo que muestre que la estrategia de Mai funciona? ¿Y qué tal uno que muestre que no funciona?” // “Who can give an example that shows that Mai’s strategy works? How about one that shows it doesn’t work?”

In this activity, students continue to consider rounding in the same context as in the first activity. Students think about why rounding the altitudes to the nearest 1,000 may make it appear that two planes are a safe distance apart while the exact altitudes may show otherwise.

As they consider different ways and consequences of rounding in this situation, students practice reasoning quantitatively and abstractly (MP2) and engage in aspects of mathematical modeling (MP4).

• Groups of 2–4
• “¿Cómo creen que los pilotos saben si su avión está demasiado cerca de otro avión cuando está volando?” // “How do you think pilots know whether their plane is too close to another plane while in the air?”
• 30 seconds: quiet think time
• Share and record responses.
• Explain air traffic controllers are a group of people whose job is to monitor air traffic, including to track the positions of all the planes and the distances between them.
• Consider showing an image of an air traffic control room and controllers.

• “En silencio, trabajen unos minutos en los primeros tres problemas. Después, compartan sus respuestas con su grupo y trabajen juntos en el último problema” // “Take a few quiet minutes to work on the first three problems. Then, share your responses with your group and work on the last problem together.”
• 5 minutes: independent work time
• 5 minutes: group work time

• Select a previously identified student to share their symbol- or color-coded table, or display the table in the Student Responses.
• Invite the class to share their responses to the question of which set of data air traffic controllers should use.
• Discuss students’ ideas on whether there were better ways to round the altitudes or to round at all.
• Explain that air traffic controllers in fact rely on technology and computers to calculate exact distances between planes.

This optional activity offers students another opportunity to round numbers and solve new problems in the context of airplane altitudes. They analyze some statements made about the quantities in the given situation and consider how rounding might have led to those conclusions. Along the way, students practice constructing logical arguments and critiquing those of others (MP3). They also engage in aspects of mathematical modeling (MP4).

• Groups of 2–4
• Explain to students that, in the United States, air passengers are not permitted to use their cell phones from take-off until landing because phone signals can interfere with flight communication signals. However, this hasn't always been true. There had been a time when phone use was allowed after the plane reached a certain altitude. Some countries still allow phone use based on the altitude of the plane.
• Display the table.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 1 minute: partner discussion

• “Tómense unos minutos para trabajar en silencio. Luego, compartan con su grupo cómo pensaron” // “Take a few quiet minutes to work on the task. Then, share your thinking with your group.”
• 5–7 minutes: independent work time
• 5 minutes: group discussion
• Monitor for students who agree with different characters because they round the altitudes to different place value units.

• Select previously identified students to share their responses and reasoning.