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 some planes (SK51 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.

If students rely on intuition or make estimates that do not attend to place value to decide if the planes’ altitudes are “about 30,000 feet,” consider asking:

• “¿Cómo sabes cuáles aviones están volando a aproximadamente 30,000 pies?” // “How do you know which planes are flying at about 30,000 feet? ”
• “¿Qué estrategias puedes usar para asegurarte de que encontraste todos y únicamente los aviones que están volando a aproximadamente 30,000 pies?” // “What strategies can you use to help you make sure that you found all of and only the planes that are flying at about 30,000 feet?”

• 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 ten-thousand.
  • Excluded numbers less than 20,000 and more than 35,000, and then rounded numbers to the nearest thousand.)
• 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).

• 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 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 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).

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