Lesson 16

This Number Talk encourages students to think about the distance of a number to a multiple of 100, 1,000, and 10,000 by relying on the structure of numbers in base-ten to mentally find differences (MP7). The understandings elicited here will be helpful later in the lesson when students round multi-digit whole numbers. It may be helpful to record students' reasoning on number lines.

• Display one equation.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep equations and work displayed.
• Repeat with each equation.

• “¿Qué tienen en común las cuatro ecuaciones?” // “What do all four equations have in common?” (They are about finding how far away a number is to a multiple of 100, 1,000, or 10,000. The number on the left side of the equation is 421 or ends with 421.)
• “¿500 es el múltiplo de 100 más cercano a 421? ¿Cómo lo saben?” // “Is 500 the nearest multiple of 100 for 421? How do you know?” (No, 421 is closer to 400. It is 21 away from 400 and 79 from 500.)
• “¿7,000 es el múltiplo de 1,000 más cercano a 6,421?” // “Is 7,000 the nearest multiple of 1,000 for 6,421?” (No, 6,421 is closer to 6,000.)
• “¿10,000 es el múltiplo de 10,000 más cercano a 6,421?” // “Is 10,000 the nearest multiple of 10,000 for 6,421?” (Yes, 10,000 is less than 5,000 away from 6,421.)

In this activity, students connect the idea of “nearest multiple” to rounding. They are reminded that to round to the nearest 1,000, 10,000, or 100,000 is to find the nearest multiples of these values. When they find all of the numbers that round to a given number, students need to think carefully about place value and may choose to use a number line to support their reasoning (MP5).

• Groups of 2
• “¿Qué saben sobre redondear?” // “What do you know about rounding?”
• 1 minute: quiet think time
• Share and record responses.
• Highlight responses that share times when students need to round numbers in their life.
• “¿Cuánto es 112 redondeado a la decena más cercana?” // “What is 112 rounded to the nearest 10?” (110) “¿A la centena más cercana?” // “To the nearest 100?” (100)
• “Redondeemos algunos números más grandes” // “Let’s round some larger numbers.”

• “Completen la actividad con su pareja” // “Work with your partner to complete the activity.”
• 10 minutes: partner work time
• Monitor for students who consider all numbers that could be rounded to 490,000, to 500,000, and to both by:
  • drawing a number line
  • reasoning numerically

Students may be able to list some numbers that can be rounded to 500,000 (or 490,000) but may be unsure how to describe all of them. Encourage them to consider the upper and lower limits of the possible numbers by asking:

• “¿Qué números no se pueden redondear a 500,000 por ser demasiado pequeños?” // “Which numbers cannot be rounded to 500,000 because they are too low?”
• “¿Qué números no se pueden redondear a 500,000 por ser demasiado grandes?” // “Which numbers cannot be rounded to 500,000 because they are too high?”

• Display the number line from the activity.
• Select students to share their responses and reasoning for the first three problems. Record their responses.
• If no students mentioned using number lines to help them identify all the possible numbers that round to 500,000 and 490,000, ask them to try showing the ranges on the number line.
• Display students’ annotated number lines or the following:

  

  

  

  

• “¿Cómo encontraron los números que se pueden redondear tanto a 500,000 como a 490,000?” // “How did you find numbers that can be rounded to both 500,000 and 490,000?” (I chose numbers that are between 485,000 and 494,999, since all of them can be rounded to 500,000. Numbers outside of that interval cannot be rounded to 490,000.)

In this activity, students round numbers to various place values. Here they encounter for the first time a number that rounds to 1,000,000 and some that round to 0. (For example, 4,896, rounded to the nearest 100,000 is 0.) Students may wonder why we might round a number in the thousands to the nearest 100,000. Make note of such ideas to discuss in the next lesson where students explore rounding in context and see that it often involves giving meaningful information.

• Groups of 2
• Display the table.
• “Con su pareja, escojan por lo menos tres números: uno de cuatro dígitos, uno de cinco dígitos y uno de seis dígitos. Redondéenlos a los múltiplos más cercanos de los valores que están en la tabla” // “With your partner, choose at least three numbers—one with four digits, one with five digits, and one with six digits. Round them to the nearest values in the table.”

• “Trabajen con su pareja en los dos primeros números y de forma individual al menos en uno de los otros. Prepárense para explicar o mostrar cómo pensaron” // “Work with your partner on the first two numbers and independently on at least one of them. Be prepared to explain or show your thinking.”
• 6–8 minutes: partner work

• Display the blank table from the activity. Invite students to share their responses to complete the table. Discuss any disagreements.
• “El número 96,500 está a la misma distancia de 96,000 que de 97,000. ¿Cómo sabemos cuál escoger si queremos redondearlo al múltiplo de mil más cercano?” // “The number 96,500 is the same distance from 96,000 and 97,000. How do we know which way to go if rounding to the nearest thousand?” (By convention, we round up.)

In this optional activity, students round multi-digit whole numbers in the context of population. They describe the effect of rounding large numbers to different places and how the rounded values may illuminate or obscure a situation, and may help or hinder problem solving. Along the way, students engage in aspects of mathematical modeling (MP4).

• Groups of 2–4
• Display this table:

  

  

• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• Share responses.
• “¿En qué ocasiones podría ser útil tener la población exacta? ¿En qué ocasiones podría ser útil tener los números redondeados?” // “When might it be helpful to have the actual population? When might it be helpful to have the rounded numbers?” (Sample responses:
  • Rounding to the nearest million doesn’t quite make sense. It portrays the population either as 0—for Oakland—or as almost twice as large—for Mesa.)
  • Counting every single person may not be possible.
  • Rounding to the nearest 100,000 or 10,000 seems appropriate.)
• “Redondeemos otras poblaciones y veamos si aún pensamos de la misma forma” // “Let’s round some other populations and see if we still think the same way.”

• “Asocien las poblaciones de Charlotte, Jacksonville y Virginia Beach con los números redondeados que están en la tabla. Después, completen la tabla” // “Match the populations of Charlotte, Jacksonville, and Virginia Beach to the rounded numbers in the table. Then, complete the table.”
• “Completen la actividad en grupo. Prepárense para explicar cómo hicieron para asociar las ciudades con las poblaciones” // “Work with your group to complete the activity. Be prepared to explain how you make your matches.”
• 6–8 minutes: group work time
• Monitor for the ways students reason about the given populations and the rounded numbers.

• Select groups to share the matches they made and their reasoning.
• Display the completed table.
• Invite students to briefly share their responses to the last problem, using the table to aid their explanations.
• Point out that, in reality, it is unlikely to accurately count the exact number of people in a state. Highlight that rounding can be a useful way to get a sense of a quantity, but it requires making some decisions about what is most meaningful in a situation.