Lesson 15

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In this case, students rely on their familiarity with number lines and their understanding of numbers within 1,000 to estimate the value represented by a point on a number line. The reasoning here prepares students to think about the halfway point between two benchmark values as a way to estimate numbers.

• Groups of 2
• Display image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• Consider asking: “¿Alguien hizo una estimación menor que 500? ¿Alguien hizo una estimación mayor que 800?” // “Is anyone’s estimate less than 500? Is anyone’s estimate greater than 800?”
• “¿Cómo lograron hacer una estimación, si en la recta solo se muestran 0 y 1,000?” // “How did you go about making an estimate when only 0 and 1,000 are shown on the line?”
• If no students mentioned using the halfway point or 500 as a reference, ask them about it.
• “¿Cómo les ayudó el punto medio, que representa 500, a hacer una estimación?” // “How did the midpoint for 500 help you make an estimate?”
• Consider asking: “¿Alguien partió el espacio que hay entre 500 y 1,000 en espacios más pequeños? ¿Cómo les ayudó eso a hacer la estimación?” // “Did anyone partition the space between 500 to 1,000 into smaller spaces? How did that help you estimate?”

This activity transitions students from reasoning visually to reasoning numerically about the nearest multiples of 1,000, 10,000, and 100,000. Students identify the nearest multiples of 10, 100, 1,000, 10,000 and 100,000 for a series of related numbers—16, 816, 3,816, 73,816, and 573,816—and use number lines to support their thinking as needed. Tables are used to highlight the idea that a given number can be closest to a smaller number or a greater number depending on the place attended to. For example, for 816, the nearest multiple of 10 is 820 and the nearest multiple of 100 is 800.

For students, rounding to the unit in the leftmost place is not usually an issue, but rounding to the unit represented by a place in the middle of a number often is, as the nearby digits can be distracting. (For example, rounding 573,816 to the nearest 1,000 is more difficult than rounding to the nearest 100,000.) This activity allows students to work with a set of related numbers that grows by an additional digit each time, and gives them a way to think of a large number as composed of smaller place-value parts, each of which they can manage to round.

• Groups of 2

• “Trabajen individualmente en la actividad durante unos minutos. Después, compartan sus respuestas con su compañero” // “Work on the activity independently for a few minutes. Then, share your responses with your partner.”
• 6–7 minutes: independent work time
• 3–4 minutes: partner discussion

• Display the blank table from the activity. Invite students to share their responses to complete the table. Discuss any disagreements.
• “¿Cómo podríamos saber cuál es el múltiplo de 1,000 más cercano a 573,816 sin usar una recta numérica?” // “How might we tell the nearest multiple of 1,000 for 573,816 without using a number line?” (Sample responses:
  • Find two multiples of 1,000 that are closest to the number, one that is greater and one that is less. For 573,816, they are 573,000 and 574,000. If the number is less than their midpoint, 573,500, then it is closer to the lower multiple of 1,000. If it is more than 573,500, then it is closer to the higher one.
  • Look at the value of the digits to the right of the thousands—the hundreds, tens, and ones. If it is less than 500, then it is closer to the lower multiple of 1,000. If it is more than 500, then it is closer to the higher multiple of 1,000.)

This optional activity gives students another opportunity to practice identifying multiples of some powers of 10 that border a given number and identify the nearest ones. As before, students may use number lines to support their reasoning, but here the number lines are unlabeled.

• Groups of 2

• “Trabajen individualmente durante unos minutos. Después, discutan sus respuestas con su compañero” // “Work independently for a few minutes. Then, discuss your responses with your partner.”
• 5 minutes: independent work time
• 2 minutes: partner discussion

• Invite students to share their responses and reasoning.
• “¿Cuáles múltiplos más cercanos fueron relativamente fáciles de identificar?” // “Which nearest multiples were you able to identify fairly easily?”
• “¿Cuáles fueron un poco más retadores, si los hubo? ¿Por qué fueron retadores?” // “Which were a bit more challenging, if any? Why might that be?”

In this activity, students encounter numbers that are exactly between two consecutive multiples and thus are closer to neither multiple (or have two nearest multiples). This offers students an opportunity to construct different viable arguments, support them, and critique the reasoning of others (MP3).

Some students may bring up the convention of rounding up that they learned in grade 3, but if not, it is not necessary to remind them during the activity synthesis. This convention is discussed in a later lesson. It is acceptable at this point for students to say that there are two nearest multiples of 100 or that there are none.

• Groups of 2
• Give students access to blank number lines.

• “Trabajen en el primer problema con su compañero. Hagan una pausa antes de continuar con el segundo” // “Work with your partner on the first problem. Pause before continuing to the second set.”
• 4–5 minutes: group work time on the first set of problems
• Pause for a whole-class discussion.
• Invite students to share their responses for part b. Discuss how 136,850 is different from other numbers they’ve seen so far and what students think the nearest multiple of 100 is.
• “Encontremos los múltiplos más cercanos de otros números y veamos si encontramos otros casos en los que no haya solo un múltiplo más cercano” // “Let’s find the nearest multiples for some other numbers and see if we come across other cases where there is not a single nearest multiple.”
• “En silencio, trabajen unos minutos en el último problema” // “Take a few quiet minutes to work on the last problem.”
• 3–5 minutes: quiet work time

Students may be unsure how to find the nearest multiple of 100 for 70,500 because they assume that a nearest multiple of a power of ten must be different than the number itself. Consider asking them to count by 100 from 70,000, recording each counted number, and stopping after 70,600 or 70,700. Ask them which of the written multiples of 100 is the nearest to 70,500.

• “¿Hubo casos en los que no pudieron identificar un múltiplo que fuera el más cercano? ¿Cuáles casos?” // “Were there times when you couldn’t identify one nearest multiple? When?” (When finding the nearest multiple of 1,000 for 70,500.) “¿Por qué no hay un múltiplo que sea el más cercano?” // “Why was there not one nearest multiple?” (70,500 is exactly halfway between 70,000 and 71,000.)
• “Vemos que en la posición de las centenas de 191,530 hay un 5. ¿Este número tampoco tiene un solo múltiplo de 1,000 más cercano?” // “We see 5 in the hundreds place for 191,530. Does this number also have no single nearest multiple of 1,000?” (It does have one, 192,000. The 30 makes it greater than the halfway point between 191,000 and 192,000.)