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.

• 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?” (Find two multiples of 1,000 that are closest to the number, one that is greater and one that is lesser. For 573,816, they are 573,000 and 574,000. If the number is less than their midpoint, 573,500, it is closer to the lower multiple of 1,000. If it is more than 573,500, 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, it is closer to the lower multiple of 1,000. If it is more than 500, it is closer to the higher multiple of 1,000.)

• 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 múltiplos más cercanos fueron un poco más retadores, si los hubo? ¿Por qué fueron retadores?” // “Which nearest multiples 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 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 IM 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.

• “¿En qué se diferencia este número de los otros números que has visto hasta ahora?” // “How is this number different from other numbers you have seen so far?”
• Invite students to count by 100 from 70,000, recording each counted number, and stopping after 70,600 or 70,700. Ask students “¿Cuál de los múltiplos de 100 que escribiste es el más cercano a 70,500?” // “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.)