# Lesson 14

Múltiplos de 10,000 y de 100,000

## Warm-up: Conteo grupal: Múltiplos de 1,000, de 10,000 y de 100,000 (10 minutes)

### Narrative

The purpose of this Choral Count is to familiarize students with multiples of 1,000, 10,000, and 100,000 and to notice patterns in the count. These understandings help students develop fluency and will be helpful later in this lesson when students locate large numbers on number lines and identify multiples of 10,000 and 100,000 that are near those numbers.

### Launch

• “Contemos a saltos usando números grandes. Haremos tres rondas” // “Let’s count by some large numbers. We’ll do three rounds.”
• Prepare to record three rounds of counting: by 1,000, 10,000, and 100,000. Record the count by 10,000 on a number line.
• “Cuenten de 1,000 en 1,000, empezando en 85,000” // “Count by 1,000, starting at 85,000.”
• Stop counting and recording at 115,000.
• “Cuenten de 10,000 en 10,000, empezando en 80,000” // “Count by 10,000, starting at 80,000.”
• Stop counting and recording at 230,000.
• “Cuenten de 100,000 en 100,000, empezando en 0” // “Count by 100,000, starting at 0.”
• Stop counting and recording at 400,000.

### Activity

• “¿Qué patrones ven?” // “What patterns do you see?”
• 1–2 minutes: quiet think time
• Record responses.

### Activity Synthesis

• “El primer conjunto de números muestra ‘múltiplos de 1,000’. El segundo conjunto muestra ‘múltiplos de 10,000’ y el tercer conjunto muestra ‘múltiplos de 100,000’” // “The first set of numbers shows ‘multiples of 1,000.’ The second set shows ‘multiples of 10,000,’ and the third set shows ‘multiples of 100,000’.”
• “¿Cómo sabemos que 85,000 es un múltiplo de 1,000?” // “How do we know that 85,000 is a multiple of 1,000?” ($$85 \times 1,\!000 = 85,\!000$$)
• “¿Cómo sabemos que 90,000 es un múltiplo de 10,000?” // “How do we know that 90,000 is a multiple of 10,000?” ($$9 \times 10,\!000 = 90,\!000$$)
• “¿Puede un múltiplo de 1,000 ser también múltiplo de 10,000? Si creen que sí, muestren algunos ejemplos” // “Can a multiple of 1,000 also be a multiple of 10,000? If you think so, show some examples.” (Yes, for example, 80,000 and 120,000 are multiples of 1,000 and 10,000. They show up on both lists.)
• “¿Puede un múltiplo de 10,000 ser también múltiplo de 100,000? Muestren algunos ejemplos” // “Can a multiple of 10,000 also be a multiple of 100,000? Show examples.” (Yes, for example, 100,000 and 200,000)
• “¿Puede un múltiplo de 100,000 ser también múltiplo de 1,000? Muestren algunos ejemplos” // “Can a multiple of 100,000 also be a multiple of 1,000? Show examples.” (Yes, for example, 100,000)

## Activity 1: ¿A qué recta pertenecen? (20 minutes)

### Narrative

In this activity, students locate five- and six-digit numbers on a series of number lines. The endpoints of each number line are multiples of 100,000, and the space between them is partitioned into ten equal intervals. As they locate the numbers, students recognize each tick mark as a multiple of 10,000 (MP7). Later in the activity, students use a number line to name multiples of 10,000 that are near given five-digit numbers.

Prior to the lesson, create number lines from the blackline master and post them around the room for students to visit during the activity.

Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from practicing with more accessible values first. For example, display a 100200 number line with ten tick marks, and invite students to discuss multiples of 10 and 100 that they see. Help them articulate a strategy to place 182 on the number line. Encourage students to draw connections to this work as you introduce the 100,000200,000 number line.
Supports accessibility for: Conceptual Processing, Language, Attention

### Required Materials

Materials to Gather

Materials to Copy

• On Which Line Do They Belong? (0-700,000 number line)

### Required Preparation

• Create number lines from the blackline master and post them around the room before the activity.

### Launch

• Groups of 4
• “Miren las rectas numéricas que hay alrededor del salón. ¿Qué observan acerca de ellas? ¿Qué se preguntan?” // “Take a look at the number lines around the room. What do you notice about them? What do you wonder?”
• 30 seconds: quiet think time
• Share responses.
• Display a number line with 100,000 at one end and 200,000 at the other.
• “¿Ven múltiplos de 100,000 en esta recta numérica?” // “Do you see multiples of 100,000 in this number line?” (Yes, 100,000 and 200,000)
• “¿Ven múltiplos de 10,000?” // “Do you see multiples of 10,000?” (Yes, each tick mark is a multiple of 10,000.) “¡Nombrémoslos!” // “Let’s name them!” (100,000, 110,000, . . . , 200,000)
• Label the first few tick marks.
• “¿Ven múltiplos de 1,000?” // “Do you see multiples of 1,000?” (No, they’re not marked.) “Si estuvieran marcados, ¿cómo se verían?” // “If they were marked, what might they look like?” (10 tiny, equal spaces between each pair of tick marks)]
• “¿Pueden estimar en qué lugar de la recta numérica está ubicado 113,500?” // “Can you estimate where 113,500 goes on the number line?” (Between the second and third tick marks, but closer to the first tick mark. Or between 110,000 and 120,000, but closer to 110,000.)
• Assign one set of numbers (A, B, C, D, or E) to each group of 4.
• Give 4 stickers and 4 sticky notes to each group.

### Activity

• “Con su grupo, ubiquen cada número en la recta numérica correcta. Usen una calcomanía para marcar la ubicación en la recta numérica y usen una nota adhesiva para escribir el número que corresponde. Después, completen el último problema” // “Work with your group to locate each number on the right number line. Use a sticker to mark the location on the number line and use a sticky note to label it. Then, complete the last problem.”
• 8–10 minutes: group work time
• Monitor for the ways students locate their numbers and how they determine the multiples of 10,000 in the last problem.

### Student Facing

Su profesor les va a asignar un conjunto de números.

A B C D 140,261 100,025 486,840 676,850 450,099 414,500 128,201 379,900 158,002 42,326 99,982 428,950 194,030 658,340 541,700 621,035 215,300 499,600 608,720 644,700
1. Hay varias rectas numéricas puestas alrededor del salón. En grupo, decidan en cuál recta numérica debe ir cada número.

Luego, estimen la ubicación del número en esa recta, pongan una calcomanía de punto para marcarlo y escriban el número debajo de la calcomanía.

2. Miren la recta numérica que representa los números de 0 a 100,000 y que tiene dos puntos.

1. Nombren los dos múltiplos de 10,000 que están más cerca de cada punto.
2. De los dos múltiplos de 10,000 que nombraron, ¿cuál es el que está más cerca de cada punto?

### Student Response

Students may decide to place all numbers in their assigned set on the same number line. Consider asking:

• “Piensa en tus números, ¿entre cuáles dos números está cada uno?” // “Between which two numbers does each of your numbers fall?”
• “Piensa en cada número, ¿cuál recta numérica tiene extremos que están cerca de él?” // “Which number lines have endpoints close to each number?”

### Activity Synthesis

• Select students to explain how they identified which number line to use and where to put the dot sticker to represent each number. Highlight explanations that are based on place-value reasoning:
• To identify the right number line, we’d look at the digit in the hundred-thousands place. If it didn’t have a digit there, it goes on the first number line.
• To locate the point, we’d look at the digit in the ten-thousands place. For 379,000, it is the 7 in the ten-thousands place, so we’d count 7 tick marks from 300,000. The dot would be close to the eighth tick mark because 79,000 is close to 80,000.
• Briefly discuss how students identified the nearest multiple of 10,000 for 42,326 and 99,982.
• Keep the number lines displayed for the next activity.

## Activity 2: Más cerca de algún múltiplo (15 minutes)

### Narrative

In this activity, students identify the nearest multiples of 10,000 and 100,000 for the six-digit numbers they saw in the first activity. They may do so by using the number lines from earlier, but they may also start to notice a pattern in the relationship between the numbers and the nearest multiples without the number lines (MP7).

MLR8 Discussion Supports. Display sentence frames to support small-group discussion: “Observé _____, entonces yo . . .” // “I noticed _____ so I . . .” and “Estoy de acuerdo / en desacuerdo porque . . .” // “I agree/disagree because . . . .”

### Launch

• Groups of 2 or 4
• Display number line with endpoints of 100,000 and 200,000.

### Activity

• “En silencio, trabajen unos minutos en la actividad. Luego, compartan sus respuestas con su grupo” // “Take a few quiet minutes to work on the activity. Then, share your responses with your group.”
• 6–7 minutes: independent work time
• 3–4 minutes: group discussion

### Student Facing

Para esta actividad, usa la recta numérica que representa los números entre 100,000 y 200,000.

1. En cada caso, nombra el múltiplo de 10,000 que está más cerca del número (por ahora no llenes la última columna).
número múltiplo de 10,000 que está más cerca $$\phantom{nearest multiple}$$
100,025
128,201
140,261
158,002
194,030
2. Esta recta numérica muestra 215,300. ¿Qué múltiplo de 100,000 es el más cercano a 215,300?

3. Marca la última columna de la tabla con las palabras “múltiplo de 100,000 que está más cerca”. Después, nombra el múltiplo de 100,000 que está más cerca de cada número de la tabla.

### Student Response

Students may see that the nearest multiples of 10,000 for a number are the two tick marks surrounding the point but may be unsure what numbers they represent. Ask them to recall what each tick mark represents on this set of number lines and urge them to count the marks.

### Activity Synthesis

• Display the blank table from the activity. Invite students to share their responses to complete the table. Discuss any disagreements.
• Invite students to share how they identified the nearest multiples of 10,000 and 100,000.
• If no students mentioned that they examined the location of each point visually and decided the closest tick marks on the number line, ask them about it.

## Lesson Synthesis

### Lesson Synthesis

“Hoy aprendimos a identificar los múltiplos de 10,000 y de 100,000 que están cerca de un número” // “Today we learned to identify multiples of 10,000 and 100,000 that are close to a number.”

Ask students to write a six-digit number.

“¿Cuáles dos múltiplos de 10,000 están más cerca de su número? De estos dos, ¿cuál está más cerca de su número?” // “Which two multiples of 10,000 are closest to your number? Of the two, which one is the nearest?”

“¿Cuáles dos múltiplos de 100,000 están más cerca de su número? ¿Cuál de ellos está más cerca?” // “Which two multiples of 100,000 are closest to your number? Which one is the nearest?”

“Intercambien su número y los múltiplos de 10,000 y de 100,000 más cercanos con los de su compañero” // “Trade your number and its nearest multiples of 10,000 and 100,000 with those of your partner’s.”

“¿Están de acuerdo en que los múltiplos de 10,000 y de 100,000 que escribió su compañero son realmente los más cercanos? ¿Pueden explicar cómo encontró su compañero esos múltiplos?” // “Do you agree that the multiples of 10,000 and 100,000 that they wrote are indeed the nearest ones? Can you tell how they arrived at those multiples?”