# Lesson 11

Números grandes en una recta numérica

## Warm-up: Exploración de estimación: ¿Qué número podría ser? (10 minutes)

### Narrative

The purpose of this Estimation Exploration is to practice the skill of making a reasonable estimate for a number based on its location on a number line. Students give a range of reasonable answers when given incomplete information. They have the opportunity to revise their thinking as additional information is provided. The synthesis should focus on discussing what other benchmarks (multiples of 10) would help make a better estimate. The actual number is revealed in the launch of Activity 1.

This estimation exploration encourages students to use what they know about place value to determine the value of the two tick marks the point lies between and then reason about where it is located between them (MP7).

### Launch

• Groups of 2
• Display the image.
• “¿Qué número está representado por el punto?” // “What number is represented by the point?”
• “¿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
• 1 minute: partner discussion
• Record responses in the table.

### Activity

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

### Student Facing

¿Qué número está representado por el punto?

Escribe una estimación que sea:

muy baja razonable muy alta
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### Activity Synthesis

• “¿Qué información les ayudaría a hacer una estimación más precisa?” // “What information would help you make a more precise estimate?” (Additional tick marks or other numbers around the point)
• Consider providing new information. “¿Les gustaría ajustar sus estimaciones?” // “Would you like to revise your estimates?”
• Record new or revised estimates.
• “¿Qué otra información necesitarían para estar más seguros de su estimación?” // “What other information would you need to be more confident with your estimate?”

## Activity 1: Ubiquemos números grandes (20 minutes)

### Narrative

The purpose of this activity is for students to use their understanding of place value and the relative position of numbers within 1,000,000 to partition and place numbers on a number line. Students place four related numbers on a number line and consider relationships between digits to determine how to partition a number line.

The numbers have the same non-zero digits but with different place values, allowing students to observe the closely related values of the tick marks (MP7) and the identical location on the different number lines of the numbers they plot (MP8).

### Launch

• Groups of 2
• “¿Qué observan y qué se preguntan acerca de las primeras cuatro rectas numéricas?” // “What do you notice and wonder about the first four number lines?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion
• “Piensen en qué lugar de la recta numérica ubicarían el primer número” // “Think about where you would place the first number on the number line.”
• “Explíquenle a un compañero cómo decidieron en dónde ubicar el número” // “Explain to a partner how you decided where to place the number.”

### Activity

• 10 minutes: independent work time
• 3 minutes: partner discussion
• Monitor for students who:
• add tick marks to show the halfway mark, and the labeled number slightly less than half on each number line in the first problem
• label the seventh tick mark on each number line for the second problem

### Student Facing

1. Ubica y marca cada número en la recta numérica.

1. 347
2. 3,470
3. 34,700
4. 347,000
2. Ubica y marca cada número en la recta numérica.

1. 347
2. 3,470
3. 34,700
4. 347,000
3. ¿Qué observas acerca de la ubicación de estos números en las rectas numéricas? Haz dos observaciones y discútelas con tu compañero.

### Student Response

If students label the number line by ones, consider asking: “¿Cómo podrías usar la relación que hay entre 100 y 1,000 para ayudarte a marcar la recta numérica?” // “How might you use the relationship between 100 and 1,000 to help label the number line?”

### Activity Synthesis

• Ask 2–3 students to share their responses and their reasoning for each problem.
• “¿Cómo partieron la recta numérica del primer problema?” // “How did you partition the number line in the first problem?” (I know that 350 is halfway between 300 and 400, so I marked the halfway point, and then estimated where 3 down from that would be.)
• “¿Cómo les ayudan las rectas numéricas a ver las relaciones que hay entre los números?” // “How do the number lines help you to see the relationship between the numbers?” (The number lines have endpoints that are ten times as much as the number line before. Also, each number is ten times as much as the number before. The place values changed, but the numbers are located in the same relative position.)

## Activity 2: Muchos números, poco espacio en la recta (15 minutes)

### Narrative

In this activity, students place a set of numbers that are each ten times as much the one before it on the same number line. In doing so, they notice the impact of multiplying a number by ten on its magnitude. Unlike before, the number lines here have no or fewer intermediate tick marks, prompting students to think about how to partition the lines in order to facilitate plotting their assigned number.

MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain their approach to the problem. Invite groups to rehearse what they will say when they share with the whole class.
Representation: Access for Perception. Begin by demonstrating the relative magnitude of numbers in the hundreds, thousands, ten-thousands, and hundred-thousands using millimeters. Invite students to examine a meter stick and notice the size of one millimeter, ten millimeters, one hundred millimeters, and one thousand millimeters. Invite students to guess the length of ten-thousand and one hundred-thousand millimeters. If time and space allow, prepare a walk outside the classroom with stops at 10,000 millimeters from the door and 100,000 millimeters from the door.
Supports accessibility for: Conceptual Processing, Visual Spatial Processing, Attention

### Launch

• Groups of 4
• Assign each student in a group a letter A–D.

### Activity

• “En silencio, piensen unos minutos en qué lugar de la recta numérica debería ir el número que se les asignó” / “Take a few quiet minutes to think about where your assigned number should go on the number line.”
• “Luego, discutan en grupo cómo pensaron y, juntos, ubiquen los cuatro números en la recta numérica” // “Then, discuss your thinking with your group and work together to locate all four numbers on the number line.”
• 3–4 minutes: independent work time
• 7–8 minutes: group work time
• Monitor for students who:
• partition the number line into hundred-thousands or ten-thousands
• use benchmarks such as 50,000, 200,000, or 350,000

### Student Facing

Su profesor le va a asignar un número a cada uno de ustedes para que lo ubiquen en la recta numérica dada.

1. 347
2. 3,470
3. 34,700
4. 347,000
1. Decidan dónde ubicar cada número en esta recta numérica. Expliquen su razonamiento.

2. En grupo, escriban debajo de cada marca el número que representa. Después, decidan juntos dónde deben ubicar cada número.

### Student Response

If students run out of room and only place some numbers on the number line, consider asking: “¿Cuál es la marca del medio?” // “What is the halfway mark?” Then, follow up with: “¿Cuáles números están antes de la marca del medio? ¿Cuáles números están después de la marca del medio?” // “Which numbers would fall before the halfway mark? What about after the halfway mark?”

### Activity Synthesis

• Ask 2–3 small groups to share their number line.
• “¿De qué manera decidieron partir la recta numérica?” // “How did you decide to partition the number line?” (I partitioned the number line by tens, hundreds, thousands, ten-thousands, hundred-thousands—not by ones.)
• “¿Cuáles números fueron más fáciles de ubicar? ¿Por qué?” // “Which numbers were easier to locate? Why?” (34,700 and 347,000, were easier to locate because they were further away from zero.)
• “¿Qué hubiera ayudado a ubicar los demás números más fácilmente?” // “What would have made it easier to locate the other numbers?” (A longer number line would have made it easier to include more partitions)
• “Hagan algunas observaciones acerca de dónde están ubicados los números en la recta numérica” // “Make some observations about where the numbers are positioned on the number line.” (Most of the numbers we located are much closer to zero than to 400,000)
• “Aquí ubicaron los mismos cuatro números que habían ubicado en la primera actividad. ¿Qué diferencias hay entre esta ubicación de los puntos y la ubicación de los puntos de la actividad 1?” // “You located the same four numbers here as you did in the first activity. How are the locations of the points different from those in Activity 1?” (Sample response: Ten times as much looks different when they are all on the same number line.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy ubicamos y analizamos conjuntos de números grandes en una recta numérica. En cada conjunto, cada número era 10 veces el número que estaba antes de él. Observemos las rectas numéricas de la primera actividad” // “Today we located and analyzed sets of large numbers on a number line. In each set, each number was 10 times as much as the number before it. Let’s look at the number lines from the first activity.”

“¿Cómo podemos usar ecuaciones de multiplicación para mostrar la relación que hay entre los puntos que están en la recta numérica?” // “How might we use multiplication equations to show the relationship between each point on the number line?”

• $$347 \times 10 = 3,\!470$$
• $$3,\!470 \times 10 = 34,\!700$$
• $$34,\!700 \times 10 = 347,\!000$$

“¿Cuál es la relación que hay entre los valores de las marcas en cada recta numérica?” // “What is the relationship between the values of the labels on each number line?” (Each new number line has tick marks that are valued at 10 times as much as the labels on the previous number line.)

## Student Section Summary

### Student Facing

En esta sección, trabajamos con números hasta 999,999, es decir, que van hasta la posición de las unidades de cien mil.

Primero, usamos bloques en base diez, cuadrículas de 10 por 10 y diagramas en base diez para nombrar, escribir y representar números de varios dígitos (menores que 1,000,000). Escribimos los números en forma desarrollada para poder ver el valor de cada dígito. Por ejemplo:

$$725,\!400=700,\!000 + 20,\!000 + 5,\!000 + 400$$

Luego, aprendimos que el valor de un dígito de un número de varios dígitos es diez veces el valor del mismo dígito en la posición que está a su derecha. Por ejemplo:

• 14,800 y 148,000 tienen un 4.
• El 4 en 14,800 está en la posición de las unidades de mil. Su valor es 4,000.
• El 4 en 148,000 está en la posición de las unidades de diez mil. Su valor es 40,000.
• El valor del 4 en 148,000 es diez veces el valor del 4 en 14,800.

Usamos ecuaciones de multiplicación y de división para representar esta relación.

$$10 \times 4,\!000 = 40,\!000$$

$$40,\!000 \div 10 = 4,\!000$$

Finalmente, analizamos la relación “diez veces” ubicando números en rectas numéricas.