# Lesson 9

Mismo dígito, distinto valor

## Warm-up: Verdadero o falso: Expresiones desarrolladas (10 minutes)

### Narrative

The purpose of this True or False is for students to consider the value of the same digit in different places. This reasoning will also be helpful later in this lesson when students describe the relationship between different places in multi-digit numbers.

In this activity, students have an opportunity to look for and make use of structure (MP7) as they use commutative and associative properties of addition to compose numbers and determine equivalent sums.

### Launch

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.

• $$4,\!000 + 600 + 70,\!000 = 70,\!460$$
• $$900,\!000 + 20,\!000 + 3,\!000 = 920,\!000 + 3,\!000$$
• $$80,\!000 + 800 + 8,\!000 = 800,\!000 + 80 + 8$$

### Activity Synthesis

Focus question:

• “¿Cómo pueden explicar su respuesta sin tener que encontrar el valor de ambos lados?” // “How can you explain your answer without finding the value of both sides?”
• “Podemos escribir números de distintas formas” // “We can write numbers in different forms.”
• “¿Qué forma se usa para representar los números en este ‘Verdadero o falso’?” // “What form is used to represent the numbers in this True or False?” (expanded form)

## Activity 1: Clasificación de tarjetas: Números grandes (20 minutes)

### Narrative

In this activity, students sort a set of multi-digit numbers and describe the place-value relationships they notice in the sorted numbers. They analyze numbers that have the same digits and write the numbers in expanded form, highlighting the value of each digit. Students then describe relationships they see between the digits in each number.

For example, students may note that the value of the 2 in 46,200 is 200, in 462,000 it is 2,000, and that 2,000 is ten times as much as 200. In the synthesis, they learn that the observed relationship can be expressed with multiplication and division equations, such as $$2,\!000 = 200 \times 10$$, $$2,\!000 \div 200 = 10$$, or other equivalent equations.

When students sort the cards, they look for how the numbers are the same and different, including their overall value or the digits that make up the numbers (MP7).

Here are the numbers on the blackline master, for reference:

186,000

375,000

18,600

37,500

499,000

3,750

49,900

1,860

4,990

Representation: Access for Perception. Synthesis: Use base-ten blocks to demonstrate the relationship between the value of the same digit in different numbers. For example, represent 200 with two large square blocks and 2,000 with two large cubes.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Required Materials

Materials to Copy

• Card Sort: Large Numbers (4 to 6 digits)

### Required Preparation

• Create a set of cards from the blackline master for each group of 2 students.

### Launch

• Groups of 2
• “Lean las instrucciones de los dos primeros problemas y explíquenselas a su compañero con sus propias palabras” // “Read the directions for the first two problems and explain them to your partner in your own words.”
• Collect explanations and clarify any confusion about directions.

### Activity

• Give each group a set of cards from the blackline master.
• 5 minutes: partner and group work time on the first two problems
• As students work, listen for place-value language such as: value of the digit, ten times, thousands, ten-thousands, and hundred-thousands.
• Record any place-value language students use to describe how they sorted the numbers and display for all to see.
• “Ahora, individualmente, escriban los números del siguiente problema en forma desarrollada. Luego, hablen con su compañero sobre el valor de los dígitos” // “Now work independently to write the numbers in the next problem in expanded form. Then, talk with your partner about the value of the digits.”
• 3 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• accurately write the numbers in expanded form
• describe the relationship between the value of the digits in multiplicative terms (“ten times”)

### Student Facing

Su profesor les va a dar varias tarjetas. Cada una tiene un número de varios dígitos.

1. Clasifiquen las tarjetas de una manera que tenga sentido para ustedes. Prepárense para explicar su razonamiento.
2. Júntense con otra pareja y expliquen cómo clasificaron sus tarjetas.
3. Escriban cada número en forma desarrollada.

1. 4,620
2. 46,200
3. 462,000
4. Escriban el valor del 4 en cada número.
5. Comparen el valor del 4 en dos de esos números. Escriban dos afirmaciones para describir lo que observan acerca de los valores.
6. ¿Cómo se relaciona el valor del 2 en 46,200 y el valor del 2 en 462,000?

### Advancing Student Thinking

Students may describe the relationship between digits only in terms of “more” or “less.” Consider asking: “¿Cómo podríamos usar la multiplicación para describir la relación que hay entre los valores de los dígitos?” // “How might we describe the relationship between the digits using multiplication?”

### Activity Synthesis

• Invite students to share their expressions in expanded form and what they noticed about the value of the 4.
• “¿Qué observan acerca del valor del 6 en cada número? ¿Y del valor del 2?” // “What do you notice about the value of the 6 in each number? The value of the 2?” (The value of the 6 is different in each number. It is first 600, then 6,000, then 60,000.)
• Students may talk about the number of zeros in each number. Shift their focus to the place value of the 6— hundreds, thousands, ten-thousands.
• “¿Cómo se relaciona el valor del 2 en 46,200 con el valor del 2 en 462,000?” // “How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?” (The value of the 2 in 462,000 is 2,000 and the same digit in 46,200 has a value of 200. 2,000 is ten times the value 200.)
• “¿Qué ecuación de multiplicación podemos escribir para representar la relación que hay entre el 2 en 46,200 y el 2 en 462,000?” // “What multiplication equation could we write to represent the relationship between the 2 in 46,200 and 462,000?” ($$2,\!000 = 200 \times 10$$)
• “También podemos escribir esta ecuación usando división: $$2,\!000 \div 200 =10$$” // “We can also write this equation using division: $$2,\!000 \div 200 =10$$.”

## Activity 2: Desarrollemos números grandes (15 minutes)

### Narrative

In this activity, students read, write, and analyze multi-digit numbers and use expanded form to describe the relationship between the digits. The numbers in the activity are designed to highlight common errors in reading and writing large numbers. Students encounter numbers with the digit zero in the ten-thousands place and think about how to represent this in expanded and word forms.

MLR2 Collect and Display. Synthesis: Direct attention to words collected and displayed from the previous activity. Invite students to borrow language from the display as needed, and update it throughout the lesson.

### Launch

• Groups of 2
• “Lean el encabezado de cada columna. Luego busquen, en la tabla, ejemplos de cada forma de expresar los números” // “Read the heading in each column and look in the table for examples of each form of number.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses from students. Clarify any misunderstanding about each number form. Record on chart for future reference if needed.

### Activity

• “Trabajen individualmente en los primeros tres problemas. Después, busquen 2 compañeros de clase y trabajen con ellos en el último problema” // “Work independently on the first three problems. Then find 2 classmates to work on the last problem with.”
• 10 minutes: work time

### Student Facing

1. Expresa cada número en forma estándar, en forma desarrollada y en palabras.

número forma desarrollada en palabras
784,003
$$50,\!000 + 9,\!000 +$$ $$300 + 60 + 1$$
ochocientos tres mil noventa y nueve
310,060
novecientos treinta y cuatro mil novecientos
2. Escoge dos números de la tabla y úsalos para lograr que esta afirmación sea verdadera:

El valor del 3 en _______________ es diez veces el valor del 3 en _______________.

3. Explícale a tu compañero por qué escogiste esos números.
4. Encuentren dos compañeros que hayan escogido números distintos a los que ustedes escogieron. Anoten sus números. Por turnos, compartan sus afirmaciones completas y expliquen su razonamiento.

• El valor del 3 en _______________ es diez veces el valor del 3 en _______________.

• El valor del 3 en _______________ es diez veces el valor del 3 en _______________.

### Advancing Student Thinking

Students may find a number with a digit that is not ten times the value of the digit in another number. Consider asking students to record the values of the 3 in each number and arrange them side-by-side. For example, the 3s in 784,003, 59,361, 803,099, 934,900, and 310,060 have values of 3, 300, 3,000, 30,000, and 300,000, respectively. Ask:

• “¿Cuál valor es diez veces otro valor?” // “Which value is ten times another value?”
• “Explica cómo esto te ayuda a decidir qué números puedes usar para completar esta afirmación: ‘El valor del 3 en _______________ es diez veces el valor del 3 en _______________’” // “How does this help you determine what numbers can you include in the statement, ‘The 3 in _______________ is ten times the value of the 3 in _______________.’?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Hoy estudiamos números de varios dígitos y describimos la relación que hay entre los valores de un mismo dígito cuando está en posiciones diferentes” // “Today we described the relationship between the same digit in different places in multi-digit numbers.”

“Compartan con un compañero algo que hayan aprendido hoy sobre la relación que hay entre los valores de los dígitos” // “Share with a neighbor something you learned about the relationship between digits from today’s lesson.” (I learned that a digit in the ten-thousand place is ten times the value of the same digit in the thousands place.)

Record students’ ideas using words and ask, “¿Qué ecuación podríamos escribir para mostrar cuántos grupos de 80,000 hay en 800,000?” // “What equation could we write to show how many groups of 80,000 there are in 800,000?” ($$800,\!000 \div 80,\!000 = 10$$ or $$10 \times 80,\!000 = 800,\!000$$)