# Lesson 8

Más allá de 100,000

## Warm-up: ¿Cuántos ves? (10 minutes)

### Narrative

The purpose of this How Many Do You See is to allow students to use place value language to describe the value of the base-ten blocks they see. Students may provide answers that indicate the number of blocks they see, while others may indicate the value of the blocks.

If students do not bring it up, ask about the value of the blocks.

### Launch

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Display image.
• 1 minute: quiet think time

### Activity

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

### Student Facing

¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?

### Activity Synthesis

• “¿Qué cantidad está representada por los bloques?” // “What amount do the blocks represent?”
• “¿Qué relaciones observan entre los bloques?” // “What relationships do you notice between the blocks?”

## Activity 1: La representación que hizo Lin (15 minutes)

### Narrative

In this activity, students use base-ten blocks or base-ten diagrams to represent large numbers in the ten-thousands and hundred-thousands. They learn that when they assign a new value, 10, to the small cube, larger numbers are more accessible and can be represented with fewer blocks. The limitation of blocks in the classroom will create a need to represent large numbers in a different way. Blocks should be made available and students should be invited to use them if needed. Students should also be encouraged to represent base-ten blocks in diagrams in ways that make sense to them.

When students interpret and use Lin's strategy, they state the meaning of each base-ten block or part of their diagram in a strategic way allowing them to represent large numbers (MP6).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Representation: Access for Perception. Provide access to labels for the base-ten blocks. Invite students to act out Lin’s strategy by shifting the labels so that the small cube is labeled 10 (ten), the long rectangle block is labeled 100 (hundred), and so on. Extra labels can also support students who need a more concrete representation of the numbers, but who may have run out of base-ten blocks. Invite students to make connections between the blocks, the labels, and the diagrams they draw on paper.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Memory

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give each group a set of base-ten blocks.
• “¿Cómo usarían bloques en base diez o un diagrama en base diez para representar 15,710?” // “How would you use base-ten blocks or a base-ten diagram to represent 15,710?”
• 2 minutes: independent work time
• “Aunque puede que aún no hayan terminado, por favor explíquenle a un compañero su plan para representar 15,710” // “Although you may not be finished, please share your plan for representing 15,710 with a partner.”
• 2 minutes: partner discussion

### Activity

• “Lin estaba trabajando en este mismo problema cuando se le ocurrió una estrategia para construir números grandes sin usar o dibujar tantos bloques” // “Lin was working on this same task when she came up with a strategy to build large numbers without using or drawing so many blocks.”
• “Lean en silencio la estrategia de Lin. Luego, explíquenle la estrategia a su compañero” // “Silently read about Lin’s strategy. Then, explain it to your partner.”
• “Con un compañero, completen el resto de la actividad” // “Work with a partner to complete the rest of the activity.”
• “Mientras completan el segundo problema, incluyan detalles, como notas o marcas, para ayudar a los demás a entender sus ideas” // “As you complete the second problem, include details such as notes or labels to help others understand your thinking.”
• 5–10 minutes: independent work time
• As students are working, monitor for students who:
• attempt to draw or represent each unit of the number
• create a key or communicate groups of each unit (for example, represent 15 thousands using 150 large squares, or draw a new image to represent 1 ten-thousand and use 5 large squares to represent 5 thousands)

### Student Facing

1. Usa bloques en base diez o dibuja un diagrama en base diez para representar 15,710.
2. Lin está usando bloques como estos para representar 15,710. Ella decidió cambiar el valor del cubo pequeño y ahora este representa 10.

¿Cuál es el valor de cada bloque si el valor del cubo pequeño es 10?

1. Cubo pequeño: 10

2. Bloque largo rectangular: __________

4. Cubo grande: __________

3. Usa la estrategia de Lin para representar 15,710.
4. Usa la estrategia de Lin para representar cada uno de estos números.

1. 23,000
2. 58,100
3. 69,470
5. Usando la estrategia de Lin, ¿cuáles bloques en base diez se usarían para representar 100,000?

### Student Response

Students may try to build 15,710 and run out of blocks to build or represent. Consider asking: “¿Qué bloques te hacen falta?” // “What blocks are missing?” and “¿Qué puedes dibujar para representar los bloques que faltan?” // “What can you draw to represent these missing blocks?”

### Activity Synthesis

MLR7 Compare and Connect

• 2–3 minutes: gallery walk
• “¿Qué conexiones observan entre las representaciones que hicieron cuando usaron la estrategia de Lin?” // “What connections do you see between the representations you made using Lin's strategy?” (I see 15 of the large squares for 15,000 and in another drawing. I see 1 large cube and 5 large squares.)
• Collect student responses and revoice connections between groups of ten that are equivalent to other units.
• “¿Cuáles bloques usarían para representar 100,000?” // “Which blocks would we use to represent 100,000?” (100 large squares or ten large cubes. We could also create a new block to represent 100,000.)

## Activity 2: ¿Qué número está representado? (10 minutes)

### Narrative

In this activity, students interpret a collection of blocks in which a small cube represents different values. They notice a pattern in the value of the digits when the small cube represents 1 and then represents 10. Although students are not required to articulate this relationship until the next lesson, the reasoning here elicits observations about the relationship between the digits in the multi-digit number and the number of each type of block (MP7, MP8).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• 3 minutes: independent work time

### Activity

• 5 minutes: partner work
• Monitor for students who can describe the relationship between the numbers in the first two problems.

### Student Facing

1. Un cubo pequeño representa 1. ¿Qué valor representan los bloques de la imagen?

2. Ahora un cubo pequeño vale 10. ¿Cuál es el nuevo valor que representan los bloques de la imagen?

3. Escribe dos afirmaciones para comparar los números de los problemas anteriores.

### Student Response

Students may say that the value of the collection of blocks remains the same even when the small cube has changed in value. Consider asking: “Si cada cubo pequeño ahora vale 10, ¿cuál es el valor del rectángulo largo?” // “If each small cube is now worth 10, what is the value of the long rectangle?”

### Activity Synthesis

• Invite students to share their answers to the last problem.
• Create a chart of statements students make when comparing the two numbers for reference in a future lesson.

## Activity 3: Construyamos unidades de cien mil (10 minutes)

### Narrative

The purpose of this activity is to remind students the meaning of expanded form so they can write numbers to the ten-thousands in expanded form.

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• “Completen el primer problema de manera independiente. Luego, lo vamos a discutir todos juntos” // “Complete the first problem on your own and then we'll talk about it as a class.”
• 3 minutes: independent work time
• Select 1–2 students to share their responses to the first problem.
• “¿Cómo se escribe este número en forma desarrollada?” // “How would we write this number using expanded form?” $$(40,\!000 + 9,\!000 + 800 +30)$$
• “Recuerden que cuando escribimos un número como una suma de centenas, decenas y unidades, estamos usando la forma desarrollada” // “Remember, when we write a number as a sum of hundreds, tens, and ones, we are using expanded form.“

### Student Facing

1. Para representar números grandes, Lin cambió el valor del cubo pequeño a 10. Ella usó los siguientes bloques para representar su primer número.
1. ¿Qué número representó Lin? Muestra o explica tu razonamiento.
2. Escribe una ecuación para representar el valor de los bloques.
2. Después, ella usó más bloques para representar otro número.
1. ¿Qué número representó Lin? Muestra o explica tu razonamiento.
2. Escribe una ecuación para representar el valor de los bloques.

### Student Response

If students lose track of the value of the blocks, consider asking: “¿Cuál es el valor de cada bloque largo cuando el cubo pequeño tiene un valor de 10?” // “What is the value of each long block when the small cube has a value of 10?” and “¿Cómo te puede ayudar esto a encontrar el valor de 10 cubos grandes?” // “How might this help you find the value of 10 large cubes?”

### Activity Synthesis

• Select 1–2 students to share equations for the second problem.
• “120,450: practiquemos todos juntos cómo se dice este número” // “120,450: Let’s practice saying this number together as a class.”
• “¿Qué dígito está en la posición de las unidades de mil en este número?” // “What digit is in the thousands place in this number?” (zero)
• “¿Por qué Lin obtuvo un 0 en la posición de las unidades de mil si ella tenía 20 bloques con un valor de 1,000 cada uno?” // “How did Lin end up with a 0 in the thousands place, when she had 20 blocks with a value of 1,000?” (Each group of 10 thousands makes 1 unit of ten-thousand. Since there are 2 groups of 10 thousands, there are 2 ten-thousands.)
• “¿Cómo podemos explicar qué número está representado por 10 bloques que tienen un valor de 10,000 cada uno?” // “How can we explain the number represented by 10 blocks with the value of 10,000 each?” (Ten groups of 10,000 is 100,000. We can also reason by counting by 10,000. Nine blocks with a value of 10,000 is 90,000, so 10 blocks would be 10,000 more than that, or 100,000.)
• Record the reasoning about the value of the blocks using equations:
$$10 \times 10 = 100\\ 10 \times100 = 1,\!000\\ 10 \times 1,\!000 = 10,\!000\\ 10 \times 10,\!000 = 100,\!000$$

## Lesson Synthesis

### Lesson Synthesis

Consider using whiteboards during the synthesis to poll the class informally.

“Hoy escribimos números de varios dígitos en su forma desarrollada. Explíquenle a un compañero qué es la forma desarrollada” // “Today we wrote multi-digit numbers using expanded form. Explain expanded form to a partner.”

Write 115,000 for students to see.

“¿Cuántas unidades de cien mil hay en este número?” // “How many hundred-thousands are in this number?” (1)

“¿Cuántos grupos de 10,000 forman 100,000?” // “How many groups of 10,000 make 100,000?” (10)

“¿Qué ecuación podríamos escribir para mostrar que 10 grupos de 10,000 son equivalentes a 100,000?” // “What equation could we write to show 10 groups of 10,000 are equivalent to 100,000?” ($$10 \times 10,\!000 = 100,\!000$$ or $$100,\!000 \div 10 = 1,\!000$$)

“¿Cómo escribimos 115,000 en forma desarrollada?” // “How would we write 115,000 using expanded form?” $$(100,\!000 + 10,\!000 + 5,\!000)$$