# Lesson 6

Diez veces

## Warm-up: Conteo grupal: 12, 15 y 24 (10 minutes)

### Narrative

The purpose of this Choral Count is for students to practice counting by 12, 15, and 24, and notice patterns in the count after 10 multiples. This understanding will help students later in this lesson when they represent quantities that are 10 times as many using tape diagrams.

When they use the words multiple, value, and place students use language precisely (MP6).

### Launch

• “Cuenten de 12 en 12, empezando en 12” // “Count by 12 starting at 12.”
• Record as students count.
• Stop counting and recording after the tenth multiple has been said.
• Repeat these steps with 15 and 24.

### Activity

• Circle the tenth multiple in from each count.
• “¿Qué observan acerca del décimo múltiplo?” // “What do you notice about the tenth multiple?” (It has a zero in the ones place.)
• “¿Por qué creen que sucede esto?” //  “Why do you think this is?” (The value is now ten times as much as the first value.)
• “¿Pueden predecir cuál sería el vigésimo múltiplo (es decir, el múltiplo número 20)?” // “Can you predict the twentieth multiple?” (It is the double of the tenth multiple and it also ends in zero.)
• 1–2 minutes: quiet think time
• 1 minute: partner discussion
• Share and record responses.

### Activity Synthesis

• “¿Cuál sería el décimo múltiplo si contáramos de 192 en 192? ¿Y de 1092 en 1092?” // “What would the tenth multiple be if we were counting by 192? 1092?” (1,920, 10,920)

## Activity 1: Diez veces (20 minutes)

### Narrative

In this activity, students are given a diagram that shows two quantities, one of which is 10 times as much as the other. They identify possible values and possible equations that the diagram could represent.

Students see that a single unmarked diagram could represent many possible pairs of values that have the same relationship (in this case, one is 10 times the other) and be expressed with many equations.

The activity also reinforces what students previously learned about the product of a number and 10—namely, that it ends in zero and each digit in the original number is shifted one place to the left because its value is ten times as much (MP7).

### Launch

• Groups of 2
• Read the first problem aloud.
• “Piensen en dos posibles parejas de valores de A y B” // “Think of two possibilities for the value of A and B.”
• 2 minutes: quiet think time

### Activity

• 2 minutes: partner discussion
• “Trabajen con un compañero en el resto de la actividad” // “Work with a partner on the rest of the activity.”
• Monitor for students who:
• use the tape diagram to reason about the equations.
• rely on numerical patterns in multiples of 10 to reason about 10 times as many

### Student Facing

Este es un diagrama que representa dos cantidades, A y B.

1. ¿Cuáles son algunos valores posibles de A y B?

2. Selecciona las ecuaciones que pueden ser representadas por el diagrama.

1. $$15 \times 10 = 150$$
2. $$16 \times 100 = 1,\!600$$
3. $$30 \div 3 = 10$$
4. $$5,\!000 \div 5 = 1,\!000$$
5. $$80 \times 10 = 800$$
6. $$12,\!000 \div 10 = 1,\!200$$
3. En el caso de las ecuaciones que no pueden ser representadas por el diagrama:

1. Explica por qué el diagrama no representa estas ecuaciones.
2. ¿Cómo cambiarías las ecuaciones para que pudieran ser representadas por el diagrama?
3. Compara tus ecuaciones con las de tu compañero. Haz al menos dos observaciones sobre las ecuaciones que tú y tu compañero escribieron.

### Student Response

Students may assume that it is impossible for the same diagram to represent multiple expressions. Consider asking:

• “¿Cómo cambia el valor de B si A representa 6? ¿Cómo lo sabes?” // “How does the value of B change if A represents 6? How do you know?”
• “¿Qué expresión podemos escribir para representar esto?” // “What expression could we write to represent this?”

Consider repeating with another number to reinforce the idea.

### Activity Synthesis

Display possible values for A and corresponding values for B in a table such as this:

value of A value of B
• “¿Qué observan acerca de los valores de cada grupo? ¿En qué se parecen y en qué se diferencian los valores de cada grupo?” // “What do you notice about the values of each set? How do the values compare?” (The values for B are all multiples of 10. Each value for B is ten times the corresponding value for A.)
• Select students to share their responses and reasoning.
• For students who used the diagram to reason about the equations, consider asking, “¿Cómo podrían marcar el diagrama para mostrar que representa las ecuaciones que seleccionaron?” // “How might you label the diagram to show that it represents the equations you selected?” (Sample response: For the equation $$30 \div 3 = 10$$ I would label A “3” and B “30.” I know that 30 is ten times as much as 3, so the diagram represents the equation.)
• For students who used their understanding of numerical patterns to support their reasoning, consider asking, “¿Cómo supieron que la ecuación podía ser representada como una comparación que incluye diez veces una cantidad?” // “How did you know that the equation could be represented as a comparison involving ten times as many?” (Sample response: I know that when we multiply a number by 10, the product will be ten times the value. I also know that division is the inverse of multiplication, so I looked for equations that were multiplying or dividing by 10 or had ten as a quotient.)

## Activity 2: ¿Qué sigue siendo lo mismo? (15 minutes)

### Narrative

In this activity, students analyze situations in which one quantity is ten times as much as another quantity. Students may use different strategies to determine the missing quantity. For example, they may rely on counting as a strategy, or use place value understanding to explain regularity in the products of numbers with 10 (MP8). The reasoning in this lesson prepares students to consider quantities that are 100 times and 1,000 times as many in the next section.

MLR8 Discussion Supports. Synthesis. Display sentence frames to support whole-group discussion: “Observé _____, entonces yo . . .” // “I noticed _____ so I . . .” and “Primero, yo _____ porque . . .” // “First, I _____ because . . . .”
Representation: Access for Perception. Invite students to use a class set of base-ten cubes to explore how the value of each digit changes when a number is multiplied by ten.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

• Groups of 2

### Activity

• 5–6 minutes: quiet work time
• 3 minutes: partners discussion
• Monitor for students who:
• use multiplication expressions to represent the relationships between values of A and B, for example, $$14 \times 10$$ to represent the first pair of quantities
• use place value relationships to determine the value of each quantity

### Student Facing

1. Usa el diagrama para completar la tabla.

valor de A valor de B
14 $$\phantom{0000000000}$$
$$\phantom{0000000000}$$ 1,000
160
850
1,000
2,070
3,900
2. Selecciona algunos valores de tu tabla para explicar o mostrar:

1. Cómo encontraste el valor de B cuando el valor de A era conocido.

2. Cómo encontraste el valor de A cuando el valor de B era conocido.

### Student Response

If students write a value other than 10,000 for B when A is 1,000, consider asking:

• “De lo que observaste de las parejas de números más pequeños que encontraste, ¿qué te puede ayudar a encontrar el valor de B?” // “What do you notice about the pairs of smaller numbers you found that can help you find the value of B?”
• “Del conteo grupal del calentamiento de hoy, ¿qué te puede ayudar a saber qué número escribir?” // “What about the choral count from today’s warm-up can help you know what number to write?”

### Activity Synthesis

• “¿Qué cosas observaron acerca de los valores de A y B?” // “What were some things you noticed about the values of A and B?” (Sample responses:
• A was less and B was always greater.) “¿Cuántas veces A era B?” // “How many times as much as A was B?” (Ten times)
• The digits of A are all in B and are in the same order, but they are not in the same places in B. There's an extra 0 at the end of the value for B. “¿Por qué creen que sucede esto?” // “Why do you think this is?” (Because they are multiplying 10 by A.)
• “¿Podemos representar tanto $$4 \times 10$$ como $$10 \times 10$$ usando el mismo diagrama de A y B? ¿Por qué sí o por qué no?” // “Could we represent both $$4 \times 10$$ and $$10 \times 10$$ using the same diagram of A and B? Why or why not?” (Yes, because A could represent 4 or 10, and B represents 10 times that value.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy usamos diagramas para representar valores que son diez veces otros valores. Observamos algunos patrones cuando analizamos los valores” // “Today we used diagrams to represent values that are ten times as much as different values. We noticed some patterns when we analyzed the values.”

Display:

“¿Qué otras afirmaciones podemos hacer sobre este diagrama que sean siempre verdaderas?” // “What are some other statements we can make about this diagram that would always be true?” (The value of B is always the ten times the value of A. If you know the value of A you can always figure out the value of B using multiplication. If you know the value of B you can always figure out the value of A.)

Focus discussion on how the diagram shows that the value of A is ten times as much as the value of B no matter what the value of each rectangle is.

## Student Section Summary

### Student Facing

En esta sección, aprendimos a usar la multiplicación y la frase “_____ veces” para comparar dos cantidades.

Primero, usamos cubos y dibujos para representar las cantidades. Por ejemplo: Andre tiene 3 cubos y Han tiene 12. Comparamos el número de cubos así:

• Dijimos: “Han tiene 4 veces la cantidad de cubos que Andre tiene”.
• Dibujamos diagramas que muestran 3 cubos para Andre y 4 veces esa cantidad para Han.
• Escribimos la ecuación $$4 \times 3 = 12$$.

A medida que los números se hacían más grandes, dibujar todas las unidades de cada cantidad era menos práctico, así que usamos diagramas más sencillos y con números para representar el tamaño de las cantidades.

Si Andre tiene 30 cubos y Han tiene 4 veces esa cantidad, podemos representar la comparación con un diagrama como este:

Al final, comparamos cantidades en las que una cantidad es diez veces la otra. También recordamos algunos patrones numéricos cuando multiplicamos un número por 10.