# Lesson 22

Dividamos números enteros entre 0.1 y entre 0.01

## Warm-up: Conversación numérica: Recordemos la división entre fracciones unitarias (10 minutes)

### Narrative

The purpose of this number talk is for students to divide a whole number by a tenth and a hundredth, presented both as fractions and as decimals. This prepares students to work with and understand quotients of whole numbers by decimals in this and future lessons.

### Launch

• Display one problem.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep problems and work displayed.
• Repeat with each problem.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$1 \div \frac{1}{10}$$
• $$2 \div \frac{1}{10}$$
• $$1 \div 0.01$$
• $$2 \div 0.01$$

### Activity Synthesis

• “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “What is the same about these expressions? What is different?” (They all show a whole number divided by a number less than 1. Some have decimals and some have fractions, but the decimals and fraction are all 1 of some unit. They all have a value that is a whole number.)

## Activity 1: Patrones al dividir entre unidades decimales (15 minutes)

### Narrative

The purpose of this activity is for students to divide a whole number by 0.1 or 0.01. Students may think about the quotients in different ways. Monitor for students who

• use the hundredths grids to visualize the tenths and hundredths in 3 or 4 wholes
• use their understanding of fractions or decimals to identify the number of tenths and hundredths in 1 whole and then multiply by the number of wholes
• build on and solidify their reasoning from the warm-up

The work in this activity focusing on the tenths and hundredths in 3 or 4 wholes prepares students to generalize in the next activity where they explain how to find the number of tenths or hundredths in any whole number.

Representation: Internalize Comprehension. Begin by asking, “¿Este problema o situación les recuerda algo que hayamos visto, leído o hecho antes?” // “Does this problem/situation remind anyone of something we have seen/read/done before?”
Supports accessibility for: Memory, Attention

### Required Materials

Materials to Copy

• Small Grids

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 5–7 minutes: partner work time
• Monitor for students who:
• use fractions
• use diagrams
• use multiplication

### Student Facing

1. Encuentra el valor de cada expresión. Explica o muestra cómo razonaste. Si te ayuda, usa un diagrama.

1. $$3 \div 0.1$$
2. $$3 \div 0.01$$
3. $$4 \div 0.1$$
4. $$4 \div 0.01$$
2. ¿Qué patrones observas?

### Student Response

If students do not find the value of $$3 \div 0.1$$, ask, “¿Cómo puedes usar un diagrama para mostrar $$1 \div 0.1$$? ¿Cómo puede ayudarte el diagrama a encontrar el valor de $$3 \div 0.1$$?” // “How can you use a diagram to show $$1 \div 0.1$$? How can your diagram help you find the value of $$3 \div 0.1$$?”

### Activity Synthesis

• Ask previously selected students to share their strategies for finding the value of the division expressions.
• “¿Cómo muestran las distintas representaciones que hay 10 décimas en 1 unidad y 100 centésimas en 1 unidad?” // “How do the different representations show that there are 10 tenths in 1 whole and 100 hundredths in 1 whole?” (With the diagram I can see each tenth and each hundredth in the whole. With fractions or decimals, I just know how many tenths or hundredths there are in 1 and can use that to figure out how many there are in 3 or 4.)
• Display equations:
$$3 \div 0.01 = 300$$
$$4 \div 0.01 = 400$$
• “¿Qué patrones observan?” // “What patterns do you notice?” (I notice that I multiply by 100 to find how many 0.01s there are in 3 or 4.)

## Activity 2: Dividamos números enteros entre decimales (20 minutes)

### Narrative

The purpose of this activity is for students to describe the patterns they notice when they divide a whole number by one tenth and one hundredth (MP7). Students have the option of using a diagram recalling work from the previous activity. The reasoning students develop here prepares them to:

• evaluate more complex quotients with decimals in future lessons.
• observe patterns when multiplying or dividing by powers of 10 which will be a focus in the next unit.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

### Required Materials

Materials to Copy

• Small Grids

### Launch

• Groups of 2
• “Van a usar los patrones que observaron para explicar cómo dividir cualquier número entero entre una décima y entre una centésima” // “You are going to use the patterns you noticed to explain how to divide any whole number by a tenth and a hundredth.”

### Activity

• 6–8 minutes: independent work time
• Monitor for students who notice:
• that when they divide a whole number by one tenth, the quotient is ten times larger than the whole number
• that when they divide a whole number by one hundredth, the quotient is one hundred times larger than the whole number

### Student Facing

1. Describe cómo puedes encontrar el valor de cualquier número entero dividido entre 0.1. Si te ayuda, usa un diagrama.
2. Describe cómo puedes encontrar el valor de cualquier número entero dividido entre 0.01. Si te ayuda, usa un diagrama.

### Student Response

If students do not describe how they could find the value of any whole number divided by 0.1 or 0.01, ask them to draw a diagram to show $$1 \div 0.1$$ and ask, “¿Cómo puedes usar este diagrama para averiguar el valor de cualquier número entero dividido entre una décima?” // “How can you use this diagram to figure out the value of any whole number divided by one tenth?”

### Activity Synthesis

MLR1 Stronger and Clearer Each Time

• “Compartan con su compañero su explicación de cómo encontrarían el valor de” // “Share with your partner your explanation of how you would find the value of”
• “cualquier número entero dividido entre una décima” // “any whole number divided by one tenth”
• “cualquier número entero dividido entre una centésima” // “any whole number divided by one hundredth”
• “Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su explicación” // “Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 2 minutes: structured partner discussion
• Repeat with 2 different partners.
• If needed, display question starters and prompts for feedback.
• “¿Puedes dar un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
• “¿Tu explicación funciona para los ejemplos que estudiamos en la primera actividad?” // “Can the examples we studied in the first activity fit in your explanation?”
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

## Lesson Synthesis

### Lesson Synthesis

“Hoy dividimos números enteros entre una décima y entre una centésima” // “Today, we divided whole numbers by one tenth and one hundredth.”

Display expressions:

$$5 \div 0.1$$
$$5 \div 0.01$$

“Descríbanle a su compañero cómo encontrar el valor de cada una de estas expresiones” // “Describe to your partner how to find the value of each of these expressions.” (To divide by 0.1 I just multiply by 10 so that's 50 and to divide by 0.01 I multiply by 100.)

“Ahora discutan con su compañero por qué su estrategia funciona” // “Now discuss with your partner why your strategy works.” (There are 10 tenths in 1 so that's $$5 \times 10$$ or 50 in 5. There are 100 hundredths in 1 so that's $$5 \times 100$$ or 500 in 5.)