# Lesson 25

Dividamos decimales entre decimales

## Warm-up: Conversación numérica: El mismo o diferente (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have to divide whole numbers by decimals. These understandings help students develop fluency and will be helpful later in this lesson when students divide decimals greater than 1 by decimals less than 1.

### 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.

• $$20 \div 2$$
• $$2 \div 0.2$$
• $$50 \div 2$$
• $$5 \div 0.2$$

### Activity Synthesis

• “¿Por qué $$50 \div 2$$ y $$5 \div 0.2$$ tienen el mismo valor?” // “Why do $$50 \div 2$$ and $$5 \div 0.2$$ have the same value?” (Because 5 is 50 tenths and I am dividing that into groups of 2 tenths so that's $$50 \div 2$$.)

## Activity 1: Dividamos entre una décima y entre una centésima (15 minutes)

### Narrative

The purpose of this activity is for students to divide decimal numbers by 0.1 and 0.01. They are given diagrams to help see that there are 10 tenths in each whole and 100 hundredths in each whole. The diagrams are not labeled with the whole so that the same diagram which shows $$1.6 \div 0.1 = 16$$ can be interpreted as whole number division showing $$160 \div 10 = 16$$. This dual way of interpreting one diagram is highlighted in the synthesis. When students interpret the diagram as representing two different equations they attend to precision in the meaning each part of the diagram (MP6).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “¿Qué lenguaje o qué otros tipos de detalles les ayudaron a entender las presentaciones?” // “What kinds of additional details or language helped you understand the displays?”, “¿Tienen preguntas sobre el lenguaje o algún otro detalle?” // “Were there any additional details or language that you have questions about?”, and “¿Alguien resolvió el problema de la misma manera, pero lo explicaría de otra forma?” // “Did anyone solve the problem the same way, but would explain it differently?”

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• Describe how Jada's diagram shows the value of $$1.6 \div 0.1$$ as 16.
• Describe how Jada's diagram also represents the expression $$160 \div 10$$.

### Student Facing

1. Jada dibujó este diagrama para encontrar el valor de $$1.6 \div 0.1$$.
1. Describe cómo se muestra 1.6 en el diagrama.

2. Describe cómo se muestran 16 grupos de 1 décima en el diagrama.
3. Describe cómo se muestra el valor de $$1.6 \div 0.1$$ en el diagrama.
4. Describe cómo el diagrama representa también la expresión $$160 \div 10$$.
2. Explica cómo este diagrama representa la expresión $$1.3 \div 0.01$$.
1. ¿Cuál es el valor de $$1.3 \div 0.01$$? Explica o muestra tu razonamiento.

### Activity Synthesis

• Ask selected students to share their reasoning for each problem.
• Display: $$1.6 \div 0.1 = 160 \div 10$$
• “¿Cómo se muestra en el primer diagrama que esta ecuación es verdadera?” // “How does the first diagram show that this equation is true?” (If each large square is a whole then the number of shaded strips is $$1.6 \div 0.1$$ and if each large square is 100 then the number of those strips is $$160 \div 10$$. The same diagram represents both expressions so they are equal.)
• Display: $$1.3 \div 0.01 =130 \div 1$$
• “¿Cómo se muestra en el segundo diagrama que esta ecuación es verdadera?” // "How does the second diagram show that this equation is true?" (If each large square is a whole then the number of small pieces represents $$1.3 \div 0.01$$ and if each large square is 100 then the number of small pieces represents $$130 \div 1$$.)

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

### Narrative

In this activity, students practice finding quotients of decimals divided by 0.1 and 0.01. Students find the value of different expressions without the scaffold of a diagram. Monitor for these approaches:

• diagrams
• whole number quotient facts
• multiples of the divisor
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were necessary to solve the problem. Display the sentence frame: “La próxima vez que evalúe una expresión de división que tenga números decimales, prestaré atención a . . .” // “The next time I evaluate a division expression containing decimals, I will pay attention to . . . . “
Supports accessibility for: Conceptual Processing, Attention, Organization

### Required Materials

Materials to Copy

• Small Grids

• Groups of 2

### Activity

• 8 minutes: independent work time
• 2 minutes: partner discussion

### Student Facing

Encuentra el valor de cada expresión. Explica o muestra cómo razonaste.
1. $$5 \div 0.1$$
2. $$5 \div 0.01$$
3. $$0.5 \div 0.1$$
4. $$0.5 \div 0.01$$
5. $$0.02 \div 0.01$$
6. $$1.53 \div 0.01$$

### Activity Synthesis

• Display:

$$0.5 \div 0.1 = 5$$

$$0.50 \div 0.01 = 50$$

• “¿Cómo podemos usar el significado de los valores que están en las posiciones decimales para explicar estas ecuaciones?” // “How can we use the meaning of decimal place values to explain these equations?” (5 tenths is the same as 50 hundredths so that’s 5 groups of 0.1 or 50 groups of 0.01.)
• “¿Cómo nos ayuda usar el significado de los valores que están en las posiciones decimales a encontrar el valor de $$1.53 \div 0.01$$?” // “How can we use the meaning of decimal place values to help find the value of $$1.53 \div 0.01$$?” (The three is in the hundredths place so there are 3 one hundredths in three hundredths. The 5 is in the tenths place and there are 10 hundredths in each tenth so that's 50 more hundredths. There are 100 hundredths in one whole. That's 153 hundredths altogether in 1.53.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy dividimos un número decimal entre un número decimal y después encontramos muchos cocientes de números decimales” // “Today we divided a decimal by a decimal and then found lots of quotients involving decimals.”

Display:

$$1.25 \div 0.01 = 125$$

“¿Cómo sabemos que esta ecuación es verdadera?” // “How do we know this equation is true?” (If we multiply the dividend and the divisor by 100, we get $$1.25 \div 0.01 = 125 \div 1$$, which is 125. We can also see that there are 100 hundredths in 1, and 25 hundredths in 0.25, so there are 125 hundredths in 1.25.)

“¿En qué se parece dividir con números decimales a dividir con números enteros? ¿En qué se diferencia?” // “How is dividing with decimals the same as dividing with whole numbers? How is it different?” (I can use multiplication in both cases. I can draw a diagram in both cases. I use place value in both cases. With decimals I need to think carefully about the meaning of each digit. I think the diagrams are more helpful to get started with decimals to visualize the numbers I am working with.)

## Cool-down: Divide entre decimales (5 minutes)

### Cool-Down

En esta sección aprendimos a dividir con números decimales. Estudiamos varias formas de encontrar un cociente como $$3 \div 0.1$$. Podemos dibujar un diagrama que muestra que hay 10 grupos de 0.1 en cada unidad. Entonces, en 3 unidades hay $$3 \times 10$$ o 30 grupos de 0.1. Así, $$3 \div 0.1 = 30$$.
También podemos pensar en el valor posicional. Sabemos que 3 representa 30 décimas y que 0.1 es 1 décima, entonces $$3 \div 0.1$$ es equivalente a $$30 \div 1$$, que tiene el valor de 30. También podemos usar la multiplicación para encontrar el valor de $$3 \div 0.1$$. Sabemos que $$10 \times 0.1 = 1$$, así que $$30 \times 0.1 = 3$$. Entonces esto también muestra que $$3 \div 0.1 = 30$$.