# Lesson 4

Situaciones de división

## Warm-up: Conversación numérica: División (10 minutes)

### Narrative

The purpose of this Number Talk is for students to interpret a fraction as division of the numerator by the denominator. The strategies elicited here will be helpful later in the lesson when students match division situations, expressions, and diagrams. In this activity, students have an opportunity to notice and make use of structure (MP7) because the divisor stays the same.

### 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.”

### Activity

• 1 minute: quiet think time
• Keep problems and work displayed.
• Repeat with each problem.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$35 \div 7$$
• $$1 \div 7$$
• $$36 \div 7$$
• $$37 \div 7$$

### Activity Synthesis

• “¿Qué patrones observan?” // “What patterns do you notice?” (The solutions for $$35 \div 7$$, $$36 \div 7$$, $$37 \div 7$$ increase by $$\frac {1}{7}$$)
• “Si continuamos aumentando el dividendo en 1, ¿cuál será el siguiente cociente entero?” // “If we kept increasing the dividend by 1, what would be the next whole number quotient?” (6)

## Activity 1: Libras de arándanos (20 minutes)

### Narrative

The purpose of this activity is for students to move back and forth between equations, situations, and diagrams. Each member of a group is assigned one of the representations to begin with and all of the representations can represent the same situation. This is the first time in this unit students have been asked to write situations that represent division equations or diagrams. As students work, ask them to explain how the diagram represents the number of objects being shared and the number of equal shares.

Students go back and forth between equations, situations, and diagrams, interpreting the diagrams and equations and creating situations that these diagrams and equations represent (MP2).

Representation: Access for Perception. Synthesis: Invite students to identify correspondences between representations. Make connections between representations visible through gestures and labeled displays.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing

### Launch

• Display image:
• “¿Dónde ven situaciones de división en esta imagen?” // “Where do you see division situations in this picture?” (The berries are split into 6 groups. )
• Groups of 3
• Assign groups. Make sure each student knows who is Partner A, Partner B, and Partner C.

### Activity

• 6–8 minutes: independent work time  (problems 1, 2, 3)
• 4–5 minutes: group discussion
• Monitor for students who write division situations, for $$a \div b$$, where $$a$$ represents the amount to be shared and $$b$$ represents the number of equal shares.

### Student Facing

1. Completa las partes que faltan en la tabla. Prepárate para explicar cómo pensaste.
2. Discutan sus soluciones en grupo. ¿En qué se parecen? ¿En qué son diferentes?

Compañero A

Ecuación Situación

$$4 \div 6 = \frac{4}{6}$$

Diagrama

Compañero B

Ecuación Situación

Diagrama

Compañero C

Ecuación Situación

Seis estudiantes comparten
4 libras de arándanos.
¿Cuántas libras de arándanos

Diagrama

### Student Response

If students do not think of a situation that is represented by the equation, show them the image from the launch and ask, “¿Cómo puede la ecuación representar una situación en la que unas personas comparten unas libras de arándanos?” // “How can the equation represent some people sharing some pounds of blueberries?”

### Activity Synthesis

• Select 2–3 students to share their problems and solutions. Show a variety of accurate student representations.
• Display equation: $$4 \div 6 = \frac{4}{6}$$.
• “¿Cómo les ayudó la ecuación a encontrar el número de objetos y de partes iguales en su historia?” // “How did you use the equation to help determine the number of objects and equal shares in your story?” (The numerator is the number of objects. The denominator is the number of shares. This story needs 4 objects and 6 equal shares.)
• Display image:
• “¿Cómo les ayudó el diagrama a encontrar el número de objetos y de partes iguales en su historia?” // “How did you use the diagram to help determine the number of objects and equal shares in your story?” (The number of objects is the number of wholes or large rectangles and the number of shares is the number of equal pieces each rectangle is divided into. There are 4 objects and 6 equal shares.)
• “¿En qué se parecen sus historias? ¿En qué son diferentes?” // “How were your stories the same? How were they different?” (There were 4 objects in each and there are divided into 6 equal shares, but our contexts were different.)

## Activity 2: Gramos de oro (15 minutes)

### Narrative

The purpose of this activity is for students to solve problems about the same context. The context for both problems is equally splitting some gold. There are three related quantities:

• the total amount of gold collected
• the total number of friends who collect the gold
• the amount of gold each person gets

In previous lessons, students solved problems where the unknown was the amount each person gets after some people shared something equally. In this activity, the unknowns are the amount of gold the friends share and the number of friends sharing the gold. Students may need to draw diagrams to interpret the unknown. During the synthesis, connect the meaning of division to the meaning of a fraction.

Engagement: Provide Access by Recruiting Interest. Synthesis: Invite students to generate a list of additional examples of amounts that they can split that connect to their personal backgrounds and interests.
Supports accessibility for: Conceptual Processing, Attention

### Launch

• Display first image.
• “Cuenta una historia sobre lo que ocurre en esta imagen” // “Tell a story about what is happening in this image.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Display second image.
• “Esto es oro en polvo” // “This is gold dust.”
• “Vamos a resolver algunos problemas sobre el lavado de oro” // “We are going to solve some problems about panning for gold.”

### Activity

• 5-8 minutes: partner work time
• Monitor for students who:
• draw a diagram to explain or show the relationship between the number of friends and the denominator.
• use words to explain or show the relationship between the number of friends and the denominator.

### Student Facing

1. Un grupo de 3 amigos pasó la tarde lavando oro. Después, compartieron el oro equitativamente. Si cada amigo recibió $$\frac{4}{3}$$ gramos de oro, ¿cuánto oro recolectaron entre todos? Explica o muestra tu razonamiento.
2. Un grupo de amigos pasó la tarde lavando oro. Después, compartieron el oro equitativamente. Si entre todos recolectaron 5 gramos de oro y cada uno recibió $$\frac{5}{6}$$ de gramo de oro, ¿cuántos amigos compartieron el oro? Explica o muestra tu razonamiento.

### Student Response

If students don’t have a way to start the problem, ask: “¿Qué sabes sobre el problema? ¿Qué te falta descifrar aún?” // “What do you know about the problem? What are you still trying to figure out?”

### Activity Synthesis

• Invite previously selected students to share their responses.
• Display the first situation that describes 3 friends sharing gold and each friend gets $$\frac{4}{3}$$ grams of gold.
• “¿Qué sabemos sobre la situación?” // “What do we know about the situation?” (There are 3 friends and each friend gets $$\frac{4}{3}$$ grams of gold.)
• Display: $$\frac{4}{3}$$
• “¿Qué representa $$\frac{4}{3}$$ en esta situación en la que unos amigos comparten oro?” // “How does $$\frac{4}{3}$$ represent the situation of friends sharing gold?” (It is the amount of gold each friend got.)
• “¿Qué representa el 3 en $$\frac{4}{3}$$?” // “What does the 3 represent in $$\frac{4}{3}$$?” (Thirds. The amount they collected was split into 3 equal parts because there are 3 friends.)
• “¿Qué representa el 4 en $$\frac{4}{3}$$?” // “What does the 4 represent in $$\frac{4}{3}$$?” (It represents how many thirds there are and also how many friends there are.)

## Lesson Synthesis

### Lesson Synthesis

Display equation: $$5 \div ? = \frac{5}{6}$$

“¿Cómo representa esta ecuación la situación en la que unos amigos comparten oro?” // “How does this equation represent the situation about friends sharing gold?” (5 grams of gold are being shared by some friends and each friend gets $$\frac{5}{6}$$ grams of gold.)

“¿Qué número hace que la ecuación sea verdadera? ¿Cómo lo saben?” // “What number makes this equation true? How do you know?” ($$5 \div 6 = \frac{5}{6}$$; The first number in the division equation is the numerator, the number of objects being divided, and the second number is the denominator, the number of equal shares.)