# Lesson 10

Multiplicación y división

## Warm-up: Cuál es diferente: Multiplicación y división (10 minutes)

### Narrative

This warm-up prompts students to compare four representations. The reasoning here prepares students to connect the previous multiplication work to the division work of this lesson. It gives students an opportunity to use precise terms such as “factors,” “product,” and “quotient” in making comparisons (MP6). During the synthesis, ask students to explain the meaning of any terminology they use.

### Launch

• Groups of 2
• Display images and expression.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2-3 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

### Activity Synthesis

• “En cada caso, ¿qué ecuación puede estar representada por el diagrama?” // “What equation can each diagram represent?” (The array represents $$4\times6=24$$ because there are 4 rows of 6 dots, and there are 24 dots in the array. The area diagram could represent $$4\times7=28$$ or $$28\div4=7$$ since the sides are 4 and 7, and the area is 28. The tape diagram could represent $$24\div3={?}$$ or $$3\times{?}=24$$ because we know the total is 24 and there are 3 groups, but we don’t know how many are in each group.)

## Activity 1: Clasificación de tarjetas: Encuentra la pareja (25 minutes)

### Narrative

The purpose of this activity is for students to relate multiplication and division using a variety of representations. Students are given a card with a base ten diagram, tape diagram, area diagram, multiplication equation with a missing factor, or division equation. Students need to find the other student who has the card that matches their card. Each pair of cards includes a division equation. After students find the student with the matching card, they work together to create another diagram and a division situation that their cards could represent (MP2).

Here are images of the cards for reference:

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Organization, Social-Emotional Functioning

### Required Materials

Materials to Gather

Materials to Copy

• Find the Match

### Required Preparation

• The blackline master has 24 cards. Copy and cut enough cards so that each student can have one card.

### Launch

• Groups of 2
• Give one card to each student.
• Display an example of each type of representation shown on the cards (division equation, multiplication equation with missing factor, area diagram, tape diagram, and base ten diagram).
• “En el calentamiento vimos que diagramas y ecuaciones distintos pueden representar la misma situación. Por ejemplo: 4 grupos de 6 o 4 veces 6” // “We saw in the warm-up that different diagrams and equations can represent the same situation such as 4 groups of 6 or 4 times 6.”
• “Piensen qué situación o qué cantidades representan sus tarjetas. Luego, piensen cuál podría ser otra representación de la situación o de las cantidades” // “Think about what situation or quantities your card represents. Then, think about what another representation of the situation or quantities might look like.”
• 1 minute: quiet think time

### Activity

• “Busquen un estudiante que tenga una tarjeta que represente la misma situación o las mismas cantidades de su tarjeta. Prepárense para explicar por qué sus tarjetas van juntas” // “Find a student whose card represents the same situation or quantities as your card does. Be ready to explain why your cards belong together.”
• 2-3 minutes: partner work time
• Invite 2-3 groups to share their matches and how they knew they matched.
• “Con su compañero, hagan un póster que incluya sus tarjetas y un diagrama y una situación que le correspondan a su ecuación de división. Vamos a usarlos para hacer un recorrido por el salón, así que organicen su trabajo para que los demás puedan entenderlo” // “Work with your partner to create a poster that includes your cards and a diagram and situation that match your division equation. We’ll use these for a gallery walk, so organize your work so others can understand it.”
• Give students glue or tape and tools for creating a visual display.
• 10-15 minutes: partner work time

### Student Facing

Tu profesor te va a dar una tarjeta que muestra una ecuación o un diagrama.

1. Busca un compañero que tenga una tarjeta que represente la misma situación o las mismas cantidades de tu tarjeta. Prepárate para explicar cómo sabes que las tarjetas van juntas.
2. Con tu compañero, haz un póster que incluya:

1. sus tarjetas
2. otro diagrama que pueda estar representado por su ecuación de división
3. una situación que pueda estar representada por su ecuación de división

Muestren cómo pensaron. Organicen su trabajo para que los demás puedan entenderlo.

### Activity Synthesis

• Display students' posters around the room.

## Activity 2: Recorrido por el salón: Encuentra la pareja (10 minutes)

### Narrative

The purpose of this activity is to reinforce students' understanding of the relationship between multiplication and division by examining different representations of that relationship.

MLR7 Compare and Connect. Synthesis: After the Gallery Walk, lead a discussion comparing, contrasting, and connecting the different representations. “¿Qué tuvieron en común las representaciones?” // “What did the representations have in common?” “¿Cómo se ve la relación que hay entre la multiplicación y la división en cada representación?” // “How did the relationship between multiplication and division show up in each representation?” To amplify student language, and illustrate connections, follow along and point to the relevant parts of the displays as students speak.

### Required Preparation

• Keep posters from the previous activity displayed.

• Groups of 2

### Activity

• Arrange for half of the groups to stand at their poster and answer questions while the other half visit their posters.
• 8–10 minutes: gallery walk
• Ask groups to switch roles after 4–5 minutes.

### Student Facing

Cuando estés viendo los pósteres con tu compañero:

1. Describe una representación que hayas visto que sea diferente a las que mostraste en tu póster.
2. Escoge un póster distinto al tuyo. Describe una forma en la que el póster muestra la relación que hay entre la multiplicación y la división.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Hoy hicimos pósteres que mostraban formas de representar la división” // “Today we created posters that showed ways to represent division.”

“¿Cómo nos muestra un diagrama de área la relación que hay entre la multiplicación y la división?” // “How does an area diagram show us the relationship between multiplication and division?” (It shows that multiplying is like finding the area of a rectangle when the two side lengths are known, and dividing is like finding a side length when we know the area and the other side length.)

“¿Cómo muestra un diagrama de cinta o un diagrama de grupos iguales la multiplicación y la división?” // “How does a tape diagram or equal-groups diagram show multiplication and division?” (Both show multiplying as a way to find the total when we know the number of groups and how many in each group, and dividing as a way to find either the number of groups or the size of each group when the total is known.)

“¿Qué aspectos de los pósteres que vieron los ayudaron a aclarar las matemáticas que usaron sus compañeros?” // “What were some aspects of the posters you saw that helped make the math your classmates used clear for you?” (Clear labels on diagrams that helped me understand their thinking. Units on their answers. When other students wrote their explanations, it helped me understand their thinking.)