# Lesson 18

Diagramas y ecuaciones para problemas en palabras

## Warm-up: Observa y pregúntate: Diagramas (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that diagrams can represent many operations, which will be useful when students connect diagrams to situations and equations in a later activity. While students may notice and wonder many things about these images, what operations the diagrams could represent is the important discussion point.

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Activity Synthesis

• “Ustedes ya conocían los dos primeros tipos de diagramas. Los vieron cuando representaron situaciones de suma y situaciones de multiplicación. En la lección de hoy, vamos a darle sentido al último tipo de diagrama” // “You’ve seen the first two types of diagrams before, when you represented addition situations and multiplication situations. We are going to make sense of the last type of diagram in today's lesson.”
• “¿Qué operaciones creen que podrían estar representadas por el último diagrama?” // “What operations do you think could be represented in the last diagram?” (It could be multiplication and addition. Like you multiply 4 times 5 and add it to 142.)
• “¿El último diagrama podría representar la suma y la multiplicación?” // “Could the last diagram represent addition and multiplication?”

## Activity 1: Clasificación de tarjetas: Situaciones, ecuaciones y diagramas (15 minutes)

### Narrative

The purpose of this activity is for students to connect two-step word problems, diagrams, and equations with a symbol for the unknown quantity. Interpreting and relating given representations prepare students to use these as tools for reasoning when they solve two-step word problems.

As students analyze written statements and other representations and make connections among them, they reason quantitatively and abstractly (MP2).

MLR8 Discussion Supports: Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “Observé _____, entonces agrupé . . .” // “I noticed _____ , so I matched . . . .” Encourage students to challenge each other when they disagree.
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: Visual-Spatial Processing, Attention

### Required Materials

Materials to Copy

• Card Sort: Situations, Equations, and Diagrams, Spanish

### Required Preparation

• Create a set of cards from the blackline master for each group of 4.

### Launch

• Groups of 4
• Distribute one set of pre-cut cards to each group of students.

### Activity

• “En estas tarjetas hay situaciones, ecuaciones y diagramas. Encuentren con su compañero los grupos de tarjetas que van juntas porque representan la misma situación. Prepárense para explicar sus decisiones” // “This set of cards includes situations, equations, and diagrams. Work with your partner to find the cards that belong together because they represent the same situation. Be prepared to explain your decisions.”
• 8 minutes: small-group work time

### Student Facing

Tu profesor te va a dar varias tarjetas que muestran situaciones, ecuaciones y diagramas.

Clasifica las tarjetas en grupos de forma que las tarjetas de cada grupo representen la misma situación. Prepárate para explicar cómo razonaste.

A Clare tenía 225 chaquiras. Una amiga le dio un paquete de 48 chaquiras. Luego, Clare usó 70 chaquiras para hacer un collar. ¿Cuántas chaquiras tiene ella ahora? $$225 - (6 \times 10) = {?}$$ Elena tiene 7 cuadernos. Cada cuaderno tiene 10 clips. Elena también tiene una caja de 225 clips. ¿Cuántos clips tiene Elena? $$225 + (6 \times 10) = {?}$$ $${?} = 225 + 48 - 70$$ Andre tenía 225 crayones. Él compró 6 paquetes más. Cada paquete tenía 10 crayones. ¿Cuántos crayones tiene Andre ahora? Diego tenía una colección de 225 tarjetas de béisbol. Un amigo le dio 35 tarjetas más. Después, Diego compró 72 tarjetas. ¿Cuántas tarjetas tiene Diego ahora? Han tenía 225 chaquiras. Después, él hizo 6 pulseras para sus amigos. Cada pulsera tiene 10 chaquiras. ¿Cuántas chaquiras tiene Han ahora? $${?} = (7 \times 10) + 225$$

### Activity Synthesis

• “¿Al clasificar las tarjetas, hubo casos en los que no estuvieron de acuerdo? ¿Cómo llegaron a un acuerdo?” // “Were there any cards whose placement you disagreed on? How did you come to an agreement?” (We went back and read the situation carefully together.)
• Choose a set of cards that belong together, such as B, D, and L, to discuss in detail. Ask, “¿De qué manera la ecuación y el diagrama representan la situación?” // “How do the equation and diagram represent the situation?”
• Attend to the language that students use to describe their matches and the situations, equations, and diagrams, giving them opportunities to describe them more precisely.

## Activity 2: Tiene sentido para mí: Un recorrido por el salón (20 minutes)

### Narrative

The purpose of this activity is for students to solve one of the problems from the card sort in the previous activity and examine their classmates' solutions to other problems. Students work in groups to create a poster of their solution. As students visit the posters, they leave comments about how they know the solution on the poster makes sense. As students make comments on the work of others, they critique the reasoning of others (MP3).

### Required Materials

Materials to Gather

### Launch

• Groups of 4
• Assign each group a situation from the previous activity.
• Give each group tools for creating a visual display and sticky notes.

### Activity

• “Ahora van a resolver en grupo uno de los problemas de la clasificación de tarjetas” // “Now you are going to solve a problem from the card sort with your group.”
• “Después de que hayan resuelto el problema juntos, hagan un póster que muestre su estrategia para resolverlo” // “After you’ve solved the problem together, create a poster that shows your solution strategy.”
• 8–10 minutes: small-group work time

### Student Facing

1. Tu profesor le va a asignar un problema a tu grupo. Juntos, resuelvan ese problema.
2. Hagan un póster con la solución de su grupo. Organicen su trabajo para que los demás lo puedan entender.
3. Cuando vayan a ver cada póster, piensen si la respuesta que ven tiene sentido y por qué.

Escojan un póster y hagan un comentario sobre la solución. Escriban en la nota adhesiva cómo saben que la respuesta tiene sentido.

### Student Response

If students try to find exact answers to determine if their peers’ solutions make sense, consider asking:

• “¿De qué formas podríamos decidir si la solución tiene sentido, sin resolver el problema?” // “What are some ways that we could determine if the solution makes sense without solving the problem?”
• “¿Cómo podríamos usar una estimación para saber si la solución tiene sentido? ¿Cómo podríamos usar redondeo?” // “How could we use estimating to determine if the solution makes sense? How could we use rounding?”

### Activity Synthesis

• Display posters around the room. If more than one group solved the same problem, consider grouping their posters together.
• Give each student a sticky note.
• “Cuando vayan a ver los pósteres, escojan un póster y dejen allí un comentario que explique cómo saben que la respuesta tiene sentido” // “As you visit the posters, leave a comment on one poster that explains how you know the answer makes sense.”

## Lesson Synthesis

### Lesson Synthesis

Display a corresponding set of cards that show a diagram, situation, and equation representing the same situation, such as B, D and L:

Elena has 7 notebooks. Each notebook has 10 paper clips in it. Elena also has a box of 225 paper clips. How many paper clips does Elena have?

$${?} = (7 \times 10) + 225$$

“¿Cómo les ayuda el diagrama o la ecuación a representar o a aclarar lo que entienden sobre la situación?” // “How does the diagram or equation help you illustrate or clarify your understanding of the situation?” (Seeing the parts of the situation in the diagram helps me understand how they go together, what we know, or what is missing. The equation helps me understand how the quantities in the situation are related to each other by addition or multiplication.)