# Lesson 2

Interpretemos representaciones de comparaciones multiplicativas

## Warm-up: Cuántos ves: Varias veces (10 minutes)

### Narrative

The purpose of this How Many Do You See is for students to use grouping strategies to describe the images they see.

In the synthesis, students describe how two images can be used to describe a multiplicative comparison and connect the images to a multiplication equation.

### Launch

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see and how you do see them?”
• Flash the image.
• 30 seconds: quiet think time

### Activity

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses. Use multiplication equations when appropriate.
• Repeat for each image.

### Student Facing

¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?

### Activity Synthesis

• “¿Cómo muestra esto que el segundo rectángulo tiene 2 veces la cantidad que tiene el primer rectángulo?” // “How does this show that the second rectangle has 2 times as many as the first rectangle?”
• “¿Cómo podemos escribir una ecuación que muestre esta comparación?” // “How could we write an equation that shows this comparison?” ($$6 = 2 \times 3$$ or $$2 \times 3 = 6$$ or $$3 \times 2 = 6$$)

## Activity 1: Representemos “varias veces” (20 minutes)

### Narrative

The purpose of this activity is for students to analyze and describe how images and diagrams can show “$$n$$ times as many”. Students generate ideas for how to use a multiplication equation to represent the comparison.

Students begin by interpreting an image in which the multiplier (3) and the numbers are given. They explain how some number of times of the smaller amount can be seen in the larger amount. Next, they create their own diagram and see different ways of representing the iterations (groups of) the smaller amount to create the larger amount.

During the activity, make connecting cubes accessible for students who may choose to use them for problem solving—either to reason about the quantities or to explain their reasoning.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display the image of Mai's cubes and Kiran's cubes.
• “¿De qué manera estos cubos representan 3 veces una cantidad?” // “How do these cubes represent 3 times as many?” (Mai has 6 cubes and Kiran has 2. Mai has 3 groups of 2 cubes. Mai has 6 cubes and Kiran has 2. Three times as many as 2 is 6, or 3 times 2 is 6.)
• 1 minute: quiet think time

### Activity

MLR7 Compare and Connect
• “Creen una presentación visual que muestre cómo pensaron en los cubos de cada problema e incluyan detalles para ayudar a otros a entender sus ideas” // “Create a visual display that shows your thinking about the cubes in each problem and include details to help others understand your thinking.”
• 6–8 minutes: independent or group work
• 3 minutes: gallery walk
• “¿Cómo muestra cada representación ‘varias veces’ una cantidad?” // “How does each representation show ‘times as many’?”
• 30 seconds quiet think time
• 1 minute: partner discussion
• Monitor for students who create diagrams that are similar to connecting cube images and discrete tape diagrams to share in the synthesis.

### Student Facing

1. Jada tiene 4 veces la cantidad de cubos que tiene Kiran. Dibuja un diagrama que represente la situación.
2. Diego tiene 5 veces la cantidad de cubos que tiene Kiran. Dibuja un diagrama que represente la situación.
3. Lin tiene 6 veces la cantidad de cubos que tiene Kiran. ¿Cuántos cubos tiene Lin? Explica o muestra tu razonamiento.

### Student Response

Students may compare additively instead of multiplicatively when using cubes or drawing diagrams. Consider asking: “¿Cómo usarías los cubos para mostrar 3 más?” // “How would you use the cubes to show 3 more?” and “¿Cómo usarías los cubos para mostrar 3 veces?” //  “How would you use the cubes to show 3 times as many?”

### Activity Synthesis

• Have selected students share diagrams and explain how they show “times as many”.
• If needed, use cubes to represent statements.
• “¿Qué ecuación escribirían para comparar los cubos de Kiran con los de Jada?” // “How could you write an equation to compare Kiran’s and Jada’s cubes?”
• “¿Qué representan en la situación los números de la ecuación?” // “What do the numbers in the equation represent in the situation?” (Four is the “4 times as many”. Two is how many Kiran had. Eight is how many Jada had.)

## Activity 2: Diagramas para resolver problemas de comparación multiplicativa (15 minutes)

### Narrative

The purpose of this activity is for students to deepen their understanding of how diagrams and multiplication equations can represent “$$n$$ times as many”. Students explain how the diagrams and equations represent the situation. In order to match situations, diagrams, and equations, students reason abstractly and quantitatively (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 asocié . . .” // “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to look and listen for examples of multiplicative comparison in their own lives. Encourage them to share these throughout the unit.
Supports accessibility for: Conceptual, Processing, Language, Attention

### Launch

• Groups of 2
• “Por turnos, lean una descripción y encuentren un diagrama y una ecuación que también representen la situación. Explíquenle su razonamiento a su compañero” // “Take turns reading a description and finding a diagram and an equation that also represent the situation. Explain your reasoning to your partner.”

### Activity

• 5–7 minutes: partner work time
• Monitor for students who connect the factors and product of the equation to the situation and diagram
• If students finish early, give them blank index cards. Ask them to make several sets of matching representations, shuffle the cards, and trade them with another group that is also creating their own representations.

### Student Facing

Estos son cuatro grupos de descripciones, diagramas y ecuaciones en los que se comparan parejas de cantidades.

Asocia cada descripción con un diagrama y una ecuación que representen la misma situación. Prepárate para explicar tu razonamiento.

Anota tus correspondencias aquí:

Grupo 1: _____, _____, _____

Grupo 2: _____, _____, _____

Grupo 3: _____, _____, _____

Grupo 4: _____, _____, _____

### Student Response

Students may consider the total amount represented on a card rather than making comparisons. For example, on card K, they might see 4 groups of 3 in the two quantities combined. Consider asking:

• “¿Cómo compararías las dos cantidades que se muestran en la tarjeta?” // “How would you compare the two quantities shown on the card?”
• “¿En qué parte de la ecuación de la tarjeta E ves que se están comparando dos cantidades?” // “Where might you see two quantities being compared in the equation on card E?”

### Activity Synthesis

• Select students to share their matches.
• Record student explanations to show how they connected the diagrams and equations (display or draw the diagrams and equations and annotate).

## Lesson Synthesis

### Lesson Synthesis

“Hoy estudiamos una forma nueva de usar ecuaciones de multiplicación. Las ecuaciones de multiplicación pueden describir grupos iguales, pero también representan una comparación multiplicativa” // “Today we looked at a new way to use multiplication equations. Multiplication equations can describe equal groups, but they also represent multiplicative comparison.”

Display a student's representation of Kiran's cubes and Jada's cubes from the first activity.

“Expliquen cómo ven $$4 \times 2 = 8$$ en este diagrama” // “Explain how you see $$4 \times 2 = 8$$ in this diagram.” (There are 4 groups of 2 cubes each or Jada has 4 times as many as Kiran does)

“En este caso, el valor del producto es 8. ¿Cómo se está comparando con 8 uno de los factores en este diagrama?” // “In this case, the value of the product is 8. How is one of the factors being compared to 8 in this diagram?”

If not mentioned by students, highlight that:

• “En el primer caso, la ecuación de multiplicación representa grupos iguales de objetos” // “In the first case, the multiplication equation represents equal groups of objects.”
• “En el segundo caso, la ecuación de multiplicación representa una comparación multiplicativa. Esto nos permite ver cuántas veces es el número de objetos que tiene una persona comparado con el número que tiene otra persona” //  “In the second case, the multiplication equation represents multiplicative comparison. It allows us to see how many times as many objects one person has compared to another person.”