Lesson 1

Varias veces

Warm-up: Observa y pregúntate: Comparemos cubos (10 minutes)

Narrative

The purpose of this warm-up is to elicit comparison language from students, which will be useful when students represent situations that involve “_____ times as many” later in the lesson. While students may notice and wonder many things about the cube images, the language used to compare the two sets of images is most important for discussion. Consider using actual connecting cubes rather than the display image.

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 compañero 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

• “¿En qué se parecen las imágenes?” // “How are the images alike?” (Both show Andre has 3 cubes and Han and Mai have more cubes.)
• “¿En qué son diferentes?” // “How are they different?” (Sample responses:
• Han has cubes in four colors and 3 cubes of each color. Mai has two different colors of cubes—3 cubes in one color and 4 cubes in another color.
• Han has 9 more cubes than Andre. Mai has 4 more cubes than Andre.)

Activity 1: El doble (10 minutes)

Narrative

The purpose of this activity is for students to build on what they know about the language of “twice” and “twice as many” to represent comparison situations.

In this activity, students are encouraged to represent the situation in a way that makes sense to them, though discrete representations (cubes or drawings) are the focus of the activity synthesis. The representations used in this activity serve as a foundation for the more abstract tape diagrams that will be used later in the section.

Required Materials

Materials to Gather

Launch

• Groups of 2
• “¿Qué significa que Han tenga el doble de cubos que Andre?” // “What does it mean for Han to have twice as many cubes as Andre?” (It means he has double. It means he has 2 times as many.)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

Activity

• 3 minutes: independent work time
• “Compartan su trabajo con su compañero. Expliquen cómo saben que Han tiene el doble de cubos que los que tiene Andre” // “Share your work with your partner. Explain how you know Han has twice as many cubes as Andre has.”
• “Si les queda tiempo, con su compañero, invéntense otra cantidad de cubos para Andre y Han” // “If you have time, work with your partner to come up with another amount of cubes for Andre and Han.”
• 4 minutes: partner discussion
• Monitor for students who organize their cubes (or drawings) into groups to show Andre’s cubes “twice” for Han.

Student Facing

Andre tiene algunos cubos. Han tiene el doble de cubos que Andre.

Usa cubos, imágenes u otros diagramas para mostrar cuántos cubos podría tener Andre y cuántos cubos podría tener Han.

Prepárate para explicar cómo se muestra en tu representación que Han tiene el doble de cubos que Andre.

Student Response

Students may represent “2 more than” and not “2 times as many” in this task. If so, consider asking “Muestra con cubos en qué son diferentes ‘2 más’ y el doble” // “Show with cubes how 2 more and twice as many are different.”

Activity Synthesis

• Select 3–4 different student examples (a mix of cubes and drawings).
• “¿Cómo muestra cada representación que Han tiene el doble de cubos que Andre?” // “How does each representation show that Han has twice as many as Andre?” (In each set, Andre’s number of cubes is repeated 2 times.)
• “Otra forma de decir ‘el doble’ es ‘2 veces’” // “Another way of saying twice as many is 2 times as many.”
• “¿Cómo saben que el número de cubos de Andre es ‘2 veces’ la cantidad de cubos de Han?” // “How do you know that Andre’s number of cubes is ‘2 times’ in Han’s amount?” (Andre’s number of cubes is repeated twice. Sometimes they are stuck together in one stick and other times they are separate.)

Activity 2: Varias veces (20 minutes)

Narrative

The purpose of this activity is to extend the intuitive idea of representing “twice as many” to represent “4, 6, and 8 times as many.” Although students are prompted to draw to represent each situation, keep the connecting cubes accessible for students to use as needed.

MLR8 Discussion Supports. Synthesis: During group presentations, invite the student(s) who are not speaking to follow along and point to the corresponding parts of the display.

Required Materials

Materials to Gather

Launch

• Groups of 2
• “Piensen cómo podrían usar cubos encajables o dibujos para resolver cada problema” // “Think about how you might use the connecting cubes or drawings to solve each problem.”
• 2 minutes: quiet think time

Activity

• 5 minutes: partner work time
• Monitor for students who:
• organize their cubes or drawings to visualize the comparison.
• reason using multiplication expressions or equations.

Student Facing

En cada situación, haz un dibujo que muestre el número de cubos que tienen los estudiantes. Prepárate para explicar tu razonamiento.

1. Andre tiene los siguientes cubos y Han tiene 4 veces esa cantidad.

2. Priya tiene los siguientes cubos y Jada tiene 6 veces esa cantidad.

3. Tyler tiene los siguientes cubos y Mai tiene 8 veces esa cantidad.

4. ¿Qué número representa “8 veces 2”? Muestra tu razonamiento.

Student Response

Students may write expressions to reason about the number of cubes in each problem. If students write addition expressions, ask them what multiplication expressions they could write. Then, ask what each number represents.

Activity Synthesis

• Invite selected students to share their representations and their reasoning.
• For each question, ask, “¿En qué parte de la representación de _____ ven _____ veces la cantidad?” // “Where do you see _____ times as many in _____’s representation?”
• “¿Que significa tener ‘_____ veces’ una cantidad?” // “What does it mean to have ‘_____ times as many’?” (It means we have that many groups of the original number of cubes.)
• For students who reason using multiplication expressions or equations, ask, “¿Cómo muestra esta ecuación que hay ‘varias veces’ una cantidad? ¿Qué representa cada número?” // “How does this equation show ‘times as many’? What does each number represent?” (In the example of Priya and Jada, the equation $$6 \times 3 = 18$$ represents 6 times as many as 3. The 3 is the number of cubes Priya has, 18 represents the number of cubes that Jada has, which is 6 times as many as Priya.)

Activity 3: Hagamos $n$ veces [OPTIONAL] (15 minutes)

Narrative

The purpose of this optional activity is for students to practice representing “$$n$$ times as many” with cubes and diagrams. Students may not have enough cubes to build each comparison. This gives students an opportunity to make sense of each quantity and how it relates to their corresponding cubes or diagram (MP2). Encourage them to draw a diagram to represent the cubes.

Action and Expression: Develop Expression and Communication. Provide access to a pre-formatted page that says, “Muestra _____ veces _____” // “Show _____ times as many as _____.” Invite Partner A to place their connecting cubes in the second blank and Partner B to place the number cube in the first blank.
Supports accessibility for: Language, Organization, Attention

Required Materials

Materials to Gather

Materials to Copy

• Times as Many Recording Mat, Spanish

Required Preparation

• Each group of 2 needs 40 connecting cubes.

Launch

• Groups of 2
• Demonstrate the activity with a student as your partner.
• Give each group a copy of the recording mat from the blackline master, a number cube, and connecting cubes.

Activity

• “¿Qué preguntas tienen sobre la actividad?” // “What questions do you have about the activity?”
• 8 minutes: partner work time

Student Facing

Trabaja con un compañero en esta actividad.

1. Compañero A: arma un grupo de cubos encajables que tenga de 2 a 6 cubos y muéstraselo al compañero B.

Compañero B: lanza un dado numérico. Llamemos al número que sacaste $$n$$.

2. Compañero A: usa los cubos encajables o un diagrama para mostrar $$n$$ veces la cantidad de cubos del grupo original. Muestra tu razonamiento.

Compañero B: revisa el trabajo de tu compañero y habla con él sobre cualquier cosa con la que no estés de acuerdo.

3. Intercambien roles y repitan lo anterior.

Activity Synthesis

• “¿Qué método usaron para representar las $$n$$ veces?” // “What was your method for representing the $$n$$ times as many?” (Sample responses: I built that many groups of the original group size. I multiplied the original number by the number rolled and counted out that many cubes.)

Lesson Synthesis

Lesson Synthesis

“Hoy aprendimos cómo representar ‘___ veces’ una cantidad” // “Today, we learned how to represent ‘___ times as many.’”

Display the images from the warm-up.

“¿Cuál de estas imágenes muestra 4 veces la cantidad de la otra? ¿Cómo lo saben?” // “Which of these pictures shows 4 times as many? How do you know?” (Han has 4 times as many cubes as Andre. He has 4 groups of 3 and Andre has 1 group of 3.)

“¿Cuántos cubos tendría yo si tuviera 6 veces la cantidad de cubos que tiene Andre?, ¿y si tuviera 8 veces la cantidad de cubos que tiene Andre?” // “How many cubes would I have if I have 6 times as many cubes as Andre? 8 times?”