Lesson 5

Problemas de comparación de uno y dos pasos

Warm-up: Cuál es diferente: Algo falta (10 minutes)

Narrative

This warm-up prompts students to carefully analyze and compare ways to represent multiplicative comparison.

Launch

• Groups of 2
• Display the image.
• “Escojan uno 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
• Record responses.

Student Facing

¿Cuál es diferente?

1.

$$6 \times 7 = 42$$

2.

42 es 6 veces _____.

Activity Synthesis

• “¿Cómo se muestra un número desconocido en cada representación?” // “How does each representation show an unknown?”
• “¿Cuál es el número desconocido en cada representación? Expliquen cómo lo saben” // “What is the unknown in each representation? Explain how you know.”
• Record student responses.
• Consider asking students to create a situation that can be described by these representations.

Activity 1: La feria del libro (20 minutes)

Narrative

The purpose of this activity is for students to represent and solve problems in context involving multiplicative comparison. The first 2 questions only require one operation, and the last one requires 2 operations. Students must interpret what values are unknown and make a plan for how to represent these values. Students may use one or more equations to represent the situation.

Students reason abstractly and quantitatively when they make sense of the given information and use it to find the different numbers of books (MP2).

Launch

• Groups of 2
• “Vamos a leer este problema 3 veces” // “We are going to read this problem 3 times.”
• 1st Read: (First paragraph) “Para la feria del libro de este año, en una escuela pidieron 16 libros de experimentos de ciencias y 6 veces esa cantidad de libros de imágenes” // “The book fair ordered 16 science experiment books and 6 times as many picture books.”
• “¿De qué se trata esta historia?” // “What is this story about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• 2nd Read: (First and second paragraphs) “Para la feria del libro de este año, en una escuela pidieron 16 libros de experimentos de ciencias y 6 veces esa cantidad de libros de imágenes.
El año pasado, en la escuela pidieron 4 veces la cantidad de libros de imágenes que pidieron este año y 4 veces la cantidad de libros de experimentos de ciencias que pidieron este año” //
“For this year’s book fair, a school ordered 16 science experiment books and 6 times as many picture books.

Last year, the school ordered 4 times as many picture books and 4 times as many science experiment books as they did this year.”

• “Mencionen las cantidades que podemos contar en esta situación” // “Name the quantities we can count in this situation.” (The number of books of each kind.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share and record all quantities.
• Reveal the questions.
• “¿De qué maneras diferentes podemos resolver estos problemas?” // “What are different ways we can solve these problems?”
• 30 seconds: quiet think time
• 1 minute: partner discussion

Activity

• “Resuelvan los problemas y compartan con un compañero lo que pensaron” // “Solve the problems and share your thinking with a partner.”
• 2–5 minutes: independent work time
• 1–2 minutes: partner share time
• Monitor for students who write equations, including equations that have a symbol for the unknown value to solve each problem.

Student Facing

Para la feria del libro de este año, en una escuela pidieron 16 libros de experimentos de ciencias y 6 veces esa cantidad de libros de imágenes.

El año pasado, en la escuela pidieron 4 veces la cantidad de libros de imágenes que pidieron este año y 4 veces la cantidad de libros de experimentos de ciencias que pidieron este año.

1. ¿Cuántos libros de imágenes pidieron este año?

2. ¿Cuántos libros de imágenes pidieron el año pasado?

3. ¿Cuántos libros de experimentos de ciencias más pidieron el año pasado que este año?

Activity Synthesis

• “¿En qué es diferente este problema a otros problemas que resolvimos en esta unidad?” // “How is this problem different from other problems we solved in this unit?” (We had to do more than one calculation to solve the problem.)
• Select 1–2 students share their solutions and equations that show multiple steps.

Activity 2: Más compras de la feria del libro (15 minutes)

Narrative

The purpose of this activity is for students to apply what they learned about interpreting and representing multiplicative comparison to solve multi-step problems. They also make connections between strategies for solving problems. The synthesis of this activity makes use of the 5 Practices. As students are working, identify students who clearly represent using multiple steps to solve the problem and those who do not show every step yet share the same answers (MP6). Compare students' samples that show the same solution in different ways. Make connections between samples that clearly show multiplicative comparison more discretely and those that use more abstract representations.

Representation: Internalize Comprehension. To support understanding of the situation, provide students with a template to draw a comic strip or storyboard. For example, the template might have two empty boxes labeled Tuesday morning and Tuesday afternoon for the first problem, and two empty boxes labeled Mai's purchases and total sales for the second problem.
Supports accessibility for: Conceptual Processing, Organization, Attention

• Groups of 2

Activity

• 5 minutes: independent work time
• “Comparen estrategias con un compañero. Puede que después de hablar con un compañero decidan reconsiderar lo que pensaron” // “Compare strategies with a partner. You may decide to revise your thinking after talking with a partner.”
• As students are sharing with partners, identify students who noticed that they missed a step and revised their thinking to include the step after talking with a partner.
• 5 minutes: partner discussion
• “¿Qué escucharon o qué vieron que les hizo reconsiderar lo que habían pensado?” // “What did you hear or see that made you decide to revise your thinking?” (I noticed that I missed a step when I heard or saw what my partners shared about Tuesday and Thursday.)
• “¿Les gustaría compartir lo que ajustaron en la discusión con toda la clase? Ese fue un muy buen ajuste” // “Would you be willing to share your revision in the class discussion? That was a great revision.”

Student Facing

1. En la feria del libro, recogieron \$13 el martes por la mañana y 8 veces esa cantidad por la tarde. ¿Cuánto dinero recogieron en la feria del libro el martes? 2. El jueves, Mai compró una biografía que le costó \$16 y un cómic que le costó \\$3. Ese día, la cantidad total de ventas de la feria del libro fue 9 veces la cantidad de dinero que Mai gastó.

¿Cuál fue la cantidad total de ventas de la feria del libro el jueves?

Student Response

If students solve only one part of the problem, consider asking: “Si sabemos la cantidad de dinero que se ganó el martes en la mañana, ¿cómo nos puede ayudar esto a encontrar la cantidad total de dinero ganado?” //  “How can knowing the amount of money earned in the morning on Tuesday help us find the total amount of money earned?”

Activity Synthesis

• See lesson synthesis.

Lesson Synthesis

Lesson Synthesis

“Hoy resolvimos problemas de comparación multiplicativa en los que necesitamos más de un paso” // “Today we solved multiplicative comparison problems that involved more than one step.”

Display work samples for students identified during the last activity.

“En la última actividad, ¿cómo encontraron la cantidad de dinero que se recogió el martes?” // “In the last activity, how did you determine the amount of money collected on Tuesday?” (Multiply 8 and 13, and then add 13 to get the result.)

“¿Cómo se representa esto en el trabajo del estudiante?” // “How is this represented in the student work?” (When 13 and 8 are multiplied or with $$13 + 104 = 134$$)

Display the work of two students who solved the problem using two different approaches.

“¿Cómo podemos describir la estrategia que usó cada estudiante para encontrar la cantidad de dinero que se recogió el jueves?” // “How can we describe the strategies each student used to find out the amount collected on Thursday?” (One strategy is to add 16 and 3, and then multiply 19 and 9. The other is to multiply 16 and 9, multiply 3 and 9, and then add the two products at the end.)

Select a student who revised their thinking. Ask them to explain what they noticed during the partner discussion and how they changed their work after noticing what they missed. Consider asking a different student to rephrase this revision in their own words.