# Lesson 8

Resolvamos problemas con multiplicaciones y divisiones

## Warm-up: Conversación numérica: Dividamos entre 8 (10 minutes)

### Narrative

This Number Talk encourages students to think flexibly about numbers to divide. The understandings elicited here will be helpful throughout this unit as students divide whole numbers and build toward fluent multiplication and division.

### Launch

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”

### Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$848 \div 8$$
• $$4,\!848 \div 8$$
• $$4,\!852 \div 8$$
• $$5,\!848 \div 8$$

### Activity Synthesis

• “¿Cómo les ayudaron las primeras expresiones a encontrar el valor de la última expresión?” // “How did you use the first few expressions to help you find the value of the last expression?”
• “¿Cómo podrían usar la multiplicación para encontrar el valor de cada cociente?” // “How might you use multiplication to find the value of each quotient?”

## Activity 1: ¿Dos verdades y una mentira o dos mentiras y una verdad? (15 minutes)

### Narrative

In this activity, students are given three situations and asked to determine which ones could be true and which are not. To do so they need to carefully make sense of the quantities in each story and how they are related (MP2). Students may explain why a situation is true by writing one or more expressions or equations to represent the relationships and perform the calculations to check. Students may also reason using estimation and mental computation when explaining why a situation must be false.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. For example, invite students to perform the following steps: select one situation to start with, read the problem out loud and visualize the situation, represent the situation with a mathematical drawing or a tape diagram, write an equation to represent the problem, solve or estimate the solution to the equation. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Conceptual Processing, Organization, Social-Emotional Functioning

### Launch

• Groups of 2–4
• 5 minutes: independent work time

### Activity

• 5 minutes: group discussion
• Monitor for different ways students represent and prove how each situation could be true or false, including using reasoning based on estimation and mental math.

### Student Facing

Estas son tres situaciones. ¿Cuáles son verdaderas? ¿Cuáles no lo son?

Muestra cómo lo sabes.

• Situación A: Un rascacielos tiene 53 pisos. El primer piso mide 17 pies de alto, pero todos los demás pisos miden cada uno 11 pies de alto. El edificio mide 589 pies de alto.
• Situación B: Un limpiaventanas tiene 600 segundos para limpiar 17 ventanas de un edificio. Se tarda 54 segundos en limpiar cada ventana. El limpiaventanas terminará de limpiar todas las ventanas y le sobrarán 11 segundos.
• Situación C: Once estudiantes se propusieron recaudar al menos \$600 para una organización de caridad. Cada estudiante recaudó \$17 cada día. Después de 3 días de recolectar fondos, al grupo aún le falta recaudar \$54. ### Student Response Teachers with a valid work email address can click here to register or sign in for free access to Student Response. ### Activity Synthesis • Select students to share their responses and reasoning. • Record the expressions or equations they wrote to represent the situations. Highlight different ways of representing the same situation. • “¿En cuáles situaciones tuvieron que encontrar los valores reales para saber si eran verdaderas o no? ¿Por qué razón?” // “For which situations did you need to find the actual values in order to tell if they were true or not true? Why is that?” (I tried to estimate on Situation A and I knew it would be close, so I did the multiplication and added to find out if the total height was really 610 feet. I needed to write equations to make sense of Situation C. I just did the multiplication and subtraction to check to see if it could really be \$54.)
• “¿En cuáles situaciones fue posible saberlo por medio de una estimación y un cálculo mental?” // “For which stories was it possible to tell by estimation and mental math?” (I could do some estimation for all of them, but for Situation B I could tell that that even if he washed 10 windows the window washer would almost be out of time and couldn’t do 17.)

## Activity 2: Buses para una excursión (20 minutes)

### Narrative

In this activity, students interpret situations that involve equal groups and require making sense of a remainder. Students solve problems using their understanding of multiplication and by connecting their solutions to the quantities in the situation (MP2). Although parts of the task could be solved by dividing a multi-digit number by a two-digit number, students are not expected to perform division with these numbers until grade 5. Students may access this task using multiplication and addition.

Encourage students to use the Three Reads routine as needed to solve problems.

MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “Te escuché decir . . .” // “I heard you say . . . .” Original speakers can agree or clarify for their partner.

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: group work time
• Monitor for the different ways students represent and solve the problem, including how they discuss how to treat any unfilled buses (the remainder).

### Student Facing

Todos en una escuela van a ir de excursión. Se necesitan buses para transportar a 375 personas.

La empresa de buses A tiene buses pequeños con 27 asientos cada uno.

La empresa de buses B tiene buses grandes con 48 asientos cada uno.

1. Encuentra el menor número de buses que se necesitarán si la escuela escoge:

• La empresa de buses A. Muestra cómo razonaste.

• La empresa de buses B. Muestra cómo razonaste.

2. ¿Cuál empresa de buses debería escoger la escuela? Explica tu razonamiento.

3. La empresa de buses C tiene buses grandes que pueden llevar hasta 72 pasajeros.

Diego dice: “Si la escuela escoge la empresa de buses C, solo se necesitarán 6 buses, pero los buses tendrán más asientos desocupados”.

¿Estás de acuerdo? Explica tu razonamiento.

### Activity Synthesis

• Select students to share their responses and reasoning. Record the different representations students used to solve the problems.
• “¿Cómo decidieron cuántos buses de cada empresa se necesitarían?” // “How did you decide how many buses you would need from each company?”
• “¿Todos los buses llevan la misma cantidad de pasajeros? ¿Cómo pueden ver eso en sus representaciones o en sus ecuaciones?” // “Do all the buses carry the same amount of passengers? How can you see that in your representations or equations?”
• “¿Cómo decidieron qué operaciones usar para responder cada pregunta?” // “How did you decide what operations to use to answer each question?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy analizamos y resolvimos muchos tipos de problemas en palabras” // “Today we analyzed and solved many kinds of word problems.“

“¿Qué estrategias debemos usar al resolver problemas para asegurarnos de que entendemos lo que nos preguntan en el problema?” // “What are some strategies we should use when solving problems to make sure we understand what the problem is asking?” (Read it carefully, think about what the numbers tell us and how they are related to one another.)

“¿De qué formas podemos descifrar las relaciones que hay entre los números?” // “What are some ways to figure out the relationships between the numbers?” (Create a representation—an equation or a diagram—and check to see if the representation matches the problem being solved.)

“¿Cómo podemos saber si nuestra respuesta tiene sentido?” // “How would we know if our answer makes sense?” (Check to see if it’s reasonable in the situation, double check our calculations, check to see if it answers the question.)