Lesson 7

Resolvamos problemas de comparación multiplicativa

Warm-up: Observa y pregúntate: Dos ciudades (10 minutes)

Narrative

The purpose of this warm-up is to elicit observations about differences in costs of living in different parts of the United States. It prepares students to look for and make sense of multiplicative comparisons later in the lesson. It also helps elicit what students know about how the different costs may be different depending on where you live.

Students may notice and wonder many things about the data in the table and both additive and multiplicative comparisons are likely to come up. Some students may note that an item costs “some dollars more” in San Francisco than in Fort Wayne, while others may say it costs “some number of times as much” based on the quantities or their estimation of the quantities. In the synthesis, focus student discussion on additive and multiplicative comparison.

Launch

• Groups of 2
• Display the table.

Activity

• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

Student Facing

¿Qué observas? ¿Qué te preguntas?

San Francisco, CA Fort Wayne, IN
población 805,235 253,691
leche (1 galón) \$4.45 \$2.14
pan (1 barra) \$3.54 \$1.82
gasolina (1 galón) \$3.70 \$2.42
boleto de cine \$15 \$12
conexión a internet (1 mes) \$70 \$50
alquiler de un apartamento de 3 alcobas en el centro de la ciudad (1 mes) \$6,000 \$1,500
costo de una casa \$1,400,000 \$160,000

Activity Synthesis

• “¿Cómo pueden comparar el costo de alquilar un apartamento en San Francisco con el costo en Fort Wayne? ¿Cómo pueden usar la multiplicación para comparar los costos?” // “How could you compare the cost for renting an apartment in San Francisco to the cost in Fort Wayne? How could you use multiplication to compare the costs?”
• As needed, “¿El costo de una casa en San Francisco es aproximadamente cuántas veces el costo de una casa en Fort Wayne?” // “About how many more times as much is the cost of a house in San Francisco compare to the cost of a house in Fort Wayne?”
• “Los precios de las cosas necesarias para vivir, como la comida, la ropa y la vivienda, se llaman el costo de vida. El costo de vida puede ser muy diferente dependiendo del lugar del mundo en el que vivan. Hoy vamos a comparar el costo de vida en distintos lugares del mundo” // “The prices of the things you need to live, like food, clothing, and shelter, are called the cost of living. The cost of living can be very different depending on where in the world you live. Today we are going to compare the cost of living in different places around the world.”

Activity 1: El más y el menos costoso (20 minutes)

Narrative

This activity prompts students to use multiplicative comparison to determine the cost of different living expenses in Bermuda and in India. Students use these costs to interpret and solve a multi-step problem. The multi-step problem can be solved in more than one way. Students might use multiplication or division. They may also choose to estimate rather than perform each calculation. Encourage students to use the Three Reads routine as needed to solve problems.

When finding the missing prices in the table, students also have an opportunity to reason about how to find a value that is 2.5 times as much. Students are not expected to multiply whole numbers by decimals in grade 4, but may use their understanding of decimal notation and decimal fractions to estimate the cost by doubling the price in India and adding half the price or by reasoning that 2.5 is close to 3.

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations. Monitor and clarify any questions about the context.

Launch

• Groups of 2
• “Vimos un ejemplo de cómo puede variar mucho el costo de las mismas cosas dependiendo de dónde vivas en los Estados Unidos. Examinemos los costos de otras cosas en otros dos países. Uno es conocido por ser muy costoso y el otro es conocido por ser poco costoso” // “We saw one example of how the same items can vary a lot in cost depending on where you live in the United States. Let’s look at the costs of some other things in two different countries—one known to be very expensive and the other known to be inexpensive.”

Activity

• 5 minutes: independent work time on the first question
• Monitor for students who use the multiplication or division algorithm, draw diagrams, write equations, or use estimation to find the missing values.
• Pause for a discussion. Select students to share their responses and reasoning for finding the costs in the table.
• Highlight that to find the missing cost in Bermuda, we can multiply the cost in India by the number in the middle column, and to find the cost in India, we divide the cost in Bermuda by the number in the middle column.
• 4 minutes: independent work time on the second question
• 4 minutes: partner discussion

Student Facing

Las Bermudas es el país más costoso del mundo y la India es uno de los menos costosos.
1. La tabla muestra una comparación de los precios de algunas cosas en los dos países. Estima o calcula los costos que hacen falta y completa la tabla.

2. En la India, el alquiler de un apartamento de una alcoba ubicado fuera del centro de una ciudad cuesta aproximadamente \$76 al mes. El alquiler de un apartamento parecido en las Bermudas cuesta 23 veces eso. Los servicios públicos (electricidad, gas, agua y calefacción) para un apartamento pequeño cuestan aproximadamente \$27 al mes en la India y 7 veces eso en las Bermudas.

Activity 2: El costo de vida (15 minutes)

Narrative

In this activity, students continue to solve multiplicative comparison problems in the context of cost of living. In addition to using multiplicative comparison to find a greater or smaller unknown value, students also find an unknown multiplier.

Students also work with larger numbers. Some questions (about the costs of utilities, for instance) could be answered by dividing one multi-digit number by another (for instance, $$153 \div 73$$), but students are not expected to do this in grade 4. Instead, they may reason in terms of multiplication and by using estimation. For example, they may reason that $$73 \times 2$$ is 146 or $$75 \times 2$$ is 150, so the missing multiplier must be close to 2. Encourage students to use the Three Reads routine as needed to solve problems.

Action and Expression: Develop Expression and Communication. Provide access to a variety of tools, such as base-ten blocks, counters, pre-formatted tape diagrams, grid paper, and mini whiteboards.
Supports accessibility for: Conceptual Processing, Organization, Attention

Launch

• Groups of 2
• “¿Alguna vez han viajado a un lugar lejano? ¿Conocen a algún amigo o familiar que lo haya hecho?” // “Have you ever traveled to another place far away? Do you know a friend or family member who has?”
• Share responses.
• “En esta actividad, van a comparar los costos típicos de cosas en los Estados Unidos con los costos de las mismas cosas en Ghana” // “In this activity, you will compare the typical cost for things in the United States to the cost for the same things in Ghana.”

Activity

• 4 minutes: independent work time on the first question
• 2 minutes: partner discussion
• Monitor for the different ways students make estimates, for example, whether they:
• opt to multiply or divide
• use exact or rounded numbers
• round before computing or compute before rounding
• Pause for a brief discussion. Select students to share their responses and reasoning for finding the costs in the table.
• 4 minutes: independent work time on the second question
• 2 minutes: partner discussion
• Monitor for different strategies students use to find how many months of rent the family could afford in each country, including strategies based on estimation.

Student Facing

El costo de vida en los Estados Unidos es más alto que en Ghana.

1. La tabla muestra una comparación de los precios de algunas cosas en los dos países. Estima o calcula la información que hace falta en la tabla.

2. Supongamos que una familia tiene \\$3,000 para gastar en vivienda y quiere alquilar un apartamento de 1 alcoba que esté ubicado fuera del centro de una ciudad. Con esa cantidad de dinero, cuántos meses de alquiler pueden pagar en:

1. Los Estados Unidos. Explica o muestra cómo razonaste.
2. Ghana. Explica o muestra cómo razonaste.

Activity Synthesis

• See lesson synthesis.

Lesson Synthesis

Lesson Synthesis

“Hoy comparamos el costo de vida de varios países” // “Today we compared the cost of living in several countries.”

“En las actividades de hoy, ¿de qué maneras compararon los precios y el costo de vida de dos lugares distintos?” // “What are the different ways you compared the prices and the cost of living between two different places in the activities today?” (We used multiplication to describe how many times as much the cost of living was. We used multiplication and division to determine a price in one country. We also used addition and subtraction to check to see how much of something we could buy in one country compared to another.)

“¿Cuándo es importante comparar el costo de vida de un lugar con respecto a otro?” // “When would it be important to compare the cost of living in one place compared to another?” (If you were planning a trip, you might want to plan for how much things will cost. If you are moving to a new place.)

“De las preguntas que tienen acerca del costo de vida, ¿sobre cuáles les gustaría aprender más?” // “What questions do you have about cost of living that you would like to learn more about?”