Lesson 18

Mucha leche

Warm-up: Exploración de estimación: ¿Qué tan grande es la caja de leche? (10 minutes)

Narrative

The purpose of this Estimation Exploration is for students to estimate a volume based on an image and on their own personal experience with cartons of milk. Students recall the meaning of volume as the number of cubic inches, in this case, it would take to fill the milk carton without gaps or overlaps. Because the carton is relatively small, students can formulate a reasoned, accurate estimate of the milk carton’s volume. They will then use this estimate throughout the lesson.

Launch

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

Activity

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

Student Facing

¿Cuál es el volumen del cartón de leche, en pulgadas cúbicas?

Escribe una estimación que sea:
muy baja razonable muy alta
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

Activity Synthesis

• “¿Cómo pueden usar lo que saben acerca del volumen para estimar el volumen del cartón de leche?” // “How can you use what you know about volume to estimate the volume of the milk container?” (I can measure to see how many cubic inches it would take to fill the carton. I can measure the length, width, and height and multiply them.)
• “¿Qué unidades usan normalmente para medir líquidos?” // “What units do you usually use to measure liquids?” (Liters, quarts, cups)
• “En una unidad anterior, aprendimos que los centímetros cúbicos y las pulgadas cúbicas también son unidades que se usan para medir un volumen” // “We learned in an earlier unit that cubic centimeters or cubic inches are also units for measuring a volume.”

Activity 1: Leche para todos (15 minutes)

Narrative

The purpose of this activity is for students to estimate products using the context of volume introduced in the warm-up. Students estimate how many cubic inches of milk different-sized groups of students might consume. For example, at first, students multiply the amount of milk they consume by the number of students in the class. Next, students multiply the amount consumed by one class by the number of classes. Because these are all estimates, the fact that not every student in one class drinks the same amount of milk or that different classes or grades or schools have different numbers of students can be overlooked. When students make simplifying hypotheses like this, they model with mathematics (MP4).

As currently structured, the activity is quite open-ended so that students can use their own school to make their estimates. There is a lot of variation in school size. The average size of an elementary school in Montana, for example, is less than 200, while in California, it is 600. Some large elementary schools in New York City have close to 2,000 students. The important mathematical part of this activity does not depend on the exact numbers for a particular school. The key is which numbers students choose as they make estimates, focusing on multiples of powers of 10.

MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they estimate the volume. On a visible display, record words and phrases such as: estimate, guess, predict, multiply, times, and product. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Representation: Access for Perception. Use centimeter cubes to demonstrate how many cubic centimeters can fit inside the milk carton so that students understand the size of a cubic centimeter.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Launch

• “¿Qué tipo de leche les gusta tomar?” // “What kind of milk do you like to drink?”
• Partner discussion
• “Van a estimar la cantidad de leche que los estudiantes de distintos grupos toman en un día” // “You are going to estimate the amount of milk that different groups of students drink in one day.”
• “Pueden usar esta estimación: en un cartón de leche hay 20 pulgadas cúbicas” // “You can use the estimate of 20 cubic inches for one carton of milk.”

Activity

• Monitor for students who select round numbers for their estimates and who use multiplication to go from each estimate to the next estimate.

Student Facing

En cada caso, estima el volumen de leche, en pulgadas cúbicas, que tú o las personas del grupo dado tomarían en un día. Explica tu razonamiento.

1. tu clase
3. tu escuela
4. 10 escuelas

Student Response

If students do not like milk and, therefore, do not have a connection to the problem, suggest they survey a few classmates to find out what their estimates were for how much milk they drink in one day.

Activity Synthesis

• Invite students to share responses and estimates.
• “¿Cómo les ayudaron las estimaciones que hicieron en cada pregunta a responder la pregunta que seguía?” // “How did you use your estimates from each question to help answer the next question?” (Once I knew how much milk I drank, I multiplied by the number of students in our class. Then I multiplied that by the number of fifth-grade classes.)
• “¿Cómo hicieron una estimación para el caso de su clase?” // "How did you make an estimate for your class?" (I think there are between 20 and 30 students in the class but not everyone likes milk. So I estimated that 20 students drink milk with lunch.)

Activity 2: ¿Qué tan grande es 1,000,000? (20 minutes)

Narrative

The purpose of this activity is for students to make estimates about how long it would take different groups of students to drink 1,000,000 cubic inches of milk. Unlike the previous activity in which students multiplied the 20 cubic inches of milk by larger and larger numbers, in this activity, students divide 1,000,000 cubic inches of milk by smaller and smaller numbers to find out how long it would take each group to drink 1,000,000 cubic inches of milk. If students attempt to calculate exact answers remind them that they are only looking for an estimate and the amount of milk consumed by each group in the previous activity is also only an estimate. Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

Launch

• Groups of 2
• “¿Cuánto creen que son 1,000,000 de pulgadas cúbicas de leche? ¿Se las podrían tomar?” // “How much do you think 1,000,000 cubic inches of milk is? Could you drink it?” (No, that's a lot of milk. I don't like milk that much.)
• 1 minute: quiet think time
• 1 minute: partner discussion

Activity

• 2-3 minutes individual work time
• 7-8 minutes partner work time
• Monitor for students who use the estimates from the previous activity and who base each successive calculation on the previous one, dividing by an appropriate number at each step.

Student Facing

Estima el número de días que tardarían las personas de cada grupo en tomarse 1,000,000 de pulgadas cúbicas de leche. Explica tu razonamiento.

1. 10 escuelas
2. tu escuela
4. tu clase

Student Response

Students may need support with initiating the task. Ask them to explain how they can use the solutions from the previous activity to help them solve the problems.

Activity Synthesis

• “¿Cómo estimaron el número de días que tardarían las personas de 10 escuelas en tomarse 1,000,000 de pulgadas cúbicas de leche?” // “How did you estimate the number of days it takes 10 schools to drink 1,000,000 cubic inches of milk?” (We estimated that they drink close to 100,000 cubic inches a day, so in 10 days that’s 1,000,000.)
• “¿Cómo usaron esta estimación para estimar cuánto tiempo tardarían las personas de nuestra escuela en tomarse 1,000,000 de pulgadas cúbicas de leche?” // “How did you use this estimate to estimate how long it takes your school to drink 1,000,000 cubic inches of milk?” (I multiplied by 10 because it takes 1 class 10 times as long as it takes 10 classes.)
• “¿Creen que, en toda su vida, lograrán tomarse 1,000,000 de pulgadas cúbicas de leche?” // “Do you think that you will ever drink 1,000,000 cubic inches of milk?” (No, 50,000 days is a lot. There are only 365 days in a year, so that would be more than 100 years.)

Lesson Synthesis

Lesson Synthesis

“En esta lección estimamos productos y cocientes” // “In this lesson we estimated products and quotients.”

“¿Cómo usarían la multiplicación para estimar cuántos días tardarían las personas de nuestra escuela en tomarse 1,000,000 de pulgadas cúbicas de leche?” // “How can you use multiplication to estimate how many days it would take your school to drink 1,000,000 cubic inches of milk?” (In 2 days we drink twice as much milk, in 3 days we drink 3 times as much. So I needed to estimate what to multiply the amount for one day by to get about 1,000,000.)

“¿Podrían también hacer esta estimación usando la división?” // “Could you also make this estimate using division?” (Yes, our school drinks about 10,000 cubic inches of milk each day, so I can find how many 10,000s there are in 1,000,000. That's $$1,\!000,\!000 \div 10,\!000$$.)