Lesson 18

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.

• Groups of 2
• Display the image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “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.)
• “What units do you usually use to measure liquids?” (Liters, quarts, cups)
• “We learned in an earlier unit that cubic centimeters or cubic inches are also units for measuring a volume.”

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.

• “What kind of milk do you like to drink?” 
• Partner discussion
• “You are going to estimate the amount of milk that different groups of students drink in one day.”
• “You can use the estimate of 20 cubic inches for one carton of milk.”

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

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.

• Invite students to share responses and estimates.
• “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.)
• "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.)

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).

• Groups of 2
• “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

• 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.

• “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.)
• “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.)
• “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.)