Lesson 9

Problem Solving with Volume: Water (optional)

Warm-up: Notice and Wonder: Cubic Centimeters and Grams (10 minutes)

Narrative

The purpose of this warm-up is for students to observe the relationship between the different types of units in the metric system. By contrast, in the standard system, it is not easy to see the relationship between inches, cups, and pounds. While students may notice and wonder many things about this image, conversions between liquid volume units (cups, gallons, liters) and regular volume units (cubic centimeters, cubic inches, cubic feet) are the important discussion points.

Launch

  • Groups of 2
  • Display the image.
  • “What do you notice? What do you wonder?”
  • 1 minute: quiet think time

Activity

  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Share and record responses.

Student Facing

What do you notice? What do you wonder?

blue cube, side length, 1 centimeter, on scale, weight shows 1 gram.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Highlight that the cube represents 1 cubic centimeter or 1 mL of water and one mL of water weighs 1 g.
  • The picture shows the relationships between length, capacity (or volume), and weight in the metric system.

Activity 1: Catching Rainfall (15 minutes)

Narrative

The purpose of this activity is for students to estimate how much water falls on the roof of a house, given a particular amount of rainfall. For this calculation, standard units work well as the area of the roof could be given in square feet, for example, and the rain in inches. A conversion would readily give the volume in cubic feet or inches. But, the standard units used to measure volume are cups, pints, quarts, and gallons so more work would need to be done in order to figure out how many gallons, for example, there are in a cubic foot. With the metric system, liquid volume units (liters) and regular volume units (cubic centimeters) are naturally connected.

MLR5 Co-Craft Questions. Keep books or devices closed. Display only the image, without revealing the questions, and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for this task and invite additional connections.
Advances: Reading, Writing
Representation: Access for Perception. Read statements aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Conceptual Processing, Attention

Launch

  • Groups of 2
  • “About how big is a square meter?” (It's about the size of my desk top.)
  • “About how many square meters do you think there are in the classroom floor?” (maybe 100)

Activity

  • 5 minutes: independent work time
  • 5 minutes: partner work time

Student Facing

Here is a diagram showing the roof of a house.

  1. What is the area of the roof?

    8-sided shape.
  2. Each month an average of 5 cm of rain falls on the house. How many cubic cm of rain is that?
  3. There are 1,000 cubic cm in 1 liter. How many liters of water fall on the house?
  4. You want to build a reservoir to catch the rain that falls so you can use the water. What side lengths would you suggest for the reservoir? Explain or show your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “When did you have to convert units during the activity?” (I needed to convert meters to centimeters since the roof is given in meters and the rain is given in centimeters. I needed to convert cubic centimeters to liters.)
  • “How did you make the calculations for the conversions?” (the conversions required multiplying or dividing by powers of 10. I multiplied by 100 to convert m to cm and I divided by 1,000 to convert cubic centimeters to liters.)
  • “Do you think that you could make a container to capture all of the rainfall?” (Yes, I think that a 2 meter cube is big but it's not so big that it would not fit near the house. Or we could use several containers that are a little smaller.)

Activity 2: How Much Water? (20 minutes)

Narrative

The purpose of this activity is for students to find out if the amount of water that falls on the house is sufficient for many of the daily household chores that use water. This will require a lot of estimation and will vary from house to house. How much of the calculations to leave up to the students is an individual teacher choice and this lesson could easily be extended for another day if the students make well reasoned estimates (some values are given in parentheses) for how much water is used for different activities such as:

  • taking baths or showers (150 liters or 80 liters)
  • washing clothes (100 liters)
  • washing dishes (100 liters)
  • washing hands (1 liter)
  • flushing the toilet (10 liters)

More estimation comes into play for how often each of these activities happens and this will vary greatly depending on the student. When students make a list of the different things they do in the house that use water and then estimate how much water is used they model with mathematics (MP4).

Consider inviting students to check their estimates by looking at the one of their monthly water bills. The bill will usually give the number of gallons of water used and there are almost 4 liters in a gallon.

Launch

  • Groups of 2
  • After students work on the first problem, pause the class and make a list of the main daily uses of water.
  • Depending on how much time is available and the modeling demand level desired, consider estimating together or providing estimates for how much water is used for each purpose.

Activity

  • 5 minutes: independent work time
  • 10 minutes: partner work time

Student Facing

  1. What are some of the ways you use water at home?
  2. Estimate how much water you use at your home in a month.
  3. How much rain would need to fall on your home each month to supply all of your water needs?
  4. What challenges might come up if you tried to use the rainwater that falls on the roof of your home? Do you think it makes sense to try to capture the rain that falls on your home?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “How can you visualize the volume of water that it uses to take a bath?” (I can picture the bathtub filled up partway with water.)
  • “How could you measure the volume of water in the bathtub?” (I could measure the length and width of the tub and the height of the water and multiply them.)
  • “Are any of the amounts of water used for different things surprising to you? Why?” (I am surprised by how much water it takes to wash the dishes. It’s almost as much as when you take a bath.)

Lesson Synthesis

Lesson Synthesis

“How is measuring the volume of water the same as measuring the volume of the Empire State Building?” (If I know the length, width, and height that the water takes up, then I can multiply them to get the volume, just like the building.)

“How is measuring the volume of water different than measuring the volume of a building?” (Water does not have a simple shape like a building. It needs to be put in a container in order to measure.)

“What is important to remember when measuring volume?” (It’s the amount of space something can hold or that something takes up. I can measure it in cubic units or in liters for a liquid.)

Cool-down: Reflection: Volume (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Section Summary

Student Facing

We investigated several different complex volume questions. For the ancient pyramids of Egypt we gave an estimate of a couple million cubic meters. Since these pyramids are not rectangular prisms, an estimate is the best we could hope for. Then we estimated the volume of the world's largest wagon, using information from a photograph. Lastly, we investigated the amount of rain that falls on a house and the amount of water our families use in a year.

In each case, we could only make estimates because the situations are all complex. In the previous section we used estimation to check the reasonableness of calculations. In this section we saw that making reasoned estimates is a vital part of applying mathematics to many real world situations.