standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
The purpose of this activity is for students to practice using precise language to describe how they estimated volumes of objects. Starting from an object of known volume, students must consider the difference in dimensions between the two objects. The focus here is on strategies to estimate the volume and units of measure used, not on exact answers or calculating volume using a formula (which will be the focus of later lessons). Notice students who:
have clear strategies to estimate volume of contents inside container
have an estimate that is very close to the actual volume
Launch
Arrange students in groups of 2.
Option 1: Bring in real containers and have students estimate how much rice each would hold, one at a time, preferably with one container whose volume is stated so students have a visual reference for their estimates. Also bring plenty of dried rice and measuring tools, such as tablespoons or cups. After collecting students’ estimates, you can demonstrate how much rice each container holds using whichever units of measure the class deems reasonable. Note that 1 tablespoon is 0.5 ounces or around 15 milliliters. 1 cup is 8 ounces or around 240 milliliters. 1 milliliter is the same as 1 cubic centimeter.
Option 2: Display images one at a time for all to see. Give students 1–2 minutes to work with their partner and write down an estimate for the objects of unknown volumes in the picture. Follow with a whole-class discussion.
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Representation: Internalize Comprehension.
Provide access to 3-D models of the objects depicted in the images for students to manipulate. Ask students to use the 3-D models to estimate the quantities in the other figures. Encourage students to place objects with known volumes inside or next to others to ensure an accurate estimate.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Your teacher will show you some containers.
If the pasta box holds 8 cups of rice, how much rice would you need for the other rectangular prisms?
If the pumpkin can holds 15 fluid ounces of rice, how much do the other cylinders hold?
If the small
cone
holds 2 fluid ounces of rice, how much does the large cone hold?
If the golf ball were hollow, it would hold about 0.2 cups of water. If the baseball were hollow, how much would the
sphere
hold?
Student Response
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Anticipated Misconceptions
Students may think there is a single right answer. Measurements are always approximate. Some of the measurements given by the authors of this activity were calculated using estimates from the photos, and may not be very precise. Measurements listed on the sides of packages are more accurate, but actual contents may vary slightly.
Activity Synthesis
For each set of containers, display the image and select previously identified students to share their strategies for estimating the volume. Once strategies for each set of containers are shared, discuss:
“How do the estimates differ if we measure using water verses rice?” (Measuring with rice leaves a bit of empty space between the grains, while water, being liquid, leaves no empty space, so it’s more accurate.)
“If the containers we used were much larger (like a water tank), would our units of measure change? Why?” (If we were measuring larger volumes, we might want to use a larger unit, like gallons. 4000 ml sounds big, but it’s only a bit more than 1 gallon, which isn’t that much water.)
Conclude the discussion by asking students to compare some other units of measure for volume that they know of. Have students recall what they know about unit conversion between some units of measure. Example:
Fluid ounces, quarts, cups, liters, milliliters
Cubic feet, cubic meters, cubic yards
Note that cubic centimeters are special, because 1 cc = 1 ml
If it comes up, here is the scoop on
ounces
: units called “ounces” are used to measure both volume and weight. It is important to be clear about what quantity you are measuring! To differentiate between them, people refer to the units of measure for volume as “fluid ounces.” For water, 1 fluid ounce is very close to 1 ounce by weight. This is not true for other substances! For example, mercury is much denser than water. 1 fluid ounce of mercury weighs about 13.6 ounces! Motor oil is less dense than water (that’s why it floats), so 1 fluid ounce of oil weighs only about 0.8 ounces. The metric system is not so confusing for quantities that would be measured in ounces, since it’s common to measure mass instead of weight and measure it in grams, whereas volume is measured in milliliters.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each strategy that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to speak as they make sense of the reasoning of others.
Design Principle(s): Support sense-making


aligns: no