Previously, students studied the relationship between volume and height of liquid when poured into a cylindrical container. The purpose of this Warm-up is to shift students’ attention toward other types of containers and to consider how the volume of two containers differs. This Warm-up is direct preparation for the following activity in which students reason about volumes of several container types and refamiliarize themselves with the language of three-dimensional objects.
Student Lesson in Spanish
Launch
Tell students to close their books or devices. Arrange students in groups of 2. Display the image of the two containers filled with beans for all to see.
Give partners 1 minute to estimate how many beans are in each container. Poll the class for their estimates, and display them for all to see, in particular, the range of values expressed.
Tell students that the smaller container holds 200 beans. Ask students to open their books or devices and reconsider their estimate for the large container now that they have more information. Give 1–2 minutes for students to write down a new estimate. Follow with a whole-class discussion.
Activity
None
Student Task Statement
Your teacher will show you some containers. The small container holds 200 beans. Estimate how many beans the large jar holds.
Student Response
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Building on Student Thinking
If students are not sure how to start estimating the amount of beans in the larger jar once the number of beans in the smaller jar is known, consider asking:
“Tell me what you notice is similar or different between the two jars.”
“How could estimating some dimensions (for example, height and width) help you?”
Activity Synthesis
Poll the class for their new estimates for the number of beans in the larger container, and display these next to the original estimates for all to see. Tell the class that the large container actually holds about 1,000 beans.
Discuss:
“How did you and your partner calculate your estimate for the large jar?” (We estimated the large jar holds 900 beans since the large jar is about 3 times taller than the smaller jar, and it’s about 1.5 times wider, and .)
“Is there a more accurate way to measure the difference in volume between the two containers than ‘number of beans.’” (Yes, we could use something smaller than beans, such as rice or water, so there is less air.)
“What are some examples of units used to measure volume? Where have you seen them used in your life?” (Cups, tablespoons, gallons, liters, cubic centimeters, etc. Drinks often have fluid ounces, gallons, or liters written on them. Recipes may use cups or tablespoons.)
12.2
Activity
15 mins
What’s Your Estimate?
Standards Alignment
Building On
Addressing
8.G.C
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
The purpose of this activity is for students to practice using precise language to describe how they estimated volumes of objects (MP6). 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).
Launch
Arrange students in groups of 2. For each option, select groups who have clear strategies to estimate volume of contents inside the container or have an estimate that is very close to the actual volume to share later.
Option 1: Bring in real containers, and ask students to 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, demonstrate how much rice each container holds using whichever units of measure the class deems reasonable. Note that 1 tablespoon is 0.5 ounce, 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.
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide access to three-dimensional models of the objects shown in the images for students to manipulate. Ask students to use the three-dimensional 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
Activity
None
Student Task Statement
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 cup of water. If the baseball were hollow, how much would the sphere hold?
Student Response
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Building on Student Thinking
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 ask previously identified groups 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 instead of 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 water tanks), would our units of measure change? Why?” (If we were measuring larger volumes, we might want to use larger units, like gallons. 4,000 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. Ask students to recall what they know about unit conversion between some of the following units of measure:
Fluid ounces, quarts, cups, liters, and milliliters
Cubic feet, cubic meters, and cubic yards
Note that cubic centimeters are special because 1 cc = 1 ml.
If it comes up, here is information about 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.
MLR8 Discussion Supports. For each strategy that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Advances: Listening, Speaking
12.3
Activity
10 mins
Do You Know These Figures?
Standards Alignment
Building On
Addressing
8.G.C
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
The purpose of this activity is for students to learn or remember the names of the figures, and some features of those figures, that they worked with earlier and to practice a quick method for sketching a cylinder. Students start by determining the shapes that are the faces of the four figures. They also determine which shape would be considered the base of each figure shown. This allows students to connect what they have previously learned about two-dimensional figures to the three-dimensional figures in this activity and will help later when establishing volume formulas for cylinders, cones, and spheres.
The last question introduces students to a way to sketch a cylinder. This is a skill they will continue to use throughout the unit when working on problems that do not provide a visual example of a situation.
Launch
It is strongly recommended that you provide physical, solid objects for students to hold and look at. If using physical objects, pass around the objects for students to see and feel before starting the activity.
Give students 3–5 minutes of quiet work time. Select students who sketch cylinders that are different sizes or drawn sideways to share later. Follow with a whole-class discussion.
Activity
None
Student Task Statement
What shapes are the faces of each type of object shown here? For example, all six faces of a cube are squares.
Which faces could be referred to as a “base” of the object?
Here is a method for quickly sketching a cylinder:
Draw two ovals.
Connect the edges.
Which parts of your drawing would
be hidden behind the cylinder?
Make these parts dashed lines.
Practice sketching some cylinders. Sketch a few different sizes and orientations, including short, tall, narrow, wide, and sideways. Label the radius and height on each cylinder.
Student Response
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Building on Student Thinking
If students struggle to visualize the shapes of the faces, position the object so that students can only see two dimensions. Ask students what two-dimensional shape they see.
Activity Synthesis
If using physical objects, display each object one at a time for all to see. If using images, display the images for all to see, and refer to each object one at a time. Ask students to identify:
Which figure the object is an example of.
The different shapes that make up the faces of the figure.
The shape that is the base of the figure.
Invite previously selected students to share their sketches of cylinders. The goal is to ensure that students see a variety of cylinders: short, tall, sideways, narrow, etc. If no student drew a “sideways” cylinder, sketch one for all to see, and make sure students understand that even though it is sideways, the height is still the length perpendicular to the base.
Tell students that we will be working with these different three-dimensional figures for the rest of this unit. Consider posting a display in the classroom that shows a diagram of each object labeled with its name, and where appropriate, with one side labeled “base.” As volume formulas are developed, the formulas can be added to the display.
Lesson Synthesis
The goal of this discussion is to remind students that the volume of a three-dimensional figure is the amount of space it encloses. Ask students:
“What are some shapes you worked with in today’s lesson?” (cylinders, cones, spheres, rectangular prisms)
“What are some different units of measure we use to calculate volume of these figures?” (cubic feet, fluid ounces, gallons, cubic nanometers, rice)
“What are some different examples of objects we see in our world that are very similar to the figures? What types of things go inside them?” (A cell phone is like a rectangular prism and is filled with wires, a battery, speakers, a microphone, and other electronics. A soccer ball is like a sphere and is filled with air. A chili pepper jar is like a cylinder and is filled with ground up dried chilis.)
Student Lesson Summary
The volume of a three-dimensional figure, like a jar or a room, is the amount of space the shape encloses. We can measure volume by finding the number of equal-size volume units that fill the figure without gaps or overlaps.
For example, we might say that a room has a volume of 1,000 cubic feet or that a pitcher can carry 5 gallons of water. We could even measure volume of a jar by the number of beans it could hold, though a bean count is not really a measure of the volume in the same way that a cubic centimeter is because there is space between the beans. (The number of beans that fit in the jar do depend on the volume of the jar, so it is an okay estimate when judging the relative sizes of containers.)
In earlier grades, we studied three-dimensional figures with flat faces that are polygons. We learned how to calculate the volumes of rectangular prisms. Now we will study three-dimensional figures with circular faces and curved surfaces: cones, cylinders, and spheres.
To help us see the shapes better, we can use dotted lines to represent parts that we wouldn't be able to see if a solid physical object were in front of us. For example, if we think of the cylinder in this picture as representing a tin can, the dotted arc in the bottom half of that cylinder represents the back side of the circular base of the can. What objects could the other figures in the picture represent?
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