12.1: Two Containers
Warm-up: 5 minutes
In the previous lesson, students studied the relationship between volume of liquid and the height of the 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 first activity of the lesson in which students reason about volumes of several container types and re-familiarize themselves with the language of three-dimensional objects.
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 these values 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.
- Processing Time. Provide the image to students who benefit from extra processing time to review prior to implementation of this activity.
- Manipulatives. Begin with realia (i.e., real containers and beans), which will provide access for students who benefit from concrete contexts.
Your teacher will show you some containers. The small container holds 200 beans. Estimate how many beans the large jar holds.
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.
- “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 $200\boldcdot 3\boldcdot 1.5=900$.)
- “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 so there is less air, such as rice or water.)
- “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: What’s Your Estimate?
Activity: 15 minutes
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
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.
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?
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.
12.3: Do You Know These Figures?
Activity: 10 minutes
The purpose of this activity is for students to learn or remember the names of the figures they worked with in the previous activity and learn a quick method for sketching a cylinder. Students start by determining the shapes that are the faces of the four shapes. They also determine which shape would be considered the base of each figure shown. This allows students to connect previously learned two-dimensional figures to new three-dimensional figures introduced here. 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. Identify students who sketch cylinders that are different sizes or drawn sideways.
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, followed by a whole-class discussion.
- 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.
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
Select previously identified 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.
12.4: Rectangle to Round
Cool-down: 5 minutes
Here is a box of pasta and a cylindrical container. The two objects are the same height, and the cylinder is just wide enough for the box to fit inside with all 4 vertical edges of the box touching the inside of the cylinder. If the box of pasta fits 8 cups of rice, estimate how many cups of rice will fit inside the cylinder. Explain or show your reasoning.
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-sized 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 sixes 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 half of the circular base of the can. What objects could the other figures in the picture represent?