Lesson 12

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.

Some students may not be sure how to start estimating the amount of beans in the larger jar once the number of beans in the smaller jar is known. Encourage them to start by estimating how the dimensions (e.g., height, width) of the two containers compare. For example, since the larger jar is more than twice the height of the smaller and has a greater width, then it must have at least twice as many beans.

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 \(200\boldcdot 3\boldcdot 1.5=900\).)
• “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.)

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.

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.

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.

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.

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.

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.