Which Container Holds the Most? An Activity for Younger Children

Most children will come to this activity with memories of filling containers with water, sand, dirt, or objects. Although they may not hold a sophisticated understanding of volume and surface area, their experiences will provide a learning base.

Within each group the members should comment on the activity, observing each container and projecting the outcome of the bead count. Ask the students to discuss their observations among themselves before sharing with the class.

The containers' constant outer surface area allows comparison of their volumes. The children may need some direction to understand this idea.

Young children may need help accurately taping the strips. Constructing the containers will help them understand that the surface area is the same for each.

Moving the beads from one container to the next helps students see that the quantity needed to fill them increases or decreases as the shapes change.

Use this activity to introduce vocabulary words such as cylinder, volume, and surface. The children can hear the words and practice using them as they explore the activity. Remember, however, that success at this level does not depend on using correct vocabulary. Continued practice with the ideas and use of appropriate words will instill the vocabulary over time.

For each small group you will need:

• Six strips of card stock or light cardboard cut into the dimensions 5" (~18cm) X 1.5" (~4 cm).
  
  
• A cup of wooden or plastic beads about the size of small lima beans.
  
  
• Tape
  
  
• A shallow plastic tub or metal tray

To introduce this activity, link the students' experiences with the concept of volume. Who has filled buckets with water or dirt? What other containers have you used? What is in an empty container? What shape of container would you predict holds the most? How can you test your hypothesis? Here is a way to build containers of different shapes and find out how much each will hold.

Divide the students into groups of three. Each group receives three equal strips of card stock and a cupful of small beads in a shallow tray. Groups are to build three containers of different shapes. Each container is constructed by joining the 1.5" (~4 cm) ends of a 5" (~18 cm) long strip, taping the edges where they meet so there is no overlap. The tray bottom serves as the container's bottom piece. For all shapes except a cylinder, creases in the strip help define the shape. The children will need to work together to tape the strips.

Ask the children to predict which shapes will hold the most material and record their responses on a class chart. Then the group fills one container with beads, counts the number it holds, and records its results for that shape. They then transfer the beads to another container, adding more or fewer beads if necessary. Then the same for the third container. Caution the groups to keep the container sides straight and not to let the walls bulge from the weight of the beads. A group member records the results of each container's capacity.

When the measuring is complete, the results can be compared across the groups and recorded on a class chart. What can be seen from the data? How does the number of container sides compare with the number of beads it can hold? Does it matter if the sides are of equal length? If we had to choose a shape that holds the most, which would it be?

An Extension: Give every student another strip. Ask them to construct another container with many sides. Have them predict, then count the number of beads the containers will hold, comparing their volumes to those of previous shapes.

Building Houses An Activity for Older Students

The Scenario
Imagine that you live in a society in which people produce almost everything they need by their own efforts. Your family is planning to build a house and your family and neighbors must gather all the materials. Some of the materials are scarce and you want to be as efficient as possible. You have collected all of the materials you can for the walls, but you are concerned that you might not be able to build a house with enough space for your family. Design a floor plan that will give your house the largest amount of enclosed space.

Even though the teacher sets up the context of this activity, the students are in charge of the exploration. The drawings and data will present evidence that they can interpret within the small groups, compare across the groups, and discuss with the entire class.

Establish what ideas the students have about the relation of surface area and internal area. Placing the activity in the context of house construction takes mathematics out of the abstract and grounds it in the concrete world. If they were building a house, would the walls be vertical, as in a traditional Western house, or slanted, as are teepees or A-frame houses? What effect would a slanted wall have on the amount of interior space?

By designing the shapes and determining the internal areas, the students have produced raw data. They can examine and explain hypotheses about the figure that encompasses the largest area within a given perimeter.

As students plot the shapes of their houses, the spaces within the images will emerge as identifiable figures. The students will be able to see the areas change and enlarge as the number of sides increases and the figure nears the shape of a circle.

The groups may need encouragement to use many-sided figures as well as triangles and quadrilaterals. This is a good opportunity to reinforce use of mathematical terms and vocabulary.

This activity works well for student groups of three. The group should assure that every member understands what is being performed and discussed.

You will need:

• String cut in 32 cm (~13.5") lengths
• Graph paper
  
• Rulers, compasses, other drawing and measuring tools
  
• Pins
  
• Cork board squares
  

To begin thinking about this activity, the class might discuss the different styles and shapes of dwellings seen around the world. Examples of teepees, tents, one- and two-story houses, apartments, yurts, igloos, and other structures indicate the variety of the world's structures. Students might be interested in the work of Buckminster Fuller or homes designed for fuel efficiency or other environmental concerns. A discussion of different dwellings would probably touch on aesthetics, weather, and local terrain as well as availability of structural materials.

Each student group will work with a string 32 cm (~13.5") long, and try to determine what perimeter shape provides the greatest internal area. Encourage the groups to try a variety of shapes they think might be interesting.

Students may find it helpful to pin the string onto corkboard to fix the shape, then transfer the dimensions to a rendition on graph paper. A variety of ways can be used to determine area including use of formulas or counting the number of graph squares enclosed in the shape. Some measuring and drawing tools, such as compasses, protractors, rulers, and t-squares should be available, but they may not be needed.

When the groups are finished they can record their data on a class chart that allows comparison across the groups. Particular attention to the number of sides and the area dimensions for the various figures will allow a pattern to emerge. Discussion can focus on the comparison of data from the various figures. What can be inferred from the results? What predictions can be made about other kinds of figures?

This activity is adapted from C. Zaslacsky, "People Who Live in Round Houses," Arithmetic Teacher, 37 (September 1989): 18-21.