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
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
- 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
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
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
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
You will need:
- String cut in 32 cm (~13.5") lengths
- Graph paper
- Rulers, compasses, other drawing and measuring tools
- 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.