LER 7630
TM
ACTIVITY GUIDE
A Hands-on Approach to Learning About Area and Volume
Introduction
The transparent Power Solids™ set includes 12 plastic three-dimensional
shapes that allow for hands-on study of volume. Power Solids can be integrated
easily with daily math lessons for introducing, teaching, and reviewing math
concepts effectively. They allow students to make concrete connections
between geometric shapes and their associated formulas for volume, and to
observe volumetric relationships between the geometric shapes as well.
Most shapes in this set are variations of a prism or a pyramid, both of which
are polyhedrons. Polyhedrons are solid figures with flat sides, or faces. Faces
may meet at a point, called a vertex, or at a line, called an edge. A prism has
two congruent bases; the remaining faces are rectangles. A pyramid has one
base and the remaining faces are triangles.
Three shapes in this set have curved faces rather than flat ones; the cylinder,
cone, and sphere. Technically, they are not polyhedrons. Even so, a cylinder
can be thought of as a circular prism: a figure with congruent circular bases
and a single, rectangular face. A cone can be thought of as a pyramid with a
circular base and a face that is a wedge. A sphere is a unique shape with no
parallel to prisms or pyramids.
At the outset, learning formulas for the volume of more than a dozen geometric
shapes may seem daunting to your students. Formulas become much easier to
remember when students recognize that only the method for calculating the
area of a base changes from formula to formula; the other variables are
calculated the same way, regardless of shape.
™
Getting Started With Power Solids
Allow students to become familiar with the manipulatives before beginning
directed activities. You may want to explore prisms and pyramids on separate
days. Encourage students to handle, observe, and discuss the Power Solids. Ask
them to write down their observations as they make the following comparisons:
How are the shapes similar? (All shapes have the same height. They are all
three-dimensional. They all have empty spaces inside them.) How are the
different? (Some have flat sides; some have curved sides. Some are box-
shaped; some are round, and some are triangle-shaped.) Where have students
seen these shapes in the world around them? (Great Pyramids of Egypt, traffic
pylons, film canisters, soccer balls, pieces of chalk, boxes, lipstick tubes, and
so on.)
Introduce and identify the following terms: face, edge, vertex or corner, and
base. Mention to students that the base of each Power Solid can be identified
by the hole in the face.
3
Ask students how they might organize the shapes into categories based on their
features. Write students’ answers on the board. Then, define pyramids and
prisms. Hold up an example of a prism and a pyramid for the class. Encourage
students to organize the Power Solids again based on this information. Discuss
and explain the cylinder, sphere, and cone as exceptions.
Work with students to create a table like this one to record their observations.
Number of Shape of Number of Number of Number of
™
Power Solids
Bases
Base(s)
Faces
Edges
Vertices
Large Square Prism
Small Rectangular
Prism
Large Rectangular
Prism
Hexagonal Prism
Triangular Prism
Square Pyramid
Triangular Pyramid
Sphere
Cylinder
Cone
Show students a cardboard box. Ask if the box is a prism or a pyramid. (prism)
Have a student volunteer identify the box’s bases, faces, edges, and vertices.
Have another student do the same for an oatmeal container. You may need to
cut the container to make identification easier.
This would be a good time for your students to make constructions of the
various models. You can construct models of toothpicks and gumdrops, straws
and yarn, or pipe cleaners. As you go through formulas, encourage students to
refer to their models to visualize why the formulas work.
4
Introducing Volume
Volume, or the capacity of an object, is sometimes confused with surface area.
At first glance, the formulas appear somewhat similar. A helpful way to
compare the concepts is to explain surface area as the amount of room on the
outside of a shape, and volume as the amount of space inside a shape. Discuss
the value of measuring volume, giving such examples as knowing how much
water a pool will hold, how much air a SCUBA tank will hold, or how much
cement a cement mixer will hold. Ask students for other examples.
Students will benefit from practice with building, measuring, and filling
containers to understand volume. The Power Solids have a removable base and
can be filled with water, sand, rice, or other materials. By filling one Power
Solid and pouring its contents into another Power Solid, students can explore
volume relationships between shapes. If you intend to have students perform
exact measurements using a graduated cylinder, be sure they are comfortable
reading the bottom edge of the water level, or meniscus.
Students can measure volume by reading sand levels in a graduated cylinder
before and after filling a Power Solid. Have your students take the average of
three trials to eliminate some errors. First, fill a large graduated cylinder nearly
to the top and take a reading. Use the sand in the cylinder to fill the Power
Solid. Take a reading for the sand remaining in the cylinder and subtract it
from the starting quantity. The difference is the volume of sand poured into the
Power Solid.
Challenge students to order the Power Solids from largest to smallest volume
by estimation. You may want to allow them to fill their Power Solids or use
cube models to make more accurate estimations. As you introduce the formulas
for finding the volume of each shape, encourage students to refer to their
Power Solids for reference. You also may wish to distribute copies of the table
on page 2 for reference. Once you have finished your discussion, students can
mathematically calculate the volume of each Power Solid to confirm the
accuracy of their initial estimations.
Explain to students that the thickness of the plastic takes away from the volume
each shape can hold. Therefore, students must measure from inside edge to
inside edge rather than from outside edge to outside edge when computing
what the shape can hold. Also, explain that the shapes are slightly larger at the
opening so they can slip out of the mold during manufacturing. This will cause
slight variations in the measurements. Tell them that the standard height is 4.6
cm, and the other measurements are derived from this to keep the shapes in
relationship to each other.
5
Volume Formulas
Prism
Finding the volume of a general prism is a matter of multiplying the area of the
base times the height of the prism:
Volumegeneral prism = A x H
Identify the variables:
A = Area of the base
H = Height of the prism
The formula for the area of the base of the
prism depends upon the shape
of the base.
H
H
Rectangular Prism
l
Volumerectangular prism = A x H
w
= (l x w) x H
Square Prism
Volumesquare prism = A x H
= (s x s) x H
s
Identify the variables:
s
A = Area of the square base
H = Height of the prism
s = Length of the side
Triangular Prism
h
Volumetriangular prism = A x H
b
1
H
= ( 2 b x h) x H
/
Identify the variables:
1
A = Area of the triangle base ( 2 b x h)
/
h = Altitude, or height, of the triangle
H = Height of the prism
6
H
Hexagonal Prism
Volumehexagonal prism = A x H
Identify the variables:
A = Area of the hexagonal base
H = Height of the prism
Explain that the area for a hexagon is
calculated as follows:
w
A = w x 3 2 s
/
Identify the variables:
w = Width of hexagon as shown
s = Length of side
s
r
H
Cylinder
Volumecylinder = A x H
2
= (π r ) x H
Pyramid
Introduce the general formula for finding the volume of a pyramid:
1
Volumepyramid
=
A x H
/
3
Ask students to identify the difference between this general formula and the
1
one for the prism. (There is one more variable: .) If students remember a
/
volume formula for a prism, it is easy to rememb3er the volume formula for a
1
pyramid with the same-size base and height: simply multiply by . You can
/
3
demonstrate this concept by pouring
H
three filled pyramids into the
corresponding prism in the Power
Solid set.
s
s
Square Pyramid
Volumesquare pyramid
1
1
=
=
A x H
/
33 (s x s) x H
/
7
H
b
Triangular Pyramid
Volumetriangular pyramid
1
1
=
=
A x H
/
33 (b x h) x H
/
H
r
Cone
Volumecone
1
=
=
A x H
/
/
3
2
1
3 (π r ) x H
r
Sphere
Volumesphere
2
4
=
π r
/
3
®
Also from Learning Resources :
™
• LER 7631 Investigating with Power Solids
• LER 7633 Geometry Template
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LRM7630-TG
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