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2
Introduction
®
The transparent Giant GeoSolids set includes 10 plastic, three-dimensional
shapes that allow for hands-on study of volume. These shapes can expand
daily math lessons while introducing, teaching, and reviewing geometric
concepts effectively. They allow students to make concrete connections between
geometric shapes and their associated formulas for volume, as
well as compare the volumetric relationship between each shape.
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 cylin-
der 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. However, 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 of a
polyhedron are calculated the same way, regardless of shape.
Getting Started with Transparent Geometric Shapes
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 shapes. Ask them
to write down their observations as they make the following comparisons:
How are the shapes similar? (With the exception of the sphere, all shapes have
the same height. They are all three-dimensional. They all have empty spaces
inside them.) How are they 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 shape can be identified
by its color.
Ask students how they might organize the shapes into categories based on their
3
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 shapes again based on this information.
Discuss and explain the cylinder, sphere, and cone as exceptions.
Geometric
Shape
Number of Shape of Number of Number of Number of
Bases Base(s) Faces Edges Vertices
Square Prism
Rectangular Prism
Hexagonal Prism
Triangular Prism
Square Pyramid
Triangular Pyramid
Sphere
Hemisphere
Cylinder
Cone
Work with students to create a table like this one to record their observations:
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 using toothpicks and gum drops,
straws and yarn, or even pipe cleaners. As you go through formulas, encourage
students to refer to their models to visualize why the formulas work.
Introducing Volume
Volume, or the capacity of an object, is sometimes confused with surface
area. At first glance, the formulas for finding each appear somewhat similar. A
helpful way to compare the two is to explain surface area is the amount of
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room on the outside of a shape, and volume is the amount of space inside a
shape. Discuss the importance of measuring volume, giving such examples as
knowing how much water a pool will hold, how much air fills a SCUBA tank,
or how much cement fits in a cement mixer. Ask students for other examples.
Students will benefit from practice with building, measuring, and filling
containers to understand volume. Each shape has openings in the base and
can be filled with water, sand, rice, or other materials. By filling one shape
and pouring its contents into another shape, 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.
Note: The bottoms of each shape are not removable.
Challenge students to estimate the volume of each shape and place them in
order from largest to smallest volume. You may want to allow students to
fill their shapes to make more accurate estimations. As you introduce the
formulas for finding the volume of each shape, encourage students to refer
to the shapes for reference. You 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 shape to confirm the accuracy of their initial
guesses about volume.
These models were built using the metric system. Although they can be used
with any measurement system, metric is easiest. Because of the thickness of
the plastic, measurements between students might be slightly “off,” depending
on if they measure from the inside edges or the outside edges. If students round
to the nearest centimeter, this will not be a problem.
Volume Formulas
Prism
Finding the volume of a general prism is simply a matter of multiplying the
area of the base times the height of the prism:
×
Volumegeneral prism = A
H
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
l
Rectangular Prism
×
Volumerectangular prism = A
H
w
5
= (l × w) × H
H
Square Prism
Volumesquare prism = A × H
= (w × l) × H
l
w
b
A = Area of the square base.
H = Height of the prism.
s = Length of the side.
h
H
Triangular Prism
Volumetriangular prism = A × H
= ( bꢀ × h) × H
ꢂ
A = area of the triangle base ( bꢁ× h)
ꢂ
h = altitude, or height of the triangle.
H = height of the prism.
H
Hexagonal Prism
Volumehexagonal prism = A × H
= (w × ꢀs) × H
A = Area of the hexagonal base.
H = Height of the prism.
Explain that the area for a hexagon is calculat-
ed as follows:
w
A = w × ꢀ s
s
w = Width of hexagon as shown.
s = Length of side.
r
H
Cylinder
Volumecylinder = A × H
6
2
= (πr ) × H
Pyramid
Introduce the general formula for finding the volume of a pyramid.
Volumepyramid = ꢃ A × H
Ask students to identify the difference between this general formula and the one
for the prism. (There is one more variable:
volume formula for a prism, it is easy to
remember the volume formula for
ꢂ
.)ꢃIf students remember a
a pyramid with the same-size base and
H
height: simply multiply by . ꢃYou
ꢂ
can demonstrate this concept by pouring
three filled pyramids into the
corresponding prism in the Geometric
Shapes set.
l
w
Square Pyramid
H
Volumesquare pyramid = ꢃ A × H
--------------
= ꢃ (l × w) × H
h
b
Triangular Pyramid
H
Volumetriangular pyramid = ꢃ A × H
_
= ꢃ (ꢀ b × h) × H
r
Cone
r
Volumecone = ꢃ A × H
2
= ꢃ (πr ) × H
_
Sphere
Volumesphere = ꢃ πr
r
3
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