Exploring
Properties of Rectangles and Parallelograms Using Dynamic Software
Dynamic geometry software
provides an environment in which students can explore geometric relationships
and make and test conjectures. In this example, properties of rectangles
and parallelograms are examined. The emphasis is on identifying what distinguishes
a rectangle from a more general parallelogram. Such tasks and the software
can help teachers address the Geometry Standard.
Task
Manipulate the dynamic
rectangle and parallelogram below by dragging the corners (vertices) and
sides (edges). You can rotate or stretch the shapes, but they will retain
particular features. What is alike about all the figures produced by the
dynamic rectangle? What is alike about all the figures produced by the
dynamic parallelogram? What common characteristics do parallelograms and
rectangles share? How do rectangles differ from other parallelograms?
[How
to Use the Interactive Figure]
[Stand-alone
applet]
Other Tasks
-
Predict whether
the dynamic rectangle can make each figure below, then check your
prediction by trying to duplicate the shape using the dynamic rectangle.
Predict whether the dynamic parallelogram can make each figure below,
then check your prediction by trying to duplicate the shape using
the dynamic parallelogram.
-
Can the dynamic
rectangle make all the shapes that the dynamic parallelogram can make?
Can the dynamic parallelogram make all the shapes that the dynamic
rectangle can make?
-
Describe how to
decide if the dynamic rectangle can make a particular shape.
-
Describe how to
decide if the dynamic parallelogram can make a particular shape.
Discussion
As students manipulate
and analyze the shapes that can be made by the dynamic rectangle and dynamic
parallelogram, they can make conjectures about the properties of the shapes.
For instance, students might initially say that both types of shapes have
"two long and two short sides" or that parallelograms don't have right
angles. Manipulating the dynamic rectangle and parallelogram can help
students check the validity of their conjectures. Students can determine
that (a) neither shape must have two long sides and two short sides
because both can make squares; and (b) rectangles always have right
angles and parallelograms sometimes have right angles. Subsequent investigations
using Shape Makers (Battista 1998) software, which includes on-screen
measurements for side lengths and angles, can help students transform
these intuitive notions into more-precise formal ideas about geometric
properties. With these features students can verify that both rectangles
and parallelograms always have opposite sides congruent but rectangles
must also have four right angles. They also see by measurement that a
parallelogram can have right angles (in the special cases of a rectangle
or square).
Research has shown
that an important step in students' development of geometric thinking
is to move away from intuitive, visual-holistic reasoning about geometric
shapes to a more analytic conception of the relationships between the
parts of shapes (Battista 1998; Clements and Battista 1992). Conceptualizing
and reasoning about the properties of shapes is a major step in this development.
Research further shows that dynamic geometry software is a powerful tool
for helping students make the transition to property-based reasoning (Battista
1998).
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What new
insights into the properties of parallelograms can students
gain as they work on activities like this?
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What relationships
between rectangles and parallelograms are important for students
to note?
-
What are
the advantages and disadvantages of having the students work
with existing dynamic figures compared with asking them to construct
their own?
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What other
pairs of dynamic figures would be interesting for students to
consider in activities like this?
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References
Battista, Michael T.
Shape Makers: Developing Geometric Reasoning with The Geometer's Sketchpad.
Berkeley, Calif.: Key Curriculum Press, 1998.
Clements, Douglas H.
& Michael T. Battista. "Geometry and Spatial Reasoning."
In Handbook of Research on Mathematics Teaching and Learning, edited
by Douglas A. Grouws, pp. 420–64. New York: NCTM/Macmillan Publishing
Co., 1992.
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