Understanding the Pythagorean Relationship Using Interactive Figures
The
Pythagorean relationship, a2 + b2 =
c2 (where a and b are the lengths of the
legs of a right triangle and c is the hypotenuse), can be demonstrated
in many ways, including with visual "proofs" that require little
or no symbolism or explanation. The activity in this example presents one
dynamic version of a demonstration of this relationship. Visual and dynamic
demonstrations can help students analyze and explain mathematical relationships,
as described in the Geometry Standard. The interactive
figure in this activity can help students understand the Pythagorean relationship
and gives them experience with transformations that preserve area but not
shape.
Task
In this task, you will
explore a dynamic demonstration of the Pythagorean relationship. Your
goal is to determine how the interactive figure demonstrates the Pythagorean
relationship. Consider the blue right triangle in the interactive figure
below. Red and yellow squares have been constructed on its legs. A square
has also been constructed on its hypotenuse. Click on the yellow square;
notice how the outline of a parallelogram appears. Drag the yellow square
to transform it into the parallelogram. Now drag the parallelogram to
transform it into the rectangle that appears within the large square.
Repeat this process for the red square. How does your transformation of
the squares into parallelograms and then into rectangles affect their
area? What relationship is demonstrated when the rectangles fill the large
square formed on the hypotenuse? Now drag the red and yellow rectangles
back to their original positions as squares on the legs of the triangle.
Change the blue right triangle by dragging it by a vertex and repeat the
transformation of the yellow and red squares. What do you observe?
[How
to Use the Interactive Figure]
[Stand-alone
applet]
Discussion
Suppose a and b represent the lengths of
the legs of the blue right triangle, and c represents the length
of its hypotenuse. By engaging with the interactive figure, students
observe that the squares formed on the two legs of the triangle (with
areas a2 and b2) can be transformed
to completely fill the square formed on the hypotenuse (with area c2).
That is, the sum of the areas of the squares formed on the two legs
is equal to the area of the square formed on the hypotenuse, or a2
+ b2 = c2. Teachers can ask students
to make a general observation about what happens as the areas of squares
on the legs of a right triangle are transformed into parallelograms
that will finally fit into the area of the square on the hypotenuse.
Students could also be asked to justify their observations by attending
to areas that remain constant throughout the process of transforming
a shape (a shear transformation applied to parallelograms). Finally,
the original right triangle can be transformed in various ways, and
the process of fitting the squares formed on the legs into the square
on the hypotenuse repeated. Teachers should encourage their students
to consider why it is important to repeat the demonstrations for different
right triangles.
Interesting extensions to this task include the following:
(a) Consider what relationships exist among the areas of similar
figures, other than squares, built on the legs and hypotenuse of any
right triangle; (b) Consider what relationships exist among the
areas of the squares built on the three sides of an obtuse or acute
triangle rather than a right triangle.
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Take Time to Reflect
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- How is doing this activity on the computer different from
working with physical models such as dot paper or geoboards?
- How can students use the approach in this activity to construct
a proof of the Pythagorean relationship?
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Acknowledgment
Based
on an idea provided by Colette Laborde, EIAH, Laboratorie Leibniz-IMAG,
at the 1999 ENC-NCTM Conference and Workshop: The Role of Technology
and Examples in The Principles and Standards Document.
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