Illuminations: Understanding the Pythagorean Relationship Using Interactive Figures

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.

Learning Objectives

Students will

analyze and explain mathematical relationships, as described in the Geometry Standard.

Materials

Computer and Internet connection

Instructional Plan

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 the 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.

Teacher Reflection

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?

NCTM Standards and Expectations

Geometry 6-8

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects
Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

References

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."