Task
Manipulate the dynamic rectangle and parallelogram below by dragging the vertices and the sides. 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?
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).
Take Time to Reflect
- What new insights into the properties of parallelograms can students gain as they work on activities like this?
- 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?
- What other pairs of dynamic figures would be interesting for students to consider in activities like this?
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.