Task
In this task you must
determine what transformation has been applied to a shape by comparing it to its
image and using what you know about transformations. Consider the red circle in
the interactive figure below. Drag it and observe the behavior of its image,
shown as a red outline. The image is the result of a single transformation
(either a reflection, a rotation, or a translation/slide) on the original shape.
The symmetry of the circle may make it difficult to tell exactly what is
happening in the transformations. A different shape may provide more-useful
information. You may also test or develop conjectures by accessing the three
transformations with the small icons at the lower left. The images for these
transformations are shown as blue outlines.
Choose a different shape
and observe the behavior of its image. Change the shape of the red square or red
triangle by dragging it by an edge or a vertex while pressing the Control key.
Change the orientation of a shape by dragging it by a vertex. What is the
transformation used in challenge 1? How can you decide? Describe the position
and orientation of the resulting image in relation to the original shape. Now
try another challenge.
Discussion
Using dynamic geometry software, teachers can pose additional
challenges for middle-grades students to develop their understanding of
transformations and congruence. In each challenge above, a red shape and
its image under an unknown transformation are shown. Students can learn
about the nature of the unknown transformation by investigating the
dynamic behavior of a shape and its image under the transformation and
analyzing the relationships that remain constant between the original
shape and its image. Using dynamic geometry software, students can
identify an unknown transformation in several ways: by comparing the
orientation of the shapes, by analyzing the trace of the image and of
the original shape or of points on them, or by finding the locus of
invariant points. Students can use the software to check their
conjecture by constructing the image of the original shape under the
transformation they identified.
Take Time to Reflect
- What new insights into transformations can students gain as they work on activities like this?
- What are some specific ways in which middle-grades students
can identify the unknown transformation used in each of the challenges
in this example?
- What can teachers do to help their students better describe transformations?