## 6.4.2 Identifying Unknown Transformations

The user is challenged to identify the transformation that has been used.

Each challenge shows a red shape and its image (outline) under a single unknown transformation (either a reflection, rotation, or translation). Drag the red shape to observe the behavior of its image. To select a different shape, click on a shape from the icons at the top. To select a different unknown transformation, click on the challenge icons on the left. Change the shape of the red square or red triangle by dragging from an edge or vertex while pressing the "Control" key. Change the orientation of a shape by dragging from a vertex. Resize the circle by dragging from any point on the circumference.

Use the reflection, transformation, and rotation buttons at the bottom to assist in determining the unknown transformation.

**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?