Visualizing Transformations

• ## 6.4.1 Visualizing Transformations

Choose a transformation and apply it to a shape to observe the resulting image.

### Instructions

To observe the behavior under the selected transformation, drag or change the red shape. To select a shape, click on the shape in the icons at the top. To select a transformation, click on the 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 the red square or red triangle by dragging it from a vertex. Resize the circle by dragging it from any point on the circumference.

• Reflection
To change the slope of the reflection line, drag the line. To change the position of the line, drag the point on the line.
• Translation
To change the translation, drag the arrow from any of its endpoints.
• Rotation
To change the angle of rotation, drag the angle symbol from the upper endpoint. To change the center of rotation, drag the blue dot at the vertex.

### Exploration

The goal of this task is to explore the effects of applying various transformations to a shape. Eventually you should be able to predict how each transformation will change the shape's image. Consider the red shape in the interactive figure below. Drag it and observe the behavior of its image, shown as a black outline. Choose a different shape, and using the same transformation, observe the behavior of its image. Change the shape of the red square or the red triangle by dragging it by an edge or vertex while pressing the "Control" key. Change the orientation by dragging the shape by a vertex. Describe the position and orientation of the resulting image in relation to the original shape. What is the relationship between the side lengths and angle measures of the original shape and those of the resulting image? Now consider the same tasks using other transformations.

### Discussion

Dynamic geometry software allows students to visualize a transformation by manipulating a shape and observing the effect of each manipulation on its image. By focusing on the positions, side lengths, and angle measures of the original and resulting figures, middle-grades students can gain new insights into congruence. Transformations can become an object of study in their own right. Teachers can ask students to visualize and describe the relationship among lines of reflection, centers of rotation, and positions of preimages and images. Using the interactive figure, students might see that the result of a reflection is the same distance from the line of reflection as the original shape. In a rotation, students might note that the corresponding vertices in the preimage and image are the same distance from the center of rotation and all the angles formed by connecting the center to the corresponding vertices are congruent in the image and the preimage.

### Take Time to Reflect

• What new insights into congruence can students gain as they work on activities like this?
•  What relationships between the original shape and its image are important for students to note in a translation?
•  What relationships between the original shape and its image are important for students to note in a reflection?
•  What relationships between the original shape and its image are important for students to note in a rotation?

### Objectives and Standards

NCTM Standards and Expectations
• Geometry / Measurement
• Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• 6-8
• Geometry