Transformations and Congruence
The Lesson Activities will help you meet these educational goals:
· Content Knowledge—In this lesson, you will use geometric descriptions of rigid motions to transform figures and use the definition of congruence in terms of rigid motions to decide if two figures are congruent.
· Mathematical Practices—You will make sense of problems and solve them.
· STEM—You will apply mathematical and technology tools and knowledge to solve real-world design problems.
Directions
You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.
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Self-Checked Activities
Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.
1. Transformations and Congruence
You can help the painter search for a stencil that matches her original pattern. In this activity, you will transform two different stencils in GeoGebra as you search for an exact overlap with the original pattern. If you need help, follow these instructions for using GeoGebra.
a. First work with stencil one. Use a combination of reflections, rotations, and translations to see whether stencil one will overlap with the original pattern. List the sequence of rigid transformations you used in your attempt, noting the type of transformation, the direction, the coordinates, and the displacement.
Sample answer:
Transformation1 / Translate stencil one 1 unit in the positive y-direction.
2 / Rotate stencil one 90 degrees counterclockwise about the point (-7, 1).
3 / Translate stencil one 3 units in the positive x-direction.
4 / Reflect stencil one about the y-axis.
b. Does stencil one overlap exactly with the original image in part a? Describe what you see. After the transformations are complete, what can you say about the points that make up stencil one and the points that make up the original image? Explain.
Sample answer:
Yes, stencil one overlaps exactly with the original image. The original pattern is now hidden behind stencil one. Each point on stencil one corresponds with a point on the original image.
c. Now, consider stencil two. Again, use a combination of reflections, rotations, and translations to see whether stencil two will overlap with the original pattern. List the sequence of rigid transformations you used in your attempt, noting the type of transformation, the direction, the coordinates, and the displacement.
Sample answer:
Transformation1 / Reflect stencil two about the y-axis.
2 / Translate stencil two 1 unit in the positive x-direction.
3 / Translate stencil two 7 units in the positive y-direction.
d. Does stencil two overlap exactly with the original image? Describe what you see. After the transformations are complete, what can you say about the points that make up stencil one and the points that make up the original image? Explain.
Sample answer:
No, stencil two does not overlap the original pattern exactly. Not all the points on stencil two coincide with the points on the original pattern. There is a subtle difference near the boundaries of the two shapes.
e. Should the painter use stencil one or stencil two for her project? Why? Explain in terms of rigid transformations.
Sample answer:
The painter should use stencil one for her project because a series of rigid transformations maps stencil one to the original pattern. The shapes are the same because you don’t have to deform stencil one to match the original shape.
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