Getting Started |
| Misconception/Error The student is unable to perform transformations (translations, reflections, and rotations) to experimentally verify the properties of rigid motion. |
| Examples of Student Work at this Level The student:
- Provides little or no work and has difficulty working with transparent paper.
- Is unable to correctly perform some or all of the transformations.
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| Questions Eliciting Thinking Do you know what it means to translate (or reflect or rotate) a figure?
What is the difference between an image and its preimage?
Can you show me how you used the patty paper in this task? |
| Instructional Implications Review the definition of each of the rigid transformations: translations, reflections, and rotations. To develop an intuitive understanding of rigid transformations, allow the student to experiment with a variety of transformations using transparent paper or interactive websites such as , http://www.mathopenref.com/translate.html or http://www.shodor.org/interactivate/activities/TransmographerTwo/.
Be sure the student understands that:
- A translation is a transformation of the plane. A translation along a vector v assigns to each point, P, in the plane an image point,
 , so that the distance from P to corresponds to the magnitude (length) of vector v and the direction of from P corresponds to the direction of vector v. Use grid paper to illustrate translations of points and to demonstrate the relationship between a point, its image, and the vector that defines the translation. Then illustrate translations of more complex figures such as segments, angles, and polygons.
- A reflection is a transformation of the plane. A reflection across line m (the line of reflection) assigns to each point not on line m, a point that is symmetric to itself with respect to line m (e.g., m is the perpendicular bisector of the segment whose endpoints are the point and its image). Also, a reflection assigns to each point on line m the point itself. Use grid paper to illustrate reflections of points and to demonstrate the relationship between a point, its image, and the line of reflection. Then illustrate reflections of more complex figures such as segments, angles, and polygons.
- A rotation is a transformation of the plane. Each point in the plane is rotated a specified number of degrees (given by the degree of rotation) either clockwise or counterclockwise (indicated by the sign of the degree of rotation) about a fixed point called the center of rotation. Use a unit circle to illustrate rotations of points about the origin. Then illustrate rotations of more complex figures such as segments, angles, and polygons.
After introducing each rigid motion through the transformation of points, introduce more complex figures such as segments, angles, and polygons. Emphasize the basic properties of rigid motions:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.
Discuss how these properties ensure that the image of a figure under a rigid motion is always congruent to the preimage.
Provide additional opportunities to experiment with translations using transparent paper and to verify experimentally the properties of translations, reflections, and rotations. |
Moving Forward |
| Misconception/Error The student is unable to clearly verify the properties of transformations. |
| Examples of Student Work at this Level The student can perform transformations of the given segment but is unable to clearly answer the questions. The student provides minimal answers (yes or no) with no elaboration. |
| Questions Eliciting Thinking Can you explain your answers in more detail?
Can you explain how your experimentation with the transformations supports your answers?
Can you demonstrate how the image of a segment is still a segment with the same length as its preimage? |
| Instructional Implications Provide additional opportunities to experiment with rigid motions using transparent paper and to verify experimentally the properties of translations, reflections, and rotations. Emphasize the basic properties of rigid motions:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.
Discuss how these properties ensure that the image of a figure under a rigid motion is always congruent to the preimage. Model using appropriate mathematical terminology to describe rigid motion and explain the properties.
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Almost There |
| Misconception/Error The student is unable to generalize the properties of transformations from line segments to lines. |
| Examples of Student Work at this Level The student answers all questions about line segments correctly, however the student:
- Is unable to generalize the results of his or her experimentation to lines.
- Confuses lines and line segments and states the line segment “will always be the same.”
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| Questions Eliciting Thinking What is the difference between a line segment and a line? Are there any similarities?
Do you think that a line will still be a line after each of the transformations? |
| Instructional Implications Discuss with the student the similarities and differences between a line segment and a line. Direct the student to observe that the image of a line segment maintains its length when it is translated, rotated, and reflected. Guide the student to understand that the image of any line is also a line no matter how it is transformed. Model for the student by using transparent paper that the image of a line is always a line. |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student experimentally verifies that segments remain segments with the same length after undergoing a translation, reflection, or rotation by performing transformations of  . In addition, the student determines that the image of  will also be a line.
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| Questions Eliciting Thinking How would you explain to someone else why the image of a segment is always a segment of the same length?
What does the term distance-preserving mean? How does it apply to translations of line segments?
Do you think dilations will also be distance–preserving for line segments? |
| Instructional Implications Have the student consider whether transformations of angles are degree-preserving. Consider implementing the MFAS task Angle Transformations and Parallel Line Transformations to assess the student’s understanding of other properties of rigid motion.
Introduce dilations and ask the student which properties of rigid motion apply to dilations. |
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