Getting Started |
| Misconception/Error The student is unable to accurately describe a sequence of rigid motions that carries one triangle onto another. |
| Examples of Student Work at this Level The student describes only a single transformation (and in insufficient detail) that, if implemented, will not carry one triangle onto the other. The students writes:
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| Questions Eliciting Thinking What are rigid motions? Can you think of other examples of rigid motion?
What does it mean to “map one triangle onto another?”
Will a single rigid motion map one of the triangles onto the other? What is meant by sequence? Can you describe a sequence of rigid motions?
How will the vertices of  correspond to the vertices of the other triangle?
What is the center of rotation? |
| Instructional Implications Review the definition of each of the rigid motions: translations, reflections, and rotations. To develop an intuitive understanding of rigid motion, allow the student to experiment with a variety of transformations using transparent paper, interactive websites such as http://www.mathopenref.com/translate.html, or the CPALMS Virtual Manipulatives Transformations—Translation (ID 11260), Transformations—Rotation (ID 11262), and Playing with Reflections (ID 11263).
Explain what it means to map one triangle onto another. Provide the student with a pair of congruent triangles that are related by a single transformation and ask the student to identify and describe the specific transformation that maps one figure onto the other. Explain to the student that describing the transformation in detail (e.g., by specifying the center and degree of rotation, the line of reflection, or the vector along which a figure is translated) and then performing the transformation is a convincing way to show that one figure is mapped onto the other. Next, provide two congruent triangles that are related by more than one transformation. Have the student identify and describe the sequence of transformations that maps one triangle onto the other. Ask the student to perform the sequence of transformations to ensure the description is sufficient.
Provide additional opportunities to describe a sequence of transformations that maps one figure onto the other. Remind the student to include all necessary components in each description: the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. |
Moving Forward |
| Misconception/Error The student provides only a general description of a sequence of rigid motions. |
| Examples of Student Work at this Level The student suggests a rotation and a translation but does not provide sufficient detail or perform the transformations.
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| Questions Eliciting Thinking Can you describe the rotation more specifically? What is the center of the rotation? How many degrees? What direction?
Can you describe the translation more specifically? What vector describes the translation?
Will all of the vertices coincide after the translation? |
| Instructional Implications Explain to the student that describing the transformations in detail (e.g., by specifying the center and degree of rotation, the line of reflection, or the vector along which a figure is translated) and then performing the transformations is a convincing way to show that one triangle is mapped onto the other. Encourage the student to be precise when describing transformations. Model a concise description of each transformation using mathematical terminology.
Provide additional opportunities to describe a sequence of transformations that maps one figure onto the other. Remind the student to include all necessary components in each description, the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. |
Almost There |
| Misconception/Error The student makes an error in describing one of the transformations. |
| Examples of Student Work at this Level The student provides a detailed description of a sequence of transformations that maps one triangle onto the other but the description contains an error. For example, the student imposes a coordinate grid on the diagram and describes the translation as  . |
| Questions Eliciting Thinking Where is after the rotation?
How did you determine the distance and direction of translation? Can you check your work? |
| Instructional Implications Provide specific feedback to the student concerning any error made and allow the student to revise his or her work. Correct any notation errors. If needed, model for the student the conventional way to describe rotations and translations.
Provide additional opportunities to describe a sequence of transformations that maps one figure onto the other. Remind the student to include all necessary components in each description: the center and degree of rotation, the line of reflection, or the vector along which a figure is translated. |
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 describes a correct sequence of transformations and in sufficient detail. For example, the student says to rotate  about point A. Then translate according to  (or five units vertically up). The student may also describe the translation in terms of the effect on the coordinates of the rotated triangle.
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| Questions Eliciting Thinking What if you changed the order of the two transformations you described? Will the result still be the same?
Must all possible correct responses contain a translation?
What does this exercise indicate about the relationship between and  ? |
| Instructional Implications Ask the student to complete the task using a single rotation around a carefully chosen center and provide the following hint: The center must be equidistant from each pair of corresponding vertices. |
