Getting Started |
| Misconception/Error The student draws an incomplete or incorrect figure and is unable to precisely define the term parallel lines. |
| Examples of Student Work at this Level The student draws:
- Lines that are not parallel.
- Two parallel segments.
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| Questions Eliciting Thinking What does it mean for lines to be parallel? Do they ever intersect?
Can you explain what you drew? Where are the parallel lines in your picture?
What is the difference between lines and line segments? |
| Instructional Implications Discuss with the student the qualities of a definition that make it precise and complete. Then offer the student a precise definition of parallel lines such as, “Two lines are parallel if and only if they are coplanar and have no points in common.” Discuss with the student the features of this definition that make it precise (e.g., the inclusion of the condition that the lines be coplanar). Also discuss what it means for two lines to “have no points in common.” Introduce the student to the concept of a counterexample. Challenge the student to find a counterexample (i.e., an example of two coplanar lines that have no points in common) that are not parallel. Indicate a quality of a good definition is that it eliminates all counterexamples.
Be sure the student understands the distinctions among lines, rays, and segments and the notation associated with naming each.
Show the student how to use embedded arrows in diagrams to indicate that lines are parallel. |
Moving Forward |
| Misconception/Error The student correctly draws an example of parallel lines but is unable to write a precise definition. |
| Examples of Student Work at this Level The student draws an example of parallel lines, but the student's definition is either incorrect or incomplete. For example, the student says parallel lines are:
- Identical or linear.
- The same or same length.
- Lines that match.
- Same size or congruent.
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| Questions Eliciting Thinking Provide a nonexample and ask, “Are these lines parallel? Why or why not?”
If one line is drawn on the floor and another on the ceiling, are the lines parallel?
Are the two sides of the same railroad track parallel? |
| Instructional Implications Discuss with the student the qualities of a definition that make it precise and complete. Then offer the student a precise definition of parallel lines such as, “Two lines are parallel if and only if they are coplanar and have no points in common.” Discuss with the student the features of this definition that make it precise (e.g., the inclusion of the condition that the lines be coplanar). Also discuss what it means for two lines to “have no points in common.” Introduce the student to the concept of a counterexample. Challenge the student to find a counterexample (i.e., an example of two coplanar lines that have no points in common) that are not parallel. Indicate a quality of a good definition is that it eliminates all counterexamples. |
Almost There |
| Misconception/Error The student’s definition is incomplete or imprecise. |
| Examples of Student Work at this Level The student correctly draws a pair of parallel lines. However, the student:
- Provides a definition that is incomplete. For example, the definition contains the condition that the lines never intersect (or have the same slope) but lacks the condition that they must be coplanar.
- Uses nonmathematical terminology.
- Includes an unnecessary or redundant condition such as the lines “go on forever.”
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| Questions Eliciting Thinking Demonstrate skew lines with two pencils and ask, “Are these lines parallel?”
Suppose in a plane with a coordinate system, two lines are vertical—will they have the same slope? |
| Instructional Implications Discuss with the student the qualities of a definition that make it precise and complete. Introduce the student to the concept of a counterexample. Challenge the student to find a counterexample for the definition that he or she wrote. For example, if the student defined parallel lines as “lines that never intersect,” then a pair of skew lines would serve as a counterexample (i.e., the skew lines satisfy the conditions of the definition but are not parallel).
Have the student work with other Almost There students to identify features of their definitions that are imprecise or incomplete. Afterward, ask the student to revise his or her definition. Then present a precise definition of parallel lines such as, “Two lines are parallel if and only if they are coplanar and have no points in common.” Have the student compare his or her revised definition to this one and identify features of this definition that make it precise. |
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 correctly draws a pair of parallel lines. The student defines parallel lines by saying:
- Lines are parallel if they lie in the same plane and are the same distance apart over their entire length.
- Lines are parallel if they are coplanar and do not intersect.
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| Questions Eliciting Thinking How do you determine the distance between two lines? How is this distance defined?
Can a line on the ceiling and a line on the floor be parallel? Can they lie in the same plane?
Can a line be parallel to itself? |
| Instructional Implications Guide the student to describe parallel lines in terms of slopes, rigid motion, and distances between lines.
Ask the student to consider the difference between parallel lines and skew lines. |
