Getting Started 
Misconception/Error The student draws an incomplete or incorrect figure and is unable to precisely define the term perpendicular lines. 
Examples of Student Work at this Level The student draws:
 Two lines that intersect. When asked, the student does not indicate there is anything special about the angles formed.
 Two segments that intersect at their midpoints.
 A pair of lines that appear to be parallel.

Questions Eliciting Thinking What does it mean for lines to be perpendicular? Do they intersect in a special way?
Can you explain what you drew? Where are the perpendicular lines in your picture?
What is the difference between lines and line segments? 
Instructional Implications Present the student with examples and nonexamples of perpendicular lines. Ask the student to identify important qualities of perpendicular lines. Introduce the student to the right angle symbol drawn at the point of intersection and make the use of this symbol a convention when drawing perpendicular lines.
Discuss with the student the qualities of a definition that make it precise and complete. Then offer the student a precise definition of perpendicular lines such as, “Two lines are perpendicular if and only if they intersect in a 90° angle.” Discuss with the student the features of this definition that make it precise. Introduce the student to the concept of a counterexample. Challenge the student to draw a counterexample (i.e., two lines that intersect in a angle) that are not perpendicular. Indicate a quality of a good definition is that it eliminates all counterexamples. 
Moving Forward 
Misconception/Error The student correctly draws an example of perpendicular lines but is unable to write a precise definition. 
Examples of Student Work at this Level The student draws an example of perpendicular lines and labels the drawing with the right angle symbol, but the student’s definition is incomplete. The student says:
 Perpendicular lines are lines that intersect through each other.
 Perpendicular lines are two lines that intersect.
The student’s definition is imprecise. The student says:
 Perpendicular lines have different slopes causing one to be positive and the other negative to make them intersect.
The student’s definition is poorly worded.

Questions Eliciting Thinking Is there anything special about the way that perpendicular lines intersect?
What does this symbol (point to the right angle symbol) mean? Should you include that in your definition? 
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 perpendicular lines such as, “Two lines are perpendicular if and only if they intersect in a 90° angle.” Discuss with the student the features of this definition that make it precise. Introduce the student to the concept of a counterexample. Challenge the student to draw a counterexample (i.e., two lines that intersect in a 90° angle) that are not perpendicular. Indicate a quality of a good definition is that it eliminates all counterexamples. 
Almost There 
Misconception/Error The student’s diagram contains a minor omission, or the student includes unnecessary conditions. 
Examples of Student Work at this Level The student’s diagram contains a minor omission. For example, the student:
 Omits the right angle symbol from his or her diagram.
 Draws a pair of perpendicular segments rather than lines.
The student’s definition contains unnecessary conditions. For example, the student says that perpendicular lines are:
 Lines that intersect at their midpoints and create a right angle.
 Two lines that intersect in a right angle and have slopes that are opposite and reciprocal.
 Two lines that intersect in a right angle which is shown with a little box in the corner.

Questions Eliciting Thinking Your definition correctly states that perpendicular lines create right angles. What marking is used to indicate that right angles are formed when perpendicular lines intersect?
What notation is used to indicate a line rather than a segment?
Do lines have midpoints?
Would a horizontal line be perpendicular to a vertical line? Are their slopes opposite and reciprocal? Do you need to say anything about slopes in your definition? 
Instructional Implications Provide direct feedback to the student on how to make his or her drawing more correct and complete.
Discuss with the student the qualities of a definition that make it precise and complete. Have the student work with other Almost There students to identify features of their definitions that are unnecessary or redundant. Then, ask the student to revise his or her definition. 
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 perpendicular lines (not segments) using a right angle symbol at the point of intersection. The student defines perpendicular lines as, “Two lines that intersect in a right angle.” 
Questions Eliciting Thinking Do you know any properties of perpendicular lines?
How are the slopes of perpendicular lines related to each other?
How are the slopes of perpendicular lines different from the slopes of parallel lines? 
Instructional Implications Introduce the student to biconditional statements and the role they play in definitions. Have the student rewrite geometric definitions as explicit biconditional statements (i.e., in the form “if p then q and if q then p” or “p if and only if q”).
Introduce the student to the concept of a counterexample. Challenge the student to find counterexamples, if they exist, for statements such as:
 Perpendicular lines are two lines that intersect.
 Parallel lines are lines that never intersect.
