Getting Started 
Misconception/Error The student draws an incomplete or incorrect figure and is unable to precisely define the term line segment. 
Examples of Student Work at this Level The student draws a line or ray, rather than a line segment, with two named points (usually A and B) and is unable to indicate which portion of the drawing includes the line segment. The student does not provide a precise definition of a line segment.

Questions Eliciting Thinking Does your line segment have a beginning and an end?
In your figure, which is the line and which is the line segment?
Are points A and B on your line segment? Are there other points on your line segment? 
Instructional Implications Review the concept of a point and then provide direct instruction on the differences among lines, rays, and line segments. Guide the student to draw and label examples of each.
Draw a line with three named points A, B, C on the line. Then ask the student to identify and name as many line segments as possible using the three named points.
Discuss with the student the qualities of a definition that make it precise and complete. Then offer the student a precise definition of a line segment such as, “Line segment AB consists of the points A and B and all of the points between A and B on line AB.” 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 find a counterexample (i.e., a figure that consists of the points A and B, and all of the points between A and B on line AB) that is not a line segment. Indicate that a quality of a good definition is that it eliminates all counterexamples. 
Moving Forward 
Misconception/Error The student correctly draws and labels the line segment but is unable to provide a precise mathematical definition. 
Examples of Student Work at this Level The student correctly draws and labels a line segment. When defining a line segment, the student:
 Describes it as a line with a certain quality such as a line with two points or two endpoints.
 Writes a definition that is incomplete or imprecise.
 Provides a circular definition such as “a line segment is a segment of a line.”

Questions Eliciting Thinking How does a line segment differ from a line?
What points are included on a line segment? How can you describe these points?
Could someone who did not know what a line segment is draw one based on 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 a line segment such as, “Line segment AB consists of the points A and B and all of the points between A and B on line AB.” Discuss with the student the features of this definition that make it precise. Explain the distinction between a line and a line segment and why it is incorrect to define a line segment as a line with a certain quality. Be sure the student understands that lines and line segments are related but that a line segment, by definition, is not a line.
Introduce the student to the concept of a counterexample. Challenge the student to find a counterexample (i.e., a figure that consists of the points A and B and all of the points between A and B on line AB) that is not a line segment. Indicate that 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 and labels a line segment. The student provides the foundation of a good definition (“a line segment is part of a line”) but is unable to precisely describe how the part is characterized.

Questions Eliciting Thinking Is a ray part of a line? How does a line segment differ from a ray? Is a line segment part of a ray?
Do line segments have length?
Can there be more than one line segment on a given line? 
Instructional Implications 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 imprecise or incomplete. Afterward, ask the Almost There student to revise his or her definition. Then present a precise definition of a line segment such as, “Line segment AB consists of the points A and B and all of the points between A and B on line AB.” Have the student compare his or her revised definition to this one and identify features of this definition that make it precise.
Introduce the student to the concept of a counter example. Challenge the student to find a counterexample for the definition that he or she wrote. For example, if the student defined a line segment as “part of a line” then a ray would serve as a counterexample. In this case, explain that the ray satisfies the conditions of the definition but is not a line segment. Consequently, this definition of a line segment is not complete. 
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 and labels the line segment. The student defines the line segment as “two points on a line and all the points between these points on the line” or “part of a line that consists of two distinct points and all of the points between them.” 
Questions Eliciting Thinking Since you have defined a line segment in terms of points and a line, do you need to define point and line? What about the term between? Do you think this needs to be defined?
What if you name a line segment instead of ? Does that name the same line segment or a different one?
What happens if you draw two line segments, and that have a common endpoint, B. Under what circumstances will AC < AB + BC? 
Instructional Implications Introduce the student to the use of notation in defining line segments. For example, offer the student the following definition, ' consists of points A and B on as well as all points between A and B on .' Ask the student to consider the number of points on a line and the number of points on a line segment.
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:
 All right angles measure 90°.
 All rectangles are squares.
 All triangles are scalene.
 For any triangle, the sum of the measures of its angles is 180°.
 All isosceles triangles are equilateral.
 If = 16, then x = 4.
