Getting Started 
Misconception/Error The student is unable to correctly identify the triangle type. 
Examples of Student Work at this Level The student incorrectly describes . For example, the student says is equilateral, equiangular, right, acute, obtuse, or some combination of these types.

Questions Eliciting Thinking How do you know that is acute (or any other triangle type used to describe )?
What must be true as a consequence of knowing that line l is the perpendicular bisector of ? What does perpendicular mean? What does it mean to bisect a segment? 
Instructional Implications Review definitions of terms used to classify triangles by angle measure (i.e., acute, equiangular, right, and obtuse) and by lengths of sides (equilateral, isosceles, and scalene). Give the student a variety of examples of triangles that contain measures of angles and sides (or congruence marks) and ask the student to identify each triangle in as many ways as possible.
Review the definitions of perpendicular lines, segment bisectors, and perpendicular bisectors. Provide opportunities for the student to use a straightedge and compass to construct perpendiculars and perpendicular bisectors. Emphasize the conclusions that can be drawn as a consequence of knowing that two lines are perpendicular, a line bisects a segment, or a line is the perpendicular bisector of a segment. Ask the student to label diagrams (with right angle symbols and congruence marks) involving lines and segments given that some lines are perpendicular, bisectors, or perpendicular bisectors of segments in the diagram.
Review the Perpendicular Bisector Theorem. Be explicit about the biconditional nature of the theorem and its assumptions and conclusions. Guide the student to apply the theorem to the diagram provided on the worksheet. Ask the student to draw and and to mark right angles and congruent segments that result from applying the theorem. Assist the student in writing a complete, concise, and clear justification of the conclusion that is isosceles.
Have the student consider what kind of given information would result in the conclusion that the triangle formed by points A, F, and B is equilateral, equiangular, right, acute, and obtuse. 
Moving Forward 
Misconception/Error The student describes the triangle as isosceles but also identifies it in another way that is inconsistent with the assumptions. 
Examples of Student Work at this Level The student describes as isosceles and:
The student makes an assumption about that is not supported by the given information. For example, the student says:
 and are congruent, but is longer than either.
 Two angles are congruent, but the third angle is not.
Additionally, the student does not provide adequate justification. 
Questions Eliciting Thinking How do you know that is congruent to ?
How do you know is acute (or any other type of triangle not justified by the assumptions)?
What does it mean to justify your work? What definitions or theorems have you used to justify your conclusions? 
Instructional Implications Review the Perpendicular Bisector Theorem. Be explicit about the biconditional nature of the theorem and its assumptions and conclusions. Guide the student to apply the theorem to the diagram provided on the worksheet. Ask the student to draw and and to mark right angles and congruent segments that result from applying the theorem. Assist the student in writing a complete, concise, and clear justification of the conclusion that is isosceles.
Provide feedback to the student concerning any unsupported assumptions made about the diagram or any unsupported conclusions drawn. Discourage the student from drawing conclusions about lengths of segments and measures of angles based on the way they “look.” Explain that any statement made must be supported by the application of relevant definitions, postulates, or theorems to the assumptions provided in the problem.
Provide additional opportunities for the student to apply theorems about lines and angles to solve problems. Ask the student to cite relevant definitions, postulates, and theorems and to be explicit in applying them. 
Almost There 
Misconception/Error The student’s justification is incomplete. 
Examples of Student Work at this Level The student appears to apply the Perpendicular Bisector Theorem and correctly identifies as isosceles. Additionally, the student does not identify the triangle in any way that is inconsistent with the assumptions. However, the student’s justification lacks some critical component. For example, the student:
 Does not cite the Perpendicular Bisector Theorem to support the conclusion that .
 Cites the Perpendicular Bisector Theorem but is not explicit about what conclusion can be drawn when it is applied.

Questions Eliciting Thinking How do you know that is congruent to ? What theorem supports this conclusion?
What specific conclusions can you draw when you apply the Perpendicular Bisector Theorem to this diagram?
What does it mean to justify your work? What definitions or theorems have you used to justify your conclusions? 
Instructional Implications Provide feedback to the student concerning any omissions in his or her justification. Explain that a complete justification includes any relevant definitions, postulates, or theorems that support conclusions drawn. Provide additional opportunities for the student to apply theorems about lines and angles to solve problems. Ask the student to cite relevant definitions, postulates, and theorems and to be explicit in applying them.
Provide feedback to the student concerning any misuse of terminology or notation and allow the student to revise his or her work. 
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 states that is isosceles. The student justifies this conclusion by clearly stating:
Since point F is on the perpendicular bisector of , Point F is equidistant from the endpoints of by the Perpendicular Bisector Theorem. Therefore, so that is isosceles (by definition of an isosceles triangle).

Questions Eliciting Thinking Can also be acute (or right or obtuse)? Would it have to be acute (or right or obtuse)?
Must AB differ from FA and FB? Why or why not?
In the plane, how many perpendicular bisectors can a segment have? In space, how many perpendicular bisectors can a segment have? 
Instructional Implications Be sure the student understands that the Perpendicular Bisector Theorem is a biconditional statement (i.e., includes both a conditional statement and its converse). Ask the student to clearly identify each conditional statement described by the theorem [i.e., in a plane, (1) if a point is on the perpendicular bisector of a segment then it is equidistant from the endpoints of the segment and (2) if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment]. Ask the student to prove the Perpendicular Bisector Theorem by proving each conditional statement included in the theorem. 