Getting Started 
Misconception/Error The student is unable to devise a method to locate point E, the midpoint of , using the given information and constraints. 
Examples of Student Work at this Level The student:
 Describes a method that will not result in locating point E.
 Suggests an alternative approach such as paper folding.

Questions Eliciting Thinking What were you given in this problem? What are you asked to find?
Do you remember any theorems about the medians of a triangle and their point of intersection?
Can you use the given information and only a straightedge and a pencil to locate point E?
Can you describe a reallife situation where it would be useful to precisely locate the midpoint of the third side of a triangle as in this problem? 
Instructional Implications Review relevant terminology (e.g., vertex, midpoint, median, concurrent, and centroid) and the Median Concurrence Theorem. Guide the student through the process of locating point E using only a straightedge and pencil. Explain and justify each step of the process. Review notation for naming points and segments. Then ask the student to write a stepbystep description of the method used. Encourage the student to write a description that is detailed enough that it can be followed by someone else who is unfamiliar with the method. 
Moving Forward 
Misconception/Error The student demonstrates a correct method but is unable to correctly and completely describe it. 
Examples of Student Work at this Level The student uses the straight edge and pencil to draw medians and locating the centroid. The student then draws the third median using point A and the centroid to locate the midpoint of . However, the student’s explanation is either incorrect, incomplete, or both.

Questions Eliciting Thinking Where is the midpoint of in this diagram? How were you asked to label this midpoint?
Do you think that someone could follow your explanation if you did not provide the drawing? 
Instructional Implications Ask the student to explain his or her method. Then assist the student with any relevant terminology (e.g., vertex, midpoint, median, concurrent, and centroid) that could be used to improve the explanation. Guide the student to use named points to assist in making the explanation more precise. If needed, review notation for naming points and segments. Then ask the student to rewrite his or her explanation to make it more complete and precise. Encourage the student to write a description that is detailed enough that it can be followed by someone else who is unfamiliar with the method. 
Almost There 
Misconception/Error The student’s explanation is nonmathematical or contains an error in notation or terminology. 
Examples of Student Work at this Level The student describes a correct method but:
 Does not indicate the location of point E.
 Does not use mathematical terminology to precisely describe the method.
 Is not specific about how the third median is drawn.
 Does not use notation correctly.

Questions Eliciting Thinking Where is the midpoint of in this diagram? How were you asked to label this midpoint?
What did you actually draw? Lines? Line segments? How many points should you identify to precisely describe this line (or line segment)?
You used the phrase “cross through.” What mathematical term could you have used instead (i.e., “intersect”)?
How are points (or angles, lines, or line segments) named? What is the proper notation? 
Instructional Implications Provide feedback to the student concerning any misuse of terminology or notation and allow the student to revise his or her work. Encourage the student to use mathematical terminology and conventional notation to make explanations 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 uses the straight edge and pencil to draw medians and and names the point of intersection (i.e., the centroid) as, for example, point X. The student then draws the third median using vertex A and centroid X. The student identifies the point where this median intersects side as the midpoint of . The student explains that the three medians are concurrent and intersect in the centroid so that the centroid and vertex A can be used to draw the third median. The third median can then be used to locate the midpoint of .

Questions Eliciting Thinking What theorem did you rely on in your explanation?
What term describes three or more lines that intersect in a point? In the case of the medians, what is this point called?
Will the centroid always be located in the interior of the triangle?
What other special lines related to triangles are concurrent? 
Instructional Implications Challenge the student to construct the centroid of a variety of triangles (e.g., acute, obtuse, and right) and devise a conjecture about the location of the centroid with regard to the triangle (i.e., the centroid is always located in the interior of a triangle).
Ask the student to prove the Median Concurrence Theorem. Consider implementing MFAS task Median Concurrence Proof (GCO.3.10). 