Getting Started 
Misconception/Error The student does not have an effective strategy for completing the construction. 
Examples of Student Work at this Level The student:
 Uses the straightedge to draw two diameters and a square in the circle.
 Constructs six arcs around the circle and draws two diameters and a square in the circle.
 Constructs six arcs around the circle and draws four chords of the circle.
 Does not understand how to use a compass and straightedge.

Questions Eliciting Thinking What is the difference between drawing and constructing?
When doing a geometric construction, what tools are typically used?
What is the difference between a straightedge and a ruler?
What is it that you are supposed to construct?
Can you describe a strategy to construct a square in a circle? What special property about the diagonals of a square can help you with your construction? 
Instructional Implications Explain to the student the difference between drawing and constructing. Show the student the tools traditionally used in geometric constructions and explain the purpose of each. Be sure the student understands the difference between a ruler and a straightedge.
Guide the student through the steps of the construction. Prompt the student to justify each step. Have the student remove any unnecessary marks or marks made in error from his or her paper. Ask the student to write out the steps of the construction and keep them for future reference.
Construction: Use the endpoints of perpendicular diameters as vertices of the square. Step 1: Use a straight edge to construct a diameter of the circle. Step 2: Construct the perpendicular bisector of this diameter. Step 3: Connect the endpoints of the two diameters to construct the square.
Have the student also use a paper folding method to construct a square:
Step 1: Given a circle on translucent paper, fold the paper so that two halves of the circle coincide. The crease of this fold is a diameter of the circle. Label the endpoints of the diameter A and C. Step 2: Fold so that point A coincides with point C. The crease of the fold produces another diameter of the circle. Label the endpoints of this diameter B and D. Step 3: Use a straightedge and pencil to construct the sides of square ABCD.
Ask the student to compare the paper folding method to the compass and straightedge construction. 
Moving Forward 
Misconception/Error The student attempts an effective strategy but makes a significant error. 
Examples of Student Work at this Level The student:
 Does not maintain the same radius setting on the compass for both arcs of a perpendicular bisector.
 Does not have the compass open enough and the arcs do not intersect.
 Makes a number of extraneous arcs and cannot accurately locate a needed intersection of two arcs.

Questions Eliciting Thinking You drew a diameter. What is the next step in the construction process?
Do you know how to construct a perpendicular bisector?
What must be true of the radius settings on your compass in order to ensure that the line is perpendicular?
How did you determine the radius setting on your compass?
You drew a number of arcs on your paper. Were all of them necessary for this construction? 
Instructional Implications Review how to construct the perpendicular bisector of a segment and then guide the student through the parts of his or her construction that contained errors. Have the student remove any unnecessary marks or marks made in error. Ask the student to write out the steps of the construction and keep them for future reference.
Explain to the student the need to precisely locate points in constructions. Help the student find a way to hold the compass so as not to inadvertently change the radius setting.
Assist the student in developing a justification for the construction strategy used. Ask the student to use postulates, definitions, or theorems to explain the results of each step of the construction and to justify why the strategy results in a square. Provide feedback as needed.
Give the student additional opportunities to construct perpendicular bisectors of segments as part of other constructions such as the constructing the midpoints of the sides of a triangle or constructing a segment whose length is the length of a given segment. 
Almost There 
Misconception/Error The student’s construction lacks precision. 
Examples of Student Work at this Level The student uses an effective strategy to construct a square inscribed the circle but:
 Does not use the exact point of intersection of the arcs when constructing the perpendicular bisector.
 Does not use the exact endpoints of the diameters when constructing the sides of the square.
 Draws large dots at the endpoints of the diameters, so the intersection is hidden and precision is lost.
 Leaves unnecessary construction marks on the paper. (Making it difficult to identify the square.)

Questions Eliciting Thinking Can you explain the steps of your construction?
How did you locate the points you used to construct the sides of the square? Did you draw the sides so that they contain these points?
Are there any adjustments you could make to your construction to make it more precise?
What do these arcs (or markings) represent? Did you use them in your construction? 
Instructional Implications Explain to the student the need to precisely locate and connect points in constructions. Emphasize that drawing large dots at intersection points reduces the precision of the construction. Ask the student to explain why large dots (or any at all) can affect the precision of a geometric construction.
Ask the student to remove any unnecessary marks or marks made in error from his or her paper.
Ask the student to use postulates, definitions, or theorems to explain the results of each step of the construction and to justify why the strategy results in a square. 
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 constructs a square inscribed in the circle using an effective strategy. The student’s construction is precise and contains no unnecessary marks or marks made in error.

Questions Eliciting Thinking Why does this approach enable one to construct a square in a circle? Why does this method work?
Can you think of another effective strategy for constructing a square in a circle?
How can you expand your construction of a square to construct a regular octagon inscribed in a circle? 
Instructional Implications Challenge the student to construct a square circumscribed about the circle.
Consider implementing other GCO.4.13 tasks. 