Getting Started 
Misconception/Error The student does not demonstrate an understanding of the relationship between the measure of an inscribed angle and its intercepted arc. 
Examples of Student Work at this Level The student:
 Is unable to determine the measure of either of the inscribed angles.
 Confuses an inscribed angle with a central angle.
 States that Â because the triangle is isosceles.

Questions Eliciting Thinking Do you know the difference between a central angle and an inscribed angle?
What type of angle is ? What do you know about an angle inscribed in a circle? What is true if the angle is inscribed on a diameter?
Why did you say that the triangle is isosceles? Was that given in the problem? 
Instructional Implications Review terminology related to angles of a circle (e.g., central angle, inscribed angle, intercepted arc, semicircle, diameter, and center) and the definitions and theorems that describe angle measures in relation to intercepted arcs. Provide the student with a variety of problems in which either angle measures can be determined from arc measures or vice versa. Model explaining the relationship between the measures of angles and their intercepted arcs. Provide opportunities for the student to describe these relationships.
Consider using Math Open Reference (http://www.mathopenref.com/semiinscribed.html and http://www.mathopenref.com/thalestheorem.html) to allow the student to further explore inscribed angles on a diameter. 
Moving Forward 
Misconception/Error The student is unable to apply what is known about the relationship between the measure of an inscribed angle and its intercepted arc. 
Examples of Student Work at this Level The student demonstrates an understanding of the relationship between the measure of an inscribed angle and its intercepted arc. For example, the student determines that mÂ =Â Â and explains that sinceÂ Â is inscribed in a semicircle, its measure isÂ . However, the student is unable to find or explain how to find the measure of . For example, the student findsÂ Â and gives this measure as the measure ofÂ .

Questions Eliciting Thinking How did you find m? What is the sum of the measures of the angles of a triangle?
Can you show me exactly whereÂ Â is located? Where Â is located? Which were you asked to find? What did you find? 
Instructional Implications Review the relationship between inscribed angles and their intercepted arcs. Provide feedback to the student concerning any errors made and allow the student to revise his or her work. Provide the student with a variety of problems in which either angle measures can be determined from arc measures or vice versa.
Consider using the Illustrative Mathematics tasks Right Triangles Inscribed in Circles I (https://www.illustrativemathematics.org/illustrations/1091) and Right Triangles Inscribed in Circles II (https://www.illustrativemathematics.org/illustrations/1093). 
Almost There 
Misconception/Error The student is unable to clearly describe the relationships among the inscribed angles and their intercepted arcs. 
Examples of Student Work at this Level The student can find each angle measure but is unable to completely and clearly describe the relationship between the angle measures and their intercepted arcs.

Questions Eliciting Thinking How did you know that mÂ isÂ ?
How is the measure of an inscribed angle related to the measure of its intercepted arc?
Can you explain to me how you found m? 
Instructional Implications Ask the student to explain how he or she found each angle measure. Correct any misuse of terminology and assist the student in revising any statements that are unclear or incorrect. Assist the student in using appropriate mathematical terminology to explain that, in general, the measure of an inscribed angle is equal to half the measure of its intercepted arc. Provide additional opportunities for the student to describe relationships between angles of a circle (e.g., central angles, inscribed angles, and circumscribed angles) and their intercepted arcs. 
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 explains that the measure of an inscribed angle is half the measure of its intercepted arc. Consequently, sinceÂ Â is inscribed in a semicircle, its measure is . Additionally, the student determines that Â since it intercepts an arc whose measure isÂ Â or the student uses the Triangle Sum theorem to calculate as follows:Â Â or .

Questions Eliciting Thinking Is there another way you could have found these angle measures?
Would mÂ still be Â if point A were not the center of the circle? Why? 
Instructional Implications Provide the student with opportunities to find angle measures of quadrilaterals inscribed in a circle.
Provide the student with the opportunity to find the center of a circle by constructing two right triangles inscribed in a circle. Ask the student to explain why the intersection of the hypotenuses is the center of the circle.
Consider using the Illustrative Mathematics tasks Right Triangles Inscribed in Circles I (https://www.illustrativemathematics.org/illustrations/1091) and Right Triangles Inscribed in Circles II (https://www.illustrativemathematics.org/illustrations/1093). 