Getting Started 
Misconception/Error The student is unable to correctly draw the diagram. 
Examples of Student Work at this Level The student does not understand the described relationship between the circle, the tangent line, and the radius and is unable to correctly draw the diagram. Consequently, the student is unable to describe the relationship between the tangent line and the radius drawn to the point of tangency. The student draws:
 The radius parallel to the tangent line.
 A secant line instead of a tangent line.
 A diameter and a radius.

Questions Eliciting Thinking What is the significance of point O in the diagram?
What does it mean for a line to be tangent to a circle?
What is the significance of point P in the diagram?
Where is radius in your diagram? 
Instructional Implications Review the following terms: radius, tangent line, and point of tangency. Illustrate each with a diagram. Then guide the student to draw and label the diagram described in the task. Provide additional descriptions of circles, radii, tangent lines, and points of tangency and ask the student to draw and label diagrams that illustrate the descriptions.
Explain to the student that if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Provide the student with several circles each with one radius drawn in a different location or orientation. Have the student sketch a tangent line to the circle at a point of tangency and label the diagram to show the perpendicular relationship. Allow the student to explore this relationship using an interactive website (e.g., http://www.mathopenref.com/tangentline.html).
Have the student solve problems that require the application of the theorem that the radius of a circle is perpendicular to a tangent line at the point of tangency. For example, provide a diagram of a circle that includes a tangent line and a radius drawn to the point of tangency. Draw a segment whose endpoints are the center of the circle and a point on the tangent line, forming a triangle. Provide the radius and the length of one other side of the triangle and ask the student to find the missing length of the triangle. 
Making Progress 
Misconception/Error The student is unable to describe the relationship between line t and . 
Examples of Student Work at this Level The student draws the diagram correctly but is unable to describe the relationship between line t and . The student:
 Says line t and create supplementary angles.
 Writes a statement that does not make mathematical sense.

Questions Eliciting Thinking Is there anything special about the way line t and intersect?
Do you know anything about the angle formed by line t and ? How would you describe the angle formed by line t and ? 
Instructional Implications Explain to the student that if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If needed, assist the student in appropriately labeling his or her diagram and ask the student to insert the “right angle box” to show the perpendicular relationship between the radius and the tangent line. Provide the student with several circles each with one radius drawn in a different location or orientation. Have the student sketch a tangent line to the circle at a point of tangency and label the diagram to show the perpendicular relationship. Allow the student to explore this relationship using an interactive website (e.g., http://www.mathopenref.com/tangentline.html).
Have the student solve problems that require the application of the theorem that the radius of a circle is perpendicular to a tangent line at the point of tangency. For example, provide a diagram of a circle that includes a tangent line and a radius drawn to the point of tangency. Draw a segment whose endpoints are the center of the circle and a point on the tangent line, forming a triangle. Provide the radius and the length of one side of the triangle and ask the student to find the missing length of the triangle. 
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 a diagram and describes the relationship between line t and as perpendicular or forming a right angle.

Questions Eliciting Thinking How many points does a tangent have in common with a circle?
How many lines can be drawn tangent to a circle at a single point on that circle? How many tangents may be drawn to a circle from a point outside the circle?
What is the relationship between two tangents drawn to the two endpoints of a circle's diameter?
How can you be certain that line t and are perpendicular? 
Instructional Implications Ask the student to prove that the radius of a circle is perpendicular to a tangent line at the point of tangency.
Ask the student to explore how to construct a tangent line to a circle from a point outside the circle (GC.1.4).
Introduce the student to the concept of a common tangent line. Have the student explore and describe the various common tangents to two nonintersecting circles, two circles that intersect in one point, two circles that intersect in two points, and two concentric circles.
Consider implementing the MFAS task Constructing A Tangent Line (GC.1.4) and/or Using a Compass To Construct A Tangent Line (GC.1.4). 