Getting Started |
| Misconception/Error The student does not demonstrate an understanding of the similarity of the circles or of how arc length and radius are related. |
| Examples of Student Work at this Level The student:
- Suggests that L, R, l, and r are equal.
- Assumes
 and  and that proportionality follows from this.
- States that since all radii in a circle are congruent and all circles are congruent, R is proportional to r.
|
| Questions Eliciting Thinking What is similarity? Is it the same as congruence?
Are all circles similar? Why?
What is proportionality? What is a constant of proportionality or a scale factor?
Does it make sense to say that, in general, and ? Does the problem state that? |
| Instructional Implications Review the definition of similarity in terms of similarity transformations. Explain that two figures are similar if there is a dilation or a dilation and a congruence (e.g., a sequence of rigid motions) which carries one onto the other. Have the student develop his or her understanding of similarity by showing two figures are similar. Discuss with the student the consequences of similarity (e.g., if two figures are similar then corresponding angles are congruent and corresponding lengths are proportional). Be sure the student understands the meaning of proportional. Have the student find the scale factor that relates corresponding lengths of similar figures and use the scale factor to find missing lengths.
Review the fact that all circles are similar (G-C.1.1). Guide the student to relate both the radii and the corresponding arc lengths in the diagram by a scale factor, k. Show the student how to represent these relationships symbolically (e.g., as R = kr and L = kl). Emphasize that the arc lengths correspond because they are intercepted by congruent central angles. Guide the student to reason algebraically to the conclusion that  .
Consider implementing the MFAS tasks All Circles Are Similar (G-C.1.1) and Similar Circles (G-C.1.1). |
Moving Forward |
| Misconception/Error The student understands that arc lengths and radii are proportional but is unable to derive the proportion . |
| Examples of Student Work at this Level The student states that since all circles are similar and the central angles are congruent:
- “They” are proportional.
- It follows that .
|
| Questions Eliciting Thinking What do you mean by “they are proportional”?
How does it follow that ?
What are the consequences of similarity? What must be true of the radii of the circles? What must be true of arcs that subtend congruent central angles? |
| Instructional Implications Remind the student that when two figures are known to be similar, corresponding lengths are proportional. Consequently, there is a scale factor that relates corresponding lengths. Guide the student to use the similarity of circles to relate both the radii and the corresponding arc lengths in the diagram by a scale factor, k. Show the student how to represent these relationships symbolically (e.g., as R = kr and L = kl). Emphasize that the arc lengths correspond because they are intercepted by congruent central angles. Guide the student to reason algebraically to the conclusion that . |
Almost There |
| Misconception/Error The student does not demonstrate an understanding of radian measure. |
| Examples of Student Work at this Level The student is able to use the similarity of circles to show that arc length is proportional to radius but does not demonstrate an understanding of radian measure. The student:
- Indicates that he or she does not understand what radian measure is.
- Attempts unsuccessfully to explain how the fact that arc length is proportional to radius leads to a definition of the radian measure of an angle.
|
| Questions Eliciting Thinking What do you know about radian measure?
Suppose the smaller circle were the unit circle. What would its radius be? |
| Instructional Implications Review the unit circle; in particular that its radius is one and its circumference is  . Define radian measure of an angle, and ask the student to calculate the radian measures of central angles of the unit circle and the radian measures of congruent central angles of circles in which  . Then guide the student to reason from r = 1 and to the definition of the radian measure of an angle.
Consider implementing the MFAS task Deriving the Sector Area Formula (G-C.2.5). |
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:
- Since all circles are similar, there is a scale factor, k, that relates the radii of the two circles and the corresponding arc lengths so that R = kr and L = kl. Consequently,
 and  so that (by substitution) .
- If the circle of radius r is the unit circle, then r = 1. Since R = kr for some value k, then
 so that k = R. Since L = kl, then L = Rl which means that  which is the radian measure of the angle.
|
| Questions Eliciting Thinking What is the radian measure of a  angle?
How could you calculate the length of the arcs in the diagram? |
| Instructional Implications Challenge the student to find the length of an arc that subtends an angle of measure one radian.
Ask the student to find the radian measure of angles given their degree measures.
Consider implementing the MFAS task Deriving the Sector Area Formula (G-C.2.5). |
