Getting Started |
| Misconception/Error The student does not understand the concept of arc length. |
| Examples of Student Work at this Level The student:
- Indicates that he or she does not understand how to complete the problem.
- Attempts to find the arc length but does not use the correct formula.
- States that the arc length is equal to the arc measure (e.g., 72 cm for problem one and 137 m for problem two).
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| Questions Eliciting Thinking What are you being asked to find? Can you show me on the diagram the length you are trying to find?
How is the measure of an arc related to the measure of its central angle?
How are the arc length and the circumference of the circle related? How do you find the circumference of a circle?
What fraction of the circle’s circumference are you looking for? How do you determine that fraction?
What is the arc measure? How are the arc measure and the arc length related?
Do you know the formula for the arc length? What formula did you use? |
| Instructional Implications Review the definitions of arc, arc measure, and circumference. If necessary, review the relationship between the diameter and the radius of a circle. Then explain arc length in terms of circumference. Describe arc length as a fraction of the circle’s circumference. Explain that the fraction is determined by the degree measure of the arc. Guide the student to write a general formula for finding arc length. Initially, use as an example an arc of measure 180º and show the student that the length of such an arc is equal to one half of the circumference of the circle. Then, use an arc of measure 90º as an example and ask the student to identify the fraction of the circumference this arc would represent. Show the student that the ratio of the degree measure of the arc to the degree measure of the circle, 360, is the fraction of the circumference that the arc represents. Model finding the arc length given the arc measure and the radius and/or diameter.
Give the student more opportunities to find arc length, giving assistance as needed. Ask the student to identify the degree measure of the arc and the radius in each problem and to highlight the arc length he or she is being asked to find. Require that the student write the formula for arc length for each problem and show all work carefully and completely.
Use an interactive resource such as Math Open Reference Arc Length (http://www.mathopenref.com/arclength.html ) and let the student explore the relationship between the arc measure and the arc length. |
Moving Forward |
| Misconception/Error The student makes an error in using the arc length formula. |
| Examples of Student Work at this Level The student identifies the correct formula for finding arc length but:
- Confuses the radius and the angle measure.
- Uses the diameter of the circle instead of the radius in the formula and does not write the final answer in the form requested.
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| Questions Eliciting Thinking What do the variables in your formula represent?
In problem two, are you given the radius or diameter? How do you convert the diameter to the radius?
How would you find the circumference of the circle?
What does “in terms of  ” mean? What does “round to the nearest hundredth” mean? |
| Instructional Implications Review the meaning of the variables in the formula and guide the student to find the appropriate values from the diagram.
Remind the student of the relationship between the circumference of the circle and the length of an arc of the circle. Be sure the student understands that arc length is a fraction of a circle’s circumference and that fraction depends on the degree measure of the arc.
If needed, explain the difference between writing an answer in exact form (i.e., in terms of ) and in approximate form by using an approximation for . Explain why it is not possible to write an exact answer as a rational number.
Provide the student with additional opportunities to find arc length. |
Almost There |
| Misconception/Error The student makes a minor computational or rounding error. |
| Examples of Student Work at this Level The student identifies the arc length formula and substitutes the appropriate values for the radius and arc measure but:
- Does not round according to the directions.
- Does not label the answer with the correct unit.
- Does not give the answer in the form requested.
- Rounds factors before multiplying producing an answer with more error than is necessary.
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| Questions Eliciting Thinking There is a small error in your work. Can you find it? How were you asked to give your answer?
Explain to me how you got your answer. Did you round during the problem or only at the end of the problem? How could that make a difference?
Did you label your answer with a unit of measure? What is the correct unit of measure? |
| Instructional Implications Review with the student that rounding several factors before multiplying introduces a greater degree of error than is necessary. Remind the student that  and guide the student to multiply 2466 by a suitable approximation of and then divide this product by 360 in one step on the calculator. Then round the final answer as directed.
If needed, explain the difference between writing an answer in exact form (i.e., in terms of ) and in approximate form by using an approximation for . Explain why it is not possible to write an exact answer as a rational number.
Provide the student with additional opportunities to find arc length. |
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 uses the arc length formula to find the length of  in terms of ,  =   cm or 2.4 cm, and the length of  in approximate form, rounding to the nearest hundredth,  21.52 m. |
| Questions Eliciting Thinking What is the difference between a major arc and a minor arc?
Which answer is in exact form? In approximate form? |
| Instructional Implications Challenge the student to derive the formula for arc length using MFAS task Arc Length and Radians (G-C.5) if not previously used. |
