Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 71060
Arc Length and Radians
Students are asked to explain why the length of an arc intercepted by an angle is proportional to the radius and then explain how that
proportionality leads to a definition of the radian measure of an angle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, arc length, circle, proportional, radian measure
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ArcLengthAndRadians_Worksheet.docx
MFAS_ArcLengthAndRadians_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Arc Length and Radians worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
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.
page 1 of 4 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
page 2 of 4 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).
ACCOMMODATIONS & RECOMMENDATIONS
page 3 of 4 Special Materials Needed:
Arc Length and Radians worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-C.2.5:
Description
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and
define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
page 4 of 4