Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 70897
Circles with Angles
Students are given a diagram with inscribed, central, and circumscribed angles and are asked to identify each type of angle, determine angle
measures, and describe relationships among them.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, circle, central angle, inscribed angle, arc, intercepted arc, circumscribed angle, exterior angle
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_CirclesWithAngles_Worksheet.docx
MFAS_CirclesWithAngles_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 Circles with Angles worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to correctly identify central and inscribed angles and determine their measures.
Examples of Student Work at this Level
The student:
Is only able to correctly identify the central angle.
page 1 of 4 Correctly finds some angle measures but is unable to identify them or describe relationships among their measures.
Incorrectly labels the three angles as either inscribed or central.
Attempts to classify the angles as acute or obtuse.
Questions Eliciting Thinking
Can you identify the angle types with respect to the circle?
Can you identify a central angle in the diagram?
Do you know how to find the measure of
?
Can you identify an inscribed angle in the diagram?
Do you know how to find the measure of
?
Instructional Implications
Review the definitions of central, inscribed, and circumscribed angles and the relationships between the measure of each angle type and the measure of its intercepted
arc(s). Guide the student to focus on the location of the vertex of the angle when identifying its type (e.g., the vertex of a central angle is the center of the circle, the
vertex of an inscribed angle is on the circle, and the vertex of a circumscribed angle is in the exterior of the circle).
Consider using one of the following sites which enable the student to explore the relationships between angle measure and arc measure:
Math Open Reference: http://www.mathopenref.com/.
Direct link to explore the Central Angle: http://www.mathopenref.com/arcangle.html. Select the box marked “Show central angle measure” to visualize how the central
angle measure changes when the arc is adjusted.
Direct link to explore the Inscribed Angle: http://www.mathopenref.com/circleinscribed.html. The student is able to move the vertex of the inscribed angle around the
circle (while the intercepted arc remains fixed) and observe how the measure of the angle remains the same regardless of its location.
Direct link to explore the relationship between an inscribed and a central angle that intercept the same arc: http://www.mathopenref.com/arccentralangletheorem.html.
Provide a variety of problems in which the student must find the measure of a central angle and an inscribed angle given information about the measures of intercepted
arcs.
If needed, provide instruction on using correct notation when naming angles, arcs, and referring to their measures. Address the differences in naming minor and major arcs
of a circle. Also address the difference between naming an angle (e.g.,
) and referring to the angle’s measure (e.g., m
).
Moving Forward
Misconception/Error
The student is unable to correctly identify the circumscribed angle and determine its measure.
Examples of Student Work at this Level
The student correctly names both the central and inscribed angles and determines their measures. The student is unable to name and correctly find the measure of the
circumscribed angle.
Questions Eliciting Thinking
What do you know about the measure of
What kind of lines are
What arcs does
and
?
?
intercept?
Instructional Implications
page 2 of 4 Review the definition of a circumscribed angle and the relationship between its measure and the measures of its intercepted arcs (e.g., the measure of a circumscribed
angle is equal to half the difference in the measures of its intercepted arcs). Guide the student to observe that the sides of a circumscribed angle are each tangent to the
circle. The points of tangency separate the circle into the two intercepted arcs. Provide a variety of problems in which the student must find the measure of a circumscribed
angle or the measures of one or both intercepted arcs given appropriate information.
Review the following theorems:
The radius of a circle is perpendicular to a tangent line at the point of tangency.
The sum of the measures of the interior angles of a quadrilateral is
Guide the student to observe that
and
.
are opposite angles of quadrilateral ABED. Ask the student to determine the measures of
guide the student to reason that if the sum of the four angles of quadrilateral ABED is
), then the other pair of opposite angles (e.g.,
relationship to determine another way to find the measure of
and one pair of opposite angles sum to
and
) sum to
and
. Then
(since
or are supplementary. Ask the student to use this
.
If needed, provide instruction on using correct notation when naming angles, arcs, and referring to their measures. Address the differences in naming minor and major arcs
of a circle. Also address the difference between naming an angle (e.g.,
) and referring to the angle’s measure (e.g., m
).
Consider implementing other MFAS tasks for G-C.1.2.
Almost There
Misconception/Error
The student is unable to describe, in general, the relationships among the angle pairs.
Examples of Student Work at this Level
The student correctly names the central, inscribed, and circumscribed angles and determines their measures. However, the student is unable to correctly describe, in
general, the relationship between
and
and the relationship between
and
.
Questions Eliciting Thinking
How did you find the measures of
and
? How do their measures compare?
What kind of polygon is ABED? What is the sum of the measures of the interior angles of a quadrilateral?
What kind of angle is
? How do you know this?
Instructional Implications
Ask the student to explain how he or she found the measures of
student to reason that if
and
and
. Then remind the student that these two angles intercept the same arc. Guide the
, then
.
Review the following theorems:
The radius of a circle is perpendicular to a tangent line at the point of tangency.
The sum of the measures of the interior angles of a quadrilateral is
Guide the student to observe that
and
.
are opposite angles of quadrilateral ABED. Ask the student to determine the measures of
guide the student to reason that if the sum of the four angles of quadrilateral ABED is
), then the other pair of opposite angles (e.g.,
relationship to determine another way to find the measure of
and
and one pair of opposite angles sum to
) sum to
and
. Then
(since
or are supplementary. Ask the student to use this
.
If needed, provide instruction on using correct notation when naming angles, arcs, and referring to their measures. Address the differences in naming minor and major arcs
of a circle. Also address the difference between naming an angle (e.g.,
) and referring to the angle’s measure (e.g., m
).
Got It
page 3 of 4 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 identifies each angle type and its measure. The student writes the following statements for
is a central angle and its measure is the same as the measure of its intercepted arc:
,
, and
:
.
is an inscribed angle and its measure is half the intercepted arc:
is a circumscribed angle and its measure is 180 minus the measure of the central angle:
.
OR
is an exterior angle and its measure is half the difference between the measures of the major intercepted arc,
, and the minor intercepted arc,
:
.
Questions Eliciting Thinking
Would any of your calculation methods change if the three angles did not intersect the same arc on the circle? Suppose the rays of
are secants rather than
tangents? What calculation(s) would change, if any? Illustrate your explanation using a specific example.
Suppose the rays of
are a combination of one tangent and one secant? Would you need any additional information to determine the measure of
? Draw a
diagram to illustrate the situation and then respond.
Does the Central Angle Theorem apply to the angles in this diagram? Explain why or why not.
Instructional Implications
Ask the student to use the diagram to prove, in general, that central
and circumscribed
are supplementary.
Consider implementing other MFAS tasks for G-C.1.2.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Circles with Angles 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.1.2:
Description
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central,
inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
page 4 of 4