Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 57538
Circumscribed Circle Construction
Students are asked to use a compass and straightedge to construct a circumscribed circle of an acute scalene triangle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, circumscribed circle, circumcenter, perpendicular bisectors
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_CircumscribedCircleConstruction_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
Note: MAFS.912.GC.1.3 refers back to MAFS.912.G-CO.4.12 which requires students to "make formal geometric constructions with a variety of tools and methods." This task
and rubric assume the use of a compass and straightedge but can be adapted for use with any tool. Throughout the geometry course, students should be exposed to a
variety of construction tools and methods.
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 Circumscribed Circle Construction worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student sketches or draws rather than constructs.
Examples of Student Work at this Level
Using the compass and straightedge, the student attempts to draw the circumscribed circle rather than construct it.
page 1 of 4 Questions Eliciting Thinking
What does it mean to construct? How is it different than drawing?
When doing a geometric construction, what tools are typically used?
What is a circumscribed circle of a triangle? What are some of its properties?
Instructional Implications
Explain to the student the difference between drawing and constructing. Show the student the tools traditionally used in geometric constructions and explain the purpose
of each. Be sure the student understands the difference between a ruler and a straightedge.
If necessary, review the definition of a circumscribed circle of a triangle and its relationship to the perpendicular bisectors of the sides of a triangle. Review important
vocabulary such as: perpendicular bisector, point of concurrency, circumcenter, and circumscribed circle.
Model finding the circumcenter of the triangle by using other construction techniques such as paper folding. Instruct the student to draw a large triangle on unlined paper
using a ruler or straightedge. Be sure the sides of the triangle are darkened so they can be seen through the paper when folding. Have the student label the vertices of the
triangle A, B and C. Then guide the student to fold the paper so that vertex A meets vertex B, making sure
student to crease the paper completely forming the perpendicular bisector of
aligns with itself when holding it up to the light. Ask the
. Have the student construct the perpendicular bisectors of the remaining two sides to
locate the circumcenter of the triangle. Then guide the student to use a compass to measure the distance from one of the vertices to the circumcenter and use this radius
to construct the circumscribed circle.
Demonstrate constructions using an interactive website such as Open Math Reference. Begin with the simpler constructions such as copying a segment or angle and then
transition to the more complex constructions. (http://www.mathopenref.com/tocs/constructionstoc.html)
To assess an understanding of basic constructions, consider implementing MFAS tasks Constructing a Congruent Segment (G-CO.3.12), Constructing a Congruent Angle (GCO.3.12), and Bisecting a Segment and Angle (G-CO.3.12).
Moving Forward
Misconception/Error
The student is unable to correctly construct perpendicular bisectors of the sides of the triangle.
Examples of Student Work at this Level
The student constructs the altitudes of the triangle and then draws a circle over the triangle.
The student’s work shows some correct marks for constructing the perpendicular bisectors but the perpendicular bisectors are not completed.
Questions Eliciting Thinking
Can you explain what you constructed?
How did you construct the perpendicular bisectors?
Why was it necessary to construct the perpendicular bisectors?
Instructional Implications
Guide the student through the parts of his or her construction that contained errors or are incomplete. Have the student remove any unnecessary marks or marks made in
error. Ask the student to write out the steps of the construction including important vocabulary such as perpendicular bisector, point of concurrency, circumcenter, and
page 2 of 4 circumscribed circle, and keep them for future reference. Be sure the student understands how to correctly answer the questions asked in the task.
Provide additional opportunities to construct circumscribed circles of a variety of triangle types. Guide the student to draw generalizations about the location of the
circumcenter with regard to the triangle depending on the triangle type.
For intermediate level practice on constructions, consider implementing MFAS tasks Constructions for Parallel Lines (G-CO.3.12) and Constructions for Perpendicular Lines (GCO.3.12).
Almost There
Misconception/Error
The student’s construction is not precise.
Examples of Student Work at this Level
The student constructs all three perpendicular bisectors and finds their point of concurrency (circumcenter), however the point of concurrency is too large and the radius of
the circle is not precise. The circle only goes through one of the three vertices of the triangle.
Questions Eliciting Thinking
Do all of your perpendicular bisectors appear to be perpendicular to and bisecting the sides of the triangle?
How do you think drawing the point of concurrency as a large dot would affect your construction?
Why do you think your circle does not intersect all three vertices of the triangle?
Instructional Implications
If necessary, explain to the student the need to precisely locate points in constructions. Help the student find a way to hold the compass so as not to inadvertently change
the radius setting.
Provide the student more opportunities to construct inscribed circles of a variety of triangle types using several different tools and methods. Use real world contexts when
possible (such as placing a park or public facility on a map so that it is equidistant from three towns represented by non-collinear points on the map) to demonstrate the
need to understand and apply geometric constructions.
Consider implementing MFAS Task Inscribed Circle Construction (G-C.1.3).
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 constructs the perpendicular bisectors of each of the three sides of the triangle and finds their point of concurrency (circumcenter). The student uses the
compass to measure the distance from the circumcenter to one of the three vertices of the triangle and constructs the circumscribed circle. In response to the questions
asked in the task, the student states that he or she constructed the perpendicular bisectors of the sides of the triangle to locate the center of the circumscribed circle and
identifies this point as the circumcenter.
Questions Eliciting Thinking
Was it really necessary to construct all three perpendicular bisectors? Could you have located the circumcenter by constructing only two of them?
Can you think of any real-world situations when this construction would be helpful?
Instructional Implications
Challenge the students to solve real-world problems using constructions, such as placing a park or public facility on a map so that it is equidistant from three towns
represented by non-collinear points on the map.
Challenge the student to prove that the circumcenter is equidistant from the vertices of the triangle.
Provide the student the opportunity to complete constructions using dynamic geometry software such as Geogebra or Geometer’s Sketchpad.
Consider implementing MFAS Task Inscribed Circle Construction (G-C.1.3).
page 3 of 4 ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Circumscribed Circle Construction Worksheet
Compass, straightedge
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.1.3:
Description
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
inscribed in a circle.
page 4 of 4