Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 71089
Area and Circumference – 1
This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and
circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of
the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, area, circumference, diameter, radius, circle, polygon, apothem, perimeter, inscribed, regular ngon, regular polygon
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_AreaAndCircumference1_Worksheet.docx
MFAS_AreaAndCircumference1_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 answer the questions on the Area and Circumference – 1 worksheet.
2. The teacher asks follow-up questions, as needed.
Note: This task is the first in a series of three tasks that assess the students’ understanding of an informal derivation of the formulas for the area and circumference of a
circle. Once the student is successful with this task, the next MFAS task in this series, Area and Circumference – 2 (G-GMD.1.1), can be implemented. For a more thorough
discussion of these derivations, please see pages 53 – 60 of this text.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not indicate an understanding of the convergence relationship between a circle and an inscribed regular n-gon as n increases.
Examples of Student Work at this Level
The student:
page 1 of 4 Associates n with the size of each figure, rather than the number of sides of the polygon.
Assumes that the area and perimeter (circumference) of both the n-gon and the circle increase as n increases.
States that as n increases, the area and perimeter of the n-gon increase, but the area and circumference of the circle decrease.
States that as n increases, the area and perimeter of the n-gon stay the same.
Does not indicate any relationship between the area and perimeter of the inscribed n-gon and the area and circumference of the circumscribed circle as n increases.
Indicates that the area of the n-gon might increase or decrease as n increases.
Questions Eliciting Thinking
What does the n represent in a n-gon? What does n represent in the diagram? What will happen to the length of the sides of the n-gon as n increases?
Assume the circle stays the same and only the polygon changes. What happens to the n-gon, in relation to the circle, as n increases?
What happens to the area between the two figures as n increases?
Instructional Implications
Assist the student with visualizing how a regular polygon inscribed in a circle changes as the number of its sides, n, increases. Focus the student’s attention on the area
between the two figures (e.g., the area in the interior of the circle but in the exterior of the polygon). If available, use dynamic software to illustrate this relationship. The
following link provides an interactive diagram offering the ability to adjust the number of sides of a regular polygon inscribed in a circle,
http://www.mathopenref.com/polygoncircumcircle.html. Use the link to illustrate that the area between the polygon and the circle decreases as the number of sides of the
polygon increases. Guide the student to also observe that the apothem of the polygon approaches the radius of the circle as n increases.
Moving Forward
Misconception/Error
The student attempts to describe a convergence relationship but lacks the mathematical vocabulary to describe it.
Examples of Student Work at this Level
The student says that as n increases:
The area of the circle stays the same while “there will be more area in the shape.”
The “open space” between the circle and the polygon decreases.
The n-gon becomes more like the circle.
Questions Eliciting Thinking
In the first diagram, what is the relationship between the circumference of the circle and the perimeter of the n-gon? Which is larger? How can you tell? What happens to
this relationship as n increases?
In the first diagram, what is the relationship between the area of the circle and the area of the n-gon? Which is larger? How can you tell? What happens to this relationship
as n increases?
In the first diagram, what is the relationship between the radius of the circle and the apothem of the n-gon? Which is larger? How can you tell? What happens to this
page 2 of 4 relationship as n increases?
Instructional Implications
Explain the convergence relationship between the regular polygon and its circumscribed circle as n increases. Assist the student in adopting and using language such as, “As
n increases, the area of the n­gon approaches the area of the circle.” Guide the student to describe the relationship between the n-gon and the circle in terms of quantities
such as area and perimeter (circumference) rather than in terms of how the figures look in the diagram. Then ask the student to consider the relationship between the
apothem of the n-gon and the radius of the circle as n increases. Again, guide the student to describe this relationship in terms of length. Model explaining that as n
increases, the length of the apothem approaches the length of the radius. The terms apothem and radius are often used to refer to both segments as well as the lengths
of these segments. However, it may be best to emphasize that the lengths of the apothem and radius are converging.
Consider implementing the next two MFAS tasks in this sequence Area and Circumference – 2 (G-GMD.1.1) and Area and Circumference – 3 (G-GMD.1.1).
Almost There
Misconception/Error
The student does not compose a complete and clear derivation of the equation relating area and circumference.
Examples of Student Work at this Level
The student explains that as n increases, the lengths of the sides of the n-gon decrease, so that the perimeter of the n-gon approaches the circumference of the circle,
and the area of the n-gon approaches the area of the circle. When deriving an equation relating the area and circumference of the circle, the student:
Is unable to begin.
Substitutes C for p and stops.
Correctly writes the equation, but does not justify the substitution of r for a and/or C for p into the formula
.
Questions Eliciting Thinking
What do you know about the relationship between the circumference of the circle and the perimeter of the polygon as n increases?
What happens to the relationship between the radius of the circle and the apothem of the n-gon as n increases?
What justifies substituting r for a and C for p into the formula?
Instructional Implications
Ask the student to consider the relationship between the apothem of the n-gon and the radius of the circle as n increases. Guide the student to describe this relationship in
terms of length. Explain that as n increases, the length of the apothem approaches the length of the radius. Explain that because of this, a can be substituted for r, and C
can be substituted for p in the formula
to derive the equation
.
A useful link illustrating the convergence of the apothem and the radius can be found at: http://www.mathopenref.com/polygonradius.html.
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 that as n increases, the lengths of the sides of the n-gon decrease, so that the perimeter of the n-gon approaches the circumference of the circle,
and the area of the n-gon approaches the area of the circle. The student derives an equation relating the area and circumference of the circle by observing that the area (
) of the n-gon is given by
. The student explains that as n increases, the apothem, a, of the n-gon approaches the radius, r, of the circle. Also the
perimeter, p, of the n-gon approaches the circumference, C, of the circle. The student substitutes r for a and C for p into the formula to obtain the equation
.
Questions Eliciting Thinking
How might you use the equation you derived?
Instructional Implications
Implement the next MFAS task in this sequence, Area and Circumference – 2 (G-GMD.1.1).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
page 3 of 4 Area and Circumference - 1 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-GMD.1.1:
Description
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder,
pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
page 4 of 4