Download Interior Angles of a Polygon

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 120654
Interior Angles of a Polygon
Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Keywords: MFAS, polygon, interior angles, interior angle sum
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_InteriorAnglesOfAPolygon_Worksheet.docx
MFAS_InteriorAnglesOfAPolygon_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 problem on the Interior Angles of a Polygon worksheet.
2. The teacher asks follow-up questions, as needed.
Note: In the rubric, the following theorem is referenced by name:
Triangle Sum Theorem – The measures of the angles of a triangle sum to 180°.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to provide a coherent explanation.
Examples of Student Work at this Level
The student:
May write an explanation that contains incorrect statements.
page 1 of 4 May reference some important aspect of a correct explanation but is unable to provide a complete explanation.
Explains how to apply the formula to specific n-gons rather than explaining the derivation of the formula.
Questions Eliciting Thinking
What do you know about the measures of the angles of a triangle?
Can you think of a way to partition a hexagon into triangles?
What does n represent? Why is two subtracted from n?
What does convex mean? What is a convex polygon?
Instructional Implications
If needed, review the definitions of polygon, convex polygon, and diagonal of a polygon. Also review the Triangle Sum Theorem. Then provide the student with an activity
involving a sequence of convex n-gons for n = 4, 5, 6, 7, and 8. For each n-gon, ask the student to draw all diagonals from one vertex to all other vertices, observe the
number of triangles formed, determine the relationship between the angles of the original n-gon and the triangles, and finally, calculate the sum of the measures of the
interior angles by multiplying the number of triangles by 180°. Assist the student in organizing the results in a table and in conjecturing a formula for calculating the sum of
the measures of the interior angles of any convex n-gon, that is, (n – 2)180°. Emphasize the relationship between the interior angles of the n-gons and the interior angles
of the triangles formed by drawing diagonals; that is, the interior angles of these triangles completely compose the interior angles of the n-gon with no overlaps or gaps.
Also, make clear why the formula only applies to convex n-gons.
Review with the student the important components of a complete explanation of the formula based on this exercise:
1. When all diagonals from one vertex to the other vertices of a convex n-gon are drawn, (n – 2) triangles are formed in the interior of the n-gon.
2. The interior angles of the triangles that are formed completely compose the interior angles of the n-gon with no overlaps or gaps.
3. Since the sum of the measures of the interior angles of a triangle is 180°, then the sum of the measures of the interior angles of a convex n-gon is given by (n – 2)180°.
Moving Forward
Misconception/Error
The student explains the formula with regard to a specific instance of an n-gon.
Examples of Student Work at this Level
The student explains the formula with regard to a specific instance of an n-gon rather than addressing the general case. For example, the student explains the formula only
in the context of a quadrilateral or a hexagon.
Questions Eliciting Thinking
Will this formula work for any convex n-gon?
In general, what is the relationship between the number of sides of a convex polygon and the number of triangles into which it can be partitioned?
If the n-gon is partitioned into triangles by drawing all of the diagonals from one vertex, what is the relationship between the angles of the n-gon and the angles of the
triangles?
Instructional Implications
Make the distinction between explaining the formula in a specific case and explaining the formula in general. Review with the student the important components of a
page 2 of 4 complete explanation of the formula:
1. When all diagonals from one vertex to the other vertices of a convex n-gon are drawn, (n – 2) triangles are formed in the interior of the n-gon.
2. The interior angles of the triangles that are formed completely compose the interior angles of the n-gon with no overlaps or gaps.
3. Since the sum of the measures of the interior angles of a triangle is 180°, then the sum of the measures of the interior angles of a convex n-gon is given by (n – 2)180°.
Ask the student to adapt his or her explanation to the general case and provide feedback.
Almost There
Misconception/Error
The student provides a general explanation that is incomplete.
Examples of Student Work at this Level
The student composes an explanation that applies to the general case of an n-gon but omits an important component. For example, the student may explain that an n-gon
can be partitioned into (n – 2) triangles by drawing all diagonals from one vertex. The student may further explain that the sum of the interior angles of each triangle is
180°. However, the student omits an explanation of the relationship between the interior angles of the n-gon and the interior angles of the triangles into which the n-gon
has been partitioned.
Questions Eliciting Thinking
What is the relationship between the interior angles of the triangles and the interior angles of the n-gon?
Why is it necessary to specify that the n-gon be convex?
Instructional Implications
Review with the student the important components of a complete explanation of the formula based on this approach:
1. When all diagonals from one vertex to the other vertices of a convex n-gon are drawn, (n – 2) triangles are formed in the interior of the n-gon.
2. The interior angles of the triangles that are formed completely compose the interior angles of the n-gon with no overlaps or gaps.
3. Since the sum of the measures of the interior angles of a triangle is 180°, then the sum of the measures of the interior angles of a convex n-gon is given by (n – 2)180°.
Have the student review his or her explanation to ensure that it addresses all important components. Allow the student to revise his or her explanation to make it complete.
Provide opportunities for the student to use the formula in the context of solving problems.
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 provides a correct explanation of the statement that the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°. For
example, the student explains that if all of the diagonals from one vertex of an n-gon are drawn, the n-gon will be partitioned into (n – 2) triangles. If the n-gon is convex,
then the interior angles of these triangles completely compose the interior angles of the n-gon with no overlaps or gaps. Since the sum of the interior angles of a triangle is
180° and there are (n – 2) triangles, the sum of the interior angles of the n-gon is given by (n – 2)180°. The student may provide a diagram to support the explanation.
Questions Eliciting Thinking
Why is it necessary to specify that the n-gon be convex?
Does the n-gon have to be regular in order to use this formula?
Instructional Implications
Review the definition of a regular polygon and an exterior angle of a polygon. Then challenge the student to conjecture formulas for finding the measure of one interior
angle of a regular polygon and one exterior angle of a regular polygon. Provide opportunities for the student to use these formulas in the context of solving problems.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Interior Angles of a Polygon worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Access Privileges: Public
page 3 of 4 License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.10:
Description
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of
interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
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