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.
 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 ngons 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 ngons for n = 4, 5, 6, 7, and 8. For each ngon, 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 ngon 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 ngon, that is, (n – 2)180°. Emphasize the relationship between the interior angles of the ngons 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 ngon with no overlaps or gaps. Also, make clear why the formula only applies to convex ngons.
Review with the student the important components of a complete explanation of the formula based on this exercise:
 When all diagonals from one vertex to the other vertices of a convex ngon are drawn, (n – 2) triangles are formed in the interior of the ngon.
 The interior angles of the triangles that are formed completely compose the interior angles of the ngon with no overlaps or gaps.
 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 ngon is given by (n – 2)180°.

Moving Forward 
Misconception/Error The student explains the formula with regard to a specific instance of an ngon. 
Examples of Student Work at this Level The student explains the formula with regard to a specific instance of an ngon 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 ngon?
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 ngon is partitioned into triangles by drawing all of the diagonals from one vertex, what is the relationship between the angles of the ngon 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. Assist the student in generalizing the explanation given to any ngon. Review with the student the important components of a complete explanation of the formula based on this approach:
 When all diagonals from one vertex to the other vertices of a convex ngon are drawn, (n – 2) triangles are formed in the interior of the ngon.
 The interior angles of the triangles that are formed completely compose the interior angles of the ngon with no overlaps or gaps.
 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 ngon is given by (n – 2)180°.
Provide opportunities for the student to use the formula in the context of solving problems. 
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 ngon but omits an important component. For example, the student may explain that an ngon 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 ngon and the interior angles of the triangles into which the ngon has been partitioned.

Questions Eliciting Thinking What is the relationship between the interior angles of the triangles and the interior angles of the ngon?
Why is it necessary to specify that the ngon be convex? 
Instructional Implications Review with the student the important components of a complete explanation of the formula based on this approach:
 When all diagonals from one vertex to the other vertices of a convex ngon are drawn, (n – 2) triangles are formed in the interior of the ngon.
 The interior angles of the triangles that are formed completely compose the interior angles of the ngon with no overlaps or gaps.
 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 ngon is given by (n – 2)180°.
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 ngon is given by the formula (n – 2)180°. For example, the student explains that if all of the diagonals from one vertex of an ngon are drawn, the ngon will be partitioned into (n – 2) triangles. If the ngon is convex, then the interior angles of these triangles completely compose the interior angles of the ngon 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 ngon 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 ngon be convex?
Does the ngon 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. 