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:
- 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.
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| 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.
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| 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 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 n-gon 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
 .
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| 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 .
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| 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). |
