Getting Started 
Misconception/Error The student is unable to describe the base and height of the rectangular shape in terms of length measurements of the circle. 
Examples of Student Work at this Level The student states that the height of the rectangular shape is:
 The diameter of the circle.
 The â€śheightâ€ť or â€świdthâ€ť of the circle.
 One circle or one triangle.
 Pi.
 Â , , or .
The student states that the base of the rectangular shape is:
 , , orÂ of the circleâ€™s circumference.
 Four, eight, or sixteen times the circleâ€™s circumference.
 Pi.

Questions Eliciting Thinking Can you explain how the rectangular shape was composed?
Can you show me a radius of the circle?
What does circumference mean?
What measurements of the circle can you identify in one of the sectors of the circle? What about in the rectangular shape? 
Instructional Implications Review the vocabulary used to describe circles and discs (e.g., diameter, circumference, radius, sector, and area), and ask the student to explain each term. Ask the student to draw a circle along with a radius, diameter, and sector of the circle with each clearly labeled. Explain the meaning of circumference and that it is a length measurement.
Model dividing a circle into sectors. Clearly label the radius of the circle, and describe each sector as being bounded by two radii and an arc of the circle. Then cut out and rearrange the sectors into a rectangular shape. Show the student that the radius of the original circle corresponds to the height of the shape. Then explain how each base of the rectangular shape is composed of congruent arcs of the circle. Assist the student in deducing that each base is half of the circumference of the circle. Decompose the rectangular shape and recompose it as a circle, if needed. 
Moving Forward 
Misconception/Error The student is unable to write an equation for the area of the rectangular shape. 
Examples of Student Work at this Level The student is able to identify the height and base of the rectangular shape as the radius and one half of the circumference of the circle, respectively. However, when attempting to write an equation that represents the area of the rectangular shape, the student:
 Writes an expression instead of an equation [e.g., ].
 Provides the basic area formula for a rectangle or parallelogram (e.g., A = bh or A = lw).
 Writes an equation that is incorrect.

Questions Eliciting Thinking What is an equation? What does an equation look like?
How do you find the area of a rectangle?
What do your answers to the first two questions tell you about the dimensions of the rectangle? 
Instructional Implications If needed, clarify the difference between an expression and an equation. Ask the student to determine if he or she wrote an expression or an equation.
Review the formula for finding the area of a rectangle and the Substitution Property of Equality. Guide the student through the logic used in the Broken Circles worksheet [e.g., because the height (h) of the rectangle is estimated by the radius (r) of the circle, and the base (b) of the rectangle is estimated by ()C, then r and ()C can be substituted for h and b in the formula A = bh]. The formula that results, A = ()Cr, now describes the area of the rectangle in terms of the radius and circumference of the circle.
Ask the student to use the Substitution Property to write expressions that represent the area or perimeter of triangles or rectangles. For example, in the following figure, the area of the rectangle can be represented in terms of its length and width as A = lw or in terms of the circleâ€™s radius as .

Almost There 
Misconception/Error The student is unable to explain what the equation indicates about the relationship between the circumference and area of a circle. 
Examples of Student Work at this Level The student is able to identify the height and base of the rectangular shape as the radius and one half of the circumference of the circle, respectively. Additionally, the student correctly writes an equation for the area of the rectangular shape. When explaining what the equation indicates about the relationship between the area and circumference of a circle, the student:
 Makes a comparative statement about the areas of the circle and rectangular shape such as, â€śThe area of the circle and rectangle are the same.â€ť
 Makes a vague statement that is unclear such as: â€śThey are the same,â€ť â€śIt compares lengths,â€ť or â€śThe area will be larger than the circumference.â€ť
 Explains the relationship between the height or base of the rectangular shape and the radius or circumference of the circle.

Questions Eliciting Thinking In the equationÂ , what does the A stand for?
What does the equationÂ say about the area of the circle and the circumference of the circle? 
Instructional Implications Ask the student to identify the meaning of each variable in the equation . Guide the student to observe that the equation is solved for A, the area of the circle and begin an explanation by saying, â€śThe area of the circle is equal toâ€¦â€ť. Prompt the student to complete the explanation by referring to the expression . Â Ask the student to explain Â mathematically (e.g., as half the product of the circumference of the circle and its radius).
Provide other familiar formulas, such as the formula for finding the area of a triangle, and ask the student to explain the formula in words. 
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 identifies the height of the rectangular shape as equal to the circleâ€™s radius and the base of the shape as equal to half of the circleâ€™s circumference. The student writes the equation ,Â and explains that the equation indicates that the area of a circle is equal to half of the product of the circumference of the circle and its radius. 
Questions Eliciting Thinking How would you describe the changes in the rectangular shape as the number of sectors into which the circle has been divided increases? 
Instructional Implications Ask the student to use the formula for the area of a circle, , and the equation from the exercise, , to derive the formula for the circumference of a circle in terms of r, the radius of the circle. Then ask the student to rewrite the circumference formula in terms of dÂ (the diameter of the circle) and explain what this formula indicates about the relationship among C, d, and .
Ask the student to rewrite the equation so that it is solved for C, circumference, and to explain what this version of the equation indicates about the area and circumference of a circle.
Ask the student to consider into how many sectors the circle must be divided before the sectors can be arranged into a rectangle in which the base is exactly equal to half the circleâ€™s circumference.
Consider administering other MFAS tasks for standard 7.G.2.4. 