When calculating the area of half the circle, students may divide the radius in half before applying the area formula, instead of first calculating the area of the entire circle and then dividing it in half. Ask these students to calculate , then to divide by , and finally to compare this to the result of .

In this activity, students get experience calculating areas of circle sectors and lengths of arcs. This will help them define general methods for these calculations in a subsequent activity, as they leverage the repeated reasoning here (MP8). In the Activity Synthesis, students observe that sectors can be used to create an informal argument for the area of a circle.

Monitor for students who use different strategies. Here are some strategies they may use, from more efficient to less efficient:

• Recognize that a 135-degree sector is of a circle.
• Divide 135 by 360 to get 0.375.

If students are stuck on the last circle, ask them how they approached the first problems and if any concepts carry over. Remind them that 360 degrees defines a complete circle, and ask if that number can help them.

The purpose of this discussion is for students to consider different approaches. 

Invite previously selected students to share their strategies for the final question. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions such as “What are the advantages and disadvantages of each approach?” (The second approach works for any central angle, while the first approach may be faster. The first approach may lead to an intuitive estimate that allows for easy recognition of calculation errors, and it always results in an exact answer that does not require approximation or rounding.)

Next, tell students that we can use sectors to help create the formula for the area of a circle. Display this image for all to see.

Ask students, “How does this image explain the formula for the area of a circle?” (If we slice the circle into many sectors and rearrange them, they take on a shape resembling a parallelogram. The parallelogram has height . The length of the parallelogram is half the circumference of the circle, or . Multiply the base and height of the parallelogram to get for its area.)

In this activity, students generalize their earlier experiences with calculating sector areas and arc lengths.

Monitor for students who use different strategies. Here are some approaches students may take to describe their method, from more abstract to more concrete:

• Give a description in words.
• Create a formula such as .

The purpose of this discussion is for students to consider different ways to describe a solution. 

Ask previously selected students to share their strategies for the first question. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How are the formula and the verbal description expressing the same process?”
• “What does the represent?” (It represents the fraction of the circle taken up by the sector.)
• “What does the  represent?” (That’s the area of the whole circle.)

If no student created a formula, invite students to rewrite their method using symbols instead of words.