Lesson 8

The purpose of this Math Talk is to elicit strategies and understandings students have for calculating areas of sectors and finding arc lengths for simple fractions of circles. These understandings help students develop fluency and will be helpful later in this lesson when students will develop general methods for calculating sector areas and arc lengths.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they combine fraction operations with area and circumference formulas.

If necessary, provide students with formulas for the area and circumference of a circle.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

When calculating the areas, students may divide the radius in half before applying the area formula, instead of first calculating the area of the entire circle then dividing it in half. Ask these students to calculate \(\pi \boldcdot \left(8^2\right)\) then divide by 2, and compare this to the result of \(\pi \boldcdot 4^2\).

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

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. In the activity synthesis, students observe that sectors can be used to create an informal argument for the area of a circle.

Monitor for a variety of strategies on the last problem, in particular for students who recognize that a 135 degree sector is \(\frac38\) of a circle and for those who divide 135 by 360 to get the decimal 0.375.

After students have read the definition of a sector, tell them that any 2 radii define 2 mutually exclusive sectors. Shading or description will denote the particular sector that students should work with.

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.

Select previously identified students to share their strategies for the final problem in this order: first, a student who recognized that a 135 degree sector is \(\frac38\) of a circle; second, a student who divided 135 by 360 to get 0.375. Ask students 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 original circle area formula. 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 \(r\). The length of the parallelogram is half the circumference of the circle, or \(\pi r\). Multiply the base and height of the parallelogram to get \(\pi r^2\) for its area.)

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

Monitor for students who describe their method using words and for those who create a formula.

Choose previously identified students to share their strategies in this order: first, a student who gave a description in words; next, one who created a formula such as \(\frac{\theta}{360}\boldcdot \pi r^2\). If no student created a formula, invite students to rewrite their method using symbols instead of words.

Challenge students to explain why the formula and the verbal description are expressing the same process. Ask, “What does the \(\frac{\theta}{360}\) represent?” (It represents the fraction of the circle taken up by the sector.) “How about the \(\pi r^2\)?” (That’s the area of the whole circle.)