Lesson 15

The purpose of this activity is for students to estimate the area of a circle by comparing it to a surrounding square.

Explain that some farms have circular fields because they use center-pivot irrigation. If desired, display these images to familiarize students with the context.

Provide quiet think time followed by whole-group discussion. 

Students might think the answer should be 640,000 m² because that is the area of the square, not realizing that they are being asked to find the area of a circle. Ask them what shape is the region where the plants are growing.

Some students might incorrectly calculate the area of the square to be 6,400 m² and therefore estimate that the circle would be about 5,000 m².

Some students might try to use what they learned in the previous lessons about the relationship between the area of a circle and the area of a square with side length equal to the circle's radius. Point out that the question is asking for an estimate and answer choices all differ by a factor of 10.

Discuss the estimation strategies students used to answer the question. Ask students what the area of the square is in square meters (\(800 \boldcdot 800\), or 640,000). Ask them if the circle's area is greater than or less than the square's area (less). Then ask them to use the picture to determine the best estimate (500,000 since the circle is close in area to the square).

In a previous lesson, students measured various circular objects and graphed the measurements to see that there appears to be a proportional relationship between the diameter and circumference of a circle. In this activity, students use a similar process to see that the relationship between the diameter and area of a circle is not proportional. This echoes the earlier exploration comparing the length of a diagonal of a square to the area of the square, which was also not proportional.

Each group estimates the area of one smaller circle and one larger circle. After estimating the area of their circles, students graph the class’s data on a coordinate plane to notice that the data points curve upward instead of making a straight line through the origin. Watch for students who use different methods for estimating the area of the circles (counting full and partial grid squares inside the circle, surrounding the circle with a square and removing full and partial grid squares) and invite them to share in the discussion.

For classes using the digital version, students can record the class data in the spreadsheet and graph points directly on the grid using the Point tool. Note: you have to click on the graph side of the applet for the point tool to appear.

Some students might be unsure on how to count the squares around the border of the circle that are only partially included. Let them come up with their own idea, but if they need additional support, suggest that they round up to a whole square when it looks like half or more of the square is within the circle and round down to no square when it looks like less than half the square is within the circle.

Invite selected students to share their strategies for estimating the area of their circle.

Next, ask "Is the relationship between the diameter and the area of a circle a proportional relationship?" (No.) Invite students to explain their reasoning. (The points do not lie on a straight line through \((0, 0)\).)

To help students see and express that the relationship is not proportional, consider adding a column to the table of measurements to record the quotient of the area divided by the diameter. Here is a table of sample values.

Remind students that there is a proportional relationship between diameter and circumference, even though there is not between diameter and area. Recall that students saw the same phenomenon when they examined the relationship between the diagonal of a square and its perimeter (proportional) and the diagonal of a square and its area (not proportional).

The purpose of this activity is for students to use what they know about finding the area of a parallelogram to develop the formula for the area of a circle. This activity builds on the work students did in grade 6 when cutting and rearranging shapes in order to calculate their areas. In this activity, students cut and rearrange parts of a circle to approximate a parallelogram. They see that the area of the parallelogram would be calculated by multiplying half of the circle’s circumference times its radius. Since students are not familiar with the process of writing proofs, it is necessary to walk them through writing the justification that uses the formula for the area of the parallelogram to develop the formula for the area of the circle.

The construction in this activity shows that the constructed parallelogram has a height at most the radius of the circle and a base at most half the circumference of the circle. Establishing equality is beyond grade level and will be addressed again in high school.

The process used to decompose the circle and recompose it into a shape resembling a parallelogram is a good example of MP8. The pie shaped wedges are successively cut in half and rearranged. Each time, the sides of the shape look more like line segments. 

Watch for students who identify that the rearranged circle pieces resemble a parallelogram as the pieces get smaller. Also watch for how they estimate the width and height of this parallelogram and invite them to share during the discussion.

Arrange students in groups of 2. Each group needs a circular object, with a diameter between 3 and 5 inches, and a thick marker with which to trace it. Also provide each group a sheet of white paper, a sheet of colored paper, a pair of scissors, and glue or tape. Remind students that in the past they decomposed and rearranged a shape to figure out its area. Demonstrate how to do the first 4 steps of the activity, and invite students to follow along with your example. Ask how the area of the new shape differs from that of the circle. Solicit some ideas on what the new shape resembles and how the area of such a shape could be approximated. Without resolving this, ask students to continue the process.

Students might not fold the wedges accurately or make a straight cut. Remind them that the halves must be equal.

Ask students which polygon resembles the shape they constructed of circle pieces and how they know. (A parallelogram or rectangle)

For a dynamic visualization, see http://ggbm.at/RUqSMrjn, created in GeoGebra by Malin Christersson, or display this image.

Ask, “If we could continue cutting the wedges in half, how would that affect the new shape?” (It would look even more like a parallelogram or rectangle. The bumpy top and bottom straighten out, and the slanted height becomes more vertical.)

Note that we are going to refer to the bumpy parallelogram-ish shape as the “parallelogram” (in quotes). Ask students to describe comparisons we can make between the measurements in the circle and the “parallelogram.” Record and display these ideas for all to see. It may be helpful to write over the actual images themselves. Students should notice the following measurements, but if they do not, prompt them to look for them:

• The base of the “parallelogram” is approximately equal to half of the circle’s circumference.
• The height of the “parallelogram” is approximately equal to the radius of the circle.
• The areas of the 2 shapes are equal.

Tell students to label these measurements on their visual display:

• “\(\text{Circumference} = \pi d\)” around the circle
• “\(\frac12 \text{Circumference} = \pi r\)” at the base of the “parallelogram”
• “Radius” on the radius of the circle (needs to be drawn in)
• “Radius” on the height of the “parallelogram” (needs to be drawn in)

If students struggle to understand these relationships through the abstract variables, consider measuring your example circle and using the numerical measurements to talk about these relationships.

Ask students to discuss the different ways we can calculate the area of the “parallelogram.” Students should share the following ways: 

• \(\text{Area} = \text{Base} \boldcdot \text{Height}\)
• \(\text{Area} = \frac12 \text{Circumference} \boldcdot \text{Radius}\)
• \(\text{A} = \pi r \boldcdot r\)
• \(\text{A} = \pi r^2\)

Tell students to record these ideas on their visual display. Display student work for all to see. If students do not bring up one of these ideas, make it explicit in the discussion. 

The purpose of this activity is for students to consider a different way to cut and reassemble a circle into something resembling a polygon in order to calculate its area. This time the polygon is a triangle, but the area of the circle can still be found by multiplying \(\frac12\) times the circumference times the radius.

In the previous activity, students had experience following along as the teacher developed the justification. This time give students the opportunity to write their own justification for the area of a circle. As students work, monitor and select students who have clear, but different, explanations to share during the whole-group discussion. In particular, select students who use the following steps:

• \(\text{Area} = \frac12 \boldcdot \text{base} \boldcdot \text{height}\)
• \(\text{Area} = \frac12 \boldcdot \text{circumference} \boldcdot \text{radius}\)
• \(\text{Area} = \frac12 \boldcdot (\pi d) \boldcdot r\)
• \(\text{Area} = \pi r \boldcdot r\)
• \(\text{Area} = \pi r^2\)

If the bands making up the circle really did not stretch, then they would not form rectangles when they are unwound because the circumference of the inner circle is not the same as the circumference of the outer circle in each band. A rectangle is an appropriate approximation for the shape in terms of calculating its area.

If students struggle to imagine the circle and how it is cut and rearranged, suggest a familiar material for the rings that bends but does not stretch (for example, a cord or chain).

Ask selected students to explain their steps for finding the area in terms of the circle’s measures. Ask the class whether they agree, disagree, or have questions after each student shares their reasoning.