Prior to this lesson, ask students to bring in several flat, circular objects that they can measure.
As a warm-up, ask students to measure the length and width of their desktops. Ask them to decide which type of unit should be used. Then, have students measure or calculate the distance around the outside of their desktops.
With the class, discuss the following questions:
- What unit did you use to measure your desks? Why?
[Because of the size of desks, the most appropriate units are probably inches or centimeters.]
- Why did some of your classmates get different measurements for the dimensions of their desks?
[Measurements will obviously differ because of the units. In addition, the level of precision may give different results. For instance, a student may round to the nearest inch, while another may approximate to the nearest ¼-inch.]
- What do we call the distance around the outside of an object?
[The distance around the outside of a polygon is known as the perimeter. The distance around the outside of a circle is known as the circumference.]
Inform the class that they will be measuring the circumference of several circular objects during today’s lesson. Also, alert them that, just as there is a formula for finding the perimeter of a rectangle (P = 2L + 2W), there is also a formula for finding the circumference of a circle. They should keep their eyes open for a formula as they proceed through the measurement activities.
Divide the class into groups of four students. Within the groups, each student will be given a different job. (If class size is not conducive to four students per group, form groups of three — one student can be assigned two jobs.)
- Task Leader: Ensures all students are participating; lets the teacher know if the group needs help or has a question.
- Recorder: Keeps group copy of measurements and calculations from activity.
- Measurer: Measures items (although all students should check measurements to ensure accuracy).
- Presenter: Presents the group’s findings and ideas to the class.
Students should measure the "distance around" and the "distance across" of the objects that they brought to school. Students will likely have little trouble measuring the distance across, although they may have some difficulty identifying the exact middle of an object. To measure the distance around, students will likely need some assistance. An effective method for measuring the circumference is to wrap a string around the object and then measure the string. To ensure accuracy, care should be taken to keep the string taut when measuring the outside of a circular object.
Students should be allowed to select which unit of measurement to use. However, instruct students to use the same unit for the distance around and the distance across.
Students should record the following information in the Apple Pi activity sheet:
- Description of each object
- Distance around the outside of each object
- Distance across the middle of each object
- Distance around divided by distance across
After the measurements have been recorded, a calculator can be used to divide the distance around by the distance across. Students should answer both questions on the worksheet. As students are working, take note of their results. Push students to note any numbers in the last column that seem to be irregular, and have them check their measurements for these rows.
When all groups have completed the measurements and calculations, conduct a whole-class discussion. Rather than present each individual object, students should discuss the average and note any interesting findings. Students should also compare their averages with those of other groups.
You may wish to use the Circle Tool applet as a demonstration tool. This applet allows students to see many other circles of various sizes, as well as the corresponding ratio of circumference to diameter.
Explain that each group has found an approximation for the ratio of the distance around to the distance across, and this ratio has a special name: π. (It may also be necessary to explain that the "distance across" is known as the diameter and that the "distance around" is known as the circumference. Because of this relationship, algebraic notation can be used to write
circumference ÷ diameter = π
or, said another way,
π = C/d
which leads to the following formula for circumference:
C = π × d
Point out that groups within the class may have obtained slightly different approximations for π. Explain that determining the exact value of π is very hard to calculate, so approximations are often used. Discuss various approximations of π that are acceptable in your school’s curriculum.
