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 = πor, said another way,
π = C/dwhich leads to the following formula for circumference:
C = π × dPoint 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.
Questions for Students
1. Why did we use the ratio of circumference to diameter for several
objects? Wouldn’t we have gotten the same result using just one object?
[If we had used just one object, an incorrect measurement
would have given an incorrect approximation for π. Using several
objects ensures that our results are correct. In addition, slight
errors in measurement may give different values of π, so using the
average of several measurements will help to eliminate rounding
errors.]
2. Were any of the ratios in the last column not close to 3.14? If not, explain what might have happened.
[The ratio of circumference to diameter is always the same, and the
ratio is always close to 3.14. If a value in the last column is not
close to 3.14, it is the result of a measurement or calculation error.]
3. Describe some situations in which knowing the circumference (and how to calculate it) would be useful.
[Bike tires are often described by their diameter. For
instance, a 26-inch tire is a tire such that the diameter is 26". Each
time the tire makes one complete rotation, the bike moves forward a
distance equal to the circumference of the tire. Therefore, it would be
helpful to know how to calculate the circumference based on the
diameter.]
Teacher Reflection
- What prior knowledge did students have of π (if any)? How did
student’s prior knowledge affect the delivery of the lesson? What
modifications did you need to make as a result, and how effective were
these adjustments?
- How precise were student measurements? How did you assist students with their measurements?
- How did students react to the use of 3.14 as an approximation
of π? Were there any adverse reactions due to conceptual
misunderstandings?
- How did students show that they were actively learning?
- Did students understand that the ratio of circumference to
diameter (i.e., π) is an approximation? Did they understand why they
had obtained different values for this approximation during the
activity?