A SLICE OF PI
Grades 7 - 9
In this lesson, students will discover pi, its history, and
some of the applications of its uses in the world today. Many students begin
the study of circles in geometry without any understanding of the number
pi and its mathematical power. This lesson gives students a feel for pi.
During the lesson, students will conduct an experiment by measuring the
diameter and circumference of various circular objects to calculate their
own approximation of pi. They can also conduct a probability experiment
that approximates the value of pi.
Students will identify uses in the real world for pi, both dealing and not
dealing with circles.
As a concluding activity, students will compare various values for pi discovered
throughout its history, determine whose value was the best, and theorize
why people used fractions instead of decimals for pi.
MATHEMATICS!: The Story of Pi
The student should be able to:
- state that pi is the ratio of the circumference of a circle to the
diameter of a circle.
- list two approximations for pi, a fraction and a decimal.
- list at least four uses for pi in the world today and discuss the
history of the calculation of pi.
Previewing Activity For Each Student:
- 1 piece of paper with large circle copied.
- scissors
- pencil or:
- 1 paper circle (precut) & pencil
Viewing Activities For Each Group:
- objects w/ circular tops/bottoms
- worksheet "A Slice of Pi"
- ruler, tape measure or string
- pencil
- scientific calculators
Postviewing Activities:
- worksheet "Pi Around the World"
- calculator
- pencil
Extension For Each Student:
- worksheet "Buffton's Toothpick Experiment"
- toothpicks
- scissors
- ruler in centimeters
- calculator
Diameter - the distance from one side of a circle to
the other side, through the center.
Radius - the distance from the center of a circle to the outside
edge.
Circumference - the perimeter of a circle.
Ratio - the comparison of two numbers, usually written as a fraction.
As a check for understanding of vocabulary, do the following
activity. Give students a piece of paper with a large circle copied on it.
(Or give them a circle that is already cut out, if time is a factor.) In
doing so, most will fold more than one diameter. Then ask them to show you
a diameter and label it. Do a radius as well. Ask them what term is used
to describe the perimeter of the circle. (Circumference) Have them label
this. Review the definition of a ratio, and recall that ratios usually look
like a fraction.
As a check for understanding of vocabulary, do the following
activity. Give students a piece of paper with a large circle copied on it.
(Or give them a circle that is already cut out, if time is a factor.) In
doing so, most will fold more than one diameter. Then ask them to show you
a diameter and label it. Do a radius as well. Ask them what term is used
to describe the perimeter of the circle. (Circumference) Have them label
this. Review the definition of a ratio, and recall that ratios usually look
like a fraction.
BEGIN the video at the main feature, after section 0,
prerequisites. The first scene will begin where the MATHEMATICS logo
is created.
About 15 seconds into the video, an interviewer is conducting a poll and
asks "What can you tell me about the number pi?" PAUSE
and ask your students to answer his question. RESUME the video, after
telling students to compare their answers with the video answers.
PAUSE after the boy in the red shirt and glasses gives his response
and ask why his answer is incorrect. RESUME the video.
PAUSE when the narrator says that conducting a poll to define pi
is not a good method, and ask your students how many of their answers were
similar to the video. RESUME the video.
After the narrator explains that there is an experiment to find a value
for pi, a girl will demonstrate measuring the circumference and the diameter
of a canister. STOP here, before pi is actually calculated.
Tell students that they will conduct the same experiment before they see
what results were found in the video. Pass out the worksheet entitled "A
Slice of Pi", and the materials needed to complete the activity. Complete
the experiment using partners or small groups, following the directions
on the worksheet. As they finish their work, have them list their results
on the board or on a transparency. After students have listed their results
on the board (or a transparency) and on their own papers, ask if they can
find any patterns in their ratios. Have them calculate the mean for all
values found by the class. See if the pattern that they found by inspection
held. (Values should be close to 3.) Ask if any group came close to pi?
(This will also tell you if they know a value for pi.) See which group came
the closest to pi, and possibly give them prize of some sort for their accurate
measuring abilities.
Tell students to watch the video to determine if their values were the same
or different than the ones in the video. RESUME the video, to show
students on the video calculating pi.
PAUSE on the number line representation of pi, and discuss the notations
involving inequality symbols, and the placement of pi on the number line.
As they begin the next section, ask students to find the main topics discussed
in the video. RESUME the video.
PAUSE at the break at the beginning of Section 2, and ask students
what topics are in the video. Tell students to watch for 4 uses for pi in
the real world in the next section. RESUME the video.
PAUSE after the representation of the torus, and ask students for
4 places in geometry where pi is used. (There are 6 mentioned.) Ask them
now to watch for 2 places where pi is used without circles. RESUME
the video.
PAUSE at the end of Section 2 to discuss these uses for pi. RESUME
the video with Section 3, 'Early History', telling students to focus on
which groups of people tried to discover a value for pi, and why they needed
one.
STOP after the Bible's value is discussed, and ask students to list
the groups. FAST FORWARD to Section 5, 'The Computation of
Pi', and RESUME the video after telling students to look for 4 ways
people compute pi.
STOP at the newspaper article that discusses the computer-generated value,
and have students list the methods used to compute pi. RESUME the
video and have students listen to determine the differences between a rational
number and an irrational number. This section may be easier for your students
to understand, if you TURN OFF THE SOUND and
narrate it.
STOP at the end of Section 5, and formulate definitions for rational
and irrational numbers. FAST FORWARD to the recap portion
of the video. Tell students to listen for a reason why people may want to
find a value for pi. RESUME the video.
PLAY to the end.
As a review for the values found in the video, have students
complete the worksheet "Pi Around the World", which reinforces
the various values of pi found by different groups or mathematicians. This
will allow them to determine for themselves which values were better than
the others.
Have the students devise a pneumonic device to help them remember the value
of pi for the first 5 or 6 digits. For example, "Yes, I know a digit"
is a device that gives a value for pi to 4 decimal places. Each word contains
the same number of letters as the corresponding digit in pi. They could
write this sentence on the back of the worksheet.
Since there will probably be a variety to the brands of calculators in any
class, have the students all use their own calculator to find it's value
for pi. Discuss why these differences occur, and which calculators round
and which truncate. Also, find a value using a graphing calculator and a
computer and include in the discussion.
The banking industry uses many numbers to code accounts. Students
can interview a person from this industry to determine if pi is used when
secret codes are created.
Students could conduct an on-line poll, similar to the one shown in the
video, to find how many people really understand what pi is.
This could also be completed at school, or in their families. A short report
of their findings could involve some statistical analysis.
Invite someone from the computer industry into the class to discuss how
pi is used in the manufacture of large computers. They could discuss the
history of computers, their own careers.
Have students conduct research on the groups or individuals
who developed a value for pi. This could be a group project with library
research or an individual report.
Have students calculate how many revolutions their own bicycle wheel will
make in traveling one kilometer.
Have students find 5 circular objects at home, measure the diameter, and
approximate the circumference.
Have students measure the circumference of a tree, and calculate its diameter.
Ask them if there is a way to approximate the age of the tree, diameter
or radius of the tree.
As a follow-up to mention of Sir Edmund Hillary's quote, have students write
a short essay on why people become explorers. They could include examples
from history, who they feel are the explorers of today, and where explorers
of the future will explore.
During a unit on probability, have students conduct a simulation of Buffon's
toothpick experiment, an activity that calculates a value for pi in non-circular
application.
Have students do the lattice-point activity explained in the video, just
before the recap section. Done as a group, they could create the large pattern.
Do the following computer activity:
a. Run the following BASIC program, using a large number in the blank.
10 SUM =
20 20 FOR N = 1 TO ____
30 TERM = 1/(N*N)
40 SUM = SUM + TERM
50 NEAR PI = SQR(6*SUM)
60 PRINT N, NEARPI
70 NEXT N
80 END
b. Write down the last line the computer prints.
c. What does the program do?
d. Try a larger number in the blank and see what happens.
WHERE TO GET VIDEO
Idaho State Library 325 West State St. Boise, Idaho 83702-6072 (208) 334-2152
California Institute of Technology Caltech 1-70 Pasadena, CA 91125
Lesson plan developed by Master Teacher Kit Parker,
South Junior High School, Boise, Idaho
Click here to view the
worksheet associated with this lesson.
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