Discover Pi:
Where did 3.14 come from?
Subject: Mathematics
Topic: Numbers
Grade Level: 8-12
Time: 20-40 min
Pre/Post Show Math Activity
Introduction:
In Show Math students explored the interesting properties of Pi. This activity allows students to discover Pi for
themselves by taking their own measurements and examining the ratio of circumference to diameter. This will
help students to understand where Pi originated as well as look at errors in measurements and the properties of
irrational numbers.
Learning Objectives:
By the end of this activity students should:
• Understand that Pi is the ratio of circumference to diameter
• Understand that it is helpful to take more measurements to prevent error and that measuring instruments
have limits
Skills:
Students should develop:
• Data collection skills
• Measurement skills
• Collaboration and communication skills
Material & Resources:
• Rulers (one per group)
• Worksheets for the class
• String
WNCP Curriculum Links:
Mathematics 9
Solve problems and justify the solution strategy using circle
properties [C, CN, PS, R, T, V]
Foundations of Mathematics 10
Demonstrate an understanding of irrational numbers
[CN, ME, R, V]
Background:
The earliest known approximations of Pi date back to 1900 BCE. Both the
Ancient Babylonians and the Ancient Egyptians understood that ratio of
the circumference to the diameter of a circle was a constant. The
Babylonians approximated Pi to be 3, however one tablet dated between
1900-1680 BCE estimated Pi as 3.125. The Egyptians calculated the area
of a circle to be [(8×diameter)÷9]² and this yields an approximation of Pi
to be 3.1605 (slightly less accurate than the Babylonians).
The Indian text Shatapatha Brahmana, which was written sometime between 8th – 6th century BCE, estimated Pi to be 3.139. Archimedes was
the first to estimate Pi rigorously in 287-212 BC. He did this by inscribing
circles in regular polygons and calculating the outer and inner polygons
respective perimeters. He used 96 sided polygon he proved that Pi was
between 223/71 (3.1408) and 220/70 (3.1429), which averaged to 3.14185.
Aorund 480 AD Zu Chongzhi, Chinese mathematician approximated by
to 3.14159265 by inscribing a 126-gon. In modern times Pi is calculated
using calculus and computers.
In this activity students will be examining the errors that arise in measuring Pi as the ratio of the circumference to the diameter using string
and rulers. The types of errors that can occur are systematic or random.
A systematic error is a bias in measurement that is not reduced by taking separate measurements. Systematic errors can be constant or can be
proportional to size of the object you are measuring. Imagine you were
measuring temperature and your proportional systematic error is 2% and
the actually temperature is 100˚C your measurement would yield 102°C.
However if the actual temperature was 50°C , then your measurement
would be 51°C . In comparison a constant systematic error are normally
caused by incorrect zeroing of your measuring device. If your thermometer was starting at 2°C instead of 0°C, then when the actual temperature
was 100°C, your measurement would be 102°C. If the actual temperature
was 50°C, your measurement would be 52°C.
In this activity, a proportional systematic error is one that will occur due
to the different sizes of the circles. When students are placing their strings
on the circles, they should find that there estimations are further from Pi
for the smaller circle and closer for the bigger circle. It would be difficult
to place the string perfectly on the circle and this will cause error in the
approximation of Pi. The proportional of string not on the circle has a
greater effect on the estimation of Pi for the small circle than it does for
the big circle.
Students will also encounter random error. Random error leads to measurable values being inconsistent when repeated on a constant attribute
(such as Pi). Random error can be caused by fluctuations in reading the
ruler or in the student’s reading of the ruler. This should be reduced by
taking multiple measurements. Therefore the class approximation of Pi
should be closer to actual value than the individual groups approximation.
Why do we call it Pi?
It is thought that the constant is
named “π” because it is the first
letter of the Greek words
περιφέρεια (periphery) and
περίμετρος (perimeter). This
is most likely because of Pi’s
relationship to the circumference
(or perimeter) or a circle.
Resources:
• http://en.wikipedia.org/wiki/Pi
Explains more about the history, properties and derivations of Pi
• http://en.wikipedia.org/wiki/Systematic_error
For more information on Systematic Error
• http://en.wikipedia.org/wiki/Random_error
For more information on Random Error
Activity Instructions:
1. Divide students into groups of three.
2. Hand out one circle worksheet, rulers, markers and strings to each group.
3. Explain to the class that each group must measure the diameter and circumference of the five
different circles on the worksheets, using the string and ruler. The diameter can be measured using
just the ruler. To measure the circumference, students must place their string around each circle
and marking the place where the string meets. Then they can determine the circumference by
measure the end of the string to the marker. Then the recorder must record the group’s measurements
in the worksheet.
4. Students must calculate the ratio of circumference to diameter for each of their measurements and find the
average ratio.
5. Have them work together in groups to discuss the answers to the discussion questions at the bottom of the
work sheet.
6. Allow each group to write their results for the average ratio on the board and have them calculate the
average class ratio.
Discussion:
This activity demonstrates that Pi arises in student’s daily lives. Discuss with students the practical applications of
Pi. Also discuss the limitations of finding Pi in this way. Discuss the errors that might occur. Ask the class if any
of their measurements were far from Pi. This can lead into a discussion of errors and how they will be greater on
the smaller circle.