Math 106 - CooleyMath for Elementary Teachers II OCC

Classroom Activity #10 – Calculating the Mystery Number 

The number  is necessary for the computation of both the area and circumference of circles. Its value has been known for the past 4,000 years and has been approximated in both decimal and fractional form. In this activity students discover an approximation to  using simple materials and realize that no matter the size or nature of the circular objects measured, the ratio of the circumference to the diameter will come out the same - !

Learning Objectives

Students will:

  • measure the circumference and diameter of a variety of circular objects.
  • organize data in a table or chart.
  • calculate  - the ratio of circumference to diameter (C ÷ D).

 Materials

  • Four-Function Calculator
  • Plain white cotton string cut in two lengths – 30 cm (approximately 1 ft) and 60 cm (approximately 2 ft) Note: Length of string pieces may vary depending on the circular objects chosen
  • Rulers (preferably metric) and/or yard sticks (preferably with metric on one side) and/or tape measures (flexible and metric preferred)
  • A variety of circular objects that would allow for easy measurement of circumference and diameter, such as canned goods (soup, beans, fruit, etc.), oatmeal container, paint can, Frisbee, cake pan, plate, coffee mug bottom, any lid to a glass jar (baby food, pasta sauce, etc.), a half-dollar or quarter, or something not mentioned that is circular in nature.

Instructional Plan

  1. In the room, make a few stations of two or three circular objects each. Put students in groups of two. Give each group a 30-cm string, a 60-cm string, and a ruler or yardstick.
  1. Distribute the Calculating the Mystery Number Worksheet to each student. DO NOT pass out the History of  Sheet. Do not mention the number  until the very end! Read the column headings with them and make sure they understand the vocabulary.
  1. Explain to students that they are to work with their group to measure the circumference and diameter of the circular objects at each station. This is a good time to have a student come up to the front of the class with you and demonstrate how to do this with the string. One of the challenges for students is estimating the center of the circle when they measure diameter. Demonstrate accurate and less accurate ways of measuring, and ask students which are best.
  1. Now, have students go to their stations. Using the Calculating the Mystery Number Worksheet, they should enter the object name and then proceed with measurements. (Time permitting, they can also take two readings for each measurement to test accuracy.) All students should record the object name, circumference, and diameter before proceeding to the next object. They are NOT to fill in any of the Mystery Numbers in the final column of the table.
  1. When finished with their initial stations, students should proceed to the next station to measure other objects.
  1. When all objects have been measured, ask students to sit with their groups and look over their data. This could be a time that you pick one or two objects and ask for data from different groups. It helps students understand the concepts when they see the variation in measurement and discuss why this happens.
  1. Next, tell students to look at the final column. Tell them they should write in the column heading, the following:
  1. Using their calculators, students should now calculate C/D for each object. Have them take the ratio to the nearest hundred-thousandths place (5 places after the decimal point). Here it would be a good time for the group to not say anything to anyone or even each other as the calculations are being made.
  1. After they have finished their calculations, ask students if they have figured out what the mystery number actually is without shouting out the answer. As students are trying to figure this out, several "Aha!" moments should start occurring.
  1. Do a quick wrap-up before class ends. Distribute the History of  Sheet. Walkthrough the information and history of  to the students. Whether or not natural objects were used, it is nice to talk about  in nature. How does mother-nature know how to "grow" a circular object?  must be a pretty important number!

Note: In text, when it is not possible to typeset the number “” it is acceptable to use the phrase “pi”.

 Extensions

Pi exists in nature. List at least five objects in the natural world that have some sort of circular

shape to them. Someone brings a huge pizza into the room and tells you that the diameter of the

pizza is 4 feet. Use your knowledge of pi to estimate what the circumference of the pizza is. (You

do not need a calculator. Just estimate.) Estimate: Would you rather have a pizza that has a

circumference of 36 inches or a diameter of 36 inches? Explain your answer.

NCTM Standards and Expectations

Measurement: Apply appropriate techniques, tools, and formulas to determine measurements – Grades 6-8

1. Select and apply techniques and tools to accurately find length, area, volume, and angle

measures to appropriate levels of precision.

 References

This lesson was created by TopCatMath.com.

History of  SheetNAME ______

Approximately 4,000 years ago people were first beginning to understand the meaning of  (read as “pie’). The number  is officially defined as being the ratio of the circumference to the diameter of a circle, which is the same for all circles.

We now know that  is an irrational number (non-fractional) that never terminates or repeats in decimal form where we cannot get at an exact numerical representation for . Thus the symbol “” is used to represent this non-terminating, non-repeating decimal expansion.

Ancient geometers knew that the value for  was slightly more than 3 and various approximations to  have been used throughout history.

Time
Frame
Used / Geographic
Location / Fractional
Approximation / Decimal Approximation
200 BC / Greece / / 3.142857143…
(overestimate)
800 BC / India / / 3.138888889…
(underestimate)
1900 BC / Babylonia / / 3.125
(underestimate)
1900 BC / Egypt / / 3.160493827…
(overestimate)
Arranged from closest approximation to  on top and worst approximation to  on the bottom.

The first 50 digits of  are:

3.14159265358979323846264338327950288419716939937510…

Today, most classroom applications involving  use either   3.14 ( rounded to the nearest hundredth) or   . For students in higher grades possessing a scientific or graphing calculator the  key on the calculator is used for more accurate results.

You can easily approximate  yourself using a string, ruler, meter/yard stick, etc. and any circular object such as a coffee can. Here’s how to do it!

Procedure for Approximating Pi

  1. Measure the circumference by wrapping the string around an object. Write down the length of the string.
  2. Then, measure the diameter by measuring distance straight across the object. Write down the diameter.
  3. Last, divide the circumference by the diameter using your calculator. Write out this ratio out to five decimal places.

Calculating the Mystery Number Activity SheetNAME ______

Fill in the entries of the first 3 columns of the table.

  • You may measure the diameter and circumference in whatever units are convenient for you to use, just use the same units for both the diameter and circumference.
  • The best results can be obtained by measuring in the metric system because we can measure very accurately in mm (which are small) with out the need to approximate fractional inches as in the American System.

DO NOT FILL IN THE LAST COLUMN UNTIL YOU ARE TOLD TO DO SO!

Object / Circumference
(indicate units used) / Diameter
(indicate units used) / Calculation of the Mystery Number
(Use your calculator to divide then record the first 5 decimal digits)

After calculation the 10 entries on the right, can you draw any conclusions about the mystery number?