Discovering the Magical Pi

Grade Levels: 6 - 8

INTRODUCTION
In this lesson plan, students will use data on the circumference and diameter of various objects to calculate pi. The exciting aspect of this lesson is 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: pi!

SUGGESTED TIME ALLOWANCE
80 minutes

OBJECTIVES
Students will:

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

MATERIALS

  • Rulers, preferably metric
  • White 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
  • 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 top, Frisbee, cake pan, coffee mug bottom, any lid to a glass jar (baby food, pasta sauce, etc.), a half-dollar or quarter. It is also very powerful to have some "natural" circular objects, such as a sand dollar, half of a well-wrapped melon or orange, or another piece of "sturdy" fruit
  • Calculators
  • Worksheets: Circular Measurements & The Mystery Ratio

PROCEDURE
This lesson is broken into two parts (each lasting approximately 40 minutes).

Part I

  1. Introduce key vocabulary:
  2. circumference = the distance around a circle
  3. diameter = the line segment that goes through the center of the circle and has endpoints on the edge of the circle
  4. ratio = a way of comparing two numbers – the ratio of a to b is a/b
  5. pi = the ratio of the circumference to the diameter of a circle; pi is approximately 3.1416 (Note: Do not introduce pi to students until after they have done the exploration.)
  6. In the room, make a few stations of three or four circular objects each. Put students in groups of two or three (try to have same number of groups as stations). Give each group a 30-cm string, a 60-cm string, and a ruler or yardstick.
  7. Give each student a Circular Measurements worksheet. Read the column headings with them and make sure they understand the vocabulary. Demonstrate circumference and diameter with an object.
  8. 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.
  9. Have kids go to their stations. Using the Circular Measurements 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 the Mystery Ratio.
  10. When finished, students should proceed to the next station to measure other objects.
Part II
  1. 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.
  2. Next, tell students to fill in the Mystery Ratio (circumference to diameter, or C/D) on the Mystery Ratio worksheet. They should enter that in the column heading.
  3. Using their calculators, students should now calculate C/D for each object. Have them take the ratio to the hundredths place.
  4. After they have finished their calculations, give each student a Mystery Ratio worksheet. The students answer the questions in their small groups. The "Aha!" moment is usually more potent in the small groups because more kids can come to it at different times.
  5. Do a quick wrap-up before class ends. Whether or not natural objects were used, it is nice to talk about pi in nature. How does mother nature know how to "grow" a circular object? Pi must be a pretty important number!

ASSESSMENT

  • Students should be able to define all terms.
  • Evaluate Circular Measurements worksheet for organization of data in the chart and calculations.
  • Assess Mystery Ratio worksheet to check answers to questions.
  • Students can write a letter to someone explaining the activity and what they discovered.

EXTENSION ACTIVITIES

  • http://www.joyofpi.com
    Lots of great pi links. Also includes lists of pi carried out to lots of digits. Kids can try memorizing digits using mnemonics and other methods.

STANDARDS CORRELATION
National Council of Teachers of Mathematics (standards for grades 6-8)
http://www.nctm.org

Student:
  • applies techniques and tools to accurately find measurements (measurement).
  • solves problems involving circumference and diameter (geometry).
  • generalizes from a pattern of observations made in particular cases and makes a conjecture (problem solving).
  • recognizes and applies geometric ideas to everyday life (connections to the world).

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