This is not a new lesson, but it is a rich one. Ask students what Pi is and they will tell you that it is “3.14” or “22/7”. You may even have a student who knows that it is irrational. But almost no one will say what it actually means. This activity will hopefully change that.

Materials:

• some kind of string or twine cut into pieces.
• a few rulers (one for each group works best)
• circular objects of different sizes (a quarter, a can, a Frisbee, a clock, a ball, a lid, etc.

Steps:

1. Start by asking students for the definition of diameter. Take the opportunity to build a definition that develops everyone’s conceptual understanding, their ability to express mathematical ideas in words, and convey the expectation that precision is important in expressing mathematical ideas. For example, if students say, “A line from one side of a circle to the other”, draw something like thisand push them to be more precise.
2. Here are a few definitions I’ve accepted from students:
   • The widest possible measurement across a circle
   • A line that divides a circle into two equal halves
   • A line that goes from one edge of a circle to another, passing through the center of the circle.
3. Ask students if anyone can explain what the word, “circumference” means. You are looking for something like, The distance around a circle.
4. Divide the class into groups of 2-3 students and give each group a circular object, some string ) and ask them to find the diameter and circumference of their objects.
5. You can create a table on the board (or on newsprint) with three columns: Name of Circle, Diameter and Circumference.
6. If you think your students might need it, ask them to talk about how they could measure the diameter and the circumference using the tools you’ve given them.
7. Walk around and engage students with the different problems that may arise from their measurements. They will ask you questions like, “Which side of the ruler should we use?” or “Do we measure the outside edge or the inside edge of the Frisbee?”. There are no right or wrong answers to those questions, but they are important questions to answer. Ask the students themselves to consider the benefits and challenges of each choice and make their own decision.
8. As students finish their measurements, ask them to repeat their measurements several times to see if they come up with the same results. If they get different results, let them decide what to do (estimate an average, actually calculate the average, use the mode, etc.). You can also give out additional circular objects, or have groups swap and compare measurements.
9. As every group finishes their measurements, ask them to add their data to the chart you created on the board. Make sure to ask them to include the unit of measurement they used with each number.
   • You’ll get something that looks like this:
     

     Object	Diameter	Circumference
     Clock	12 ½ inches	43 1/8 inches
     Quarter	1 inch	3 ¼ inches
     Coffee Tin	7 7/8 inches	24 inches
     Tupperware	12 inches	36 inches
     Plastic Lid	4 ½ inches	13 ¼ inches
     Large Plastic Lid	6 ¼ inches	24 inches (18 inches after repeating measurement)

10. After every group has a measurement for the diameter and circumference for at least one circular object, ask them to reflect on the measuring.
    • What challenges came up while you were making your measurements? Who else had similar challenges? How did you work through those challenges?
11. Next hand out the data collection chart and give them a few minutes to add the classes collective data onto their own chart.
12. Ask them, What do you notice? What patterns do you see? and give them a few minutes to look and write down some ideas on their tables.
13. Ask them to share their ideas. If it doesn’t come up, ask if anyone notices a relationship between the diameter and the circumference. Give them sometime to discuss this with the person sitting next to them.
14. If no one sees anything (unlikely), start going through the table, “I see the clock has a diameter of 12 inches and a circumference of 43 inches. I see the quarter has a diameter of 1 in and a circumference of about 3 and a quarter inches, etc.)
15. The goal is for a student to say:
    • Student: “The circumference is about 3 times than the diameter of each circle”
    • Teacher: “Exactly three times?”
    • Student: “The circumference is a little more than three times more than the diameter.”
16. They may also say:
    • “The diameter is about one-third of the circumference” or
    • “If you divide the circumference by 3, you get a number close to the diameter.”
17. Once an idea has been made, ask other students to restate it. Ask other students if they agree. Ask if it seems to work for all of the data on the table (Teacher Note: It will! The only exception that sometimes comes up is when students record the diameter and circumference in different units of measurement (i.e. inches and centimeters. If this happens, see if students can figure out why that data doesn’t fit the patterns.)
18. Ask students to make predictions:
    
    • If the diameter of a circle was 20 feet, about how large would the circumference be?
      • Multiply by 3. It will be about 60 feet. A little more than 60 feet.
    • If the circumference of a circle was about 150 inches, about how long would its diameter be?
      • Divide by 3. The diameter would be about 50 inches.
19. Teacher: There is a name for this special relationship between the circumference and diameter of any circle in the universe. Does anyone know what it is? It is called Pi.
    • Teacher: What do you know about Pi?
    • Student: It’s 3.14.
    • Teacher: What does that mean?
    • Teacher: It is an estimate of a more precise measurement than we are able to do with our rulers and twine. In our observations we said, “The circumference of a circle is about 3 times more than it’s circumference.” 3.14 is a little bit more than 3. A more precise way to describe that relationship is to say, “The circumference of a circle is about 3.14 times more than it’s circumference.”  We keep the about because even 3.14 is an estimate. 
20. Here’s a short video to help students reflect on what they’ve discovered. It’s a great chance to ask them what they wonder.

Bubbles!

There are so many lessons for discovering Pi, but here’s one I thought was interesting. If you are not afraid of things getting a little sticky, it’s a fun way to have students discover Pi through measuring bubbles (from Fawn Nguyen):

• Friday Bubbles
• Follow-up to Friday Bubbles

Depending on your goals for the class, you can connect this activity to rate of change, graphing, geometry, the scientific method, etc.

Let us know how you use it or if you have your own version of having students discover Pi.