Celebrating Pi The Rational Way, Teaching Kids About Irrational Numbers
Dale LaFrenz, one of the founders of REAL, is also my esteemed Co-Blogmeister. You can see us at the bottom of this blog page. Coincidentally, we were both high school math teachers in our early days. Think back to the ’60′s, or over 50 years ago. Why would you ever want to be a math teacher, many ask.
One of the major reasons I chose mathematics as my teaching major was that I loved seeing the lights come on in the eyes of the students when they understood. As you shall soon learn, so did Dale.
Our RE@L Blog today is a short and true story about my colleague, Dale LaFrenz and the common-sense he brought to his teaching way back when the only technology we had in a math classroom was colored chalk and the textbook. Here’s a photo of Dale from those early teaching days.
Technology as we know it today, hadn’t arrived yet. What we knew about technology back then is now found in Wikipedia’s definition:
“Technology (from Greek τέχνη, techne, “art, skill, cunning of hand”; and -λογία) is the collection of techniques, methods or processes used in the production of goods or services or in the accomplishment of objectives, such as scientific investigation.”
Yes, scientific investigation! The root of the STEM movement as we know it today. STEM stands for Science, Technology, Enigneering, Mathematics. This story about Dale and his classroom happened long before STEM, but it shows that even way back then, good math and science teachers understood scientific investigations and often used them in their teaching. So, here’s the “rest of the story”:
Dale was teaching a General Math course to a group of students who, like most kids, had a difficult time with mathematics, in truth, pretty much with anything remotely mathematical. The class had covered the area of squares, rectangles, even right triangles, without too much difficulty, but then came the chapter on Circles, the perimeter (circumference) and the area thereof. A very different scenario to be sure.
The teacher usually leads to a discussion of Pi (or π), as we know it from the Greek letter P for perimeter. But that gets to be a little heady stuff for many kids in a general math course. Sure, you could just tell the kids it’s an irrational number approximated by 3.14 and forget about the why and the wherefore. Not Dale. He thought long and hard: “How can I help these kids solve this problem and understand what π is.”
Good teachers often come up with good answers. Dale did. He sent a group of the kids to a nearby used tire dealer and they borrowed a half dozen different-sized tires. They then proceeded to measure the outside of the different tires with a tape measure, and then next measured all the way across the the center of the tire, tread to tread.
Then they went to the blackboard and divided those two numbers, the perimeter by the diameter, for each of the six different-sized tires.
Lo and behold, and what to their wondering eyes should appear was that the answer to all their divisions came out to be the same no matter how big or how small the the tire was. In other words, about 3 and 1/8, as closely as they could measure. They argued back and forth about this phenomeon and eventually came to the conclusion that for any tire or circle, that ratio of distance around, (perimeter) divided by distance across (diameter), was always the same, and it was known as π. Or as it’s written a bit more accurately these days: 3.14159265359. What a discovery for young and inquiring minds!
No, they may not have known what an irrational number was, but they did conclude that, for all circles, perimeters could be found by multiplying π times the diameter, and the area could be found by π times half the diameter (radius) squared. Wow!
It was a “Eureka!” moment for the students, and an early example of Project-Based Learning way before they had that name. They may not have known an irrational number from a rational algorithm, but they did know how to find how many square inches there were in their Apple Pi.
And that’s the story of how an irrational number was made rational by a very rational teacher.
What Dale can’t recall is if they celebrated their finding with that Apple Pi for a reward.
Knowing Dale as I do, I’m betting they did! After all , it was a just dessert….
As always your comments and questions are welcome at: firstname.lastname@example.org
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