Hands-on with pi
Free lesson idea from HomeschoolMath.net
Pi seems to be such a simple thing - it's just two letters, and after all, you just peek into any math book to find out its value. Everybody knows about it, "Pi? Ah, yes, they taught us that in school, so it's something pretty simple."
BUT have you ever given pi some more thought? And, if you let your students give pi some more thought, it might just make math a little more interesting and 'lively' for them, too.
Think thought about it - how come the ratio of any circle's circumference to its diameter can be an irrational number? After all, this is not something that would naturally come to mind if you've never heard of irrationals before... If you start measuring lots of circles and calculating ratios, yes, you would soon notice that in every circle the ratio is the same. You would notice it's a little over 3. But would you just GUESS on your own that this ratio isn't a rational number?
If you have the time (and homeschoolers might), give your student a project to find what is the ratio of a circle's circumference to its diameter by measuring. Tell them they can play an ancient mathematician now and let them work at it - don't even tell them it's called pi and hide all math books for a while...
Some kids might just love such a hands-on activity.
See if they get so far as to realize it's the same constant in all circles. And congrats if they get 3.14 or even get it between 3.1 and 3.2.
During all the measuring your student is not likely to think that this number would be irrational - he probably hasn't even heard the word before!
And mathematicians didn't find that out real soon either. Ancient cultures were aware of the fact that the ratio of circumference to diameter was a constant. The formulas they used for the area of circle indicate the equivalent value for pi as 377/120 or 3 1/8 or 256/81.
Archimedes found out that this ratio was between 223/71 and 22/7. Or, to be exact, he was finding an approximation for the area of the circle via this simple method: he drew a regular polygon inside the circle and outside the circle, and calculated the areas of the polygons-- obviously the area of the circle would have to be between those two numbers.
Archimedes wasn't afraid of calculations and used a polygon with 96 sides...! You might try giving 8th graders this as a project using just hexagons or octagons.
It makes for a nice exercise in geometry, plus it should be interesting since it ties in with history. Here's a nice slideshow about Archimedes' method.
Over time, mathematicians were able to calculate Pi more accurately - find better rational approximations to it. I'm sure they started to guess that maybe Pi is irrational - in fact, my guess is they suspected it to be so for a long time.
Proof of pi's irrationality came in 1768 by Johann Lambert. After that, mathematicians have been sure of that - but the quest for finding pi's decimals still continues.
Guess how many decimals they have found thus far?
1,241,100,000,000 decimal digits - it's more than 1 trillion digits!
This was done in 2002 by Yasumasa Kanada and 9 other individuals in Tokyo.
If you'd like to start memorizing it, here's a link to Pi trainer :)