Our esteemed House of Representatives passed a resolution earlier this week, designating today - March 14th - National Pi Day. This is an American-centric 'holiday,' as we colloquially denote our dates in MM/DD format, hence March 14th is 3.14, the first three digits of Pi. Most other countries start with the least significant date part and work their way to the most significant - DD/MM/YY - which would mean the International Pi Day should fall on the 3rd day of the 14th month (hrm) or the 31st day of the 4th month, i.e., April 31st (hrm, again). Of course, God-fearing computer scientists denote dates from the most significant to the least, always using four digits for the year - YYYY/MM/DD - which would place Computer Scientist Pi Day on May 9th, 3141 if you were being flexible on parsing the month and day and allowing either one or two characters.
As we all learned in elementary school, Pi is the ratio between the area of a circle and its radius, squared, or the ratio between the circle's circumference and its diameter, and approximates 3.14159. One of the shortcomings of mathematical education in our schools is that math is initially introduced in a way that suggests that its rules and results are handed down by some oracle. Furthermore, children are taught that learning math involves mindless memorization. I think this turns a lot of kids against math; they end up thinking that it's some stuffy, boring pursuit that is of little interest. But there are interesting ways to present math, and ways in which the child discovers first hand why certain things are the way they are. And explaining why Pi approximates 3.14159 is a great example of a more involved and interactive form of math education.
My daughter is 6 months old, but one day I plan on sharing the following math discovery with her. I don't know when this would be age appropriate. The first part - computing Pi as a ratio of the circumference and diameter - only requires understanding of division, and I imagine it could be introduced around age seven or eight. The second 'experiment' involves areas and necessitates understanding exponentiation and square roots, and would probably have to wait until age 10, or so.
Start by drawing some circles (with a compass) and talking about the components of the circle - the circumference, the area, the radius, the diameter. It's pretty clear that there's some relationship between these components. A bigger radius means a larger area and a longer circumference. Next, get out some string and measure some of the circles' circumferences. It becomes clear that a circle with twice the radius has twice the circumference. But given just the radius, can one arrive at the circumference? There is some missing number that we need to arrive at this, and this can be approximated empirically: using the string, measure out the circumference for a circle and then hold the string up to a ruler. Divide this measurement by the diameter and you should arrive at a value that approximates Pi. Using this newly discovered number, tackle the problem the other way around - starting from a known diameter, what should the circumference be? Compute it, then measure it with the string and see how the values align.
Can we devise some ratio between the radius and the area? When in doubt, get out the graph paper! Start by deciding on a radius and then determine what points on the graph paper are bounded by a circle with said radius. I'm not sure the best way to explain this to the pupil in a way that isn't along the lines of, "This is the way it is," but the square at coordinates (x,y) on the graph paper is bounded by the circle if the square root of x2 + y2 is less than or equal to the radius. Using that knowledge, visit each square on the graph paper and apply the formula. If the square root is indeed less than x^{2} + y^{2} then shade in the square, otherwise leave it empty. After enumerating the squares, count up the shaded ones and divide that total by the radius squared. And, lo!, the result is eerily similar to the number we were getting when dividing the circumference by the diameter. Repeat the graph paper process a couple times with larger radiuses. As the radius gets larger, the area is computed more accurately as the edges of the circle are less jagged. Consequently, the ratio of the area and the radius gets more and more refined and close to the true value of Pi.
And if my daughter enjoys computers and likes spending time giving them instructions (fingers crossed), the final part of the lesson would be to have the computer do the grunt work for us. Why spend all afternoon shading in graph paper squares when a computer can shade in millions of squares for us in seconds? Here's a snippet of C# that approximates Pi.
const int r = 10;
int A = 0;
for (int x = -r; x < r; x++)
for (int y = -r; y < r; y++)
if (Math.Sqrt(x * x + y * y) <= r)
A++;
Console.WriteLine("Enjoy a slice of {0}", Convert.ToDouble(A) / Convert.ToDouble(r * r));
You can adjust the value of r to see how the approximation becomes more refined. For example, when r = 10 the approximation is 3.15. With r = 100 it's 3.1415. Of course, too large of a radius results in an overflow, which is a great segue into a discussion on how computers represent numbers!