Let's Take Another Look at Pi Day
The main portion of what appears below was published in the March 2002 issue of Mathematics Teaching in the Middle School, pp. 374-375.
In our earnest desire to educate our students about the great importance and significance of the number "pi" by celebrating with special mathematical activities on March 14, we just may be teaching a concept that we had not intended, or that we are overlooking altogether. I am referring here to the fundamental concept of the "place value structure" of our decimal number system.
To begin with, pi is an irrational number that begins 3.141592653
and those are merely the first nine digits of a theoretically infinite string of digits with no apparent pattern or repeating cycle. So to bring things down to a manageable size, especially when one is not using a calculator with a pi key, we have traditionally rounded the value to the nearest hundredth: 3.14. From there, we have altered that number into a common style of writing dates - 3/14 - meaning 3rd month (March), 14th day. Hence, the big day for pi is March 14.
This all seems fine and logical until we look deeper into the matter. The great tendency these days for reading such "decimal" numbers as 3.14 is to say: "three-point-fourteen", or "three-point-one-four", whereas its true name is "three and fourteen hundredths." In the latter form its place value meaning is clearer, meaning we have 3 full units (which is the diameter of a circle, of course) with an additional 14/100 of another (diameter) unit.
On the other hand, writing the 3 and 14 as a date involves two different bases. The "3" is based on the 12 months of the year, and the "14" is based on the 31 days of the month of March. Interesting to note that while each year has 12 months, the number of days in a given month is not uniform (31, 30, 29, or 28). And as a means of measuring a quantity of time it is not even equivalent to such mixed base times as 4:15 p.m., indicating 4 full units of hours (base of 12) followed by an additional 15 units of minutes (base of 60). The latter case is consistent with other measurements in math such as a board that is 2 feet 5 inches long, or a new-born baby who tips the scale at 6 pounds 10 ounces. They all share the "addition" concept even though they have dual bases for their numbers. The date of 3/14 lacks this essential property if it is to be considered related to the common value of pi - 3.14.
Perhaps by now you are asking yourself: "What is the big concern here with this?" And I will respond by citing a simple story problem, and giving the way many students might react to it.
Randy can walk at a speed of 4 miles per hour. He needs
to walk a distance of 9 miles. How long will it take him
to do this?
John promptly picks up his calculator and begins pressing keys. First, the 9, then the division key, followed by the 4, and finishes with the equals key. He quickly looks at the display, and seeing 2.25, proudly raises his hand and announces, "Two hours, twenty-five minutes." Of course, we gently point out to John his misinterpretation of the fact that "twenty-five hundredths", or "25/100", is really "one-fourth of an hour", which is then "1/4 of 60 minutes", or just "15 minutes". So the correct response should be "2 hours, 15 minutes." Sheepishly, John says, "Oh yeah, I just forgot."
Has that ever happened to you in your class? I'm almost certain it has. I've observed it many times in my career. I feel it's largely a result of our inconsistent manner of reading "decimals" these days. In fact, it seems that the "N-point-M" style is winning the battle. Therefore, we need to devote a little time in our classes to point out this distinction. And what better time to do it than on Pi-Day, March 14.
I'm not really against celebrating about pi at all. In fact, I'd like to suggest some additional ways to observe it. Let's consider these alternatives:
- Once a month we could have Pi-Hour; the 3 means 3rd day of the month, and the 14 means the 14th hour (2 p.m.) of that day. The only drawback here is that many students don't have their math class scheduled at that hour, so they would miss out on all the fun.
- Once a day we could have a Pi-Minute, 3:14 p.m.; the 3 means the 3rd hour after 12 noon, and the 14 is the 14th minute of that hour. However, this is often after many schools have dismissed the students to go home. But perhaps the math staff could meet in the teachers' lounge and share a piece of pie and a cup of coffee. Unfortunately it would be necessary to eat it within 60 seconds, because soon it would be 3:15, and no longer be Pi-Minute. [By the way, 3:14 a.m. doesn't sound too promising for celebrating as most folk are sleeping at that time.]
- Finally, we could have Pi-Second, 3:14. Here the 3 means the 3rd minute after every hour, and the 14 is the 14th second of that minute. Now what could you do in one second? My suggestion is that the teacher could play the role of a choir conductor. As the big moment approaches, he/she raises a ruler/baton in the air and asks the students to stand beside their desks. At the proper moment by the teacher's watch, the ruler is lowered and everybody calls out in chorus, "I like pi!" It would appear to me that this type of celebration has the advantage of probably reaching almost every student.
Maybe we could even find other days to have Pi Day. If we interpret the additive aspect of the value 3.14 as 3 full months, then have the 0.14 pass to the following month (0.14 x 30 = 4.2, rounded to 4), we obtain April 4 as Pi Day. Or, let's convert pi to other bases. In base-5, it is 3.03232214303
. Rounding this to two places after the "point" yields 3.04. This tells us that, converting by the dual base method, March 4 is Pi Day (base-5). For those who love number trivia as I do, notice the 143 embedded in the sequence of digits. If we write this as "14/3", we have "14 of March" as it is commonly expressed here in Latin America, where I live. I can only surmise that may be the reason I don't hear any talk about celebrating Pi Day in our schools. But this alternate method of writing dates might suggest we observe Pi Day on July 22, because the reverse notation, 22/7, doubles as the famous fraction form of pi. (In fact, some places do just that, calling it Pi Approximation Day.)
Please don't interpret this little essay as advocating that we not celebrate Pi Day on March 14. Just that we show our students the mathematical liberty that we are taking when we convert 3.14 to 3/14, thus making them more aware of the essential difference in our place-value notation system and dual-base systems. Then maybe John won't make the error that he committed in the problem about Randy's walking time.
Footnote: If Pi Day this year (March 14, 2002) is written in digits like this - 31402 - then those digits in that order can be found embedded in pi beginning at the 219770th position.
The website for finding digit strings in pi is http://www.facade.com/legacy/amiinpi/
(5/(27-30)/01)
Note: I'm not certain about rounding in base-5. Is 3.032 properly rounded to 3.03 or 3.04? With the five base-5 digits {0, 1, 2, 3, 4}, the 2 is exactly in the middle. In base-10 rounding with digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, the 2 sets are equal in size. So {0, 1, 2, 3, 4} cause a truncation effect, while {5, 6, 7, 8, 9} bring an increase to the digit preceding them.
(5/31/01) In an email note from Ted Alper of Stanford University about rounding in base-5, he writes:
(B) rounding in base 5.... note that 1/2 =
0.22222222222222222222222222222222222222222222222222222.....
Consequently if you are rounding, you should look at ALL the following digits...
until you hit one that is not a 2.... if the first non-2 you hit is a 0 or 1,
round down; if the first non-2 you hit is a 3 or 4, round up.
another way to approach the problem:
add (1/2) the last digit-location to the number, then drop all trailing
"decimal" places.
so to round to the nearest 125th... if you have
3.21324
that rounds to 3.214
but
3.21321 rounds to 3.213
equivalently... add 1/250 = 0.000222222222222222222222222222222222222222
to the number, then drop all but the first three places after the
"decimal" point.
Additional information about Pi Day celebrations can be found at the Math Forum's T2T FAQ by clicking here.
For more discussion about Pi Days in different cultural or religious calendars, go to this site in the Math Forum:
http://mathforum.org/t2t/thread.taco?thread=3298
My initial contribution was:
From: Terry Trotter
To: Teacher2Teacher Public Discussion
Date: 13 Jan 01 21:16:30 -0500 (EST)
Subject: Pi day & Other Calendars
Here's another idea I propose... use calendars from other cultures and religions to find their Pi Days! Then the math task becomes finding the equivalent date in the Gregorian calendar currently in use.
For example, there are Jewish, Islamic, Chinese, and Bahá'í calendars, and probably more. The Jewish and Islamic are lunar ones, which brings in added factors for problem solving. Their Pi Days fall on different Gregorian dates over time. I'm not sure about the Chinese calendar, but surely internet research would be a good idea.
But I do know the Bahá'í one quite well. [I'm a Bahá'í, in fact.] You see, we have a solar calendar, so Gregorian dates always correspond 1-to-1 with Bahá'í dates. We have 19 months of 19 days each [4 intercalary days, 5 in leap years, tucked between the 18th and 19th months, so all comes out the same]. Our New Year's Day is March 21, the first day of spring [symbolizing a new beginning and it is related to a specific astronomical event, too: the spring equinox].
So here is the problem... What is the Gregorian date that corresponds to Pi Day in the Bahá'í calendar? It's fairly easy to do.
My 2nd contribution was:
From: Terry Trotter
To: Teacher2Teacher Public Discussion
Date: 15 Jan 01 12:10:49 -0500 (EST)
Subject: Re: Re: Pi day & Other Calendars
Thanks, Roya. Now we have another flavor (calendar) for pi day to add to our smorgasbord from my previous posting. Like any good restaurant presenting many kinds of pies to eat, this Pi Day project also deserves more than one calendar.
Roya also has found some nice websites containing valuable information related to this topic. She writes:
------------
This is some info I found about the Persian calendar:
http://www.projectpluto.com/calendar.htm#jalali
http://www.treasure-troves.com/astro/IranianCalendar.html
And this page has info about our new year:
http://www2.cy-net.net/~kreza/iran/norooz.html
Happy reading.
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Happy Extended Pi Day to everybody.
Terry
Update April 3, 2002
The following letter was received from its author, Lisa Fiorante. It was originally sent to the editors of MTMS.
I am writing in response to the very timely article, "Let's Take Another Look at Pi Day" by Terrel Trotter Jr. (March 2002). At the time, our school had recently completely many Pi activities. Mr. Trotter brings up many interesting ways in which we could extend the traditional celebrations of Pi. I think that elementary school students would first need a solid background and some understanding of the concept of Pi. Perhaps it would be beneficial to use some of the ideas mentioned in the article after first celebrating Pi in the usual manner.
I would like to share some of our activities celebrating Pi Day 2002. We decided to involve all of the students in the school (Pre-school to Grade 8) on some level with our celebration. We set out to break the world record of longest Pi Chain. We found this activity on Christopher Herte's website mentioned in the NCTM Bulletin. (mathwithmrherte.com) We used a different color for each digit and each grade was assigned one color and cut the strips to use in the chain. Our chain, which had 2400 links, was hung around our entire parking lot for all to see. We currently hold the Illinois record for longest chain. Algebra students went to each room earlier in the week and demonstrated the calculation of Pi using a circle and gave the students some background information on the meaning of Pi and how it is used. We also had a song contest where over 200 families submitted song lists with songs that had Pi related words (circle, world,
around, etc.) in their titles. (The top families received gift certificates for pies) Our winning family found over 1000 such songs. It was truly a great experience for the entire school and for our families. The Algebra students, who actually assembled and hung the chain, finished the celebration by eating "pie".
Thanks to Mr. Trotter's article, we have several new ways to celebrate Pi in the upcoming years. Imagine the fun in celebrating Pi in a different month, during a particular hour, or even a particular second. Perhaps the students could even determine their own new times to celebrate Pi.
Lisa Fiorante
St. Eugene School
Chicago, Illinois
