1E:
||WiEli
;j.
X
....r' i E -....
:.'iE.,'Si
" '.........I
................
..
g...
i; the dtiaersenaion at it,i ws fsinae
tha i
Teach"" ::0000000"Sthem
it00000000as straightfowr:000-:'""::
thestate0
at that
of_tesearch
die into he.number
and teachl t&m the formulas for
i or00i23/7,
=
0i 3.14
$
and
findinlg
ig 00the-talea
circumfe-ence of a circ le, i
by their accomlplishmenlts. I also rh'*cov red that i
turn-s up in mlany situationls that have nothing to dit
wit.h circles, a fact that was ll<)t maentionled in any of:
my undergrad.uate mnathematics courses, andi thatl
people have beenl aware oft t and itls rclation to CiPm
iif;poured in th}e informnatiolt tn:t a lot of it spilled
(t andl was ilost. Here :are two exaplr@es of msy
an0
"veBll 20sil came ovut at :the other end of the
"wi4da;
t
MyostXinfIht reslinlg dmsX
likesteaches~
fractions ~ eles
nbbody
3.14 since
| spout: ()
~ for
~ severalmilidemlia.
~~~~~~~~~........
MARGARE
I'TtNT
htc-nt@altamoMschoatorg,
~ istudents:
~ ~ fouwl;
~ ..........
~covefliy~ was ~t:hat tmy
fifthand
sem&th-gn
at
thedoAmathematics
tamont
the:txr(n&rpipiformulas
that :you:~
are
twno
t anyway" an4 (@) Thiere
~
~~~~........
~
~
~
~
~
School,
Iiinnfllgharn,
enloy.
AL
expior-~
35213.::$h
nlings of manithemlatics fasc.inating gtoo O. tlIcei-wvi
fi have to usg::nrr ard nd. Soprietimles you use one
h are
rclhmais
hehi trs othkg
gwystoiicud
..........
understood wvhere the iconcepts efasne fronm ;leurn.other. I::fn you
anj*d somletime
ifi formula
............
c....cI--rgthavc sinice ibePn:
ingXthe essenti:alideas was caski
:
i:: lucky:you will gueRss right.";i;:
tlid that more than0onle class was¾wazeso)me!':iS
solmy
understanding-unlsatisfactoty,
:;ifounld:this:
..........
452ii;
MATHEMATICS
TICACKINO IN VHF MIDDLE
:t.-g ,:
:
i
flrt
ka recourse::was :to: try to dritt-the knowle dge into
00
_,
iul
trw studenWsh:arde r. Thre rtilt was frustration for
::_
them and ine. Thiey hadal.aready had lots of e.xperi111c
:
00 enice inl mlleHIOriziig.i atrd the:niore no<nsenlsical the
_
inft0
1onnataion::.was, thfe: more difficulty the studenots
more__
was
:j:: :000 hadi getting-<it rirght. Bec.atisf:nothing
f"k
; ;00; SSt gettingniiwhere.0-t
I knew-th:it::the solution to ithis problem was not
fi::::: tll0:
to teach1 sthudenits t-o memorige be.tter. I needed to
:::
illi;
teiols for: understan~ding the concep)ts in-__:_
igf give thele
i; stead. jAt abaout that timue ar: article called "Thce}
a pt" by Richardd 3 Preston((1992), ".
M0
Vountainls
;t
;000 peered in Th> Newp Vorker. TIc article described
itstattetne o anprog9am of reading about
ll'.ACHIts
hIstory t'Ha has not s
mathematic2 M:,4TIIFMAndCS
it,0and
0_m
0B:-g
._
_
.$/
"'
/4
_
i
Activities for Learning aboiut 7t
f
OVEIR THE YEARS, I HAVE INVOLVED MYS- IJ I)ENT
in the Inumber- it: in a variety of activities. The first
involves a carpenter's 25-foot steel tape measure
arid a classroom set of 6Winch tape measures. We
make a tour of the school, measuring circles as we
go. Having discovered that the circumiference and
diameter are the onily two parts of tie circle that we
need to measure, we have devised a char`t for thc
class scribe to fill in as we go. On our return to the
classroomi, we measure a variety of circles that I
have accumulated there, including oatmeal, salt,
and baking powder packages; soime borrowed pots
from the kitchen; and so on WVith the two left-hatid
columns on our charts for circumnference and dimneter completed, everyone sets to work with calculators to fill in the riglht two columlis: the ratio of cir-
i4
'I
I
cumferenice to diameter as a fraction for each circle
and the decimal equivalent of the fraction. Usually a
student conmnents that he or she keeps getting almIost the same number for every circle. Someone
else agrees, observing that the number always
se.ems to be a lUttle mnore than d3.Wlhen everyone
concurs, I give that niumber a name: it.
Then I ask my students to consider the "ball"
that they carty on top of their shotilders, that is,
their heads. People buy hats in different sizes. If
students go home and ask their parents or grandparents what hat size they wear, they will say
soimething like 6 3/4, 7, or 7 1/2 . Where do those
numbers come fromn? We then use a tape mLeasure
to record samrple circumferences of heads in
inches. When we divide those numbers by it, we
get suchI ntumbers as 6 3/4 or 7. Even milliners use
the numiiiiber it!
The History of
t
iIOW LONG HAVEl WE KNOWN ABOUT ic? TIIAI
question takes us back in history to find that the ancient Babylonians and Egyptians and many other
early civilizations had a number that they used for
it. They recognized that the ratio of the circumference anid diameter of a circle was constant and that
it allowed them to figuire out the circumierence of a
circle if they knew its diameter. My classes follow
the development of it, from the ancient Babyloiriains, who used 3 1/8, or 3.125, in about 2000 B.C.; to
the early Greeks, who used 410, or about 3.162; to
Archimedes (287-212 B.C.), who found that i was
between 3 10/71 and 3 1/.7.
Then I show students a selection from 1 Kinigs
7:23 in the Old Testamient: "And he mLade a molten
sea, ten cubits from the one brim to the other: it
was round all about, arid its height was five cubits.
And a line of thirly cuibits did compass it round
about." 'IlAe calculation in the quotatioi makes it
about 3. -The Old Testamient measurements are not
as accurate as ours, but people at the timie did inot
have steel measturing tapes! In fact, when I Kings
was written, sometime before 500 B.C., a miore precise value for it was generally known. Nehemiah
was asked about this discrepancy in about 150 A.D.,
and he said that it was the result of the difference
between the inside measure and the outside of the
vessel, since clay is thick.
In Chiina in about 500 _un., 7t was known to be ap-
proximately 3.1415929, and in Persia in 1424, it was
known correctly to sixteen decimal places. 'The
Dutchmuan Ludo]ph van Ceutlen found n correctly to
twelnty decimal places in 1596, for that d3iscovery,
the number ni is often called the Ltudolphian constant in Europe. Since then, mathematicians have
foundi millions and billions m3ore digits of the decimal expansion of nt, more than anyone couli possibiv find useful.
Student Interest in ir
WHIEN WET ALK ABOUT' THIS TOPIC, USUALLY AT
least one student wants to memorize some of the
digits of it. Almost always, another student voltunteers informationt about someone who has memorized r o.t: to 100 or 500 or 1000 digits to the right
of the decimal point. My students are impressed by
suchl a feat, and some of them start memorizing
ighlt then, although I am quiick to point out that
such an exercise serves no useful purpose. Every
calculator has a value of ri that is good enougyh for
any computation that they will ever do. I mnention,
VOL 6DNO. P
APRIL. 2001
453
however, that a couple of years
ago I funfld a Web site fthat had
ftnaemon<lic de.vice>s fo,r remembiler-
ingiit . i includes the classic "See,
7 l -0:000~ lihave 0a num}ber. .:. i." By: simply
__
coutgthe letters in each word
'Of this phrase and utsing the
comma as the decimal point, stu_ den}ts can reermber 3.1416,
which is ri rounded to the ten-
__
*
that have nothing at aMl to do with circles. In the seventeenth arid eighteenth centuries, mathematicians
began experimenting seriously with infinite series.
'llie following formulas for the infinite series of
Wallis, Leibniz, and Euler are intelligible to( middle
school students:
X
2
2'2-4'4'6'6'8'8'...
1'3'9'5'5'7'7;9
(John Wallis in 1655)
thousandths place.
OOne question that my students
ask about.tis "How do they find
l those digits?" After all, we tried
pretty hlard to measure precisely,
l biutthie be-t we could do was ti
find ithat tis probably between
l;and 3.2.A.rchimedes did much
_ better thani that, and he did not
have a s~teel measuring tape.
'Their question leads, first, to an
explanation of Archimedes'
r- 4 ( 1K 3 .5
7
+
9
11
IS
a constant,
used
su d
itit iis
method, which leads to the limit
as one of the foundationis of calcuhlus: the exhaustion of the circle.
My students t ry to figure the area
of a circle by getting closer to the
to check area using rectangles that become progressively narrower
computer 2see fig. I) or by trying to catch
chkps
t
the area between two polygons
(see
§ fig. 2). By the sixteenth and
seventeenth centuries, people
were rearranging the sectors of
two circles to form a rectangle that approximated
twice the area of a circle (see fig. 3). Convincinig
studenits that precision in measurement has limits
is not difficul. We could not use either of these
methods to get a truly precise value for 7t.
Thie answer to the students' question is that the
number 7r also tums tip in number theory-in ways
it2
6
1
1
292
1
32
1
42+
~-"
-J-LL_
The greater the number of rectangies, the less area is
wasted inside the circle and the closer the enclosed area
comes to the actual area of the} circle.
Fig. I Archimedes' method of exhaustin
454
MAr'irEMATlCS TEACIIIN(
IN THE Ml)lI)DE Sl!1001I
1
-
(Euler in 1736)
I let my students play with these formulas, using
calculators long enough to discover that the formulas do lead to i. I also mention tLhat one of these
formulas for i is used to check the finctiotning of
computer chips, because iTtruly is a constant. In
1997, a pentiumrl chip was recalled because it failed
to find it. One class several years ago programmed
my classroomn Apple 11 computer to follow one of
these foriiulas for a weekend. When we returned
on Monday, we found that it is just a littlc mnore
than 3.14.
Eventually, in every class, someone asks why we
call this number it. In my introduction, I have already told them that it is the letter P in tle Greek alphabet. The obvious auestion is why P? Students
also wonder whether Lhe ancient Greeks were the
first to use that symbol for 3.14158978.
The answer to the first question is purely conjecture: Periin Greek mieans arouind, and ic was first derived
fromt the distance aroumd a circle. However, the
Greeks did not use the symbol a tor the number.
William Jones, an Englishman, was probably the
first to use it in 1675.11te symbol gradually becamne
accepted only a little more than 300 years ago.
"Proving" the Fonnulas
I-lOW ARE MY STI)DENrS SUPPOSED
'---'
*
(Leibniz in 1673)
'3
BecauSe
-
-t
[) RElMEMBER
which forrlmula, itr2 or nui,to use whien? Once more,
we embark on a discovery process. tUsing a comipass, we construct circles of radius 2 cl, 4 cm,
6 ct, andd 8 cm1. With the correct formulas in hand,
we find the circumference aind area in terms of it
for all four circles and combine the results in a
chart (see fig. 4). As a group, we conclude that
when we start with a radius of 2, the circumference
and the area are the same riuniber. Wheni we dou-
ble the radius to 4. we doub* the circumference -,
_
but inultiply the areat by 4, not by 2. Wlen wo mulL
tiply the original radius of 2. b 3, we' Iltiply the
circumference by 3 but multiply the area by 9/
When we nultiply the original radius of 2 by 4, we
multiply the circumference bv 4 but multiply the
area by 16hMost classes first notice thiat the area
grows much fastr thadi the tirctnfmerenje After
all, we started out witlh.aradius, of 2, a.d the cir_ \
I..
cutiferernce and areawere fthe samne numnber, 4i_.
By the time we miultiplied the radius by4, thWearc
As the number of sides: af the inscribed and circumscribed
had practkaily ex4liidPd
i
polygon grows, the closer tle two areas becomne the area of
With s6omie classes, I do not take this lesson any
the circle is caught between the areas of the two polygons.
further, but other elassesexplore theifadt that. when
"onwe mnultiply the radius by 2, weo multiply the area by
fg.2 The area of the circle is trapped between the areas of the two powgons
22. wheIlewe multiply the radit by 3 w6e multiply
the ana by3; and wheni we multiply the fradius byi
_
_
4, wemrultiply tihearea by 4tF0or someglzroups, aLn
, T_
interestilg extenasionis t jexplore tie resultsowhen
a
we multiply the radius by a fraction, suchli as 2/3 .
_
Students are oftein pleasedl to discover that the.area
Is multiplied by 0@/8)1or 4A9 f
We have used the formilais to get these results,
but can we believe thoe? .After all, ty studetits are
taking those formulas on faith. They will niot be able
to prove the formulas untilt they take calculus,
wlichsl is a few years in the future. Our next step is
to make the results look reasonable. My students
/ \
radius
take out their compasses again and try to fit circles
of radius 2 inside the bigger circles The students
\
,
,
- <
_-J
find that they canniot fit too many whole circles in_
__
__
side the bigger circles, but with a little imagination,
they begin to be convinced (see fig 5).
Circumference = 2im (radius)
Next we consider thle prices that pizza restauTo perform this investigation, cut two congruent circles into
rants charge for pizzas of different sizes. When
sectors, and fit the sectors together with one circle's cutoujts
Pizza Hlut doubles the price, does it dcouble tlhe raon top of the other, alternating sectors as you go. Then find
dius? No, just an extra inch or two in radius may
the area of that figure, whose area you can find by multiplydouble the price. Is the restaurant taking unfair ading length times width: r * 2itr - 2itr 2. Note that as the, size
vantage of us? No, because the area, or the amotut
of the sectors gets smaller and sinaller, the figure becomes
of pizza, grows fiaster titan the i-adius; therefore, it
virtually a rectangle. Because you started with two circles, digrows faster than the circumference.
vide by 2 to get the foirmula for finding the area of a circle:
Once m}y students believe that the area grows
A -Itr2
faster than the circumference, we start thinking
_-_
______
_____
-_
about, the formulas. When rtd tells IIs to tiultiply the
Fig. 3 Rearranging the wedges of two cirdes to form a rectangie that
diameter by Ft, we watch the result of the formula
appreximates twice the area of a drcle
grow at the same rate; that is, whern we double the
(liameter, the result of the formula doubles. 'he forRAIITAS
CIRtJMFERENCE
AREA
mula that asks us to multiply the radius timies itself,
_
4
4r. = 4ir
nr, behaves differently. 'Tie bigger the radius is,
2
2
222 *
the faster the result grows. We have already discov4-=-___
2 *4
2
ered that the area grows faster than the circumfer6 =3 * 2
12 = 3 *-39
* 4i
7t
ence; therefore, the formula for area must be the
=4 *2
l = 4 * 4i
16w
64 =16 * 4c - 42 *4
one that involves multiplying the radius times itseltf,
_
__
nr2. Because circumference grows slowly, we can
Fig. 4 Aconmpadison of the circumference and area of a circle as the radius
first find the diameter by doubling the radius, then
is multiplied by 2, 3, and 4. The area becomes larger at a faster rate while
(Cantinuedon page 457)
the cirumference grows at the same rate as the radius.
_
._
.
.
[t
.
t4
___
VOL. 6, NO. S
APRIL 2001
455
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lives. But gradilng 35 assignments, niglht
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(Continuedfrom page 45.5,)
multiply the d'iamneter by m. With that understanding, students can see that area rtr2 and that circumference = itd. Any time they forget, they can
think the formulas through again quickly. Approachinlg this idea firom another angle, students rememuber that the way we foutnd.rfromn ou. circles.
was by tmeasuring the circumference and the diarn- .
eter of tle circles. Ifctrct-iiklre: ce
(
circuunli&xence
\,/
/
diameter
then circumference IC x diameter. We certainly
did not measure the area of the circle to find our ap:
proximation of :t.
'The number t is a constant, as opposed to a variable, such as x or y. fin othier words,
tuiev.¾.
changes. I return to this comparison between constants and variables later in th e year when we are
dealing with variables. Some of my studenl s think
.n
Fi. S Acircle witlhalfthe radus Eothe largecrcle has.one-fo.ilh*e
area oftIheiarger circle..
that c is infiniitely large because it catn be written
using so mnany digits to the right of the decimal
Point. They have to be reminded that it is really not.
very big at al, oanly slightly mnore than 3, Iti class,
we use a variety of approximiations for i, ranging
from 'lust plain 3"; to 3.1, 3.14, andl 3.14159, to that
wonderful fraction. 22/7.
A liuge tunie ult of wonderful. readily accessible iafotrnation oni a
wiile variet fmnathemadtial topics.
Preston, RiThari. 'lte Moun}tains of P."'.The New Yorker
6.8 2 March 1992): 36.
Along artikle uescrTbing the statusofresearch into the value of als of
1992 A)
Conclusin 0 : IN A SENS;E, it IS A M{AGIC:AL NUMBER, ALD000.SS
lOWing Us tO deal with circles inl a way tilati000 make.s .seemingy difficultproblems: fairly: 0
easy. The onlIy re:.4iffulty iisremeinbering..
how to use it so thatLit is a reliable:he.lp. By::. :- : li
the timle that we ave fini:s-hed the ecourse,
it. New
Yolk:andf
A History
St
mo.st of mty studlentsl3eckmann,
can oftfind
;dieI-Nt.
area
> -tj-i
circumference of a crcle c.onsistently be-................................
cause they have leared :tO tink about cir-
3
A rather nirthosloxbhistoy19')7~~~~~~~~~~~~~~~~~~~~~..
of tutietmatics with sonic
F§0
'.
strong opintiont boutl the rolexe offmillsueles
vr ioi0. cultures but fa-.......
-matindg information about the Shistor af threm inu,e, a.
i
1
:
Blatnefr, David. The Jy of Pi. New York: Walke 0:;tl0 0;:: Sff+w1
& Co., 19.97. 0;0z0w: isE0..fnlFd. -iRe-:f-|::
Alttlevolue wvith a great deai of information presented in :n :a:
Glioberg, Jan. Matheatitcs froel tia Bir.th ofe
Numbers. New York: W. W. Nor.on.
Co,............................-:
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ifo
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fftbt.
ta
-
tl
tn
RC m r¼
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~~~~~~~~~~~~~~~~~.............
-............i.. .
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s
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..........................
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makfrmtesemyn
b
stde
ffitseoyraing. uprbermspoe onri
3t WX ld itsin8oonydlgijsO:
Annouaiod
Biblioaphyteaytehae tefrtonl
e n,llo
digficty s o, a.treatmet.t..t.................to
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...........
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Vli,NO; 8 0-APRI ''il
-:-:-:-.:-:a
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.
.
457
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;00t
COPYRIGHT INFORMATION
TITLE: Circles and the number p
SOURCE: Mathematics Teaching in the Middle School 6 no8 Ap
2001
WN: 0109105891004
The magazine publisher is the copyright holder of this article and it
is reproduced with permission. Further reproduction of this article in
violation of the copyright is prohibited.
Copyright 1982-2001 The H.W. Wilson Company.
All rights reserved.