Wed Dec 11 18:57:09 1996
Newsgroups: alt.algebra.help
Subject: Talking about fractions
Date: Wed, 11 Dec 1996 10:07:59 -0500
Organization: MindSpring Enterprises
Whenever I first introduce fractions, I make a point of stressing the
following "vocabulary" lessons:
1. Avoid the term "over", as in 2/3 is "2 over 3". Instead,
say "2 divided by 3". This helps emphasize that fractions are division.
2. Avoid the words "cancel" and "reduce", as in, "cancel out
the 2 in 2x/2y and reduce it to x/y". Instead, be on the lookout for
common factors in both numerator and denominator. If and when common
factors are found, use the rule that:
ab/ac = (a/a)(b/c)
and feel free to "ignore" the factor of 1 in computing, and
re-expressing, the product.
It may be fine for those of us who understand and appreciate
fractions to use words like "over", "cancel" and/or "reduce" amongst
ourselves, but we should be very careful about passing on words that might
lead students to believe that math is a system of arcane rules and
formulae which have to be memorized.
From owner-math-history-list@ENTERPRISE.MAA.ORG Fri Jan 2 18:34 EST 1998
Approved-By: rickey@BGNET.BGSU.EDU
Date: Sun, 28 Dec 1997 20:46:00 -0500
Reply-To: Discussion List on the History of Mathematics
,
"Randy K. Schwartz"
From: "Randy K. Schwartz"
Subject: Wordwise on Fractions
Comments: cc: Rheta Rubenstein
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Content-Type: text/plain; charset="iso-8859-1"
Content-Length: 7615
Status: RO
Here is the fourth installment of our Wordwise column.
The article below is from _The Right Angle_ (Volume 5, No. 4,
December
1997), the monthly newsletter of the Department of Mathematics,
Schoolcraft
College, Livonia, Michigan, USA. Permission is granted to reproduce its
contents provided that the names of the authors, publication, and
institution
are cited.
To facilitate transmission over the Internet, apostrophes have been
replaced with asterisks, quote marks with double asterisks, and
italicized or
bolded text has been delimited with underline characters.
Your comments will be appreciated. Please bear in mind that the
article
is written in a style intended to appeal to American youth.
Wordwise
------------
Unfracturing Fractions
by Rheta Rubenstein and Randy Schwartz
Numerator? Denominator? Which is which? How can I remember these?
People
are often puzzled by the words that describe the **top** number
(numerator)
and the **bottom** number (denominator) of a fraction. But knowing the
origins of these words might help make the vocabulary- and the
mathematics-
easier for you.
Naming names, numbering numbers
Consider the fraction 3/5, three-fifths. **Three** is the
_numerator_ or
_number_ of fifths we are considering. The numerator, literally
**numberer,**
3, tells the number of items of interest (in this case, fifths).
**Fifth** is the _denominator_ or _name_ of the fraction. It tells
the
number of equal parts in the whole. The denominator gives the **name**
of the
fraction, or what pieces of this size are called: it is three _fifths_,
as
opposed to three _fourths_ or three _Cokes_ or three _Rosie O*Donnell
tickets_.
The most important part of the word **denominator** is **nom**,
which
means **name** in French. Some related English words are:
nominate - to name someone to run for office
nominal rate - the named rate of interest charged by a bank, which
is
lower than the real or **effective** rate of interest
denomination - the name of a religious sect, of a playing card,
etc.
Fractions don*t have to be difficult, if you just distinguish their
names (denominators) from their numbers (numerators). Just do it! That
way,
you*ll avoid the most common mistakes, and master a very important
skill. As
our own Professor Sandy Kerr often says, **Give me someone who can do
fractions, and I can teach them how to do anything.**
Graphic No. 1:
3/5 + 1/5 = 4/5
Just as three Cokes plus one Coke equals four Cokes, so also three
fifths
plus one fifth equals four fifths. **Fifth** is the **brand name** or
_denominator_ of the fraction.
Graphic No. 2:
3/5 + 1/6 not = 4/11
3/5 + 1/6 = 18/30 + 5/30 = 23/30
Just as three Cokes and one Pepsi can only be added if you think of them
both
as **colas**, or as **beverages,** etc., so also three fifths and one
sixth
can only be added if you think of them both as **thirtieths,** or as
**sixtieths,** etc.
Graphic No. 3:
2 x 2/5 not = 4/10
2 x 2/5 = 4/5
Two times two Cokes equals four Cokes: the _number_ of Cokes changes,
but the
_name_ of the Cokes does not change. The same is true for fractions: two
times two fifths equals four fifths, not four tenths.
More Broken Bones
Here is how numbers were described in an algebra text printed in
London
in 1557: **Some are whole nombers ... Others are broken nombers, and are
commonly called fractions.** (Robert Recorde, _The Whetstone of Witte_)
In
fact, the word _fraction_ comes from the Latin root **fractus**, meaning
**broken,** because fractions are parts broken out of a whole.
Interestingly,
the English verb **break** and its Latin equivalent, **frangere**, both
come
from the older Indo-European word **bhreg**. In English we have these
related
words:
fracture - the result of a broken bone
fractiles - the equal parts into which a sorted list of scores can
be
broken, such as quintiles (fifths) or percentiles (hundredths).
fractal - an object whose dimension is a fraction, such as a
1.5-dimensional curve
diffract - to break a light beam into separate colors
fragment - a broken-off piece
fragile - easily broken into pieces and damaged
frangible - able to be broken into any quantity for shipment.
Note that sugar is frangible but not fragile, while sugar cookies are
fragile
but not frangible. Christmas trees are neither fragile nor frangible.
Fractions are often thought of as **parts of a whole.** The word
_part_
comes from a root meaning **piece**, also seen in such words as
**portion,**
**parcel** and **particle.** The _whole_ from which a fraction is
derived is
**unbroken.** It comes from a root meaning **uninjured,** also seen in
such
words as **hale** and **healthy**, the conditions of our bodies when
they are
not broken.
Fractions make PERfect CENTs
A _percent_ alone is nothing but a type of fraction. For example,
2% is
just another way of saying 2/100, or 2 divided by 100; in decimal form,
0.02.
In fact, the Latin words **per cent** literally mean **per one hundred**
or
**for each 100**. You can think of 2% as 2 parts per 100 parts, such as
two
cents for every dollar, or two years for every century.
People often get confused by the words **percent** and
**percentage.**
They are two different things! Tacking on those few extra letters
**-age**
takes you a long way: it means the _final result_ of the process. For
example, if you break something, the final result is breakage; if you
marry
someone, the final result is a marriage (or maybe a broken marriage!).
Other
examples are leakage, shrinkage, and package.
Okay, so you go to a bank that offers car loans at a nominal rate
of 2%
interest per month. Your junker costs $2100, while the fellow ahead of
you in
line has his eyes on a Lamborghini for $132,000. You both get a 2%
interest
rate; the percent is the same for everyone. Is the final result the
same? It
better not be! When the bank applies the 2% rate to $2100, it means
you*ll
pay 2/100 x 2100 = 42 dollars of interest during the first month. When
the
same bank applies the same percent to Mr. Moneybags, it means he*ll be
paying
2/100 x 132,000 = 2640 dollars of interest in that same month. The
percent is
the same, but the final result- the percentage- is vastly different.
When you
take a percent of something, the _percentage_ or final result is some
portion
of the whole, so its size depends on how big the whole is.
And you can take that to the bank!
Sources:
+ Steven Schwartzman, _The Words of Mathematics _(Washington, 1994).
+ MAA Archives in Math History,
+ assorted English dictionaries
A challenge:
Certain numbers are said to be **irrational**, and certain
people are also
called **irrational**. Why do we use the same word for these things?
_The
Right Angle_ will publish the first rational explanation that we
receive.
=====================================================================
Randy K. Schwartz, Chairman email rschwart@schoolcraft.cc.mi.us
Department of Mathematics voice 313/462-4400 extn. 5290
Liberal Arts Building
Schoolcraft College "In the Inn of the world there is room for
18600 Haggerty Road _everyone_. To turn your back on even one
Livonia, MI 48152-2696 person, for whatever reason, is to run
USA the risk of losing the central piece of
fax 313/462-4558 your jigsaw puzzle." - Leo F. Buscaglia
=====================================================================