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 =====================================================================