Recent editions of Monday Math have seen us working pretty hard. So how about we lighten the mood a bit and think about fractions.
Let us start with the obvious. Fractions have tops and bottoms. Got that? Numerators and denominators exist only in elementary and middle school math classes. Their sole purpose is to make mathematics as offputting and unpleasant as possible. If you refer to the top of the fraction and the bottom of the fraction everyone knows what you mean. Say numerator and denominator (which come respectively from Latin words meaning roughly “one who numbers” and “that which is named”) and you may as well be speaking a different language.
When mathematicians look at a fraction like 1/x we say, “One over x” At least, that's what American mathematicians say. British mathematicians use the far more elegant phrase, “One upon x.” They do that because they speak the language much better than we do. At any rate, we certainly do not say, “one xth” or some such thing as that.
And why is it that every kid, when learning how to add fractions, is told of the terrible things that occur upon failing to use the least common denominator? (Yes, FINE, here I shall allow the term, on account of “common bottom” sounding ridiculous.) What, pray tell, is the BFD? It is bad enough they must struggle with the whole idea of a common denominator, a notion invariably presented as a rule from on high possessing neither rhyme nor reason. But then they are told, having figured out the trick where you just multiply together the given denominators, that such techniques are not acceptable. Why? Because let's say you are adding one sixth to one fourth. If you naively multiply the denominators you will get twenty-four as the common denominator, when actually it should be twelve. I mean, really, check out the carnage:
\[
\frac{1}{6}+\frac{1}{4}=\frac{4}{24}+\frac{6}{24}=\frac{10}{24}.
\]
Are your eyes bleeding yet? Have you fully appreciated the horror? Your answer is not even reduced! Kids actually get points taken off for this. And then we wonder why they don't like math.
From where did we ever get the notion that when adding fractions it is terribly important that your sum be pre-reduced? If it bothers you to have a non-reduced fraction as your answer then just reduce it after adding. It's OK, really.
But while we're at it, let's ruminate about this miserable obsession with reducing fractions. 10/24 is a perfectly good fraction. If you have some specific reason for wanting the smallest numbers possible then by all means go ahead and reduce. But don't do it because you think the reduced fraction is necessarily simpler, or because you think the state of the world is happier when tops and bottoms (!!) have no common factors, or anything along those lines.
Let me tell you what happens when you are morbidly obsessed with reducing fractions. A while back I had to install a new doorknob on a closet door in my house. This meant buying one of those small, hand-held power drills, since new holes were needed for the screws that would hold the small metal dingus to the door. When I prepared to begin drilling I discovered that the five thirty-seconds bit was too fat, but the three thirty-seconds bit was too thin. So I went hunting around for the four thirty-seconds bit. Do you think I found it? Of course not, because whoever stamped the widths onto the bits came from the zombie-like, “must reduce my fractions” school of fraction manipulation, and therefore I had to find the one eighth bit. That's right! My bits were stamped in halves, fourths, eighths, sixteenths and thirty-seconds, all because of an educational system that thinks reducing fractions is an inherent good, like charity and kindness to others. Of course, as a highly trained math professional I can do these conversions in my head. But I shudder to think of all the missized holes that have been drilled in the nations door jambs over this.
And where this really stops being funny is when you get to calculus. Suddenly you are asked to find the derivative of
\[
f(x)=\frac{x^3+7x-13}{x^4+5x^2}.
\]
You remember the jingle, “If the quotient rule you want to know, it's low dee high minus high dee low,” and since no one ever has any trouble remembering what goes on the bottom (!!) of one of these goobers we get
\[
f'(x)=\frac{(x^4+5x^2)(3x^2+7)-(x^3+7x-13)(4x^3+10x)}{(x^4+5x^2)^2}.
\]
How many times have I explained to students that they should stop at this point, seeing as how the problem asked them to find the derivative and they have, in fact, found it? That, with probability approaching one, they will mess it up if they set their miserable algebra skills to the task of multiplying everything out? And how many times have they ignored me, multiplying away with the same enthusiasm with which a small child offers to “help” you with some household chore, in the bizarre but sincere belief that they are somehow “simplifying” the fraction? And then mind you, after wasting ten minutes in this futile and totally unnecessary effort, they turn to me and complain they did not have enough time to finish the test!
But you know what makes this really not funny? The fact that if they multiply it out then I have to do likewise, on the off chance that they actually did it right.
Highly vexing.
And while we're at it, who came up with mixed numbers? WTF? As far as I'm concerned if you write
\[
2 \ \frac{7}{9}
\]
you're talking about fourteen ninths. The proper way of expressing two and seven ninths is
\[
\frac{25}{9}.
\]
Improper fraction my ass! That is how fractions should be written. It is how they would be written if the world was the product of a just and loving God, which it is not. Instead of that perfectly clear, impossible to misinterpret fraction, we get the hellish, confusing miasma of mixed numbers. Try breaking a college student of that little habit!
And stop using diagonal lines for your fraction bars, people! Major yuck! A proper faction bar, in nearly all cases, is a horizontal line perfectly parallel to the edge of the paper. (Notice, by the way, that “parallel” has two ells, then one ell. Just mentioning it, even though common math spelling errors is actually a different post.) The only exception is when you are dealing with very complicated fractions with elaborate tops and bottoms (!!), in which case for typographical reasons it is sometimes convenient to use a diagonal line.
And if you insist on using decimals (whose primary pedagogical purpose, so far as I am concerned, is for teaching people that clever little trick for turning a repeating decimal like .347347347347... back into a rational number), then at least make sure your decimal point is thick and dark and easy to read. Don't leave me wondering whether that smudge on the paper is a decimal point or not!
Oh, I could go on. You know I could. But since it is barely conceivable that I am trying your patience, and since the season premiere of House is less than ninety minutes away, I shall call it a day. See you next week!
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My own rule for the deriv of a fraction is bottom dee top minus the rest, itâs alphabetical, after all.
In calculus classes, I would tell them Donât Simplify once you have something that is an answer. Some times this worked.
Another âruleâ that should be abolished: the one that tells them always to rationalize the denominator (oops, bottom) of a fraction.
Cheers to that!!!
However, I must protest, slightly, to your last points about decimals and mixed fractions being the work of the devil. In a math class I wholeheartedly agree. However, in the outside world where you have to measure things 2 and 7 9ths means more, to more people than does 25 9ths.
In a similar way, decimals are nice for those in the world using a metric system of measurement. If I need to cut a rope in 7 pieces, I want the rounded decimal equivalent length of each piece, not the fraction when I measure.
The exception to the rule of reducing terms shows how dumb the rule is: $1.98 reduces to $99/50.
Well said! We were taught to reduce fractions but at least our teacher made it very clear that the reduction was an optional and additional step i.e. not a compulsory part of the problem solving.
Leaving fractions unreduced and unmixed (e.g. 10/24 and 25/9) can help children to spot the patterns in numbers.
I teach high school science and bang my head against the wall trying to explain exactly the things you talk about, only to have my students berated by their math teachers when they try to use the skills I have taught them (and find myself ridiculed). If anyone has suggestions, I appreciate it and will keep up on the thread. (Oh, I am also a certified math teacher)
I am 75 years old. Therefore I reject all your arguments and comments against doing things right. LOL
I think the problem is not so much these actions, but the idea that there are Rules by which students Must Abide regarding these. It's more helpful to understand that sometimes a fraction should be reduced, other times it shouldn't be. Sometimes you want to rationalize a denominator, but other times not. And sometimes you do want a mixed number, but if you're just doing math, leaving it unreduced and unmixed is a blessing.
What really matters is what you intend to do with that number. 1 over root two is a perfectly reasonable answer. However, sometimes root two over two is more useful for additional things you're going to do with it. Have the tools, but don't simply hammer the nail because there is a nail. Know that sometimes you want that nail sticking out because you're going to hang a picture off of it.
I swear this is not an apocryphal tale - it really did happen to me: I went into a Builders' Emporium to get some 8' lengths of 2x1 for a little construction project I had. The young lad who served me apologized for not having any 2x1 in stock. Then, after a few moments of apparently quite difficult thought, he tentatively said " but we do have some 1x2 - will that do instead?"
On the subject of mixed fractions: as long as we continue to refuse to go fully decimal, rulers will still be marked in integer feet and inches with fraction marks between the inch marks (tho why my ruler stops at feet and does not mark yards is beyond me). I see the continuing difficulty people have even going as far as feet in how furniture measurements are given as mixed fraction in inches (72" not 6' for instance).
That whole business with the drill bits is one of my favorite reasons for supporting the metric system. Drill bits, sockets, wrenches... all of 'em are easier to work with in millimeters.
For the derivative of quotients, I always use the product rule. After all, quotients are just "derived" from products anyway. ;)
British mathematicians use the far more elegant phrase, âOne upon x.â
British person here. Never, ever, ever, ever heard anyone say "upon". It's always been "over".
Cheers,
Jon.
I did have a teacher who complained that millimetres were too small and fiddly, and he preferred the British system of thirty-seconds of an inch....
I tend to agree, but with one caveat: if any of the calculating situations you describe occur as an intermediate step in some larger calculation, it is probably a good idea to reduce/multiply out before moving on. Imagine if the poor student had to find the limit of that derivative you wrote up, ugh...
You're ten tenths correct about it being better (more logical, more sensible, easier both to understand and do) to cross-multiply fractions when adding rather than find an LCM and do fiddly divisions.
Reduction? I think your example of thirty-seconds of inches shows the important point: when using fractions, often it's good to pick a denominator and stick with it, in effect to do the calculation in units (!) of 1/32 inch. So reduce 10/24 if (and only if) twelfths are more interesting than twenty-fourths. Even for ageing UK engineers who have grown up with inches but use millimetres nowadays (we are Yurrpeons, y'know), it can be a jarring pain when the occasional need to calculate in fractions of inches arises.
The one thing I disagree on: 25/9 feels pretty much like (say) 28/9, but one is less than 3, the other is larger - so it is useful to make a mixed fraction if the fact that 25/9 is two-and-a-large-bit is meaningful, which is instantly obvious from 2 7/9. The message though is again: horses for courses.
You realize that this is only a fraction of the topics you'll have to cover for us, don't you?
If you find an old set of carpenters' wood bits (the square-shanked augers made for hand braces) their sizes will usually be stamped on the shafts, from 3 to 13, meaning 3/16 to 13/16.
The half-inch bit is always marked 8.
Grade-school teachers also have the annoying habit of failing to carry units through their calculations, leaving high-school chemistry and physics teachers to teach this. If you divide 120 miles by 3 hours, you really should state the answer as 40 miles/hour, not just 40.
I always prefer to write the mixed numbers using a plus sign, and I think they should teach that in school. It's clear, it's correct, and it's consistent with the notation you learn later on if you take more math classes.
On the topic of finding the LCM before adding or subtracting fractions, I agree with you: it's the kind of thing that will very quickly make kids hate math. I know I did, until we started learning about algebraic fractions, and I finally figured out what all the crap I'd been told to do to fractions was for, and how some of it was entirely useless.
Fortunately, I don't recall ever having lost points due to doing things properly on my own, but I could very well have if my teachers had been stupider, or my schools incompetently managed.
This is the first post I've read from this blog...so hilarious!haha
Ah Jason, I'm going to have to disagree with you on many points here. Lets take them one at a time.
(1) Least common denominators before adding, or cross multiply, add the fractions and find the greatest common divisor to reduce. The first deals with smaller numbers, and is thus preferable. Arithmetic is always better with smaller numbers!
(2) Reduce, reduce always reduce! OK, your drill bit example is one where we go too far, but that is simply naming the fractions. In terms of arithmetic, its always better to reduce. For realistic problems beyond the trivial 1/6+1/4, do you really want to end up with a solution with ten, twenty or more digits when it could be written with three? If you've ever tried doing arithmetic without a calculator recently, you'd know that the smaller the numbers the better.
(3) 2 (7/9) is much better than 25/9 if you want to compare fractions to integers, or add. The reverse is better if you want to multiply. Neither is superior, both are useful.
(4) You really think decimals are only useful for showing periodicity for fractions? Umm, the rest of the the number using world, particularly those who have to interact with scientists, engineers and anyone who uses a computer or electronic calculator, uses decimals (or their base two equivalent on a computer). They make addition much easier (just add columns) if you are willing to approximate numbers to a given number of places. They also make comparing numbers substantially easier.
(5) Writing 1/2 rather than with the horizontal line makes typing numbers a whole lot easier if you don't have an equation editor of some kind.
Just my 3 cents (rounded for inflation) worth.
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Fun post. I must say I do not mind the terms numerator and denominator. But I have a visceral hatred of the terms abscissa and ordinate.
Hear, hear!
My guess is that the tradition of subdividing English lengths in halves, quarters, eighths, 16ths, 32nds, and even 64ths (my drill bit set has some of these) dates to a time when precision machining was not widely available, and it was easier to obtain an intermediate size by making it halfway between two sizes. Nowadays, there is no reason to do this. Metric sizes FTW.
I do disagree on decimals, because they are convenient for dealing with currency. As recently as the 1990s stock prices on the NYSE were quoted in fractions (eighths were usual, and some stocks were quoted in 16ths). Conversion to decimals was a win all around: tighter bid-ask spreads, and a guarantee that your odd lot transaction would come to an integer number of cents. Decimal currency is so much easier to deal with than the British system of shillings and pence--so much so that the UK converted to a decimal system themselves, and there are now 100 pence in a pound rather than 240 of the old pence.
I also second the comment about keeping units attached to numbers. For one thing, it's a check on whether your answer makes sense--did the units come out right? Recall the Stonehenge scene from This Is Spinal Tap, in which Nigel specifies inches instead of feet. Not to mention the Mars mission that was lost because one group was using pounds and another was using newtons.
I'm not sure who your target audience is here. If you're talking about fourth or fifth graders learning to work with fractions, you might have a point about the reduction.
But at the high school level and beyond (I teach at a university, but in the fall I teach "College Algebra", otherwise known as "Algebra You Should Have Learned in High School, Had You Been Paying Attention"), this doesn't make a lot of sense. You generally want to reduce first because, as I tell my students, it's easier to multiply small numbers than to divide large ones. Also, students learn that fractions are just a special case of the more general class of polynomials. And while it is possible to reduce fractions with the touch of a button on a calculator, polynomials are just a bit harder to do that automatically :-) Iow, you reduce first because in general division is harder than multiplication (otherwise a lot of crypto wouldn't work.)
The rest of your points, especially the one about not "simplifying" your results when differentiating, are generally good ones. I can't tell you how many times I've sighed and gone through the tedious "simplification" to check the student's results before simply adding the sentence "do not reduce" in the instructions.
I would say that the argument for teaching LCD fractions ultimately is for rational functions, specifically for showing removable discontinuities. It's a lot easier at an elementary level to factor a quadratic than a cubic, for example.
On the other hand, "improper" fractions should be the preferred way to write fractions; you skip the silly step of conversion every time you actually have to do something with them.
And simplifying denominators has always seemed to me to be the work of some nut who decided people should be introduced to elementary field concepts early on, that is that Q(b) where b is algebraic over b can always be written as a vector space over Q. Which is as useless as a $120/40 bill to me.
Eric Lund: I think you're referring to the Mars Climate Orbiter, which was lost because one correction was made from calculations using yards, but it had been designed using metres. Yet another argument for the US to go metric!
I'm a British mathematician and I also never use, and have (very occasionally but) hardly ever heard anyone say "upon" instead of "over" when reading fractions.
Totally agree about "improper" (bah!) fractions.
Excelent text, I have to show it to my daughter, who is learning fractions right now.
You hole drilling problem would be easier if you got metric. Our drils are in millimeters, for the smaller ones half millimeters are usual. A simple linear series to look for.
My students' problems with fractions were probably my biggest aggravation as a high school math teacher. You're mostly right on here.
But I think "/" as fraction bar is fair game:
1. It means the same thing as division, often represented on calculators as "/"
2. It's how you do fractions when using Google as a calculator
3. It's how you (by which I mean I) represent fractions in plain text ASCII
In my experience, the "rules" of algebraic manipulation as learned by my students in junior high and earlier made a hash of teaching them any algebra. They remembered the mnemonic devices and slogans, but frequently applied them incorrectly and only rarely had any clue of why it was the correct way to approach the problem.
In response, I went backwards a bit and made them present every step as an operation performed on both sides of the equation. If I was to do it again, I'd also spend time looking at why the mnemonic devices, shortcuts, and algorithms work, and how they're derived from operations performed on each side of the equation.
I think the mechanical approach to problems they teach in lower grades also contributes to the sense that mathematics is a boring yet borderline incomprehensible thicket of arbitrary rules.
English teacher? (I hope I hope I hope...)
Hilarious either way, but a little scary if he was math/science/engineering.
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Ah, this reminds me of my Almost Completely Fair Grading System. The point was simple: a problem set was made up of N problems. Each problem was worth a point (hence *almost* completely fair; in a *completely* fair system, each problem would be worth 1/N points). Each problem was divided into S_n sub-problems, which could be divided into S_S_n sub-sub-problems, etc. Easy right?
So the final grades were these huge some-odd-hundred-blah-ty-blee over ecks-thousand-why-hundred-zee-ty-eh fractions. There were complaints to the professor I was grading for. I was kindly asked not to try that experiment again.
I still have a fond place in my heart for the ACFGS.
I had an absolutely, ridiculously, horrible time when I was in school learning math. (I suspect possibly undiagnosed dyscalculia on top of terrible teachers, my mother suspects ADD, but I will probably never know. :P) But of all the horrors I faced in math class, fractions (and converting them to into decimal form) was one of the few things I actually managed to intuitively grasp.
Calculus on the other hand... *shudder* o_O
Oh, and btw, when I was in school we always pronounced it "one sixth", almost never "one over six." And that's how I still read them.
Also, I'm sure the reason why kids are told they must always simplify their fractions is so the teacher can simply look at the correct answers in the book instead of having to actually figure the problems out themselves. Some of the teachers I had growing up probably couldn't have done half the problems they were expecting us to do!
I agree with the points on nomenclature. We do not need latin terms for everything in mathematics. This also holds for things like contravariance and covariance.
Teaching the method of finding the least common denominator is useful once fractions like 5/13 + 8/39 are encountered. However, these are rarely if ever encountered in practice and the reality is that the formal process of finding the LCD of 13 and 39 would take longer and introduce more complication than just ending up with 299/507. At THIS point, it now makes sense to introduce the method of greatest common denominators in order to simplify the fraction; not before.
"Simplifying" algebraic expressions indeed depends on context. Technically, what the students feel compelled to do is "expand" the fraction above and below the line. This can be advantageous but often isn't. A very simple example is something like (n+1)(n+2)=n^2+3n+2. The RHS is expanded, but it may not be simpler. In the context of evaluation, especially on a computer, the LHS has 2 additions and 1 multiplaction. The RHS has 2 additions and 2 multiplications(6.1*7.1 vs 5.1^2+3*5.1+2). Sometimes expansion is bad; and this is before we get to factoring polynomials.
Mixed fractions made sense back in the day when the bossed needed to know the nearest integer to his 1234/23 bales of hay. Now we have decimals(and calculators to find themâYou can find out what 9/14 is nowadays) and so mixed fractions have become entirely redundant.
Sticking with fractions rather than decimals is fine so long as you are using the exact numbers which are found in mathematics.
Those of us who aren't in pink ties have to deal with numbers that are of limited precision and a decimal is far easier to trim off to what is actually known.
As an example, in the Math Faculty 8.2 * 8.1 = 66.42, in the Science Faculty it's 66.
in my experience, i have found that 5 out of 4 students have a problem with fractions.
I'm an engineer by trade, but I love tutoring math, and I'm currently going through some of these discussions with my 8th grade student.
I've explained that she just needs to do it how the teacher wants for now, and it's good practice either way. I've also told her that in the 'real world', it is usually only important to make your answers clear.
I wonder how many people here even know terms like subtrahend and minuend.
I'm a freshmen in high school and I agree with a lot of what you are saying. It's really annoying to get points taken off on a test or a quiz just for not reducing. You have to right answer don't you? But once you get it in you head it's hard to forget it. Each math teacher I have had since 6th grade has been drilling in my head how important it is to reduce and now I can't help but reduce. Of course it comes in handy sometimes, but other times it really isn't all that important, I'm just happy someone else realizes this other them me.
What is the obsession with the lowest common denominator you described? When we learned in class how to add fractions, I asked whether instead of the lowest common denominator, I could just multiply the 2 numbers. The teacher said that this can lead to big numbers but was okay.
And thanks for clarifying the pronunciation. They never taught us that in high school (but before you lament about the education system, the language of it was German and there, such a handy way of saying does not (seem to) exist so 1/x really is "ein xtel" for complicated ones, we list top and bottom as such: Im Zähler ist... und im Nenner...).
@37: *raises hand* We learned these terms in the first week after primary school. IMHO they are handy since they are understood in German and English. Some terms are not and that always confuses me (you say "limit" instead of "Limes"...)
I strongly disagree about simplifying derivatives, and simplifying fractions in general. When you have some complicated expression and need to take a derivative, unless you're in a math class, there's going to be something important in that derivative, or you wouldn't have taken it. Simplifying is what helps you find that important thing and see patterns. If you end up with a bunch of terms in the numerator and all but one end up canceling out, for instance, that's important, even if the less simplified form is still technically correct.
Furthermore, while mixed numbers are silly in general, they're pretty useful when comparing things, which is why we use fractions in real life anyway. If we have an apple and a half, it's much more understandable to say so than to say that we have 3/2 apples. When a rational number is used as an approximation to a real number, for instance, including in decimal fractions, we're generally fairly interested in the integer part of the number.
From a mathematician's point of view, I see your point, but from a physicist's, there are some good reasons to simplify expressions and use mixed numbers.
"With probability approaching ONE, they will mess it up..." that is about the funniest thing I have ever read! Kudos!
It is getting better...I have helped my nieces/nephews in recent years. I always reduced the work to "simplest" terms, then when we look up the answers in the back, lots and lots of books are going to leaving them "unsimplified". I say "About time!"