Dividing Fractions

I'm on an email list on which the topic today is a video showing a silly way of doing division.  Literally, it involves drawing columns of dots.  It is no more understandable than the standard division algorithm, and certainly less efficient.  Here's the video:

 Here's what I posted on the maillist:

What I kept thinking while watching that video was "why not just draw 245 dots and circle them in groups of 5?"  It would have been only slightly less efficient but far more understandable.

Finding nifty techniques for math, techniques specifically designed to avoid the standard (and extremely efficient) algorithms, is often a case of mistaking cause for effect.  People like us, who understand math, can often see very interesting patterns or ways of viewing things that make math vibrant. If only everyone saw these things, math would be vibrant for them as well, yes? NO.  We see these things BECAUSE we already grok the underlying math: showing our patterns and techniques doesn't help people LEARN the underlying math.  "If a then b" does not imply "if b then a".  It amazes me how many people in education, how many people who should know better, make that mistake.

The division algorithm is plenty efficient already, and students should learn it.

Now let's shift gears a bit.  Knowing there's an exception that proves the rule, there is one non-standard-algorithm technique that I kinda sorta like.  It is slightly more difficult than the standard algorithm, but it combines some math that students already know with some simple understanding.  It's dividing fractions.

Here's the standard "invert and multiply" algorithm.  

 

It's easy enough to teach why/how this works (and I always did), but it still leaves some students confused.  Some cannot follow the explanation as to why it works, and some forget which fraction to invert.  Practice, though, should cement the process, even if they don't understand why it works.

Some teachers would get all worked up at this point--"understanding it is the whole point", they say.  To which I reply, no it's not.  I doubt most of those teachers could explain the 4 strokes of the internal combustion engine that powers their cars, but that isn't the point of driving.  The car is a tool, not an end in itself--and elementary math is exactly the same.

I won't get into how I demonstrated that the "invert and multiply" rule works, those many years ago when I taught junior high.  What I want to do here is show a slightly different way of doing the same problem.  Yes, it has a more difficult step than the process above does, but the payoff is a little more understanding of what the answer means.

So let's start with the original problem:

If students are learning how to divide fractions, they should already know how to add fractions--and hence, to get a common denominator.  So, let's rewrite the problem using a common denominator:

When students learned how to multiply fractions, they learned to "multiply straight across".  Now that we have a common denominator, it's easy to divide straight across.  No, you don't need a common denominator to divide straight across, but it doesn't buy you anything to do that step without a common denominator (try it if you don't believe me).

Since any number divided by 1 is that number itself, then

Generally speaking, students know that the problem above is asking "how many times does 5/8 go into 2/3 ?"  The number 16/15 doesn't seem to relate to that answer, even though it's correct.  By getting a common denominator, students can better understand the new question, "how many times does 15 (24ths) go into 16 (24ths)?"  Now, the 16/15 makes a little more sense.  Fifteen goes into 16 one time with a 15th left over.

This process has the advantage of using skills students have already learned (common denominators, "multiplying straight across", and dividing by 1) to get an answer to a new type of problem (division of fractions).  It required no new skills or rules, and it allows for an understanding of what the answer means.

If I were teaching division of fractions, this is what I'd teach first.  Then, after students become proficient with this interim process, I'd show them the "shortcut"--the "invert and multiply" process.

This video shows another method but doesn't at all explain why it works, when it's really just a variation of "invert and multiply":

Easy, but just a "plug and chug" with no understanding.

This video shows one of those ohmygawd methods--the process is so tedious that any understanding goes out the window:

Wouldn't 3 3/4 divided by 2 1/2 be unnecessarily difficult using this method?

The standard algorithms are standard because they're the most efficient way of to get answers manually.  The vast majority of the attempts to "make math clearer" fall prey to the point I made on the maillist--they are only clearer to those who can already do the math.

And that doesn't help anyone except the person trying to make a name for him/herself with some "new" method.