What’s Hiding Inside Lengthy Division?

 

Look acquainted? For most individuals, in the event that they ever realized to do lengthy division correctly, they needed to “grasp” one thing just like the above process/algorithm. What was not required was having the vaguest notion of what’s occurring, why anybody ought to care about figuring out the way to do it, or why, when you did have some thought about what the reply was alleged to imply, this mumbo-jumbo received you to that reply accurately. One other method of describing the state of affairs is “a black field.” You are taking two numbers, throw them into the “lengthy division field,” flip the deal with as many instances as obligatory, and – voila! – out pops the magic quotient, with a the rest, if any, in some kind or different. Fabulous! Who wants conceptual understanding when you can memorize a process and carry it out precisely? Oh, and shortly: don’t neglect that “math” (effectively, faculty math, anyway), is a race! The “good” college students are like machines! The sluggish, the error-prone, the confused, will probably be left by the wayside.

 

Properly, that doesn’t work for me anymore with arithmetic. I can’t be happy with being a flesh-robot or coaching others to be like that (or abetting academics to make kids view math like that). Procedural fluency sans understanding just isn’t higher than utilizing an digital black field for computation, although traditionalists reward the previous and decry the latter. There’s a deep dialog (or pointless, screaming argument full with epithets that may make individuals in a Trump vs. Clinton partisan on-line combat blush) available right here, however not in the present day. As an alternative, I’m going to attract again the curtain and present you the wizard hiding behind it.

To start, let’s recall from final month that a method to consider computing a multiplication downside on the earth of integers (Z) is as “repeated addition.” Whereas there are issues with this phrase as a definition of multiplication, one I counsel that academics eschew, there isn’t a downside getting the product of, say, 18 x 6 by summing six addends of eighteen or vice versa. Not perfect for effectivity, after all.

By analogy, since division is the inverse operation of multiplication and subtraction is the inverse operation of addition, we should always be capable of compute division in Z as repeated subtraction (so long as we’re snug with remainders. Within the rational numbers (Q), we have now “fractions” or decimals when a division of integers doesn’t give a “good” integer quotient). Right here’s an instance of how that may work in apply:

So, we may actually do with out lengthy division if we’re keen to do repeated subtraction as many instances as essential to get the quotient of two integers, proper? And if we aren’t in a rush for outcomes or fearful that including greater than the minimal calculations obligatory will increase the chance of error, we’re high-quality. We actually DO have that possibility.

However for individuals who favor one other method, there’s lengthy division. Why does it work? That’s the lacking piece for many of us, together with, I strongly suspect, most Ok-6 math academics. I’d be keen to wager that when you requested folks to elucidate why the long-division algorithm works – not merely what the steps are, however why and the way they make sense – few may give a coherent, appropriate response.

So, let’s attempt to unpack lengthy division by making use of the usual algorithm to one thing just like the earlier downside (we will danger a much bigger dividend right here):

I belief your reply agrees with mine. However how do we all know we’re appropriate? Properly, 78 x 22 = 1716. Add the rest, 14, and we get  1730, the unique dividend. Math works, life is nice.

Besides that we don’t know why we received that end result, precisely, solely that we adopted the recipe and nearly miraculously, out popped the proper cake.

One factor value noting that might simply be missed: if we add 1540 + 176, we get 1716. That may’t be sheer coincidence, can it? What are these numbers, anyway?

Let’s undergo the steps used to get our reply, however a bit extra completely than you’re most likely used to doing. First, you possible mentioned, “Hmm, 22 received’t go into 17, so let’s strive 22 into 173. Ah, that goes, um, 8 instances. Wait, no, too massive, so strive 7; yeah, cool, that’s 154.”

All effectively and good, however largely lies. First, that’s not “17,” it’s 1700. The query is NOT actually “What’s 22 into 17?” however moderately, “what number of teams of 1000 x 22 are there in 1730?” Oh, that surprises you? Properly, why do you’re taking as a right that you just don’t must ask that? The very fact is that solely expertise, quantity sense or the senseless following of what academics, dad and mom, or friends instructed you enables you to skip that query as unneeded. And equally, most likely with out figuring out it, you miss that the primary query about 22 into 17 now turns into “What number of teams of 100 x 22 are there in 1730?”

Once more, the reply is 0, since 1 x 100 x 22 = 2200 which is just too massive. So lastly, we ought to ask, “What number of teams of 10 x 22 are there in 1730?” And the reply is 7. However please be aware effectively: that 7 represents “70,” which is 7 x 10, which is what we actually multiply 22 by to get the primary “partial product” of 1540 (see that I appended a zero to the 154 you calculated to clarify what is de facto being subtracted from 1730 on the primary cross). Additionally, notice that utilizing our methodology, that’s the most we will justify subtracting, however we’d have subtracted much less! (See the dialogue of partial quotients under).

So after we subtract 1540, we have now not 19, however 190 remaining. The remaining is fairly apparent (a harmful phrase I’ll danger utilizing right here). The subsequent step doesn’t require mendacity: we actually do ask,”What number of teams of 1 x 22 are there in 190?” The reply is certainly 8. Any extra would put us previous 190. Our second partial product, from 8 x 22, is 176. We subtract that from 190 and are left with 14, too small for any extra teams of twenty-two to be eliminated. That’s our the rest, then (alternatively, we will say that there are 14/22 of a gaggle left, the place a complete group has 22 in it).

So that’s nearly all we have to be aware save that what’s lacking from the method you have been taught at school takes as a right one essential thought: place worth. We’re first taking 70 teams of twenty-two from 1730 suddenly, moderately than doing it 70 instances by way of particular person subtractions of twenty-two at a time. And that’s an excellent factor certainly. However as a result of this means of compressing 70 subtractions into one estimation, one multiplication (or extra if we estimate badly) and one subtraction hides info from us. “Higher” algorithms are designed for effectivity, not for transparency. You snooze, you lose.

Besides, we just about all snooze as a result of it’s nearly extraordinary for academics to take the time to unpack this course of, not to mention educate it as repeated subtraction, then as partial quotients utilizing place worth, after which lastly going to the extra environment friendly however opaque commonplace algorithm. So for youths who don’t decide up the process, apply it to “mastery,” after which transfer on to the subsequent disconnected process, lengthy division is continuously a stumbling block. Even some college students who can carry out the algorithm accurately discover it off-putting, I consider, given how negatively they react to the introduction of polynomial lengthy division, a process that’s, in apply, far simpler to carry out.

I’d be remiss to conclude this essay with out taking a look at one different method, partial quotients: a mix of repeated subtraction and the usual lengthy division algorithm.  Let’s take the second instance once more, 1730 ÷ 22.

What if we weren’t positive concerning the precise process for the usual algorithm, however we felt that we knew the purpose of division (which is definitely much more difficult and context-dependent than I’ve chosen to discover on this put up): what number of teams of twenty-two are there in 1730 with a leftover quantity of from 0 to fewer than 22? (Usually, we may name 22 the divisor, 1730 the dividend, 78 the quotient, and 14 the rest, and in symbols, b = a*q + r, the place b is the dividend, a is the divisor, q is the quotient, and r is the rest, a, b, q, and r in Z, and 0 ≤ q < a).

So maybe we do that:

What’s THAT all about? Properly, I assumed, “Hmm, 10 x 22 = 220, so 20 x 22 = 440.” I then subtracted 1 x 20 x 22 and was in a position to repeat that twice extra. However that left me with 410, too small for an additional subtraction of 20 x 22, however large enough to subtract 1 x 10 x 22. That left 190, too small for subtracting one other 10 x 20 (predictably, given the earlier evaluation), however large enough to remove 1 x 5 x 22, or 110. That left 80, and I may see that 3 x 22 = 66 was sufficiently small, whereas 4 x 22 = 88 can be an excessive amount of to subtract. And with 14 left, I needed to cease subtracting. Including the partial quotients gave the entire quotient of 78 (teams of twenty-two) plus the rest, 14.

Earlier than the forces of righteousness assault me for “dumbing down” America’s youth, I’m not lobbying for “partial quotients” to interchange the usual algorithm. Nor am I arguing in opposition to having college students study the usual algorithm, since, in any case, calculators, computer systems, and smartphones do arithmetic, and far more, sooner and extra precisely. As an alternative, I’m advocating for the top to black packing containers when understanding is quickly offered. How we get there, nevertheless, I’ll go away for an additional column.