How long should long division take?

[Not_a_Number]

Say something like 22428 divided by 63 — we’re not currently doing remainders. What’s a reasonable time goal? We’re still drilling and I’m curious when we should declare victory. 

[EKS]

For stuff like this, I do it myself and then multiply by some number.  For my high school students, it's four when the material is new (like for homework) and two when it has been practiced (for exams).

I just did this problem by writing everything out and not rushing at all, and it took 1 minute 35 seconds (ETA: I tried it several hours later, still writing everything out, but going at a faster pace, and did it in one minute).  For a lower elementary aged student, I'd be very happy with double that, and I'd be satisfied with triple that for a student with a pencil allergy or some slowness (or attitude) with facts.

Edited September 25, 2020 by EKS

[kirstenhill]

I never did a time goal on long division, but my general experience so far is it was just proportional to their general computation speed on other types of subtract/multiply/divide.  My one kid that is super fast at computation was super fast at long division, and kids #1 and #3 who are slower processors in general just could never do it nearly as fast because they just move more slowly (yet still methodically) in those kinds of problems.  I also teach short division first, which seems to be the best-kept math secret hardly anyone teaches, so at least that is lightning fast when it is usable. 

I am teaching DS8 long division right now, and he wants to do it the way BA teaches it.  I dislike their method.  My oldest is too old to have done BA, and my next two DSs thought BA's way of long division was silly and ditched it as soon as I taught them the regular algorithm.  DS8 doesn't even want to hear about the regular algorithm and I really think BA's method is slower. 

[EKS]

4 minutes ago, kirstenhill said:

I also teach short division first, which seems to be the best-kept math secret hardly anyone teaches, so at least that is lightning fast when it is usable. 

Me too!  I have never understood why it's not in every math textbook (or, really, any math textbook).

[kirstenhill]

8 minutes ago, EKS said:

Me too!  I have never understood why it's not in every math textbook (or, really, any math textbook).

RightStart math teaches it! I used RS math with my DD in grades K-4 and we both learned it there. 🙂

[Not_a_Number]

What’s BA’s method and what’s short division?

Questions like this take me considerably under a minute, so my goals for DD8 may be too ambitious. We did LOTS of intuition-building for division, so she’s conceptually solid. She also has all the prerequisite skills solidly in place. I’m just curious what a reasonable time goal may be. Right now, I’m asking for 2 minutes... 

[EKS]

12 minutes ago, Not_a_number said:

Right now, I’m asking for 2 minutes... 

How long is she taking?

[EKS]

I will add that with long division I never required them to do problems in a particular amount of time.  My goal was for them to get through a few problems each day without experiencing gridlock or having a meltdown (or both!).  Once they could do that reliably, I called it good and allowed them to use a calculator.  

[Not_a_Number]

12 minutes ago, EKS said:

How long is she taking?

A bit more than 2 minutes now 🙂 . We’ve identified issues like not writing down the estimated digit and not using mental math to multiply. (Technically, we’re doing 22.428 divided by 6.3, so she also has to find the decimal point. But that doesn’t take long.)

It was more like 7 minutes before, so there’s been a definite improvement. We  do use these skills for other math, and it’s been slowing her down, hence the drill. I’ve been asking her to tell me her steps so that I can see what’s taking a long time, which has been helpful.

[Not_a_Number]

12 minutes ago, EKS said:

I will add that with long division I never required them to do problems in a particular amount of time.  My goal was for them to get through a few problems each day without experiencing gridlock or having a meltdown (or both!).  Once they could do that reliably, I called it good and allowed them to use a calculator.  

We got to long division quite organically, so she doesn’t mind it. And we’ve been using it in our current unit, which is where I noticed it took way too long! (If each long division takes 7 minutes, it’s hard to prime factor fast!!)

[kirstenhill]

39 minutes ago, Not_a_number said:

What’s BA’s method and what’s short division?

Questions like this take me considerably under a minute, so my goals for DD8 may be too ambitious. We did LOTS of intuition-building for division, so she’s conceptually solid. She also has all the prerequisite skills solidly in place. I’m just curious what a reasonable time goal may be. Right now, I’m asking for 2 minutes... 

It would take me much longer than a minute to do that problem...LOL.  😄  I am pretty slow at computation.  (As an aside, my DS13 who is super fast at computation and has done well in a speed-computation-related math competition used to sometimes race with me on a calculator and him doing a problem in his head.  He would win at least half the time on mid size problems.  I can't even use a calculator very quickly I guess!).

Here's short division. I mean, it's basically a short cut.  So, conceptually I probably shouldn't have taught it before long division, but it doesn't seem to hurt anyone.  I just think it is the coolest thing so I am eager to tell everyone about it.

 

BA's method of long division is something like asking yourself to make an estimate on how many times the divisor fits in the whole dividend.  Write that on top where you would normally be writing your digits one at a time.  Whatever estimate you want to make is fine as long as it is not too big.  BA suggests picking something easy to multiply by like a multiple of 10, 100, etc.  Then multiply your estimate by the divisor and see how much is left.  Then make a new estimate of how many times your divisor will fit in what is left, and write that on top of the last estimate.  Later, Rinse, Repeat until you only have a remainder left.  Then add up your stack of estimates to get the final quotient.  If that doesn't make sense I can try and post a photo of a problem done that way later. 

I can see how it can theoretically be faster if you make really excellent estimates...but our experience so far is that DS ends up doing many more subtraction problems because he "plays it safe" with small estimates, then having done an easy multiplication for each estimate, still has to subtract every time to get down to the remainder or lack therof.

[EKS]

47 minutes ago, Not_a_number said:

(If each long division takes 7 minutes, it’s hard to prime factor fast!!)

What sorts of numbers does she need to find the prime factors for?  I've never (or rarely) encountered problems where I've needed or wanted prime factors where the numbers were so large that I needed hard core long division to figure things out.  That said, I guess if that were an issue, as an adult I'd just use a calculator, so I might not be remembering correctly.

[Not_a_Number]

8 minutes ago, EKS said:

What sorts of numbers does she need to find the prime factors for?  I've never (or rarely) encountered problems where I've needed or wanted prime factors where the numbers were so large that I needed hard core long division to figure things out.  That said, I guess if that were an issue, as an adult I'd just use a calculator, so I might not be remembering correctly.

We’ve done pretty big ones, because I wanted her to practice... but actually, it also came up with combinatorics. There, it’s often a good idea to just do fraction simplification, though, but she wasn’t fluent at that, either. We’re practicing that, too. 

[wendyroo]

That problem took me 1 minute and 12 seconds.

My 9 year old took just a bit longer at 1 minutes and 41 seconds.

My 11 year old took 3 minutes even, but he started by writing out the multiples of 63 in a list like Math Mammoth encourages.  Once that was done, the problem itself was very fast.  I think pre-writing multiples can reduce careless errors while long dividing, but it obviously slows down the process.  It is probably more worthwhile on longer long division problems or when continuing on to calculate an answer to a certain number of decimal places.  Or, I suppose, in any situation where accuracy is more important than speed.

 

[Not_a_Number]

4 minutes ago, wendyroo said:

That problem took me 1 minute and 12 seconds.

My 9 year old took just a bit longer at 1 minutes and 41 seconds.

My 11 year old took 3 minutes even, but he started by writing out the multiples of 63 in a list like Math Mammoth encourages.  Once that was done, the problem itself was very fast.  I think pre-writing multiples can reduce careless errors while long dividing, but it obviously slows down the process.  It is probably more worthwhile on longer long division problems or when continuing on to calculate an answer to a certain number of decimal places.  Or, I suppose, in any situation where accuracy is more important than speed.

Thank you! Yes, those seem like the kinds of goals we'd reasonably meet. 

I just tried it myself and it took 31 seconds. That's about what I get if I do it myself without much rushing (although rushing would probably not make me faster and might, in fact, make me slower.) 

I told DD8 she'd earn a prize if she makes all her goals, and people's answers make me feel that I should probably make the goal a bit more than 2 minutes, especially since I've been really doing questions with decimal points, which adds 10 seconds or so for her. 

[domestic_engineer]

Because learning is about improving one’s self, why not base her goal on a percentage-improved from her current time?  

[Not_a_Number]

Just now, domestic_engineer said:

Because learning is about improving one’s self, why not base her goal on a percentage-improved from her current time?  

Because I really just want to get it as functional as possible for future math, and because my judgment is that she's ready to do it quite quickly. She just needs to do it without many pauses and to use more mental math tricks she already knows. 

[domestic_engineer]

I think I understand what you’re saying .... and perhaps I’m inferring that you want to give her a   “Numerical” time goal rather than a percentage goal. But what I’m suggesting is that you calculate the percentage and then set that time as her goal rather than basing her goals on what you or other kids can do.   And what goal you give her now doesn’t have to be your final goal. 
 

For example if I have a struggling reader who’s reading 35 WPM, I’m not going to say “you need to read this list at 60 WPM” like all the experts say you should. Sure 60 WPM would be my end goal for this child, but I might set the goal of 38 WPM before the child, which is ~10% improvement for this child. 

[Not_a_Number]

8 minutes ago, domestic_engineer said:

I think I understand what you’re saying .... and perhaps I’m inferring that you want to give her a   “Numerical” time goal rather than a percentage goal. But what I’m suggesting is that you calculate the percentage and then set that time as her goal rather than basing her goals on what you or other kids can do.   And what goal you give her now doesn’t have to be your final goal. 
 

For example if I have a struggling reader who’s reading 35 WPM, I’m not going to say “you need to read this list at 60 WPM” like all the experts say you should. Sure 60 WPM would be my end goal for this child, but I might set the goal of 38 WPM before the child, which is ~10% improvement for this child. 

I'm definitely setting the goal based on what she can do 🙂 . I've adjusted a few of the goals up when I saw she couldn't do it as quickly as I thought. I may adjust long division up a bit as well. 

She's quite close on all of the goals, though, so none of them are frustratingly out of reach. I was just curious how quickly other kids do them! 

[Not_a_Number]

3 hours ago, kirstenhill said:

BA's method of long division is something like asking yourself to make an estimate on how many times the divisor fits in the whole dividend.  Write that on top where you would normally be writing your digits one at a time.  Whatever estimate you want to make is fine as long as it is not too big.  BA suggests picking something easy to multiply by like a multiple of 10, 100, etc.  Then multiply your estimate by the divisor and see how much is left.  Then make a new estimate of how many times your divisor will fit in what is left, and write that on top of the last estimate.  Later, Rinse, Repeat until you only have a remainder left.  Then add up your stack of estimates to get the final quotient.  If that doesn't make sense I can try and post a photo of a problem done that way later. 

I can see how it can theoretically be faster if you make really excellent estimates...but our experience so far is that DS ends up doing many more subtraction problems because he "plays it safe" with small estimates, then having done an easy multiplication for each estimate, still has to subtract every time to get down to the remainder or lack therof.

 

Ah, got it. We actually did this with DD8 for a while, because I don't do algorithms for a while, and this is what she came up with. I thought it was helpful, because it cemented the connection between multiplication and division, and because it let her work on her intuition. 

I wound up having to prod her to use multiples of multiples of 10 for quite a while... she wasn't ready for something like a year, and she would do what your son did. Eventually, though, she understood the idea with the powers of 10 and was ready, and then we did that for a few weeks and jumped into long division. 

Anyway, this is all to say I bet you'll be able to do long division in a bit, and your son's grasp of the algorithm will probably be deeper than it would be otherwise. 

[Not_a_Number]

29 minutes ago, OneThoughtMayHideAnother said:

Just gave this to my son. It took him a little under four and a half minutes, and that's with me reminding him to focus. 🙂 I should probably set a reminder for myself to time him again in a year or two once his handwriting is faster and more fluent. 

He's so much younger that DD8, though! And that was definitely the kind of time she was getting until we started drilling, even with my reminders. He sounds he's doing great 🙂  .

[daijobu]

16 hours ago, Not_a_number said:

What’s BA’s method and what’s short division?

 

BA uses what I believe is called "partial quotients."  I like the partial quotients method, especially for when you are dividing by numbers that I don't know a lot of multiples.  Like .  I don't know a lot of multiples of , maybe just  and .  Unfortunately I make more errors because I have the additional step of adding all the partial quotients at the end, and I often silly that part.  So I'll consider the traditional algorithm first and then partial quotients if I know I'll have difficulty.  

I also like teaching partial quotients because I like the BA story about the pirates and gold coins.  I can't remember if it's in the book or if I just made it up, but I also appreciate that the one doing the actual work of division gets to keep the remainder as a fee.  

Edited September 26, 2020 by daijobu

[Wheres Toto]

23 hours ago, kirstenhill said:

It would take me much longer than a minute to do that problem...LOL.  😄  I am pretty slow at computation.  (As an aside, my DS13 who is super fast at computation and has done well in a speed-computation-related math competition used to sometimes race with me on a calculator and him doing a problem in his head.  He would win at least half the time on mid size problems.  I can't even use a calculator very quickly I guess!).

Here's short division. I mean, it's basically a short cut.  So, conceptually I probably shouldn't have taught it before long division, but it doesn't seem to hurt anyone.  I just think it is the coolest thing so I am eager to tell everyone about it.

 

BA's method of long division is something like asking yourself to make an estimate on how many times the divisor fits in the whole dividend.  Write that on top where you would normally be writing your digits one at a time.  Whatever estimate you want to make is fine as long as it is not too big.  BA suggests picking something easy to multiply by like a multiple of 10, 100, etc.  Then multiply your estimate by the divisor and see how much is left.  Then make a new estimate of how many times your divisor will fit in what is left, and write that on top of the last estimate.  Later, Rinse, Repeat until you only have a remainder left.  Then add up your stack of estimates to get the final quotient.  If that doesn't make sense I can try and post a photo of a problem done that way later. 

I can see how it can theoretically be faster if you make really excellent estimates...but our experience so far is that DS ends up doing many more subtraction problems because he "plays it safe" with small estimates, then having done an easy multiplication for each estimate, still has to subtract every time to get down to the remainder or lack therof.

I've seen this called Chunking.   It definitely seems easier if you don't have all your multiplication/division facts down solid.

[Not_a_Number]

7 hours ago, Where's Toto? said:

I've seen this called Chunking.   It definitely seems easier if you don't have all your multiplication/division facts down solid.

Yeah, we were definitely using it before the facts were solid. And we were still working on using place value to multiply 🙂 . It was good practice. 

[kiwik]

I have never been required to do long division fast.  It is useful to know but only dinosaurs like me use it instead of their phone.

[Not_a_Number]

1 hour ago, kiwik said:

I have never been required to do long division fast.  It is useful to know but only dinosaurs like me use it instead of their phone.

Hahahah, I definitely use my phone if I need to divide!! But I don't think my 8 year old should be doing that yet 😉 

[melbotoast]

On 9/25/2020 at 8:12 AM, kirstenhill said:

RightStart math teaches it! I used RS math with my DD in grades K-4 and we both learned it there. 🙂

Us too! I'm sad I didn't learn short division earlier! DD's has low processing speed (per eval) so I feel like math problems always takes her forever. I have her doing xtramath to practice her basic math facts to help increase her speed. She also still uses a multiplication chart. And....we haven't gotten to dividing by 2-digits yet. At our house it will take as long as it takes. 🤷

[Moonhawk]

For more data: DS9 did it in 2'08", but he had an issue with his dry erase marker and had to find a new one, it was probably closer to  2' or right under that.

DD11 took 2'13".

These were surprise runs where I interrupted their drawing time, they weren't warmed up. I'll be interested to see if on Monday if I give it to them (or a similar question) in the middle of math class, if that would change the results.

Decimals would probably throw DD11 into a panic, DS9 wouldn't mind. 

[Not_a_Number]

On 10/4/2020 at 12:17 PM, Moonhawk said:

For more data: DS9 did it in 2'08", but he had an issue with his dry erase marker and had to find a new one, it was probably closer to  2' or right under that.

DD11 took 2'13".

These were surprise runs where I interrupted their drawing time, they weren't warmed up. I'll be interested to see if on Monday if I give it to them (or a similar question) in the middle of math class, if that would change the results.

Decimals would probably throw DD11 into a panic, DS9 wouldn't mind. 

Thanks!! 

Hmmm, I need to give her a test that's actually exactly like this one, and not a decimal. She's generally been coming in between 2 minutes and 3 minutes, and the decimal does take a bit of time. She hasn't done it in under 2 minutes yet, so I'm probably going to move her goal up to 2:30 so we can finally move on from arithmetic practice, lol. It's been very good to get everything fluent and integrated, but we're getting booored. 

[Little Green Leaves]

This thread has done me good!

I taught my son division a few years ago, and I made sure that he really knew what he was doing. Fine. But recently, I have been indulgently turning a blind eye while he does division (and multiplication) in his head, rather than on paper. This process can be either weirdly fast or surprisingly slow. It is mostly very accurate, which is why I've allowed it to go on so long. It is also kind of cute and he gets a kick out of it.

Anyway this week, with this thread in mind, and after he made a few mistakes in his mental division, I've made him spend lots of time working long division problems on paper, and checking them on paper. At first there were lots of grumbles. He had half-forgotten how to do things on paper, but it was actually a lot of fun refreshing his memory. He'd been basically doing it in his head the way you'd do it on paper -- so he understands the concept, but he had forgotten the method, if that makes sense. 

Editing to add that I haven't timed him at all. Maybe after a little more practice I will, I don't know.

Edited October 8, 2020 by Little Green Leaves

[Not_a_Number]

3 hours ago, Little Green Leaves said:

This thread has done me good!

I taught my son division a few years ago, and I made sure that he really knew what he was doing. Fine. But recently, I have been indulgently turning a blind eye while he does division (and multiplication) in his head, rather than on paper. This process can be either weirdly fast or surprisingly slow. It is mostly very accurate, which is why I've allowed it to go on so long. It is also kind of cute and he gets a kick out of it.

Anyway this week, with this thread in mind, and after he made a few mistakes in his mental division, I've made him spend lots of time working long division problems on paper, and checking them on paper. At first there were lots of grumbles. He had half-forgotten how to do things on paper, but it was actually a lot of fun refreshing his memory. He'd been basically doing it in his head the way you'd do it on paper -- so he understands the concept, but he had forgotten the method, if that makes sense. 

Editing to add that I haven't timed him at all. Maybe after a little more practice I will, I don't know.

I'm glad it was helpful! 

We're done our drilling for this round as of TODAY 😄 . We wound up with long division taking around 2 minutes, a bit longer if you include "finding the decimal point." Most operations are now taking under 3 minutes, with some much faster than that, which is a relief. 

I've gotten pretty tired of drilling for the last month, but I'm really hoping that it'll make everything faster from now on! I'll report back...