Long division with zeros in the quotient

[Farrar]

Help! One kid got it fine. The other one it's like pulling teeth. I've tried a bunch of things. Whenever we take it back to partial quotients, he gets it. I feel like he understands the place value, but the algorithm is leading him astray.

 

Any thoughts on worksheets or resources specifically about this?

[Farrar]

ETA: Found a better image to illustrate the second algorithm...

 

By the way, this is what I mean about the algorithm messing him up...

 

I learned long division with this algorithm:

 

PICTURE removed

 

When you "bring down" the next number, that's how you know you've advanced a place value and need to put a zero in the quotient if your divisor won't "fit in" the number you have.

 

But ds's text taught him more like this way:

 

PICTURE removed

 

I understand the advantage. It helps kids see the place value and is a better lead in from the partial quotients method. You see that you're really not dividing it in 2 times, but 20,000 times and that you're not "taking out" 6, but 60,000. But it's harder to see when you have a zero in the quotient because you never "bring down" the numbers. They're just down on every single row and you keep looking for the place that the number will fit and take out that part... Argh, I'm not explaining well with good math terms, but I see why it helps conceptually, but it's really leading him wrong.

 

His twin sort of learned with the same algorithm but it somehow never threw him off for zeros. Triple argh. I tried to teach ds the first way, but he was really married to the second. His twin let me teach him the second way as quicker and was fine with it.

[Lori D.]

Ug. I feel your pain. Long division took DS#2 a number of tries over an entire year and into the next school year before it clicked. I think for him, it just took time for the math portions of his brain to mature enough for the concept to "click".

 

Always hard to know whether to keep pushing through, or whether it's better to set it aside for a week or two and work on something completely different and let the problem topic "simmer on the back burner". I tend to lean towards the "set aside" option, which also helps defuse the student's frustration -- I had one who could really work himself up about not getting a topic, and then he was "locked out" brain-wise for being able to move forward.

 

Just me… I'd probably find something completely other to do for two weeks -- Geometry Labs (Picciotto), Working with the Geoboard, Hands-On Equations, Mathlink Cube Activities for gr. 3-6 ...

 

When you come back to it, you and DS might check out the "Division Woman" explanation in this video -- she visually shows the need the "Zero the Hero" to step in at that point so you not only hold the place value in the quotient, but also to help you do all the steps as usual:

 

divide -- the divisor is too big to go into the next part of the dividend, so, write zero in the quotient

multiple -- zero times the divisor, which equals zero

subtract -- when you subtract zero, you end up with the same number you started with

bring down -- bring down the next digit of the dividend and start the process again with dividing

 

My guess is that trying to combine steps right now (by just writing zero as the place holder in the quotient) and not following thru with the multiple, subtract, bring down steps, is throwing DS off in the process.

 

 

BEST of luck! Warmly, Lori D.

 

[kiana]

I like the second method a lot better, and Lori D's response about doing all the steps with the 0 rather than just saying "Ah, unnecessary" is spot on.

 

When your KID notices that you don't have to do all the steps with 0, that's when going straight to "Ah, unnecessary" works. 

 

Funny thing is that I didn't learn how to do long division until we did it with polynomials in algebra class. Suddenly a light bulb went off, and I said "Oh! This is what they were trying to make me do before! Just all the x's were 10s before!"

[mom31257]

Let me get straight what his problem is exactly. 

 

I watched the video that Lori linked (Division Woman). If your son were doing that same problem, he would write down 2700 and subtract leaving 48, right? Is his problem that he's not seeing to put the 0 in the tens place and the 5 in the ones place? Is he trying to put the 5 in the tens place?

 

 

If that is the case, I would show him the result of putting the 5 in the tens place. He'd actually have to write 450 underneath the 48, which is obviously way too big. 

 

I understand the second method actually showing the place value and how it works, but I still prefer the 1st method. My ds did much better with long division once I allowed him to use the 1st algorithm. 

 

 

 

 

 

 

 

 

[Farrar]

Let me get straight what his problem is exactly. 

 

I watched the video that Lori linked (Division Woman). If your son were doing that same problem, he would write down 2700 and subtract leaving 48, right? Is his problem that he's not seeing to put the 0 in the tens place and the 5 in the ones place? Is he trying to put the 5 in the tens place?

 

 

If that is the case, I would show him the result of putting the 5 in the tens place. He'd actually have to write 450 underneath the 48, which is obviously way too big. 

 

Yes, exactly. The problem is that he sees it when I show him like that, but he never catches it himself until he's done it wrong.

 

I had him doing one of the Middle School Math with Pizzazz worksheets (huzzah for the people who suggest this free resource!) since he clearly needed more practice (and he loves puzzle type worksheets for math). The answers were in the puzzle below so when he went to find them he saw every time he'd messed up. And there were a bunch of them. 

[Tanaqui]

Funny, when they taught me at school I never understood the first method at all and had to teach myself to divide by canceling out prime factors to get anything done! The second, newer method would've made much more intuitive sense to me as a child, I can see that clearly.I wish I had advice for you, though.

[Farrar]

Always hard to know whether to keep pushing through, or whether it's better to set it aside for a week or two and work on something completely different and let the problem topic "simmer on the back burner". I tend to lean towards the "set aside" option, which also helps defuse the student's frustration -- I had one who could really work himself up about not getting a topic, and then he was "locked out" brain-wise for being able to move forward.

 

I feel like it's a push through moment though. We already have side fun math stuff to work on (we're doing this book and some supplemental stuff about ratios and so forth). And he already worked all the division in MiF so he's already moved on to the second part of the chapter to do order of operations. I feel like he needs to keep up some level of momentum and practice the long division stuff. Like, he's almost got it...  it's just these stupid zeros.

 

I like your fun suggestions though. We've done two of them previously!

 

The video is good. I'm going to have him watch it at math tomorrow. It's basically what I've been telling him, but sometimes another voice, you know?

 

Thanks!

[mom31257]

I tried to come up with a way which might help him. How about doing the division like this example? 

[Roadrunner]

My little one forgets the standard algorithm every time I teach it. He has no problem with partial quotients. My solution? I decided if partial quotients is good enough for Beast, it's good enough for us. No help here, just commiserating.

[Lori D.]

We already have side fun math stuff to work on (we're doing this book… 

 

Oh.my.goodness! Just the title alone had me! I SOOO would have gotten that to use here!  :laugh:

 

 

 

The video is good. I'm going to have him watch it at math tomorrow. It's basically what I've been telling him, but sometimes another voice, you know?

 

Yes, exactly. My sons, too. :) Now why IS that?? Kind of personally frustrating… BUT, ultimately if DS gets it, then that's what you're after, right?! :) Good luck!

[Miss Tick]

I just can't bring myself to "like" this problem, but I'm paying attention as I somewhat expect one of my twins to have similar issues when we get there.

 

At least your ds had a working method, no? You just want him to become fluent with the standard method? Does he do the "multiply to check" step?

[Farrar]

I just can't bring myself to "like" this problem, but I'm paying attention as I somewhat expect one of my twins to have similar issues when we get there.

 

At least your ds had a working method, no? You just want him to become fluent with the standard method? Does he do the "multiply to check" step?

 

Well, it isn't really working. If there's a zero in the quotient, such as 405, he'll end up with 450 as his answer. He can check, but a lot of these are getting to be really big numbers so if he estimates to check (which is the method that MiF uses the most), sometimes the estimate is reasonable for the answer even when it's wrong. And he can multiply to check, but when there's a remainder, there's that extra step. I may make him do it more.

 

I think I'm resolving to force him to stop doing it the way MiF teaches it and make him use the standard algorithm so he doesn't mess it up any more. Maybe MiF gets to the more traditional method in grade 6? I'm not waiting that long.

[MiMi 4under3]

When my son gets "stuck" on a particular concept, I find that if I leave it for a while (say, for several days, or even a few weeks) and re-visit again later, he usually gets it. If not, I wait some more...

 

It seems that his brain has to process it on some level before real understanding occurs.

[MiMi 4under3]

 How about doing the division like this example? 

 

That's interesting. I've never seen it explained this way. 

[sbgrace]

I'm using MIF 5th grade. Is this 5th or 4th? Can you give an example of a problem type he would miss?

 

I did have a struggler, both initially in 4th and then again in 5th when we started with 3568 divided by 15 and similar.

 

If it's any help, it appears that MIF switches to the standard approach in 5th grade, because I've seen nothing like your 2nd example this year. I looked back in 4A, though, because I don't remember teaching that at all! I do see it there. 

 

I'd just switch now. In fact, I think I may (must) have taught my kids that standard approach in 4th grade at some point. One son actually draws those arrows (still), and I didn't teach him  to do that in 5th grade, he brought it with him from 4th. With that child we did daily problems with a divisor of 2 for a really long time to get the steps down. I moved on to other things while he worked on 1-2 long division problems a day.

 

 

 

 

 

 

[mom31257]

That's interesting. I've never seen it explained this way. 

 

I haven't either. I just tried to come up with a way he would think of the 5 in the one's place instead of putting it next. 

[threedogfarm]

My daughter had the same exact problem.  I didn't like teaching division the second way.  I think that doing it the first way makes so much more sense.  Bring down the number from the top--still to small to divide?  Then put a zero in the quotient. . .and bring the next number down.  Each time you bring a number down, a number must be placed in the quotient. . .

 

I finally had to tell my daughter (who was determined to use the second way) that sometimes we just have to change to a different method that will work better for us.  It was easy for me to sway her in this way because she rides ponies and often times what worked with one pony will not work with another.  When presented this way, she was much more amenable to the change.

 

Another option would be to consider looking at Education Unboxed videos using the rod but I personally found it cumbersome to use the rods for the bigger division problems. 

 

 

[Kathie in VA]

I like to teach that long division is really a short cut! Kids love short cuts. Basically division is subtracting by groups. So 12 / 6 is really asking how many times can I subtract 6 from 12? 12-6=6. That's one. 6-6=0. That's two times. So 12/6=2. Simple but now try it with bigger numbers that can't rely on basic facts. It becomes a longer, tedious problem to work out... But it's fun to see how it relates when you compare the methods. In long division you do many of the subtractions in one move

[Farrar]

If it's any help, it appears that MIF switches to the standard approach in 5th grade, because I've seen nothing like your 2nd example this year. I looked back in 4A, though, because I don't remember teaching that at all! I do see it there. 

 

You're right. He is doing the fifth grade book. I think they had a sort of reminder and then just dove into the new algorithm - I think with the assumption that kids would find it easier or more intuitive? I don't think I focused much on it. They don't show the arrows most of the time if ever. I should go back and look... I have tried to convince him to switch but he refused. I kept thinking, well, it doesn't really matter much, but as we get to bigger problems and zeros, that's a problem.

 

My daughter had the same exact problem.  I didn't like teaching division the second way.  I think that doing it the first way makes so much more sense.  Bring down the number from the top--still to small to divide?  Then put a zero in the quotient. . .and bring the next number down.  Each time you bring a number down, a number must be placed in the quotient. . .

 

I finally had to tell my daughter (who was determined to use the second way) that sometimes we just have to change to a different method that will work better for us.  It was easy for me to sway her in this way because she rides ponies and often times what worked with one pony will not work with another.  When presented this way, she was much more amenable to the change.

 

Another option would be to consider looking at Education Unboxed videos using the rod but I personally found it cumbersome to use the rods for the bigger division problems. 

 

Yeah, I think I just have to require him to switch. We did start with the Education Unboxed method awhile ago and I thought it was great. But better as just an introduction. If I was still trying to do it with things like 45,747 Ã· 9 then it really would get unwieldy. I only have one thousand cube after all - not 45. ;)

[Farrar]

I haven't either. I just tried to come up with a way he would think of the 5 in the one's place instead of putting it next.

I may use that briefly to get him to switch. It's actually really similar to the first image I found of the second method where you include the zeros and subtract the whole dividend. Except when I looked at it, I was like, huh, that's not really what he's doing. But it's closer to what you did (though in the example below, the student made a mistake - though this algorithm would allow for corrections). See:

 

PICTURE removed

 

 

I'm remembering all the various switchovers now in algorithms. Like when my other kid didn't want to stop regrouping the whole number into columns Miquon style every time he subtracted. Oyvay.

[MrSmith]

I may use that briefly to get him to switch. It's actually really similar to the first image I found of the second method where you include the zeros and subtract the whole dividend. Except when I looked at it, I was like, huh, that's not really what he's doing. But it's closer to what you did (though in the example below, the student made a mistake - though this algorithm would allow for corrections). See:

 

 

 

I'm remembering all the various switchovers now in algorithms. Like when my other kid didn't want to stop regrouping the whole number into columns Miquon style every time he subtracted. Oyvay.

 

I teach a sort of oddball method that seems to work for my kid.  It's easier for me to show than actually put into words, but here is how I would have approached the example you provided.

 

In the example I write like this: XX (yy) where XX is the number and yy is the value of the number.

 

0. Start with the highest place value in the divdend (in this case ten thousand).

 

1. How many 37 ten thousands fit into 8 ten thousands?  None so skip this place since it's at the beginning of the number.

2. How many 37 thousands fit into 85 thousands? 2 so put 2 on the 5 (thousand) and subtract 74 (thousand), leaving 11 (thousand).  Bring down the 4 (hundred)

3. How many 37 hundreds fit into 114 hundred? 3 so put 3 on the 4 (hundred) and subtract 111 (hundred), leaving 3 (hundred).  Bring down the 3 (tens).

4. How many 37 tens fit into 33 tens?  None.  But we can't skip this place because it's in the middle of the number, so put 0 on the 3 (tens).  Bring down the 4 (ones).

5. How many 37 ones fit into 334 ones? 9, so put 9 on the 4 (ones), and subtract 333 (ones), leaving 1 (one).

6. Oops we ran out of places in the whole number, so what's left is the remainder.

 

At higher levels the method continues: how many 37 tenths fit into 10 tenths, and so on.  Note also that the written form is closer to the traditional method (write 74 not 74000).

 

By this reasoning in Step 4 the student sees why there should be a 0 in the tens place.

[Paige]

I teach a sort of oddball method that seems to work for my kid.  It's easier for me to show than actually put into words, but here is how I would have approached the example you provided.

 

In the example I write like this: XX (yy) where XX is the number and yy is the value of the number.

 

0. Start with the highest place value in the divdend (in this case ten thousand).

 

1. How many 37 ten thousands fit into 8 ten thousands?  None so skip this place since it's at the beginning of the number.

2. How many 37 thousands fit into 85 thousands? 2 so put 2 on the 5 (thousand) and subtract 74 (thousand), leaving 11 (thousand).  Bring down the 4 (hundred)

3. How many 37 hundreds fit into 114 hundred? 3 so put 3 on the 4 (hundred) and subtract 111 (hundred), leaving 3 (hundred).  Bring down the 3 (tens).

4. How many 37 tens fit into 33 tens?  None.  But we can't skip this place because it's in the middle of the number, so put 0 on the 3 (tens).  Bring down the 4 (ones).

5. How many 37 ones fit into 334 ones? 9, so put 9 on the 4 (ones), and subtract 333 (ones), leaving 1 (one).

6. Oops we ran out of places in the whole number, so what's left is the remainder.

 

At higher levels the method continues: how many 37 tenths fit into 10 tenths, and so on.  Note also that the written form is closer to the traditional method (write 74 not 74000).

 

By this reasoning in Step 4 the student sees why there should be a 0 in the tens place.

 

This is kind of how I do it as well except I draw pictures or use base 10 blocks at first as we go. I say how many can we take out, and then erase or remove the appropriate blocks. I know partial quotients works, but I find it slow and feels it leaves more room for errors. I try to get my kids using the standard algorithm as soon as possible. I would force your DS to use the standard algorithm but sit by him and do the method above visually with blocks until he got the hang of it. He would fill in the algorithm normally, but on the side we'd have blocks or pictures on the white board as we went through it. If he gets partial quotients, he should catch on pretty quickly- long division is essentially partial quotients with the math organized more efficiently. It's kind of like multiplication- the standard algorithm for multiple digits is really the same thing as multiplying in expanded form- it's just organized better.

 

[MrSmith]

This is kind of how I do it as well except I draw pictures or use base 10 blocks at first as we go. I say how many can we take out, and then erase or remove the appropriate blocks. I know partial quotients works, but I find it slow and feels it leaves more room for errors. I try to get my kids using the standard algorithm as soon as possible. I would force your DS to use the standard algorithm but sit by him and do the method above visually with blocks until he got the hang of it. He would fill in the algorithm normally, but on the side we'd have blocks or pictures on the white board as we went through it. If he gets partial quotients, he should catch on pretty quickly- long division is essentially partial quotients with the math organized more efficiently. It's kind of like multiplication- the standard algorithm for multiple digits is really the same thing as multiplying in expanded form- it's just organized better.

 

 

When written out, the method I described looks exactly like the standard algorithm.  I just placed an emphasis on place values while working.

 

My son does fine with division.  His problem now is that some numbers look like letters :p

[Farrar]

Yes, I've definitely talked out the standard algorithm a bunch similar to what you're saying there, Mr. Smith. If I'm there talking it out, or if we back it up and do partial quotients or show it with the rods or something, he seems to get it fine. And if there's no zeros in the quotient, then he is decent with them (he's still in the careless errors stage, but he's basically got it). It's just when left to his own devices, he misses those zeros every single flipping time.

 

If anyone knows of any problem sets that practice this specifically that would be awesome.

[kiana]

Yes, I've definitely talked out the standard algorithm a bunch similar to what you're saying there, Mr. Smith. If I'm there talking it out, or if we back it up and do partial quotients or show it with the rods or something, he seems to get it fine. And if there's no zeros in the quotient, then he is decent with them (he's still in the careless errors stage, but he's basically got it). It's just when left to his own devices, he misses those zeros every single flipping time.

 

If anyone knows of any problem sets that practice this specifically that would be awesome.

 

You can make them by taking a number with zeroes in it as the quotient, picking any number you desire as the divisor, and multiplying them together. If you specifically want a remainder, just add any number less than the divisor.

 

Example: q = 405, d = 17, qd = 6885 -- let's have a remainder of 4 -- ok, kiddo, divide 6889 by 17. 

[Plink]

Have you tried the graph paper and index card trick? Write the equation on graph paper so that the place value is extremely visible. Then, start with the card revealing only the highest place value digit, keep the rest of the problem covered. Divide multiply subtract start again. He should be able to see his skipped steps much more easily if he uses the card faithfully, although it will probably drive him bonkers at first.

[sbgrace]

It really sounds like you need to switch him over.

 

Could you go back to the 4th grade type problems, with a single digit divisor, but use the standard algorithm just to get accustomed to using it? Do you think that might help? The larger divisors add a bit more to the whole thing, at least they did for my two.

[8filltheheart]

Would a visual aids work? I'm thinking visual cues like colors.

 

Taking Kiana's example:

Example: q = 405, d = 17, qd = 6885 -- let's have a remainder of 4 -- ok, kiddo, divide 6889 by 17.

what if you wrote out a bunch of similar problems with color cues like 68 in one color, 8 in another, and 9 in another. For every color, there has to be a matching number in the corresponding quotient answer.

 

My kids would definitely latch on to that type of concept and not mind doing the work on a dry erase board with lots of markers.