Long Division Ideas from the Living Math Forum
Long division is a common area of struggle, this is an accumulation of ideas from members of th Living Math Forum.
Long division woes may take the lead for the most-often discussed topic on the Living Math Forum. This is a compendium of postings related to several threads on long division struggles posted on the LivingMathForum email list.
[LivingMathForum] Re:My son is having trouble with long division
Question: My 10 year old has just started long division in school and he's lost. He can complete the steps with lots of help, but he doesn't 'get it'. I've broken the dividend down and he sees it better, but not fully.
Are there any other ways to explain it than the standard school explanation? Anything like the lattice method of multiplying – he really took off with that and over time was able to adjust to the "standard way".
My youngest daughter learned what we called "Break It Up" division because she hated the long division method (I believe this is called the partial quotients method, but don't “quote” me on that ;o). She simply "broke up" the number to be divided into pieces that could easily be divided by the divisor. For example, 1757 / 15 could be:
1500 / 15 = 100 What's left after this piece is broken off? 257
150 / 15 = 10 What's left after this piece is broken off? 107
75 / 15 = 5 What's left after this piece is broken off? 32
30 / 15 = 2
The 2 leftover is the remainder
Add the 100, 10, 5 and 2 together and you get 117, R2.
This makes it easy to do with mental math skills, the child breaks off pieces he/she can easily divide. I would, for example, have divided 105 when 107 was left, but my daughter sees that 75 is half of 150, so she breaks that piece off since it's easy for her to see 15 goes in five times. This just works pretty well for her.
This is one resource that was sent around:
Re:My son is having trouble with long division
I feel your pain greatly - I have 4 boys - all really math minded but Long division is slow and boring (their words) -
I'm not sure if there is a name for this method or not - but my sons call it short division - you write the problem as you normally would let's say 531/4- ask the question what x 4 gets you close to 5 - the answer of course is 1 - ask with what remainder? The answer of course if 1 - write a small 1 before the 3 as you would if you're were borrowing or carrying and then start the process over with what times 4 would get you close to 13 - 3 with remainder 1 - write a small 1 next to the 1 - what times 4 gets you close to 11 - 2 with a remainder 3. This might not make any sense as it's hard to describe in an e-mail but instead of going down in a column you just jot the "remainders" in the original problem. This really helped my sons BECAUSE I realized that their main issue was too many steps and loosing track of what they were to do next. Conceptually they get division but I got questions like "why do I have to write down a subtraction problem when I'm doing division?".
We use Singapore and love it - they've been taught through their mental math methods to get it done in your head if possible and the "short division" way really helps them to visualize it instead of making all the notations.
I loved Double Divison for myself... it made so much more sense! But neither of my boys liked it. I thought it would appeal to them since they both love lattice multiplication.
Sometimes it's hard to figure out what will work and what doesn't. But I love that I am learning new methods; it helps me understand so much of the math I just didn't get when I was in high school and college.
This is the only way we were able to make it through long-division.
It's a really BORING game, but my son at least stopped writhing on the floor. It teaches long-division just by rote memorization of the algorithm. Not sure any real learning took place at the time, but I think it may click for him now.
Re: [LivingMathForum] Long Division Help
There are both good ways to break down division into phases so that what's really going on becomes more evident, and alternate algorithms involving estimation, partial quotients, etc. And of course, the two are intimately related.
Consider a simple division problem such as 54 / 6.
Have your son repeatedly subtract 6 from 54, keeping track of the number of subtractions until he has either 0 remaining or less than 6 remaining. This number is, of course, the quotient, and in this case he will be able to do this 9 times and have 0 left, hence the answer is 9 r. 0
Do another example (say 55 / 7) where there is a remainder, to be sure he sees why, and what the two numbers he winds up with (7 r. 6) really mean. Make sure he can "prove" that this is right by multiplying 7 * 7 and adding 6 to get 55, the dividend.
(Of course, you can use other examples, smaller numbers, physical models with chips or plastic bears or whatever feels right for you and your son).
Then have him do something like 132 / 33, to get into the notion that we can subtract larger numbers. But we keep the quotient manageable, in this case it is 4.
Next, go to something "not so manageable, like 1242 / 54. It's not outlandish, but repeatedly subtracting 54 from 1242 takes 23 steps.
This raises the same question about division that he hopefully has considered about multiplication as "repeated addition": is there a faster way than simply repeatedly subtracting the divisor, one at a time, from the dividend?
The answer, of course, is yes, because we can subtract multiples of powers of 10 times the divisor. Pause and make sure you understand what that means.
For example, in the previous problem, 1242 / 54, what is the biggest power of 10 we can multiply 54 by without "going over" 1242? The answer is 10 (that is, 10^1), because if we went to 10^2 = 100 and multiplied that by 54, we'd get 5400, which is already too big. But then, we can "do better" than just 10 x 54 = 540. We can in fact multiply 2 such groups without going over the dividend. So we say 2 and put that over the 4 in the dividend. But why? Because we're really multiplying not 2 x 54, but 2 x 10^1 x 54 or, more compactly, 2 x 10 x 54, or most compactly, 20 x 54 = 1080. But what we actually WRITE is 2 x 54 = 108, write the 108 under the 124 in the dividend, and subtract. This is one of the places the real confusion sets in. We're "hiding" in the notation what's really going on, because place value is secretly at work in this "compressed" or "compact" algorithm.
How do we get kids to see this? By having them write out the COMPLETE multiplication, and realize that the 2 really represents 20, and that the complete partial product we subtract from the quotient is real not 108, but 1080, and they should get in the habit of writing that extra 0. No harm can come from this.
From there, we subtract as previously suggested, and have a remainder of 162.
(note that in the traditional approach, we'd have a remainder of 16 and "bring down the 2," but this is a completely mysterious and mechanical step. By writing that trailing zero, it's OBVIOUS (sort of) what's going on: 20 x 54 = 1080, and when we subtract that from the original dividend, we have 162 left. That 16 is really 160, and then we take the remaining 2 and add it to get 162.)
From there, we ask if we can take any powers of 10 times 54 away, but even 10 x 54 is too big, so our next step is, technically, 10^0 power (the next smallest power of 10), which is 1, times 54. Can we take multiples of that? Yes, exactly 3 times 54. Which makes 162 exactly, which we subtract from the remaining 162 to get remainder of 0. Our quotient is 23. But that's 20 groups of 54 + 3 groups of 54, which makes 1242.
Again, check with multiplication.
I wouldn't necessarily go into the exponent thing with the powers of 10, but the student has to be able to thing in terms of 1, 10, 100, 1000, etc, in reverse order, times the divisor, and then see which power is the greatest s/he can use, and then check to see if there are multiples between 1 and 9 of that result which won't be too big to subtract from what's left.
If this is confusing, and without diagrams it very well may be, my apologies. But it's really what is going on in our standard long division algorithm.
Let's take the same problem with a "looser" approach: the student may look at 1242 times 54 and estimate that 100 x 12 = 1200, so 50 x 24 should also equal 1200. That's not too big, so we subtract 1200 from 1242, leaving 42. But we still have to multiply 50 x 3 = 150, which is bigger than 42. So 24 is too big for our initial quotient. The student may go down by 1 or so, or may go down to 40, or a host of other strategies, but any of them can lead to the correct solution as long as the individual arithmetic steps are right, and the student keeps track of the partial quotients, products, and remainders.
Another student might not be so bold and would use groups of 10 or 20 x 54, but that's okay. The point is to reduce the number of steps from the maximum, which is 23 individual subtractions of 54, and the closer we get to only needing basically TWO steps as in the first approach, the better. But accuracy and understanding are worth sacrificing a little speed for, in my opinion.
Re: Long Division Help
I teach grade 6 math and once the students hit division by 2 digit numbers they often hit the wall.
Looking at the problem1242/54 i suggest the student round the number up to the nearest 10....1240/60...rounding up gives the student the chance to estimate the first number to multiply...1240/60 (which also is the same as 120/6) gives them 20...they are then able to continue with the original question an multiply 54 x20 (1080) and and then subtract 1240-1080 (160) and then continue on from there.
RE: [LivingMathForum] Long Division Help
My daughter, who I thought understood it, demonstrated to me a few days ago that she had simply followed steps without understanding it, when we were working a Singapore math problem that required division. I WISH there was as simple a method as the Lattice Method for multiplication, my kids love this method.
What I realized, however, is that we could do something similar with the long division format if we were creative in understanding what it was about the lattice method we liked. Well, we liked the fact that the process was structured and you almost couldn't make a mistake. You could do different operations in a different order and it wouldn't affect the answer. It was aesthetically pleasing :o). This is what we came up with the past couple days.
On a problem she had already started and was having trouble with, I showed with colored pencils what was going on, matching colors to results. She had not caught on to the fact before that it could be an infinite process. She asked me to make a longer problem and show her how it worked, and then she kept asking me to add digits. Again, I made a division "lattice" in a sort of way, showing each step, and had her orally narrate to me each step as we went along. You can see what we created here: http://ph.groups.yahoo.com/group/LivingMathForum/photos/view/58f3?b=1 after which she got the answer to her question in the Singapore book (that's the "344" part).
Then she asked me to make her a super long problem to do, and help her do the "division lattice." That is in this photo: http://ph.groups.yahoo.com/group/LivingMathForum/photos/view/58f3?b=2 She did all the math in two sessions after taking a break in between :o). The thing is, it showed her that each step of the process has a specific beginning and end that she could *see*. Something about the color blocking caused it to stick for her. Now if she wants to, we can create the lattice before she puts in all the "data", just like with lattice multiplication. We did one today where we color blocked the areas before filling them in. But she didn't need this anymore after that one time (it's a lot more effort to do that). She did obviously see how important it was to line things up with the blocking, those are her dotted lines going all the way down.
Note this was a method – she actually did understand division, but couldn't seem to find a way to do long problems keeping track of the steps. If a firm understanding of what division is is not there, I'd go back to concrete representations like dividing up blocks with and without remainders.
She understood that if she had a zero remainder she could have just carried down the next number, but she didn't want to alter the symmetry of her blocks, so she just did the zero steps without that shortcut.
She has a math notebook, and when she forgets some things, she runs to the notebook for her best visual examples of things, so this is now filed away in her math notebook to refer to if she forgets the process again. The other day we were doing some word problems involving converting recipes and she was trying to remember fraction conversions such as two fourths equal four eighths, but we had done an exercise where we'd colored graph paper into a half, quarter, eighths, sixteenths, and 32nds, and just looking at that totally refreshed her memory on these.
I don't know if your son will respond to this idea or not. For us, because it was a stumbling point, I put the Singapore book away and we just focused on the process for a couple days. Usually they can't do that in school unless the whole class is struggling, so it's an afterschooling activity. If you have a good working relationship with your son and he likes the visual effect of this, it might help, it made all the difference for my daughter.
Re: [LivingMathForum] Long Division Help
From a mathematical viewpoint, I don't see this adding anything, and like lattice multiplication, there's the danger of mistaking an effective organizing tool being used as a black box for a method that promotes understanding.
You have to make a choice: do you care about whether kids understand why long division algorithms work, or does it suffice that they be able to do the process and get the right answer? If the latter, then your method will be attacked by educational conservatives as gimmicky and needlessly time-consuming (at least the ones who attack lattice multiplication that way are likely to do so). If the former, then I will criticize the method as failing to add conceptual understanding, just as lattice multiplication, the way it's taught in a lot of classrooms, fails to add such understanding.
You want to be sure that a method doesn't turn out to be more than a "neat trick." See my blog (rationalmathed.blogspot.com) for comments on the error of teaching lattice multiplication as a black box. On my view, mathematics should never be taught as black boxes, but it so frequently is as to be truly harmful for kids.
M., you are exactly right. I posted this expecting it would be used the way I used it - in a one-on-one contact setting with a child, with the parent / tutor monitoring the entire process and adapting it to the child. This was a creative process that evolved as I know my daughter, what attracts / motivates / triggers her to remember and engage, and it is working well for her to that end. It's an idea for a parent to try - if it doesn't fit the individual child, you try something else. It's a very different approach than teaching a class, because you have total flexibility to adjust to your child's signals of understanding or lack thereof. If you ever take the time to view the archived discussions here, you'll see that this is the perspective ideas are offered from.
Everything I do with my daughter is geared toward process and understanding rather than product and only "right" answers. She chooses lattice method as her preferred method after learning three different methods. But even as we work our way through Singapore and Primary Grade Mathematics by Zacarro, she looks at each problem for the best strategy to solve it, enlisting my help and feedback as she needs it.
For example, in a word problem involving multiplication of 59 by 12, she doesn't use the lattice. She adds the products of 59 times 10 and 59 times 2, because she can see that multiplying by 10 and doubling is much easier to do than setting up the lattice. She also recognizes this is exactly what she is doing if she does the lattice method, which she reserves only for longer multiplication problems she can't break up other ways, and similarly, it doesn't matter if she multiplies the 59 by 10 first, or by 2, as long as she does both, and adds both, she gets the right answer. My middle son, who is really amazing me with his ease in working through a formal algebra course, had similar likes. No black boxes here, although I certainly see the danger that a child can see it that way if all they are taught is the process.
My daughter is creative and aesthetics motivate her. We have done many art and math projects that she saves in her notebook, and the most visual and colorful projects she has created on her own are the ones that have seemed to be the kinds of things that drive lasting learning and that she refers back to. This was what gave me the idea, not that it was a one size fits all solution to teaching long division to any child.
For my daughter, the boxing was what caused her to see the regularity of the process; prior to that she got lost in a sea of numbers by the time she got to a third or fourth digit. As we talked through it on the smaller problem, she understood she wasn't dividing three into 10, she was dividing three into 10-hundred. But she didn't seem to grasp that the carrying down was of the next place until we repeated this over and over. The looonnng problem which she specifically requested that I set up for her, and which she asked me to add digits to even after she finished the initial problem, totally engaged her in the process. She only repeated the color boxing herself one time, and since then she does not seem to need or even want the boxes. So I see the exercise as something similar to what I do in my own homework when I color code and copy things down in a way that facilitates what I want to understand and remember.
I saw that spending the time to do this with her accomplished several things. The repetition of the process helped her remember where we put the elements of the process, which she had trouble remembering before. She could have done this by doing pages of practice problems, but this was far more interesting to her. She was figuring remainders in her head before this, and occasionally making mistakes - by showing the subtraction she could see where she was making mistakes. It *motivated* her to practice, and for me, this is a big positive. She can do shorter division problems pretty easily in her head, and has a good head for mental math, so she has tended to not willingly practice an algorithm like this until she can either see the reason for it or become proficient enough at it that she can enjoy the process. And the fact that I was willing to sit with her and work with her as long as she needed to understand the process meant a lot to her. I had no expectation that she "get it" by any deadline.
In our home educating environment which differs from a classroom, I allow my children to see learning as something that occurs all the time, and that we (meaning the child and I working together collaboratively) have choices in how, when and with what materials they learn. You can go with whatever works, and you don't have to be concerned that every base is covered in terms of the offered teaching and curriculum, because all you are doing is providing what they *need* based on your observation of what they have already learned or are effectively teaching themselves. This is of course not very easily duplicated in a classroom, the teacher must select what he/she believes to be the best presentation of concepts to the group as a whole.
My goal is to gradually move my children toward taking full responsibility for their own educations, and many homeschoolers have this goal for kids much younger than the more typical high school graduation date. Therefore much of my efforts are put into helping my children learn how to use resources to teach themselves, to find enjoyable ways to learn, and – what is probably most different from classroom situations - to be a participant in assessing how effective the materials, processes and timing is to their own learning process, using that collaborative feedback to make changes in materials, processes or timing as we see fit. This was an example of my daughter and I recognizing the explanations in her book and my explanations were not adequate, and our further work together to make these ideas comprehensible to her. Then, when my daughter and I do sessions like this, and she files work like this away in her binder to refer to later, it has a lasting benefit to her, so we deem it to have been valuable, even as it may not work for other children. Having taught groups of 15 to 20 or more at a time, I understand how different this approach is to classroom teaching.
The value to me of facilitating a child's learning in a way that is uniquely individual to them is the primary reason we homeschool. It fosters 24/7 learning attitudes in my children. I got home today from my kids martial arts class, sat down at our kitchen table with a cup of coffee to talk to my husband, and my 7-1/2 y/old climbed up in my lap with her Singapore math book and began working problems while we talked. She looked to me to give her feedback on her answers to some word problems, and after about 20 minutes put the workbook away to read - this was Saturday afternoon, no "school hours" mentality exists. My children's attitudes toward learning are developing in a self-driven way, and my goal is to be there when they need help, to provide them all the materials, support, facilitating and teaching they need when they need it. It is a way of looking at education that we have evolved to over nearly 11 years of homeschooling that really works well for us.
It is from this perspective that everything I suggest is offered, unless it's specifically stated otherwise, for example, ideas for a homeschool co-op class wouldn't be offered in the same way.
I'm interested in offering help to anyone interested in exploring more deeply how our "standard" algorithms work and to offer a functional (not in the technical sense), rather than historical view of how these algorithms developed. That is, we can readily imagine that multiplication derived from wanting faster repeated addition, and that division developed from wanting faster repeated subtraction (and that exponentiation derived from wanting faster repeated multiplication). There's a way of exploring finding square roots by hand that comes from the fact that the sum of the first n odd integers always yields the nth perfect square (e.g., 1 + 3 + 5 = 9 which also equals 3^2). You can therefore subtract consecutive odd integers from a perfect square to find its square root. Of course, if you knew the square root in advance, you could do this differently, but then you wouldn't need an algorithm.
What I find interesting about all of the above is the issue of compacting and loss of information. When we use any of the "standard algorithms" for multiplication or division, for example, we compact the process of the lower operation by grouping and using place value (hence, the left-shift of each partial product in "long multiplication" and the right-shift/bringing down process in long division). Once you understand what's going on, you don't need to think about it, and you just develop a certain level of automaticity. Only if you're confused or forget the algorithm (or have intellectual curiosity) is it necessary to get into the guts of the procedure. Then, you need to recapture the lost information, lost due to compacting and "assuming" place value. Kids go nuts with these algorithms for a host of reasons, but the shifting is one of the biggest "black boxes" in both algorithm, I believe. It's taught as a magic rule, not as a matter of sense-making. And my belief is that the more we allow students to think of mathematics as magical rather sensible, the more damage we do, at least potentially.
So for elementary teachers and home-schooling parents, I strongly advocate sense-making mathematics over procedural mathematics. Parents who never liked or got basic math may amaze themselves at how much sense it really can make. And once a child approaches mathematics from that perspective, I believe s/he will always want to ask good questions, will always demand understanding over mere facility, and will have fewer difficulties grappling with challenging problems and ideas.
But that's simply MY belief and I can't tell anyone else what he or she MUST do. Nor would I care to be in that position.
In this forum particularly, you are preaching to the choir when you say: "So for elementary teachers and home-schooling parents, I strongly advocate sense-making mathematics over procedural mathematics. Parents who never liked or got basic math may amaze themselves at how much sense it really can make. And once a child approaches mathematics from that perspective, I believe s/he will always want to ask good questions, will always demand understanding over mere facility, and will have fewer difficulties grappling with challenging problems and ideas."
Absolutely true in my experience. Algorithms are only taught well after conceptual understanding is there. My oldest son did not understand long division until 6th grade, and I was a little uncomfortable with that given he was my oldest, but I kept backing off the algorithm and focusing understanding. My middle children get math quicker, so it's been more like 4th or 5th grade for them.
My daughter seemed to understand conceptual division very well, she just wasn't getting the algorithm, in part I am guessing this is due to the fact we simply don't focus on algorithms, and only use those we see are truly worthwhile. And even then, I sometimes wonder why we do, my practical experience teaches me that when we get to the point where we'd be reading for the tool of lattice method or long division, in real life, we'll have a calculator or computer at hand to do the gruntwork of computing, freeing us up to focus on how to set up and solve the problem.
> Once you understand what's going on, you don't need to think about it, and you just develop a certain level of automaticity. >
Exactly. My oldest is a great case in point again. He was taking some homeschool classes in 2nd grade where he attended math classes three days a week as I was working part time when we decided not to go the traditional school track for him. The math text was teaching the long division process in 2nd grade, and he was simply in knots trying to remember the steps. That experience along with the trauma he experienced from "speed drills" was the point I realized that we could do better if we just took him home and homeschooled him full time without the support of the classes. The first thing I did was drop the procedure-based math text and get a curriculum that was highly regarded for teaching math sense. Now, years later, it's just natural for me to explain and teach this way with all my kids.
My daughter was having a difficult time understanding long division. I taught her short division with one-digit divisors and she was able to do that with little problem. Once she had that mastered she was able to understand the steps and see the need for long division with two- or more digit divisors. I know that short division is not usually taught in school, but it is what I use all the time, even with two-digit divisors. It is so much more useful than long division (less paper, less chance of error in keeping everything lined up). Here's a link that explains short division:
I found this through the MathingOff yahoo group.
Does anyone recommend a workbook on long division that has good explanation? I have general work books but the process is not good in mine. Thanks L
Re: [LivingMathForum] long division
I'm finding this to be a much easier thing to explain. There are also some links to other division pages here.
RE: [LivingMathForum] long division
Long division is discussed in a lot of upper level math readers http://www.livingmath.net/Library/MultiConceptLists.html - The Essential Arithmetricks comes to mind.
I just read Divide and Ride, a MathStart reader, to my 6 y/old the other day, she loves the book, and I realized that it is a great set up to long division, because it's all about taking a group of 11 kids and dividing them up on the rides to fill the seats. Of course, with eleven, there is always a remainder when you divide by 2, 3, 4, 5 or 6. There is a good visual representation of the division being done as they show each ride's seats being filled. They fill the remainder with kids they grab from the side to make new friends. Neat book.
The JUMP workbooks cover basic concepts in very small, incremental steps. Long division is covered in the grade 6 workbook that we are using (it may be covered in gr 4 or 5 as well but I haven't used those levels) and my son had no trouble catching on - the workbooks are a bit dry but my son enjoys them because they are so straightforward. You can see more about them at http://www.jumptutoring.org
RE: [LivingMathForum] long division
My 10 y/old and I were watching The Teaching Company's Basic Math series together. When the instructor got to long division, he emphasized again and again that the student estimate what they expect the answer to be *before* performing the long division algorithm. He does this by showing you how to round to numbers that will go into each other. You can guess whether you think the final answer will be more or less than what you rounded.
To estimate, you have to understand what long division is doing. It's sectioning off a big number into a certain number of "compartments" and indicating how much is left over when you are done filling these compartments up.
What it does is leave long division for what it really is designed for, a process to get an *exact* answer if you want one, vs. an estimate. Which makes me absolutely certain that long division generally shouldn't be taught to a child until they can comprehend this process and estimate closely. Otherwise, long division becomes a sort of black box where you put things in, and something comes out, and you don't know why or how to judge the accuracy of the process.
Hi, I fell like such an idiot asking this, being 16 and all, but I have a real problem with long division of any kind. can anybody help me with this?
We’re a Math-U-See family, so I’ll give credit to Steve Demme for this explanation, but it helps my son to understand it as an “area” problem.
If you have to find the area of a rectangle that is 2 yards by 10 yards, you know that the total area will be 20 sq yards. Easy.
Now, to turn this into a division problem, take that same 2, but put the area in the middle of the rectangle, and now try to find the amount you had to multiply the 2 by to get 20. 2 goes into 20, 10 times.
Now erase the lines and you actually “see” the long division.
All you are trying to do is find the other number to multiply by to get the area. If there is a remainder, then you know that it would be like finding the area of someone’s backyard and instead of a perfect rectangle, you have to work around their deck and so the “area” is the rectangle part, plus the small side next to the deck.
Area of grass: 61 sq. yards. You know that one side is 10 yards, so how long is the smaller side? 10 goes into 61, 6 times plus one yard. So you have a backyard that is 6 yards, by 10 yards, with one little sq. yard on the other side of the deck. If you break it down simply like this, and get a grasp on what you are trying to do, you can apply it to all sizes of back yards :-) or all sizes of long division!
Re: long division
I would like to suggest Math Matters, Understanding the Math You Teach, by Suzanne H. Chapin which has often been recommended on this site. It is aimed at teachers for grades K-8. It shows several ways that kids understand and work long division. I only recognized one method as traditional... But I understood a couple of the other methods very easily.
I would highly recommend ordering the book through your local library or consider buying it for yourself. It discusses all elementary math topics, so it is very appropriate. I have used it several times teaching my two boys.
The best explanations for long division I have seen are in the book "Math Matters." The authors show several ways that different people approach long division. The one that made the most sense to me was NOT the one that I learned in school. Here it is:
Lets say you want to divide 1967 by 15.
-1500 = 15 X 100
-150 = 15 X 10
-150 = 15 X 10
-150 = 15 X 10
-15 = 15 X 1
Add up all of the products on the right hand side (100 + 10 + 10 + 10 + 1) and you get 131 remainder 2.
You only have to estimate how much the smaller number goes into the bigger number to a certain point. If you guess too low, it's okay and you keep going. With the traditional method, if you guess too low, you have to go back and try again. You do have to know some basic times tables but you can choose the easiest ones for you with this method. I could have done 15 X 30 instead of 15 X 10 three times, but I don't have to. I can choose the numbers that work best for me. And it is really easy to multiply by 100's, 10's, and 1's.
If this doesn't work for you don't fret. Try to check out the book "Math Matters" and take a look at the division section.
My 12 year old is struggling with understanding long division. He knows all his division facts.. and he does fine doing the problems with me- spits out the answers very quickly. but he just doesn't understand it- or comprehend it to do the questions himself.. any tips or ideas? thanks, debra
Debra, my oldest son struggled to understand long division, and the Math U See trick of looking at a division problem as just a representation of area seemed to work for him.
To think of it with easy numbers first, say you have 32 blocks and you want to build a rectangle. If you make one side of the rectangle 4, how many would be in each row?
This is exactly what you are doing when you draw:
4 | 36
MUS even introduces it like this, setting it up just like an area problem earlier in the series:
4 | 36 | (there should be a complete box around the 36 but I can't make my email do that :o)
You can concretely build it out - if you have a rectangle of an area of 36, and make 4 rows, each row would have to have 8, and there are none left over.
This is exactly what you are doing with long division. If you pick a number like 38, and decide to build a rectangle out of 4 rows (i.e., divide 38 by 4), each row will have 8 and you will have 2 odd pieces left over - this is your remainder.
To go to more esoteric numbers, if you were dividing 137 by 12, what you are saying is, if I have an area of 137, and I wanted to build a rectangle that had 12 rows, how many would be in each row . . . Well, here is where I could use the process of dividing 12 into 130 first = 10 times with a remainder of 10, add the 7 and get 12 goes into 17 one time. You'll be able to build a rectangle 12 by 11, and you'll have 5 blocks left over.
This seems to be the best concrete way I know of representing long division before leaping to the abstract or process.
I thought Math U See does a great job of explaining the 'why' by completing the box formed by the long division lines and then looking at how the two factors that you have are multiplied to equal the 'area' inside the box....probably not very clear and someone better versed in MUS can explain better.... I'm going by memory here, since we don't use math u see (yet!).....we're so looking forward to starting it after the summer break.
My son could not understand long division until he had a thorough understanding of decimals. Over several years he would try long division and give up in exasperation. As his understanding of decimals increased, he suddenly "got" long division, in one 10 minute process after being so frustrated. I am not sure of the exact connection, but the concept of "remainders" had never worked for him and it had something to do with getting beyond that.
Long division has 4 steps for each digit in the dividend. We repeated the steps out loud and noticed a rhythm. Repeating this has helped my daughter do long division on her own.
"Divide, multiply, subtract, bring it down! Divide, multiply subtract, bring it down!"
This does not explain the why of division, but it does help them remember the sequence of steps. Hope this helps.
Here is a PDF file that will open a wonderful paper on the algorithms of mathematics. You will find many different ways (algorithms) to not only teach but to solve math problems - including your issue with long division. The traditional way you were taught of multiply and carry down does not help them with mental math - which is the number one way adults do math in the workforce. Go down to page 18 and 19 of this PDF file. We often use Stage 3; 'Expanded' method, when we solve long division. It is like doing partial products with multiplication. Pretty soon you'll find your kids are doing long division and big multiplication numbers in their heads. If you would like practice or other math resources, let me know what grade and I'll post them.
Here's a link that explains short division: http://www.themathpage.com/ARITH/divide-whole-numbers.htm
Long division is the bane of our life.
We used Miquon and then Singapore and both boys did an about face when we hit long division.
With the first child, I took a break and we explored living books. When we returned to the long division, it was no better. He started doing math with his Dad and his Dad just kept insisting that he use the formula. It took a while but he memorized and was able to manage. I did try using other long division methods with him (much like using the lattice method for multiplication), but they didn't work for us. He can now do long division comfortably.
With my second child, we introduced it once, he fiddled with it and didn't do real well. Then we moved on in the math book. When it came up the second time (just a couple of weeks ago), we took a break from Singapore and worked only on long division. We only worked on it about 10 minutes a day. We took it in easy strides. I broke it down into the smallest pieces possible. Two weeks later, he not only understands but he is very competent in using long division.
So, no "living math" solutions for this one, I'm afraid. At least not in this family. I think that's why we had the longest struggle with this one! We all wanted a different way but this is a sequential method and it means that it will challenge the non-sequential learner. But it doesn't mean they can't memorize it. It just means they will need more time.
Re: My son is having trouble with long division
This was a hard spot for us with Singapore math, too, because it seemed like they "skipped a step"---or rather, tried to teach too much at once. It would have been better, I think, if they had given the kids practice with the notation (putting the number to be divided inside the long division symbol and writing the answer on top) before they taught division with remainders. My son stumbled over the notation, especially the idea of writing an answer ON TOP, more than over the concept.
If you're looking for videos, Khan Academy has some very good ones, starting here. It starts out very simple, but he introduces the long division symbol near the end, then the follow-up videos get to more difficult problems. The second one stresses place value, etc.: http://www.youtube.com/watch?v=MTzTqvzWzm8&feature=youtube_gdata
Sometimes, the point of long division is easier to see if you work with a very big number to start with, rather than some small number your son could figure out in his head. Something like 67,589÷3=?
I used a "cookie factory" metaphor to help my students understand the steps. Pretend you are running a cookie factory, and you produced 77,582 cookies today. You need to ship them out to 3 stores, and to be fair, each store must get the same number of cookies. You can ship cookies as:
boxes of 10
cases (10 boxes = 100 cookies)
pallets (10 cases = 1000 cookies)
truck loads (10 pallets = 10,000 cookies)
How many cookies can you send to each store? Well, you will obviously want to pack the cookies in the biggest chunks you can, so first check to see if you can send whole truck loads. Can you send a whole truck (or two or more trucks) to each store?
2 trucks per store x 3 stores = 60,000 cookies
(Write it with all the zeros at first, to emphasize the place value.)
17,582 cookies to go...
Then, if you have truck loads that you can't send out (since you have to send the same to each store!), you will have to open those trucks and share out the pallets. So here we have 17 pallets:
5 pallets per store x 3 stores = 15,000 cookies
2,582 cookies to go...
And if you have any pallets left over, you will break them open and send out cookies by the case load:
8 cases per store x 3 stores = 24,000 cookies
182 cookies to go... And if there are any cases left over, you will send out so many boxes per store. And finally you get down to divvying up the individual cookies.
And if you get that far and have a few cookies left over, I'd say it's snack time!
Here's another video, a good introduction to the notation, and a bit shorter than the Khan Academy one: http://www.mathplayground.com/howto_longdivision.html
[LivingMathForum] Long Division Woes
Hi. I have a 12 yo dd who hates math. We use MUS primarily. Although she claims she hates math my dd loves to read and has read most of the 'Murderous Math' series (all the ones we own anyway).
Two years ago we did MUS long division. It was difficult - dd didn't like MUS's method of piling the nos. on top so I taught her the method I was taught which is to write the nos. down the side. This was OK and we got there in the end. But now we're doing long division of decimals and we're having a very teary time over them.
The problem with the long division appears to be that although she can do the method she is not keeping the nos. in the right places. For example her answer might be .0210, whereas the correct answer is .002001. She can also give very bizarre answers and be totally unaware that her answer, just intuitively, does not make sense. She doesn't look at her answers and think 'does this sound right?'.
Some of the problem seems to be that she makes her division way too long. Even in normal long division I notice this. So, instead of looking at a problem like 5689 divided by 83 and saying 'OK, I'll try and multiply by 70 first and see what I get since 7 x8 = 56 " she'll start with a small number like 20, and then she'll end up doing 20 twice more and adding the whole lot up at the end. I hope that makes sense. I've tried explaining to look at the first couple of digits and think about what goes in them to find a better number to start with, but she says this is too hard (this doesn't make sense to me as she knows her math facts better than us all). Any suggestions to make this easier ?
I've also wondered, since she can do the method, if we just forget the whole thing ? But then I remember that this is what happened to me at school with math - if I didn't get something we moved on and while at the time it didn't make a lot of difference by the end of high-school I was struggling.
RE: [LivingMathForum] Long Division Woes
P., long division seems to be such a common woe! I recall that it was literally the LAST STRAW that caused me to take my son out of a homeschool center math class and take over his curriculum etc. (we started out with a homeschool center that told us what books to use, etc. - It just took a while for me to figure out it wasn't that much different than school and I could do better!)
I can understand your dilemma. I have no problem using any sort of method for dividing larger numbers that works, my daughter is using a method she made up. But I can personally see how the more traditional method works well with decimals by keeping the place of the decimal straight!
A couple comments though about the problem you said she struggled with. I know, I know this makes some people uncomfortable, but honestly, in the real world no one in their right mind would do this by hand unless they loved doing math. I'd ROUND both numbers, estimate an answer, talk about whether the real answer is lower or higher than the estimate and do we think it's a lot lower, or a little . . . how close you feel you need to get if you are dividing something up for various purposes, and then use a calculator after that.
For example, what is 5,600 divided by 80? 70 of course, and that's close enough to the calculator answer of 68.542 for probably 95% of life's applications of this problem. That estimation process is the real life skill she'll need for the rest of her life, we will rarely of ever need to be more accurate than that without using a calculator or computer. Seriously, beyond dividing one digit into a longer number, which only means you have to recall your basic multiplication facts to do the problem, who would be doing this without a calculator today?? If she knows *how* to do it, I really wouldn't go on with the algorithm which seems to actually be doing her more harm than good, given what you wrote about her confusion and lack of ability to *see* the numbers. I wrote a review about a book called Arithmetic for Human Beings http://groups.yahoo.com/group/LivingMathForum/message/14627 here that you might want to take a look at if you have any trouble understanding why I am saying this, it is a GREAT book for understanding how this kind of blind application of arithmetic skills can and does kill any potential interest or enjoyment in math for so many kids especially in the pre-algebra phase. Maybe if you do this FIRST, and use the calculator afterwards, you can get her to begin thinking more the way you seem to understand she needs to, and not come up with nonsense answers. In fact, I'd be dealing with why she is doing that to begin with, and focus on that first.
You wrote: < But then I remember that this is what happened to me at school with math - if I didn't get something we moved on and while at the time it didn't make a lot of difference by the end of high-school I was struggling. >
But here is the beauty of homeschooling! Your daughter will never be "condemned" to the same fate unless you follow that same school model in how you go about learning. My own 12 y/old son and I sat down with an algebra text he worked part way through last year. We went back to the beginning, and we're reviewing the majority of it orally, just discussing and refreshing. It's customized to him - zero to minimal time reviewing things he remembers well, as much time as he needs on areas - I just earlier shared how he'd forgotten the right way to divide fractions, something he learned 2+ years ago. He just didn't use the skill for a while, and needed a refresher. We never got the time we needed in school targeted this way, nor were we ever taught how to teach ourselves, we were taught to rely on teachers.
Most of us didn't learn how to do manage our own learning until college, when we had a lot more choice in the content, teachers and schedule of our learning. My son does not see this effort as some indication he was "behind" - in fact, the pre-algebra portion of the review took us a mere 4 days. We've already skipped whole sections of the beginning algebra and celebrated the fact that he did in fact remember 90% or more of it. You can't do this in a classroom of kids, but you *can* do this homeschooling, you can use a text, or curriculum as a tool for you, not be a slave to it.
Re: Long Division Woes
I certainly am not an expert. But, I have an 11 yr old who is math challenged and I thought I would share what worked for us. We also use MUS, but got stuck in division.
I ended up going back to the basics with him on division. I used Maria Miller's Math Mammoth. I know I have seen her post on here. She has two websites. One is www.mathmammoth.com and the other is www.homeschoolmath.net. We used the blue series book on division. She explains the concepts differently than Steve Demme. It was what my son needed to be able to understand it. I still use alot of her worksheets for review. Thank you, Maria!
Just a quick note on MUS & children's learning differences. We met Steve Demme this summer at a homeschool conference. He is such a nice man. My son told him he was behind. Mr. Demme emphatically (in a nice way) said "No, you are not." He basically told him each child is different and nowhere does it say you must know certain math facts by a certain age. Some people may disagree with this. But, it helped my math challenged child to think more positively about his own math abilities.