[Photo by scubadive67.]
Help! My son was doing fine in math until he started long division, but now he’s completely lost! I always got confused with all those steps myself. How can I explain it to him?
Long division. It’s one of the scariest of the Math Monsters, those tough topics of upper-elementary and middle school mathematics. Of all the topics that come up on homeschool math forums, perhaps only one (“How can I get my child to learn the math facts?”) causes parents more anxiety.
Most of the “helpful advice” I’ve seen focuses on mnemonics (“Dad/Mother/Sister/Brother” to remember the steps: Divide, Multiply, Subtract, Bring down) or drafting (turn your notebook paper sideways and use the lines to keep your columns straight).
I worry that parents are too focused on their child mastering the algorithm, learning to follow the procedure, rather than on truly understanding what is happening in long division.
An algorithm is simply a step-by-step recipe for doing a mathematical calculation. But WHY does the algorithm work? If our students could understand the reason for the steps, they wouldn’t have to work so hard on memory tricks.
[Photo by norwichnuts.]
Prerequisites to Long Division
There are a few things any student needs to understand before the long division algorithm will make sense. Don’t try to teach long division unless your student has mastered these concepts:
- Dismantling a Number
The student must know the basics of place value, that the number 77,582 is a shortcut for the addition problem “70,000 + 7,000 + 500 + 80 + 2”.
- Renaming a Number
This place value concept is more subtle, but it’s extremely important: 7,500 can be thought of as “7 thousands and 5 hundreds”, or as “75 hundreds”, or as “750 tens”, etc.
- The Concept of Division
The student must understand how division is related to multiplication and recognize the two basic models of division — sharing and measuring. Here is a clear explanation: Division (Parker & Baldridge).
- Times Tables Are Optional
A student does not need to have fully mastered the multiplication facts before studying long division. Let him keep a times table chart handy for reference, if needed.
[Photo by Plutor.]
Setting the Stage
I use a “cookie factory” metaphor to help my students understand the long division algorithm. Pretend you are running a cookie factory, and you produced 77,582 cookies today.
Yes, that many!
Most textbooks teach the long division algorithm with relatively small numbers that could be solved more easily with mental math. By working with large numbers instead, students repeat the pattern of steps several times, fixing them in memory.
So, you produced a huge batch of cookies. You need to ship them out to three warehouses — and to be fair, each warehouse must get the same number of cookies.
How will you split the shipments?
What Are Your Options?
You can ship cookies as:
- individual ones (a single cookie)
- boxes (1 box = 10 cookies)
- cases (1 case = 10 boxes = 100 cookies)
- pallets (1 pallet = 10 cases = 1000 cookies)
- truckloads (1 truckload = 10 pallets = 10,000 cookies)
Remember, there are three warehouses, and each must receive the same amount. How many cookies can you send to each warehouse?
[Photo by hjordisyr.]
Cookies by the Truckload
Obviously, you want to pack the cookies in the biggest shipments you can. That is the most efficient way to send them, and it will cost the least in shipping fees.
Can you send a whole truck (or two or more trucks) to each warehouse?
3 warehouses × 1 truck per warehouse = 3 trucks.
3 warehouses × 2 trucks per warehouse = 6 trucks.
We have seven truckloads, so we can send two trucks of cookies to each warehouse.
We will “build” our answer on top of the long division sign, so that we have room down below for subtraction. After all, we can’t send out the same cookies more than once. We have to mark them off as soon as we send them.
Remember, each truckload is 10,000 cookies. Be sure the “2” goes in the correct place value column.
Below, subtract the 60,000 cookies you just sent out. Don’t use a shortcut yet! Later, when you are sure the student fully understands each step, you can explain the “subtract, bring down” shortcut. You may even want to teach short division. But for now, until the process is mastered, you need to emphasize the full 60,000 with all the zeros.
17,582 cookies to go…
[Photo by norwichnuts.]
Next Come the Pallets
We had one truck we couldn’t send (since you have to send the same to each warehouse). So let’s open it up and unload those ten pallets, adding them to the other seven pallets we have ready to go. That makes a total of 17 pallets.
How many per warehouse?
3 warehouses × 1 pallet per warehouse = 3 pallets.
3 warehouses × 2 pallets per warehouse = 6 pallets.
…
3 warehouses × 5 pallets per warehouse = 15 pallets.
That is the best we can do. We can’t send those last two pallets to any warehouse, or the others will feel cheated.
Remember, we are building the answer on top. Add your five pallets per warehouse as a “5” in the thousands column, and subtract those cookies from the total we still need to ship out.
2,582 cookies to go…
[Photo by gemsling.]
Cases and Cases of Cookies
Now we open the remaining pallets and send out cookies by the case load. Two pallets = 20 cases, plus we had five cases already.
25 cases. How many per warehouse?
3 warehouses × 1 case per warehouse = 3 cases.
3 warehouses × 2 cases per warehouse = 6 cases.
…
3 warehouses × 8 cases per warehouse = 24 cases.
We can send eight cases per warehouse, which will use up 2400 more cookies. Remember, write how many each warehouse gets on top (in the correct place value column!), and take away the cookies you sent out.
182 cookies to go…
Break Out the Boxes
Okay, we had one case left, which is ten boxes of cookies. Plus we had eight boxes already, so that’s 18 boxes we can ship.
How many per warehouse?
3 warehouses × 1 box per warehouse = 3 boxes.
3 warehouses × 2 boxes per warehouse = 6 boxes.
…
3 warehouses × 6 boxes per warehouse = 18 boxes.
We can ship six boxes of cookies to each warehouse. Remember to subtract the 180 cookies that you send out.
That brings us finally down to the level of individual cookies — of which, in this problem, we can ship zero per warehouse.
And if you get that far and have a couple of cookies left over, I’d say it’s snack time!
[Photo by Tomi Tapio.]
Advanced Long Division
While the shipping metaphor helps the student understand the value of working with large chunks first and gradually moving to smaller amounts, such non-standard usage could become a stumbling block if we keep it up too long.
After a few problems, I switch from the shipping terminology to talking in terms of place value: How many hundreds (or thousands, or tens) per warehouse, rather than cases and pallets.
This process works the same with large divisors as with single digits. The only difficulty in dividing by a larger number, such as 24, is that the “how many per warehouse?” step is harder to do in one’s head.
Therefore, when working with a large divisor, I like to begin by jotting down a counting-by list in my margin: 24, 48, 72, …, adding 24 to each number until I reach ten of that number.
That makes it easy to see that 24 warehouses × 5 hundred cookies per warehouse = 120 hundred cookies, and that six hundred cookies per warehouse will be too many.
[But note my Erratum update below.]
When a student has reached this stage, there is no need to write all the place value zeroes and subtract every column each time, as we did in the beginner problem above. Instead, we work with just the relevant columns and use the “bring down” step, which is equivalent to unpacking the next smaller shipping unit.
Drafting tip: Notice how the arrows keep place value columns lined up (with no need to turn the paper sideways). As the answer is built up column by column, you must place a number up top for every arrow before bringing down the next digit. If you weren’t able to send out any packages of a certain size, don’t forget to write the zero!
Erratum
As Alex pointed out in the comments, the answer to that last division problem is wrong. Sorry! I’ll correct it one of these days, when I have more time. Or perhaps I should leave it alone, as proof that even teachers make mistakes…
Any math problem with multiple steps offers multiple places for a small error to mess up everything else. In my case, the error was “5 − 2 = 2.” Oops! Even though I proofread the text of this article multiple times, I missed the arithmetic mistake.
Which is a warning: When you are working this level of problem with your students, limit them to just a few problems per session.
It is so easy for anyone to let a little mistake slip through, but too many wrong answers in a row (each due to a different small error) can discourage a student.
It’s much less frustrating to work only a couple of problems, search out any errors — as I should have done by checking with a calculator — and correct them until they are perfect.
* * *
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“The Cookie Factory Guide to Long Division” copyright © 2010 by Denise Gaskins.
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What a great way to teach long division…It explains it clearly and in a way that will capture most kids’ attention. Interesting to me, because I am a 3rd grade teacher (and I don’t teach long division yet) and I use cookies to teach multiplication. It’s just one strategy I use to help kids, but it is a favorite for many kids. After they have an understanding of what multiplication is (repeated addition, groups of objects), then I have them roll a die. The first roll tells them how many cookies they get and they draw them. The second roll tells them how many chocolate chips in each cookie. They can then count the chocolate chips to get the total or the product of the multiplication problem. Of course, they each get a couple of real cookies to eat. Many kids use this strategy throughout the year to help them with the problems they have not yet mastered.
Great idea!!
I wish I had seen this about four years ago! What a great idea! ( We love cookies.) We just went through the long division struggle with my second math-phobic child. And for the second time, what saved us was doubledivision.org. It doesn’t exactly teach the why’s of division, but it gets them through it without tears. (Yay.)
Thanks. I’ll be back in four years, when child #4 is in fifth grade. :o)
Thanks. I’m going to print this out and study it and then have my boy study it (the one that should be past it by now).
Blessings,
Carol
This is such an excellent reinforcement of place value and it makes so much sense out of the division algorithm. I find that many middle school students have a very limited understanding of the place value system and multiplying/dividing by 10. I think your lesson would set the stage for converting among any other unit of measurement. Thanks for the great idea!!
I find it very helpful to use grid paper (graph paper, with squares that are at least 1cm in size so the younger ones have enough space to write one digit per box clearly). That keeps things lined up much more nicely than most kids are able to do with other paper.
Thanks for the comments, everyone!
Joshua, I have heard other people recommend grid paper, too. I’ve never tried it, since drawing the arrow has been enough to keep my kids in line (and easier than finding special paper). But for anyone who would like to experiment, here’s a link to grid paper with choose-your-size squares:
Plain Graph Paper PDF Generator
I wonder what you think of this step-by-step approach to teaching long division:
http://www.homeschoolmath.net/teaching/md/how_teach_long_division.php
It concentrates on the “how” first, not on the “why”. In the end of the article I have included links to the “why” part, for example
http://www.homeschoolmath.net/teaching/md/long_division_why.php
and a link back to this post.
Hi, Maria!
I especially like your Step 1 and Step 2, which I would consider mental math rather than long division. It’s a good way for students to get used to the long division symbol (the goesinta?) and to writing the answer on top.
Once the student gets to your Step 3, writing the subtraction and bringing down the next digit, I think the problems need to be big enough that mental math isn’t easy. My kids get (understandably) impatient with writing steps that they could do in their heads, so the problem needs to be big enough that they can’t keep it all in mind. That gives them a reason for writing it out. And, as I said above, working with a large number emphasizes the repetitive nature of the algorithm.
That might be just my kids, however. Your progression is logical, and the little steps will surely help people who are not so comfortable with math.
Error in your advanced long division. 3565 R22
Drats! And I proofread this a dozen times. 😦 How did I manage to miss that? Thank you for pointing out the error, Alex!
Thank you so much for this, Denise. It’s exactly what I needed! It makes so much difference to my understanding not to skip to the shortcut too soon. I like that you didn’t correct the arithmetic error – it helps enormously to know everyone’s human and makes mistakes! (Especially when multiple steps are involved!)
Re Advanced Long Division sum above. It’s wrong….85 minus 72 makes 13 not 12…! Jesus wept.!
Yes, you’re right! Check the Erratum section at the end of the article. 🙂