[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.]
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.]
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. By working with a large number instead, we force our 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 3 stores — and to be fair, each store must get the same number of cookies. How will you split the shipments?
What Are Your Options?
You can ship cookies as:
- individual ones
(1 box = 10 cookies)
(1 case = 10 boxes = 100 cookies)
(1 pallet = 10 cases = 1000 cookies)
(1 truckload = 10 pallets = 10,000 cookies)
Remember, there are 3 stores, and each must receive the same amount. How many cookies can you send to each store?
[Photo by hjordisyr.]
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 store?
1 truck per store 3 stores = 3 trucks.
2 trucks per store 3 stores = 6 trucks.
We have 7 truckloads, so we can send 2 trucks of cookies to each store. 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.
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.]
We had one truck we couldn’t send (since you have to send the same to each store). So let’s open it up and unload those 10 pallets, adding them to the other 7 pallets we have ready to go. That makes a total of 17 pallets.
1 pallet per store 3 stores = 3 pallets.
2 pallets per store 3 stores = 6 pallets.
5 pallets per store 3 stores = 15 pallets.
That is the best we can do. We can’t send those last 2 pallets to any store, or the others will feel cheated.
Remember, we are building the answer on top. Add your 5 pallets per store 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.]
Now we open the remaining pallets and send out cookies by the case load. 2 pallets = 20 cases, plus we had 5 cases already.
25 cases. How many per store?
We can send 8 cases per store, which will use up 2400 more cookies. Remember, write how many each store 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 10 boxes of cookies. Plus we had 8 boxes already, so that’s 18 boxes we can ship.
How many per store?
We can ship 6 boxes of cookies to each store. 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 store.
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 two or three problems, I switch from the shipping terminology to talking in terms of place value: How many hundreds (or thousands, or tens) per store, 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 store?” 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 10 24. Then it will be easy to see that, for instance, 5 hundred per store 24 stores = 120 hundred, and that 6 hundred per store 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 I use arrows to keep my place value columns lined up. This is especially useful when the problem involves a zero in the quotient, which students so easily miss. As the answer is built up column by column, the student must place a number up top for every arrow before bringing down the next digit.
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 . Oops! Even though I proofread the text of this article multiple times, I missed the arithmetic mistake.
That reminds me: 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 the errors, and correct them until they are perfect.