Home School Life Journal From Preschool to High School

Home School Life Journal ........... Ceramics by Katie Bergenholtz
"Let us strive to make each moment beautiful."
Saint Francis DeSales

Advanced Division: Long Division Theory

This place value board shows small green circles which can totally hold 10 single counters in the ones column, 10  medium blue circles which can hold 10 cups each with 10 counters in them for the tens column and (off the picture)10 large red circles for 10 bowls, each of which can hold 10 cups of 10 counters for the hundreds column. A gallon sized aluminum can can hold 10 bowls for the thousands column, when they get to that point.
We have been using a place-value counting board for some time now, but it has been a while since I have talked about it. It has grown to inlude new columns to the left as number concepts have grown, and we have even been able to dispose of the actual board. They now make their own columns on a piece of scratch paper to make a "board" as they need it, so now it is available for them to use whenever and where ever they need it.

One of my boys has been struggling with the switch from simple division to long division and it was because I switched from giving him the theory behind divison to supplying him with rules of steps to follow. He didn't understand why the steps were there or what they were doing, so he couldn't keep them in his mind. Some students are able to memorize the steps without this theoretic background, but eventually even these students develop trouble with math because eventually there are too many steps to memorize. It is these students that suddenly have trouble in math when they hit higher math, when they didn't seem to have any difficulty prior to that point.

So, I backed up and went back to the place value counting board. First we used it as a tool to complete simple division problems, since that is where we left off. For example, if we had the problem 316 divided by 3, I have him put the appropiate amount of counters in the seperate columns. We have been doing this so long, he understands that the counters in the ten's column really represent cups of 10 counters, and that the counters in the hundreds column really represents bowls of 10 cups (with each cup holding 10 counters). If he did not understand this concept, I would need to start there and actually have these sets available to use. This process is time consuming at the time, but is invaluable in teaching the theory behind mathematical operations.

We used the system of division steps, not as a means to an end, but as a way of recording the manipulation of the tokens.

He was able to draw row to make a grid under the columns representing the divisor. In this case, since the divisor is 3, he made 3 rows. First he divided the three tokens (which represent the bowls of 10 cups, each holding 10 tokens) into 3 equal groups, in this case making 1 token for each square. Then he moved onto to cups. Since he only had one cup, he couldn't divide them into 3 rows, so he exchanged one of the tens pieces for 10 in the ones column (When we first did this, months ago, we would dumped out all the beans from the cup and put them into the beans column), and dividied them evenly into the 3 boxes under the ones column. This left one bean left over. This is the remainder. That was pretty easy for him to do. Now, we tried long division.

So, I gave him an easy one to start with. 4,321 divided by 5. We started with the thousand's column. Four counters cannot be divided into five piles, so he put a zero over the thousand's place. Now we will take the thousands and the hundreds columns, or 43. He could easily see that when 43 was divided into 5 piles, he had eight tokens in each pile with 3 left over. So, he put 8 at the top for the 8 tokens and 40 under the 43 for the number of tokens that were used in the piles and 3 under it for the remaining tokens.
Now we moved on to the tens column, which had two in it. We added it to the leftover 3 from the hundreds column to make 32.  He then divided thirty two into 5 piles, which made 6 in each piles, with two left over. We continued on in the same manner with the ones column, using the remaining two in the tens column with the 1 from the ones column, making 21. This was divided into 5 cups, making 4 in each cup, with one left over, which is the problem's remainder.

"With the aid of the abstract system of division to which they have been introduced, each student can now produce answers to division problems that are impractical to solve using materials."
-Robert Baratta-Lorton, Mathematics...A Way of Thinking
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  1. A has been struggling with understanding the purpose of the steps of long division too - I'll have to see if this works for her - thanks!

  2. I love this post. I was taught to follow the steps and I did well in math, but I recently saw an explanation of my 8th grader's math program and suddenly everything made sense. I understood why, rather than just it is what it is if you follow the steps. That is so important!

  3. It's worthwhile taking the time to demonstrate how and why something is done. You may have to show it once and then everything makes sense, including the procedural steps. It's no longer something to memorize but just steps to get to the solution. Then, math becomes enjoyable rather than a chore.

    But once they understand the reasons behind long division, I like to teach the short division and see if connection is made. If so, there's no need to continue doing long division. Better yet, it's better if a child is able to come up with a way to do division mentally by breaking down the numbers that makes sense to the child. Then you know, the child has a good number sense.

    Great exercise, by the way!


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