I was super stoked a few weeks ago to figure out how to model multiplying and dividing fractions (I'll show you that another time!), but it took a Google search and this guy to teach me how to model long division.
I read his post on modeling long division probably 50 times, and watched his video maybe 50 more before I got it, but now I got it! And you should get it, too--it makes so much sense. It's so logical. You can SEE how the algorithm works.
And it's easy!
Although you really need to go check out that post for yourself and see the magic, I'll walk you through a couple of problems that will clear up the parts that I was confused about for a while, and you can see how it works for a big kid who's in the process of learning the long division algorithm.
The biggest thing that I had to wrap my head around is the way that you set up the model. I did NOT want to set it up this way, and it took a loooong time before I understood why it's best. You start by laying out your Cuisenaire rods in a rectangle, and then you put the long division sign over it, to show that you're measuring these sides of your rectangle:
Here's what I didn't like at first:
Can you see what I don't like?
In area and perimeter models, and any type of measurement, the number on one side illustrates the measurement of THAT SIDE. So in the problem above, I want the 3 and the 4 to be reversed.
But that's NOT how the model works. The 3 measures across, as the arrow shows, and the 4 measures down. You put the arrows there to help you remember.
You have to do it that way because of the way that you model the algorithm. Say that you're starting out with a model that looks like this:
Always build the model with the hundred flats first, then add the ten bars to the right and bottom ONLY, then fill in the last rectangle at the bottom right with Cuisenaire rods.
Have the kid count the total (good reinforcement of counting strategies and skip counting!) and the number of units across, then set up the algorithm so that it's next to, but a little higher than, the bottom of those hundred flats. Don't forget the arrow!
Your kid is going to want to immediately just count down to find the quotient, especially if she's calculated area before, but keep her focused on the fact that with this model, you're going to count how many 34s are contained within this number by subtracting out groups of 34.
The first thing that the kid does is count down to see how many whole tens there are of 34s, then separate them out. We'll count them first:
Syd has separated out the whole tens by moving the rest of the model down, and I've drawn a line under the whole tens to model that we're counting those first. See how the line extends to the algorithm? It's beautiful how much sense that makes!
The kid now counts how many whole tens there are. There are 30. Review place value, and review that 30 is the same thing as three 10s. We can write 30, and just replace the zero when we know how many units' worth we'll have, or we can just write 3 in the tens column. We've done it both ways:
After she knows that there are thirty 34s within the number we've separated, she needs to count the total number of units. Again, more skip counting and adding big numbers! Syd likes to count the hundreds and write the answer down, then the tens, then the units, and then add on paper.
That answer, of course, gets plugged into the algorithm. If you've got an older kid like Syd, you can ask her to double-check the model with calculations, if she seems game. It's a way to reinforce the calculations that she'll actually have to do when she's only got the algorithm, not the model. It reinforces that they both work exactly the same way.
Next, the kid counts the total number of units left below the line in the model, and plugs that number below the line in the algorithm--that's the number still left to divide. She can double-check the algorithm to see that subtraction will give her that answer there, as well.
The only limit to this is how many blocks you have to build models with!
It's fun to have the kid build models of her own to solve--
--but I'll also give her problems with the dividend and divisor, and she gets to figure out how to build the rectangle and then calculate the quotient.
Mind you, Syd does NOT love these lessons, which we've been doing all week. This is NOT play-based learning. However, each day that we've sat down together for no more than half an hour to work these models and do the calculations together, I can see not just how her understanding of how long division works growing, but also her overall number sense. Putting your hands on math, having to group it and count it and keep track of it and organize it, AND having to regroup it and count it and subtract it and count it again, AS WELL AS having to organize and note that on paper, is some hard-core math to do on a typical Tuesday morning.
Come Monday, we'll also likely be doing this!