Although by now my kids well know that with almost every math problem there's an algorithm hiding in there somewhere, and that the algorithm will be even quicker and easier than the model and so they want to know it RIGHT NOW, I still try to whip out the mathematical models whenever possible, and especially whenever a kid is having trouble remembering or utilizing the algorithm. If, for instance, a kid can't consistently remember the algorithm for dividing fractions, that's because she doesn't understand dividing fractions. If she understands how it actually looks to divide a fraction, then she'll have a better idea of the answer that she's looking for, and that will likely remind her of the algorithm that she needs to use to get that answer.
You know how when you're trying to spell a word that you don't fully know how to spell, or trying to grammar check some tricky grammar, you often know when that word or sentence "looks" right, or when it doesn't? That's because you've read so much that you understand how words and sentences are formed, even if you don't completely remember the algorithm for your particular word or sentence. It's the same with math. If you understand how to physically divide fractions, and you're presented with, say, the expression 1/2 divided by 1/3, then even if you don't completely remember the tricky algorithm involved (which is to invert the divisor and then multiply), you should at least know that the answer isn't 1/6. You'll probably, in fact, be able to look at the two terms and know that the answer will be greater than 1, and that might be enough to remind you of the algorithm.
Here, then, is a quick-and-easy way to model fraction division using Cuisenaire rods. It takes longer than simply calculating using the algorithm, but it makes dividing fractions make SENSE, which, if you learned how to do it using only the algorithm, is a pretty big deal and might blow your mind even now.
Take this problem:
Syd's math curriculum asks her to work the following: 2 divided by 5/6. I asked her to interpret it in this way: "How many times will the fraction 5/6 fill into two wholes? Or, how many 5/6s are in two wholes?"
Step 1, then, is to wrap your head around one possible interpretation of a fraction division problem: how many times will the divisor fit into the dividend? You can work some whole number problems to illustrate that interpretation, if you like. How many times will 7 fit into 35? How many times will 10 fit into 120? You can even work those with the Cuisenaire rods first, using the method that I'm about to show you, to prove that the problems are essentially no different.
Step 2 is to set up the problem with Cuisenaire rods. Since we're being asked to divide 5/6, then the whole is clearly 6/6, or a six bar, and there are two of them, so set up two six bars. You need to divide that by 5/6s, so gather up some five bars to represent that, and some single units in case you're figuring out remainders. The five bars make up 5/6 of the six bars, and those singles will represent fractions of the five bar. One single unit is 1/5 of a five bar.
That alone is going to tell you that if you do have a fraction remainder, it's going to be in fifths. The remainder, if you have one, is always going to have a denominator the same as the numerator of the divisor, because that numerator is what you're actually dividing. It's not something you notice when you divide whole numbers, because when you divide whole numbers in our society, you're using the Base Ten system, and so denominators are ALWAYS in tenths (or hundredths, or thousandths, etc.). Not so when you divide fractions!
Step 3 is to notice how many of the five bars, which are 5/6 of the six bar, it takes to equal the two six bars, which are each 6/6. When you line them up together, you can see that to equal two whole 6/6s, it takes two whole five bars and two single units. Those two units represent 2/5 of the five bar.
2 divided by 5/6 = 2 2/5
Note that this model takes a LOT longer to explain than it does to do. I'd recommend just modeling the actions with the kid, not necessarily all of my blather that engulfs the simple and clear actions with a bunch of verbal baggage. The kid can see perfectly well what you're doing, and doesn't have to hold all the explanations in her mind for her mind to *know* it, if that makes sense. It's the way that when you read to a kid, you're not all, "Here's the subject of the sentence. It's a pronoun, but that's just because there's a noun that we just saw in the previous sentence, so we don't want to see that again so soon. And here we have a helping verb, which comes before the action verb. You know that next you'll see the direct object, unless you have an indirect object first. Oh, no, actually we've got a prepositional phrase next! How fun!", but nevertheless the kid internalizes all of those patterns well enough that when she's old enough, she'll know that something is wrong with a sentence that lacks a verb.
If your kid is just beginning to learn dividing fractions, you can do lots of these models, as many as the kid can stand, before teaching the algorithm as the tricky shortcut that gives you the same answer. Syd, however, had already been presented with dividing fractions, and was just struggling to remember the algorithm, so every time she completed a problem using the model, I then had her rework the problem with the algorithm to show her that they give the same answer, and to reinforce the connection between the two. It still takes more time and more practice until the algorithm is second nature, because it is NOT an intuitive one, but it all goes easier when it makes logical sense!