So we’ve been working on operations with fractions in my seventh grades classes of late, and in particular division.

I’m all about helping kids to think flexibly about fractions—to have multiple strategies of computation that can be used according to the particular problem at hand. Also, I think it’s important for kids to have mental models for operations, including/especially when they’re extensions of models students are already confident with.

So we talk about scaling, and inverses, and think about “how many of these would fit into that”? In the last mode, 6 ÷ ½ can ask how many halves are in 6. The fancy term for this concept is “quotative division” (I had to look that up.) This is in distinction with “partitive division”—as in, “if I broke this into that many piles, how big would each pile be?” For more on this distinction, you can check out this Google Books preview. Basically, the distinction arises because we think of multiplication as groups of things, so that 4×5 can either mean 4 groups of 5 or 5 groups of 4.

In partitive terms, 6 ÷ ½ asks how big each pile would be if we put 6 into a half of a pile. This sounds a little weird and clunky, but sense can be made of it. If a half of a pile is 6, then a whole pile would end up being 12—even though we don’t end up making a full pile with the stuff we’ve got. So 6 ÷ ½ = 12.

Something neat that’s going on here, I think, is that often word problems introduce either quotitive or partitive contexts that can then be solved by the single operation of “division.” Here, I’m turning that framework on its head—division isn’t a single thing, but rather a bunch of related concepts and their corresponding models and strategies. To solve the single division problem 6 ÷ ½, we can pick the model that works best for us. I’m sure folks have thought about this before—reference in the literature, anyone?—but it’s an idea that feels personal and exciting to me at the moment.

Anywho, we were chatting about some fraction division problems in class, and I hadn’t planned on talking about this “half of a pile” stuff. Frankly, there are lots of other ways to help kids think about fractions, and while I’ve wrapped my head around the partitive way of looking at fraction division, it’s not something I usually put out there. It feels more like an interesting oddity that I’ve picked up in my mathematical wanderings.

But then I started talking about it, and some kids just ran with it and started thinking that way as they approached new problems. One kid in particular really took to it—the same kid as here—and I wish I had snapped a photo of her work. I’ve done a rendering of it below. So cool.

The thing I really want to share isn’t actually so much about dividing fractions at all. It’s about how surprised and excited and even a little stunned by how readily some of my students took up this way of looking at things that I had always regarded as tricky, tough, and esoteric. And, maybe most of all, weird and unorthodox.

The beautiful thing is that it didn’t seem weird and unorthodox to them. They didn’t arrive at that way of thinking somewhere in adulthood in some sideways fashion after having many previous understandings or techniques of dividing fractions. To them, it was just a model that made sense and that they could use.

I don’t know that there’s any better feeling than having my students stand on my shoulders. (:

Wow, Justin, this is really cool! I love the fact that when you say “6 ÷ ½ asks how big each pile would be if we put 6 into a half of a pile,” it matches the left to right (6, then ½) written form. More often, and I do this too, we’d ask how many halves are in 6, but to me this always seems as “backward” as how we set up a division problem in this the US.

I’ve been using pattern blocks for multiplying and dividing fractions with some success, and it’s nice for the kids to physically work with the pieces. But I definitely want to try this partitive division. It’s a little bit trickier to do something like 2/3 ÷ 4. I’d be asking, “how big would each pile be if we put 2/3 of a pile into 4 piles.” Still very cool to see this with visual model.

Thanks so much, Justin.

Not that relating the diagram to the standard invert and multiply algorithm makes this example of student thinking any more beautiful, but this diagram does help make sense of it. To multiply 6 by 8/3, I could divide 6 into 3 piles and then make 8 of these piles. This is the kind of stuff that came to my mind recently when Dan Meyer called for promoting square pegs. Thanks for sharing.