Division of fractions part 3: why invert and multiply?

We ended the previous post with a bit of a cliffhanger, with two possible diagrams to represent $1\frac34 \div \frac12$:

The first of these diagrams is more familiar to students because it reflects their past work, but the second is more productive for understanding “dividing by a unit fraction is the same as multiplying by its reciprocal.”

Why is the first one more familiar? In grades 3 and 4, students study both the “how many in each (or one) group?” and “how many groups?” interpretations for division with whole numbers (see our last blog post for examples). In grade 5, they study dividing whole numbers by unit fractions and unit fractions by whole numbers. But, as we mentioned in that post, in grade 5 the “how many groups?” interpretation is easier when dividing whole numbers by unit fractions because students do not have to worry about fractions of a group. Going from $3 \div \frac12$ to $1\frac34 \div \frac12$ using this interpretation feels fairly natural:

The main intellectual work here is seeing that $\frac14$ cup is $\frac12$ of a container, but because the structure of the problem is the same and that structure can be easily seen in the diagrams, students can focus on that one new twist. The transition also helps students see that “how many groups” questions can be asked and answered when the numbers in the division are arbitrary fractions.

So the “how many groups” interpretation is useful for understanding important aspects of fraction division and has an important role in students’ learning trajectory. It enables students to see that dividing by $\frac12$ gives a result that is 2 times as great. But it doesn’t give much insight into why this should be the case when the dividend is not a whole number.

The “how much in each group” interpretation shows why. Here are diagrams using that interpretation showing $3 \div \frac12 = 2 \cdot 3$ and $1 \frac34 \div \frac12 = 2 \cdot 1 \frac34$.In fact, the structure of this context is so powerful, we can see why dividing any number by $\frac12$ would double that number: $$x \div \frac12 = 2 \cdot x = x \cdot \frac21$$

This is true for dividing by any unit fraction, for example $\frac15$:In the diagram above, we can see that $1\frac34$ is $\frac15$ of a container, so a full container is $1\frac34 \div \frac15$. Looking at the diagram, we can see why it must be that the full container is $5 \cdot 1 \frac34 = 1 \frac34 \cdot \frac51$.

With a little more work to make sense of it, we can use this interpretation to see why we multiply by the reciprocal when we divide by any fraction, for example $\frac25$:In the diagram above, we can see that $1\frac34$ is $\frac25$ of a container, so a full container is $1\frac34 \div \frac25$. We can see in the diagram that $\frac12$ of $1\frac34$ is $\frac15$ of the container, so our first step is to multiply by $\frac12$: $$1\frac34 \cdot \frac12$$

Now, just as before, to find the full container, we multiply by 5:

$\left (1\frac34 \cdot \frac12 \right) \cdot 5 = 1\frac34 \cdot \frac52$

This shows that dividing by $\frac25$ is the same as multiplying by $\frac52$!

There is nothing special about these numbers, and a similar argument can be made for dividing any number by any fraction. Now students, instead of saying “ours is not to reason why, just invert and multiply,” can say “now I know the reason why, I’ll just invert and multiply.”

Next time: Beyond diagrams.

You may have noticed that I am back to publishing regular blog posts! My goal for now is a blog post every second Wednesday. I am now also trying to answer forum questions promptly. I want to thank the readers who took up the slack for the last year and a half in answering questions in the forums. In particular, I’d like to call out abieniek, Alexei Kassymov, and Lane Walker, whose answers were always spot on.

Now to the topic of this post. There has been a lot of talk since the standards came out about what they say about multiple methods for arithmetic operations, and I’d like to clear up a couple of points.

First, the standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply and divide. But this does not mean that you have to solve every problem in multiple ways. Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn’t mean you have to go to school by car, then by bus, then walk, then bike—every single day! The point of having multiple methods available is to encourage students to think strategically about what might be the best method for a given problem, not force them to solve every problem four times.

Second, the different methods are not unrelated; they form a progression, with the ultimate goal being the standard algorithm. For example, when students are first learning to multiply two digit numbers, they might use a rectangle to represent a product such as $42 \times 71$.

This shows the fundamental role of the distributive property in multiplying multi-digit numbers. You have to multiply each base ten component into each other one. Indeed, the same rectangle representation provides a visual proof of the distributive property itself.

At some later point students might just start writing down all the partial products, without using the rectangle to derive them.

Note the correspondence between the rectangle method and the partial product method, indicated by the colors. The first row of the rectangle and shows all the products by the 2 in 42 (in red); the second row shows all the products by the 40 (in blue). The products in the partial product method are grouped in the same way. There are many ways you can order the partial products, but if you group them as I have here, going from right to left in each two-digit number, as in the standard algorithm, you make an amazing discovery: you can add up all the partial products in each group (blue group or red group) in your head as you go along. That’s because, in each case, adding the 2 to the 140 or the 40 to the 2800, there are enough zeroes in the second addend to accommodate the first, so it is easy to write down the sum right away, without writing the addends separately.

OK, so it’s not always quite this easy, because every now and then you will have to keep in mind a bundled unit from the previous step (aka carrying), but you will never have to remember that for more than one step at a time, because each bundled unit gets used up at the next step. So if you invent a notation for remembering the bundled unit (what we used to call “little 1 in the corner” when I was growing up) then you can still avoid writing down all the partial products, and just compute the sum within each group as you go along. You have just created the standard algorithm.

The different methods are not isolated different ways of doing the same thing; they are steps towards fluency with the standard algorithm, fluency that is not fragile because it is supported by understanding.

When the Standard Algorithm Is the Only Algorithm Taught

Standards shouldn’t dictate curriculum or pedagogy. But there has been some criticism recently that the implementation of CCSS may be effectively forcing a particular pedagogy on teachers. Even if that isn’t happening, one can still be concerned if everybody’s pedagogical interpretation of the standards turns out to be exactly the same. Fortunately, one can already see different approaches in various post-CCSS curricular efforts. And looking to the future, the revisions I’m aware of that are underway to existing programs aren’t likely to erase those programs’ mutual pedagogical differences either.

Of course, standards do have to have meaningful implications for curriculum, or else they aren’t standards at all. The Instructional Materials Evaluation Tool (IMET) is a rubric that helps educators judge high-level alignment of comprehensive instructional materials to the standards. Some states and districts have used the IMET to inform their curriculum evaluations, and it would help if more states and districts did the same.

The criticism that I referred to earlier comes from math educator Barry Garelick, who has written a series of blog posts that aims to sketch a picture of good, traditional pedagogy consistent with the Common Core. The concrete proposals in his series are a welcome addition to the conversation math educators are having about implementing the standards. Reading these posts led me to consider the following question:

If the only computation algorithm we teach is the standard algorithm, then can we still say we are following the standards?

Provided the standards as a whole are being met, I would say that the answer to this question is yes. The basic reason for this is that the standard algorithm is “based on place value [and] properties of operations.” That means it qualifies. In short, the Common Core requires the standard algorithm; additional algorithms aren’t named, and they aren’t required.

Additional mathematics, however, is required. Consistent with high performing countries, the elementary-grades standards also require algebraic thinking, including an understanding of the properties of operations, and some use of this understanding of mathematics to make sense of problems and do mental mathematics.

The section of the standards that has generated the most public discussion is probably the progression leading to fluency with the standard algorithms for addition and subtraction. So in a little more detail (but still highly simplified!), the accompanying table sketches a picture of how one might envision a progression in the early grades with the property that the only algorithm being taught is the standard algorithm.

The approach sketched in the table is something I could imagine trying if I were left to myself as an elementary teacher. There are certainly those who would do it differently! But the ability to teach differently under the standards is exactly my point today. I drew this sketch to indicate one possible picture that is consistent with the standards—not to argue against other pictures that are also consistent with the standards.

Whatever one thinks of the details in the table, I would think that if the culminating standard in grade 4 is realistically to be met, then one likely wants to introduce the standard algorithm pretty early in the addition and subtraction progression.

Writing about algorithms is very difficult. I ask for the reader’s patience, not only because passions run high on this subject, but also because the topic itself is bedeviled with subtleties and apparent contradictions. For example, consider that even the teaching of a mechanical algorithm still has to look “conceptual” at times—or else it isn’t actually teaching. Even the traditional textbook that Garelick points to as a model attends to concepts briefly, after introducing the algorithm itself:

Brownell et al., 1955

This screenshot of a Fifties-era textbook is as old-school as it gets, yet somebody on the Internet could probably turn it into a viral Common-Core scare if they wanted to. What I would conclude from this example is that it might prove difficult for the average person even to decide how many algorithms are being presented in a given textbook.

Standards can’t settle every disagreement—nor should they. As this discussion of just a single slice of the math curriculum illustrates, teachers and curriculum authors following the standards still may, and still must, make an enormous range of decisions.

This isn’t to say that the standards are consistent with every conceivable pedagogy. It is likely that some pedagogies just don’t do the job we need them to do. The conflict of such outliers with CCSS isn’t best revealed by close-reading any individual standard; it arises instead from the more general fact that CCSS sets an expectation of a college- and career-ready level of achievement. At one extreme, this challenges pedagogies that neglect the key math concepts that are essential foundations for algebra and higher mathematics. On the other hand, routinely delaying skill development until a fully mature understanding of concepts develops is also a problem, because it slows the pace of learning below the level that the college- and career-ready endpoint imposes on even the elementary years. Sometimes these two extremes are described using the labels of political ideology, but I have declined to use these shorthand labels. That’s because I believe that achievement, not ideology, ought to decide questions of pedagogy in mathematics.

Jason Zimba was a member of the writing team for the Common Core State Standards for Mathematics and is a Founding Partner of Student Achievement Partners, a nonprofit organization.