In her book Knowing and Teaching Elementary Mathematics, Liping Ma wrote about this question and how teachers responded to it:
Write a story problem for $1 ¾ \div ½$.
[Pause here and think about the answer yourself.]
Many people find it hard to come up with a story problem that represents fraction division (including many math teachers, engineers, and mathematicians). Why is this hard to do? For many people, their schema for dividing fractions consists almost entirely of the “invert and multiply” rule. But there is much more to thinking about fraction division than that. So much in fact, that we can’t say it all in a single blog post. This is the first of several musings about fraction division.
The trouble with English
Consider this problem:
If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?
You probably know instantly that this is a division problem and that the answer is 6, because you know your times tables, and specifically you know that $2 \times 6 = 12$. If we say that $a \times b$ means $a$ equal groups of $b$ things in group, then a division problem where $b$ and $a\times b$ are known but $a$ is unknown is called a “how many groups?” problem. Here are some other questions that ask “how many groups?”
- If you have 1 ½ liters of tea and a container holds ¼ liter, how many containers can you fill?
- If you have 1 ¼ liters of tea and a container holds ¾ liters, how many containers can you fill?
- If you have ¾ liter of tea and a container holds 1 ¼ liters, how many containers can you fill?
Some people think that the last one feels like a trick question because you can’t even fill one completely. Because we know the answer is less than one, we could also ask it this way:
- If you have ¾ liter of tea and a pitcher holds 1 ⅓ liters, how much of a container can you fill?
So a division problem that asks “how many groups?” is structurally the same as a division problem that asks about “how much of a group?”, but because of the way we speak about quantities greater than 1 and quantities less than 1, the language makes the structure harder to see.
What other ways might we see the parallel structure?
Diagrams:
Equations: $$? \times2 = 12, \quad ? \times \frac14 = 1\frac12, \quad ? \times \frac34 = 1\frac14, \quad ? \times 1\frac14 = \frac34.$$ The diagrams don’t have the language problem. In all cases the upper and lower braces show the relation between the size of a container and the amount you have. Whether a whole number of containers can be filled (diagrams 1 and 2), a container plus a part of a container can be filled (diagram 3), or only a part of a container can be filled (diagram 4), the underlying story is the same.
Many people think of diagrams primarily as tools to solve problems. But sometimes diagrams can help students see structure or reveal other important aspects of the mathematics. This is an example of looking for and making use of structure (MP7).
The equations have an even clearer structure, but more abstract. They all have the structure $$\mbox{(quantity of containers)}\times\mbox{(size of a container)} = \mbox{(how much you have)}.$$
The intertwining of the abstraction of the equations and the concreteness of the diagrams is a good example of MP2 (reason abstractly and quantitatively).
Coming up next week: what else are diagrams good for?
Really like your points and illustrations. Thanks!
I am wondering about this problem and English. If I have a container, I think I can only fill one container…. English can really be problematic!
If you have 12 liters of tea and a container holds 2 liters, how many containers can you fill?
Is the last equation supposed to be equal to 3/4 instead of 1 3/4?
Yes, thanks, fixed now.
Limping Ma was one of my muses when I started to understand fractions! Thank you for that reference. Susan Lamon is the other muse. So helpful!
We do change the language rules when writing math problems. It’s as if we forget that thing about language’s purpose being to communicate….
“Many people think of diagrams primarily as tools to solve problems.” This point could be worth elaborating someday. Curricula sometimes position mathematical representations as just another way to get the answer, rather than using the representations to develop mathematical understanding along a coherent progression. The result seems to be a syndrome of relying on the representations both too much and too little – too much, as calculation aids; and too little, as objects of mathematical discourse.
This is from the 2001 NCTM Yearbook on representation:
All of us have an intuitive idea of what it means to represent a situation;
we do it all the time when we teach or do mathematics. We represent numbers by points on a line or by rows of blocks. We use equations and geometric figures to represent each other. We talk about numerical, visual, tabular, and algebraic representations. And we think about things using “private”
representations and mental images that are often difficult to describe.
But what do we mean, precisely, by “representation,” and what does it
mean to represent something? These turn out to be hard philosophical questions
that get at the very nature of mathematical thinking.
I believe that as mathematics itself evolves, new methods and results shed
light on such questions—that mathematics is its own mirror on the very
thinking that creates it. And sure enough, there is a mathematical discipline
called representation theory. In representation theory, one attempts to
understand a mathematical structure by setting up a structure-preserving
map (or correspondence) between it and a better-understood structure.
There are two features of this mathematical use of the word representation
that mirror uses of “representation’’ in this book:
• The representation is the map. It is neither the source of the representation
(the thing being represented) nor its target (the better-understood
object). When a child sets up a correspondence between numbers and
points on a line, the points are not the representation; the representation
lives in the setting up of the correspondence.
• Representations don’t just match things; they preserve structure. Entering
on a calculator an algebraic expression that stands for a physical interaction
is not, all by itself, a representation. If algebraic operations on the
expression correspond to transformations of the physical situation, then
we have a genuine representation. Representations are “packages’’ that
assign objects and their transformations to other objects and their transformations.