Delaying "Invert and Multiply"

“Ours is not to reason why. Just invert and multiply.”

As tempting as the old rhyme is, we recommend delaying this recipe for dividing fractions until students understand both division and fractions better. The danger is that students apply the rhyme haphazardly, where it doesn’t apply. Perhaps more importantly, “invert and multiply”—like cross-multiplying—is really a special-purpose tool. It’s better to understand more fundamental, more broadly useful tools.

This is not to say that students should never learn simply to “invert and multiply.” Neither should we insist, at this stage, that they be able to prove why “invert and multiply” works. Rather, we want to make sure, as we do with students using a calculator, that they understand the fundamentals so that when they get to algebra they will be well-equipped.

The burden on the teacher is this: you need to understand thoroughly why “invert and multiply” works, and you need to have various alternatives at your fingertips, so that when students start to use them (and misuse them) you will recognize the procedures—or fragments of procedures—they are using.

Alternatives to Division

Suppose Maria has 29½ inches of extra-long licorice whip, and she is cutting it up into 5½-inch pieces. How many pieces can she make?

You could say, after one piece, there’s 24 inches left; after 2 it’s 18½, and so forth, until you have made five pieces and there are 2 inches left over. That’s repeated subtraction.

Or you could say, one piece is 5½, two is 11, four is 22, six is 33 but that’s too much, so five is 22 + 5½ or 27½ inches, leaving two inches remainder. That’s essentially repeated addition with corrections.

Or you could say, 5 x 6 = 30, so 6 pieces must be too many. Lets try five: That would be 5 × (5½) = 25 + 5/2 = 27½. With two inches left over. That’s a kind of multiplicative guess and check.

Preparing for Division

It’s key to recognize that these are all good strategies. The problem is that when the numbers get messier, or the contexts get more complex, it will help to use division. And for division, and for students’ general good, it will help if they get comfortable with two seemingly arbitrary practices. But they aren’t arbitrary:

  • Convert mixed numbers to improper fractions. The terminology is unfortunate; improper fractions are just fine. When students start using variables, they will never use mixed numbers. Furthermore, the format of a mixed number invites misinterpretation. The moment they start using implied multiplication, students will often evaluate something like 5 2/3 as 10/3. So we should look at the licorice problem above as 59/2 ÷ 5/2.
  • Stop using slanty fractions during calculations. Instead, use more space and horizontal fraction bars. This does two things: it eliminates the danger of thinking that 2/3 + 2 = 2/5; and it sets students up for comparing and performing operations on numerators and denominators more directly. For example, when I see , it’s easier to see the 8 and the 15 than when I see (2/3)×(4/5).

In addition, and maybe obviously, students need to have a solid grasp of some fraction basics, including multiplying fractions, in order to understand division. In particular,

  • Anything (except zero) divided by itself is one.
  • Anything times one is itself.
  • Anything divided by one is itself.
  • Multiplication is commutative and associative.
  • To multiply fractions, you may “multiply across” in the way that is completely forbidden for addition.
  • A unit fraction times a whole number puts the new guy in the numerator, e.g., (1/7)×4 = (4/7). (This follows from the others, but it’s so often forgotten it’s worth noting on its own.)

With that in mind, let’s address a division-of-fractions problem without any context: What’s ? We’ll be focusing on computation here, not meaning.

As you watch students, be alert for work that starts to use any of the following approaches to fraction division. You might encourage them, either individually as needed or in a mini-lecture, and give them hints to keep them going.

Common Denominators

If students create a common denominator for the dividend and divisor, then they can divide “across”:

Thinking about it this way may be dangerous—dividing “across” only works if you have a common denominator—so it may be better to emphasize the underlying structure of this technique. We’ll begin by making the fractions tall:

Notice how we multiplied by one? And did you see how we used the “multiply across” rule to separate the fractions at the end?

“Units” Approach

In the previous approach, students might get here:

 

One way to think about this is that you have 45 twentieths over 8 twentieths. That’s 45 of a thing divided by eight of this same thing. And that’s just 45/8.

 

A reasonable student might ask, “But isn’t it 45/8 twentieths?”

 

Here it might help to think of this problem in terms of groups. These twentieths could be anything. Let’s call them oranges. You have 45 oranges, and they’re going into boxes of 8 oranges each. How many boxes can you fill? The answer is 45/8 (which is five and five-eighths boxes). Not 45/8 oranges. It’s just a number—of boxes.

 

In any case, thinking about the twentieths almost as a “unit” may help students get to the next step where they can see how the (1/20)÷(1/20) giant fraction equals one.

 

Clearing Fractions

Here is an important principle from algebra: denominators are icky. When you don’t know what else to do, it often improves things to get rid of some denominators. The basic technique is what some teachers call “clearing” fractions. This will seem ridiculously long, but the thinking behind it is fantastic preparation for algebra.

 

Let’s get rid of some denominators, starting with the 5 on the very bottom. To get rid of it, we want to multiply the bottom fraction by 5. To do that legally, we have to multiply the top by 5 as well. Why? Because we are allowed to multiply by one, and 5/5 is one:

 

 

We’ve done this one step at a time here. See how these steps require all that knowledge about multiplying by one and the details of fraction multiplication? Be alert for “holes” in student understanding.

 

This is progress, but we can go further, clearing the 4—by multiplying by one. We’ll move more quickly:

 

This has been hard, but any student who does it clearly understands fractions. And they’ll quickly move to combining steps so it doesn’t take as long.

More importantly, it’s strategic mathematics—doing operations (like multiplying by one) with a purpose in mind, in this case, clearing fractions, and getting rid of denominators.

The best for last: clearing fractions twice as fast

Instead of doing the two “little” denominators one at a time, you could do them both at once by multiplying by 20/20. (Try it yourself and see.)

Or—and this is the really good one—you could multiply by a fraction equal to 1 that will clear the whole “big” denominator (2/5) at once, like this:

A proof

Since there was nothing special about the numbers we used, we could do this with symbols. For any numbers a, b, c, and d, with b, c, and d not zero,

That is, to divide fractions, “invert and multiply.”

One of the biggest challenges for students with mathematics is that they don't think they are supposed to understand, they think they're supposed to just learn the tricks—and get the right answer. Helping them see why invert and multiply works can reinforce the idea that math should make sense.

 

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