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For example, dividing by 2, means we cut something in two.

But dividing by 0.5, can only be explained with multiplying something by 2.

So, is there a "physical" explanation of dividing by 0.5? Is it "I divide by an entity that internally multiplies' or something as so bizarre?

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I once heard it explained as asking the question how many times does m/n fit into 1. For example, how many times does 1/2 fit into 1, that is 1/(1/2) = 2, or 2 times. – Amzoti Dec 1 '12 at 1:41
That's a very good answer, similar to what Peter Tamaroff gave below. – j riv Dec 1 '12 at 1:43
Also, read this, maybe as good an answer as you'll get (has some nice things): – Amzoti Dec 1 '12 at 1:43
up vote 5 down vote accepted

I once heard: "If you feed a kid $Y$ grams of chocolate he would have a density of $X$ $\rm gr/cm^3$ of chocolate in his blood. But if you feed half a boy chocolate he will have $2X$ $\rm gr/cm^3$ of chocolate in his blood".

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It sounds good and simple enough. – j riv Dec 1 '12 at 1:33
Chocolate gets into the stomach first, and the concentration of it will vary with time. I guess dissolving the same amount of table salt in water of different volumes is a better example. – Frenzy Li Dec 1 '12 at 2:37
@FrenzYDT. It is odd you took my example seriously. – Pedro Tamaroff Dec 1 '12 at 2:39
Well, as folks say $1+1=2$ doesn't always necessarily hold. AAMOF, I like your explanation. – Frenzy Li Dec 1 '12 at 2:41

$$\frac{1}{0.5} = \frac{2(\not{.5})}{\not{.5}} = 2.$$

$$\frac{1}{\frac12} = \frac{2\cdot\not{\frac{1}{2}}}{\not{\frac12}} = 2.$$

In words, "how many times does one-half fit into the whole?"

And more generally, "how many times does $\;\dfrac1n\;$ fit into $\;1\;$?"

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It sounds good mathematically but I don't know if it's complete in a physical sense since it might have to explain the meaning of 0,5/0,5 = 1. – j riv Dec 1 '12 at 1:39
I guess it can be "I multiply 2 by a number that was divided by that same number hence I multiply 2 by 1". – j riv Dec 1 '12 at 1:49

Along the same lines as Pedro Tamaroff's answer:

If you have ten ounces of vodka and each cocktail requires one ounce, then you can make ten cocktails.

If you have ten ounces of vodka and each cocktail requires only half an ounce, then you can make twenty cocktails. In other words, halving the vodka doubles the number of drinks. (Unfortunately, nobody will be happy with these drinks.)

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[This answer is essentially identical to, which I think deserves to be on Math SE as well.]

What do beginners know?

Multiplication and division of fractions is, in the United States at least, typically first encountered in either third or fourth grade; that is, at age 8 or 9. Benjamin Dickman, in his answer, has discussed various representations of division (partitive/equal sharing, quotitive/measurement, missing factor). I want to take a minute to talk about representations of fractions.

Given the age of the students in question, I do not think there is much value in formal explanations, i.e. explanations that rely on symbolic manipulation, whether or not they are "algebraic" in nature. They may learn that the algorithm is the correct one, but I don't think they will be convinced by it, in the sense of believing that the algorithm has to be that way and makes sense. To convince someone that $4 \div \frac27 = \frac{4 \times 7}2$ you have to guide them to figure it out for themselves, in a way that makes the result seem completely inevitable and unsurprising. And to do that you have to try to see things from the perspective of the learner.

The fractions students most often (perhaps exclusively?) encounter are proper fractions, i.e. fractions $< 1$; quantities larger than $1$ are usually expressed as mixed numbers rather than as "improper" fractions. Students at that level typically understand fractions in the language of part/whole relationships; the most commonly used representations are:

  1. a circle, divided into $n$ sectors of equal area, with $k<n$ of them shaded (corresponding to the "real-world" context of cutting up a pizza or other circular food);
  2. a collection of $n$ objects (often drawn as a group of small circles contained in a rectangular frame), with $k<n$ of them shaded (corresponding to the "real-world" context of sharing a box of cookies or cupcakes);
  3. a collection of $k$ units of measurement (often drawn as small scoops or cups), drawn with some kind of scale or labeling that indicates that $n$ of those units would comprise a single larger unit (corresponding to the "real-world" context of using, e.g., between three "quarter-cup scoops of flour" to measure $3/4$ of a cup of flour).

The order in which I list the representations above is not accidental; it corresponds roughly to the order in which these representations actually occur in the classroom. That is, the most commonly used representation is the "circle cut into wedges", and the least commonly used representation is the "small scoops that make up one big scoop". Since the question is about conviction, I think it is important to realize that not all representations are equally valuable for that purpose. In particular, I think the third representation above, combined with the measurement interpretation of division, is actually the most useful for the purpose of explaining how division of fractions works.

With all of that established as preamble, here is my strategy for convincing a novice that division of fractions works the way it ought to:

1. Announce the problem, but do not ask them to try to solve it yet.

Write the problem $6 \div \frac23 = ?$ in large letters on a piece of paper and tell the student that we are going to figure this out, but we're going to work out way up to it by warming up with a few simpler problems first.

2. Begin by emphasizing the measurement interpretation of division.

Since most students reflexively think of division in terms of equal-sharing, it is a good idea to start by explicitly activating the measurement interpretation, which is less common. Ask:

Suppose you want to measure six cups of flour, and all you have is a two-cup scoop. How many scoops do you need?

Most 8- to 9-year-old students will be able to answer "Three" immediately. To do so, they do not need to explicitly translate the problem into "Six divided by two equals what?", nor as "Two times what equals six?" In fact, making such translations may seem like an unnatural complication to a simple problem of counting. So you have to ask them to make that translation explicit, with prompts like:

  • How did you know that?
  • What kind of arithmetic operation are you using? (Many will say "addition", because they are simply thinking "two plus two plus two".)
  • What other kinds of arithmetic can you use to describe what you just did?

This conversation does not have to be a long one, but it does need to happen, and the goal is to activate (and keep active in the student's mind) that "$6 \div 2$" can be understood as the question "How many 2-cup scoops do you need to make 6 cups?". Now you are ready to move to the next step:

3. Consider divisors that are unit fractions.

Now we want to vary the task, just slightly:

Suppose you want to measure six cups of flour, and all you have is a $1/3$ cup scoop. How many scoops do you need?

Again most students (at this age level) will answer "18" almost immediately, at least if they have some degree of automaticity with the multiplication fact $6 \times 3 = 18$. Students who do not have such automaticity may need to count by threes. Either way they will get the answer very quickly. Ask them how they know. Guide the conversation to the following, very important summary:

Because the scoops only hold $1/3$ cup, you need three scoops for each cup, so you have to multiply three times the number of cups you need.

To the extent possible, try to get the student to be the one who says this, or something like it. Don't say it for them, but do revoice their statement of it to make it more succinct and coherent, if necessary. Once students agree with this basic idea (and once said it usually seems completely obvious to them, so much so that they wonder why you were making such a big deal about it), you can move on to the final variation:

4. Consider divisors that are not unit fractions.

One last tweak to the task:

Suppose you need 6 cups of flour, and all you have is a $2/3$ cup scoop. How many scoops do you need?

Some students will immediately try to do some kind of paper-and-pencil computation, whether they know an algorithm or not. Discourage this. Ask them instead to just think about the previous problem. We already know that it will take eighteen $1/3$ cup scoops to make six cups. What if the scoops are $2/3$ cups instead? If they are still stuck, prompt: How do the new scoops compare to the old scoops? Usually students will eventually come up with an answer like this one:

Because the scoops are twice as large, we only need half as many of them, so we need just 9 scoops instead of 18.

Once they have said this, or something like it -- but not before! -- write down "$ 6 \div \frac23 = 9$".

We are not done yet, though. The most important step is the next one:

5. Summarize and generalize

It's time to look back at the expression $6 \div \frac23$ and think about what happened when we tried to solve it. Notice that there are two basic principles interacting here:

  • The 3 in the denominator of the divisor acted by multiplication, because it takes three $1/3$ cup scoops to fill a single cup, so if you are using scoops of that size, you need three times as many scoops as you do cups.
  • The 2 in the numerator of the divisor acted by division, because a $2/3$ cup scoop is twice as large as a $1/3$ cup scoop, so you need half as many of the larger scoops as you do of the smaller ones. In other words:

To find $6 \div \frac23$, you multiply by $3$ and divide by $2$.

Or, transcribed symbolically,

To find $6 \div \frac23$, you can compute $\frac{6 \times 3}2$

Ask some questions to get the student to restate this result in their own words:

  • Where does the numerator of the $2/3$ go? (Into the denominator of the answer.)
  • Where does the denominator of the $2/3$ go? (Into the numerator of the answer.)

At this point, the "rule" that "To divide by a fraction, you invert the divisor and multiply" should not seem like a "rule", but merely a summary of something that should have been obvious all along.

Final Thoughts

I've gone into a lot of detail about this, and I think it's worth pointing out that it probably takes more time to read the above description than to implement it. I've tutored dozens of kids in precisely this manner; some of them are elementary school kids learning fractions for the first time, others are high schoolers who "learned" how to divide fractions back when they were 8 or 9 but no longer remember what to do, or remember it imperfectly. I've had very good success with this method. If working with a student one-on-one, it usually takes no more than five minutes, start to finish. I think there are two reasons why it is effective:

First, it begins by announcing the problem, and then immediately putting the problem on hold and instead considering simpler problems: first division by whole numbers, and then division by unit fractions. This models an important problem-solving heuristic: When you encounter a hard problem, consider a simpler one and see if you can get any insight from it.

Second, it concludes by looking back at a single example and trying to understand the general principles that make it work. This models a second important problem-solving heuristic: When you have solved a hard problem, take a moment to look back at it and see if the perspective of hindsight reveals any general arguments to you.

Both of these heuristics are, of course, taken directly from Polya's How To Solve It. And while these heuristics are common to most problem-solving contexts, and may even be naturally-occurring for many students, it is worth recognizing that conventional school instruction (which I would caricature as "Teach the rule, then do examples, then have students do many exercises, then provide an explanation of the rule") does not provide a lot of space for this kind of slow, reflective consideration.

One final thought: You may have noticed that the OP's example was $4 \div \frac27$, and I changed that for purposes of this explanation to $6 \div \frac23$. That change was not accidental. First of all, if you want to follow a narrative like the one I sketch above, asking about scoops that hold $1/7$ of a cup is just silly. Nobody makes or uses measuring scoops like that; the artificiality of the problem stands out and is distracting. On the other hand, $1/3$ cup measuring scoops are fairly commonplace (at least in the United States). I imagine that this whole instructional sequence would have to be reconsidered from the ground up (and may be completely unworkable) if one were teaching in a context in which the metric system is consistently used. The reason for changing the $4$ in the original problem to a $6$ is that the dual roles played by $2$ in the equation $4 \div 2 = 2$ can actually be confusing. Which $2$ represents the size of groups, and which represents the number of groups? Changing the $4$ to a $6$ eliminates that symmetry, and hence the ambiguity about which number represents which quantity.

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Well done for noting that examples need to be "rigged" so that there is no confusion between "$2$" and "$2$". It is always worth remembering that even before algebra is explicitly introduced, symbols such as "$2$" have a dual meaning for the reader, both as "the number two" and as pure algebraic symbols, like "$a$" but of a different shape. Not taking into account that second meaning causes confusion in the student's mind, at a depth at which it cannot be dredged up, addressed, and explained. So it is always worth re-working examples with that second meaning in the back of your mind. – Martin Kochanski Jul 20 at 11:13
@MartinKochanski: Thank mweiss, not me! That's why I marked it as community wiki. =) – user21820 Jul 20 at 12:56

Modeling 12 / 4,

If 12 marbles fit perfectly into 4 boxes, how many marbles fit into 1 box?

versus modeling 12 / (1/2).

If 12 marbles fit perfectly into half a box, how many marbles fit into 1 box?
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