*photo by Scott Robinson via flickr*

A comment on my post Fraction Division — A Poem deserves a longer answer than I was able to type in the comment reply box. Whitecorp wrote:

Incidentally, this reminds me of a scene from a Japanese anime, where a young girl gets her elder sister to explain why 1/2 divided by 1/4 equals 2. The elder girl replies without skipping a heartbeat: you simply invert the 1/4 to become 4/1 and hence 1/2 times 4 equals 2.

The young one isn’t convinced, and asks how on earth it is possible to divide something by a quarter — she reasons you can cut a pie into 4 pieces, but how do you cut a pie into one quarter pieces? The elder one was at a loss, and simply told her to “accept it” and move on.

How would you explain the above in a manner which makes sense?

## Division is Backwards Multiplication

Division is the inverse of multiplication, which means it is like doing multiplication backwards. When you multiply using the mad scientist’s ray gun, you have:

[*scale factor*] “times the size of” [*original amount*] = [*final size or amount*]

Children typically learn to multiply whole numbers by thinking of groups:

[*number of groups*] [*size of group*] = [*total amount*]

Now, division turns this around. You start from the end, the total amount or the final size, and divide by EITHER the scale factor or the original amount. Your answer will be the other number — the one you didn’t use.

Again, children typically learn division by thinking about whole-number groups:

[*total amount*] [*number of groups*] = [*size of group*]

or

[*total amount*] [*size of group*] = [*number of groups*]

In the comment question, the younger sister is only thinking of the first option: the total amount divided by the number of groups. She has (subconsciously) defined her “group” as one piece of pie — that is, as one serving — and easily sees how to divide the pie into 4 “groups” or pieces.

“But,” she asks, “how do you cut a pie into one-quarter pieces?”

Like any ill-posed problem, this question stumps her big sister. How can anyone answer a question that doesn’t make sense? Of course the big sister cannot divide a pie into one-quarter servings, because each piece, no matter what size, is by (unstated) definition a single serving.

Let’s look at two common mental models — partitive division and measurement division — to see how the sister could have divided her pie…

## Partitive Division: What Size Are the Groups?

In partitive division, we imagine cutting our total amount into a given number of parts:

[*total amount*] [*number of groups*] = [*size of group*]

This is easy to understand when we work with whole numbers:

20 4 groups = 5 in each group.

Or, in the little sister’s first example:

pie 4 parts = pie in each serving.

But life gets more confusing when the *number of parts* is a fraction:

20 groups = ?

or

pie parts = ?

In this case, we are saying that *our total amount is only a fraction of the entire group or part*, and we want to find out *what size is the whole thing*. And once we understand this, it is relatively easy to see what a whole group (or serving) must be:

20 groups = 20 is of *what size* group?

or

pie parts = pie is of *what size* serving?

Notice the word “of” in the second part of each statement. Do you remember that “of” means multiplication? Division is the inverse of multiplication, which means that every division problem is asking a multiplication question backwards.

20 is of *what size* group? = 20 is what? = 80 in the group.

or

pie is of *what size* serving? = pie is what? = 2 pies make a serving.

## Measurement Division: How Many Groups?

In measurement division, we imagine measuring out groups of a certain size:

[*total amount*] [*size of group*] = [*number of groups*]

To my mind, measurement division is intuitively easier to understand than partitive division when working with fractions. I often measure ingredients, or fabric, or pizza in fractional units. It makes an easy mental picture, and I used this picture of division in answering the original comment:

pie = pie cut into pieces of size = how many servings?

Or, in simpler words:

“How many ’s are there in half a pie?” = 2 servings.

## Update: For Further Study

Gary Davis and Catherine Pearn have written a wonderful pdf e-book:

Mr. Koh from Whitecorp (whose comment originally launched this post) shared this link in the comments below, but I want to make sure every reader has the chance to see it.

Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

Good lord what have I done? LOL.Thank you for this spinoff piece (or sequel), I am greatly enlightened by your explanation, especially the bit where you mentioned and emphasized the importance of correctly recognising what constitutes an item as a whole.

This may be completely irrelevant, but damn that pie looks soooooo good.

You’re welcome!

I agree, the pie looks delicious — and the recipe (on the photo page) looks very easy.

A simple way to pose the question in a realistic measurement division context would be:

I need 1/2 cup of sugar for some lemonade. I only have a 1/4 cup measuring spoon. How many times will I need to fill up the 1/4 cup measuring spoon to get 1/2 cup of sugar?

Yes, exactly!

And it even works to visualize more difficult fraction divisions, if you take it step by step:

5 1/2 ÷ 3/8 = ?

Imagine having only an 1/8-cup measure. First, find out how many 1/8’s you could measure out. Well, in each whole cup, there are 8 of them, so in 5.5 cups:

5 1/2 x 8 = 44 eighth-cup measures in all.

BUT you didn’t really want 1/8-cup scoops. You wanted 3/8-cup scoops, so now you have to group these 44 scoops into sets of 3:

44 ÷ 3 = 14 with two scoops left over

And those two leftover 1/8-cup scoops get us 2/3 of the way to our next size-we-want 3/8-cup scoop.

So in the end, we have 14 full-size 3/8-cup measures, plus 2/3 of another one:

5 1/2 ÷ 3/8 = 14 2/3

Whew! That’s some interesting analysis of Partitive Division and Measurement Division. But for a “young” one I would try to explain that each of the four pieces is one-quarter (1/4) of a whole pie. Consequently, cutting (dividing) the pie into four equal pieces is the same as cutting the pie into four one-quarter pieces.

As for the arithmetic techniques I’d state the same rule as did the elder sister. “Accept it” and move on is probably a good advice for now.

But JC, the kid probably already has the measurement model of division in mind when she does whole-number division. 98 divided by 7 is often phrased as “how many 7’s in 98?” So isn’t it reasonable to say in this situation that 1/2 divided by 1/4 similarly asks “how many fourths in a half?”

(and by the way, can we throw some love in the direction of the common denominator algorithm for division of fractions? In which 1/2 ÷ 1/4 becomes 2/4÷1/4 and then we divide the numerators?)

I am using a math program with my young children (6 & 7) that has division by fractions from the beginning. (How many 1/2 inches in 2 inches?) and they have no trouble with the concept when explained in small numbers in those terms. We’ll see how it translates once they start working with more abstractions.

Christopher, the common-denominator method is pretty much what my mental-math version of measurement division does. As abstract methods go, the common-denominator method is probably the easiest for students to understand WHY it works. And in my book, understanding why something works always trumps “Just accept it”!

I agree that it makes more sense to ask how many 1/4’s are in a pie, rather than just take for granted that you flip the fraction and multiply. I always wondered why the reciprocal? I also wondered why the answer to a fraction division problem was greater than the number you’re dividing, if that makes any sense. I have loved how homeschooling has afforded me the opportunity to go back and fill in holes in my own understanding.

Think about division as “how many of these are in my total?” Then you can see that if you divide by one, the answer will be exactly the SAME as your total. If you divide by something bigger than one, then there will be LESS of them in your total. But if you divide by a fraction smaller than one, then there will be MORE of them in your total, because you are counting smaller pieces.

“So isn’t it reasonable to say in this situation that 1/2 divided by 1/4 similarly asks “how many fourths in a half?”

Well, Christopher, that sounds perfectly reasonable to say that to any adult and experienced reader here. But I responded to the question posed, which asked, “How would you explain the above in a manner which makes sense?” I thought that I was to explain the above to a young girl who doesn’t understand the “½ divided by ¼” concept to begin with.

I see the dilemma as one illustrating the difficulty or inability of abstract reasoning for some young minds. It’s even presented in the story as the young girl speaks in terms of a material/concrete object (the pie) as opposed to the abstract and pure numeric terms that her older sister used.

So I think it may be reasonable in this situation to ask the yound girl, “how many one-quarter pieces are in one-half piece of a pie?” But only as a follow-up to the explanation I outlined before.

And yes, love to the common denominator algorithm. I think that once the girl would see that 1/2-piece of a pie is the same as two 1/4-pieces then she can see how 1/2 becomes 2/4.

Correction to my last comment: I see the dilemma as one illustrating the difficulty or inability of abstract reasoning for some inexperienced minds.

There is a free e-book on fractions made available for download at the Republic of Math site:

http://republicofmath.files.wordpress.com/2009/12/division_of_fractions_davis__pearn3.pdf

Thought it was pretty good. Peace.

That e-book is an excellent resource! Thank you for sharing.

No problem-it’s all about sharing and learning 🙂

Hi Denise!

Love that you are in favor of explaining the “why”s. They are always so important to my son. My daughter couldn’t have cared less.

Any good resources out there for explaining the “whys” to middle & high school students?

Thanks,

Kym

Hi, Kym! Did you look at the e-book that Whitecorp linked to? It is very good. A high school student could surely follow it, and a middle school student could understand the ideas with only a bit of help. (For instance, encouraging them not to be scared away by the words “partitive” and “quotitive” in the first section.)

Another approach would be to use the common denominator approach recommended by Christopher earlier in the comments. It is very easy to understand. Just put your fractions into a common denominator, and then figure out how many of the second one there are in the first:

6/7 ÷ 1/4 = 24/28 ÷ 7/28

And really, all we need to look at now are the numerators. How many 7/28ths are there in 24/28ths? The same as the number of 7’s in 24:

24/28 ÷ 7/28 = 24 ÷ 7 = 3 R 3

BUT we are working in fractions now, so we can’t leave our answer with a remainder! We have to divide those extra 3 pieces up, too:

3 ÷ 7 = 3/7

So our final answer is:

6/7 ÷ 1/4 = 3 3/7

The common denominator method takes just a little bit longer than the invert-and-multiply rule, but it is a whole lot easier to understand.

We just released an inforgraphics that explains partitive division vs measurement (or quotative) division https://www.matific.com/static/pages/quotative-fraction-division.html