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How do I divide fractions?

How do I divide fractions?

Valorie works as a teacher's aid in a 6th grade classroom at a local elementary school.

They've been working on dividing fractions recently, and she spent about two hours yesterday working with one student trying to explain exactly how division of fractions works.

So I figured I'd toss it out to the blogsphere to see what people's answers are.  How do you explain to a 6th grader that 1/2 divided by 1/4 is 2?

Please note that it's not sufficient to say: Division is the same as multiplication by the inverse, so when you divide two fractions, you take the second one, invert it, and multiply.  That's stating division of fractions as an axiom, and not a reason.

In this case in particular, the teacher wants the students to be able to graphically show how it works.

I can do this with addition and subtraction of numbers (both positive and negative) using positions on a number line. Similarly, I can do multiplication of fractions graphically - you have a whole, divide it into 2 halves.  When you multiply the half by a quarter, you are quartering the half, so you take the half, divide it into fours, and one of those fours is the answer.

But how do you do this for division?

My wife had to type this part because we have a bit of, um, discussion, about how simple this part is....

How can you explain to 9-11 year old kids why you multiply by the reciprocal without resorting to the axiom? It's easy to show graphically that 1/2 divided by 1/4 is 2 quarters because the kids can see that there are two quarters in one half. Equally so, the kids can understand that 1/4 divided by 1/2 is 1/2 of a half because the kids can see that only half of the half is covered by the original quarter. The problem comes in when their intuition goes out.  They can solve it mathematically, but the teacher is unwilling to have them do the harder problems “on faith“ and the drawing is really confusing the kids. Having tried to draw the 5/8 divided by 3/10, I can assure you, it is quite challenging. And no, the teacher is not willing to keep the problems easy. And no, don't get me started on that aspect of this issue.

I'm a big fan that if one method of instruction isn't working, I try to find another way to explain the concept. I visited my usual math sites and found that most people don't try to graph this stuff until 10th grade or adulthood. Most of the sites have just had this “go on faith“ response (show the kids the easy ones, and let them “go on faith“ that it will hold true for all cases). I really wish I could figure out a way to show successive subtraction, but even that gets difficult on the more complicated examples.

What I am hoping is that someone out there can provide me with the “aha!“ I need to come up with a few more ways to explain this. What this has been teaching me is that I've been doing this “on faith“ most of my life and never stopped to think about why myself.

Any ideas/suggestions would be much appreciated.

• If you have 4 apples and 4 people, dividing the apples to the
people results in one apple for one person (4/4=1)

If you instead give 1/2 the apples (2) to 1/4 of the people (1); that
one person gets two apples.
• Thanks for all the different ideas!

Don shares Larry's "issue" with the units which is why we had to ask the greater public for help. All I can say is that this problem only seems to happen in problems with fractions. If I say that 1 gross divided by 1 dozen equals 12 dozens, nobody gets upset. The "dozens" in the answer is the unit of the divisor. If I take a dollar, and divide it by 1/4, I have 4 quarters. Nobody gets upset there because I'm talking about money. If I say that 3/4 divided by 1/4 equals 3 quarters, we have word overlay problems and heated discussion. Yes, I understand the problem and why it is so frustrating. The kids don't have an issue with this word overlay; it's only the adults that have this problem so I'm trying to ignore that part of the issue.

Cameron's idea of taking shares is more like the successive subtraction idea I keep trying to work out. I think I just have to plan out my problems and have the chips pre-counted so I don't waste lesson time.

While most adults (and junior high students) have no problem with the idea that if a * b = c then a = c/b, that is not well understood at the fifth/sixth grade level. The kids will do it, but it is another "faith based" exercise. That understanding doesn't come in until real abstract thinking does (usually around age 13). I think this is why most of the explanations I've found have been oriented to the older kids. These 5/6th grade kids' brains have not yet had that abstract thinking jump that will happen soon.

Part of this fraction unit is working on the LCDs. I think that is another way to explain to the kids what to do (and why). I know the kids are pretty strong on their understanding of integer division, but LCDs are a terror to them.

It's times like this I get a little frustrated. Ask any 4 year old to give you half of something, and they'll understand what you want. Show them broken cookie parts, and they can tell you if they add up to a whole cookie. If you show them a cake, and ask them to share it among their friends, they have no problem figuring out how many parts they need. Kids don't fear fractions until later in school. What are we adults doing to bollox their innate understanding of what fractions are? I keep thinking it's just another example of math phobia that some parents pass down to the kids. I used to work in the 1st/2nd class. I'd wait until the kids were really good at solving simple algebra problems (no variables, more like __ * 3 = 12) and then tell them they were doing algebra. The kids thought this was cool; the adults got panicky that they couldn't even do 2nd grade math. I'd point out to the parents that they'd been doing algebra too and half of them would be insistent that they didn't know how to do algebra. What a difference a word makes!

Thanks again for all the help and ideas.
• Larry, I'm not sure this will help. It's a variation on Scott's advice, and it works well when the two fractions divide evenly. But there's a conceptual problem that's difficult to explain when there's a remainder in the fractional division itself. Anyway, here goes:

Take the two fractions, and find the LCD. For 5/8 and 3/10, the LCD is 40 (2x2x2x5 or 8x5). Express the two fractions in terms of the LCD, i.e. 5/8 = 25/40 and 3/10 = 12/40.

With a piece of graph paper, mark out a rectangle that consists of the number of squares corresponding to the LCD. In this case, a 4x10 rectangle should do, though 5x8 would likely be best (there's probably a rule, there, for figuring out how best to mark out the rectangle that correspondes to a whole unit, but it eludes me at the moment).

Now, mark out an area within the rectangle corresponding to the numerator. In this case, that's 25 squares. I'd shade this area using a colored pencil or crayon.

Now, start counting out the number of squares, within the area shaded for the numerator, that correspond to the denominator. In this case, that's 12. Using a different colored pencil or crayon, shade each collection of 12 squares that you can count out of the 25 squares you shaded in the previous step.

For this problem, you'll get two blocks of these 12 squares, and you'll have one square left over. The question is, what's the meaning of the one square that's left over? We might be tempted to say that it's 1 of 40, but that's not right. We're counting out in blocks of 12, so the square that's left over is 1 of 12, not 1 of 40. This is the conceptual problem that I can't quite make clear other than to point out that, in this case, we're counting out in blocks of 12 squares, so the 1 square left over is 1/12 of the size of the block that corresponds to the divisor fraction.

Hope this helps. Bob-math is always challenging and interesting. I miss helping out in that class (and had hoped to do so this year until having to ship a product got in the way).
• Very interesting stuff.

I think the kids do have a problem with the "word overlay", that it is this overlay which makes the answer counter intuitive.

(1/2)/(1/4)->2

Two "one quarters" fit into "one half".
A quarter will fit into a half twice.
• I was so surprised about that "essay math" that I googled for WASL and read an example test. Well, the choice of problems looks really good to me (it really requires that children think about the problem), but I consider the essay part to be a bad idea. It is trying to mix oral and written exams and takes the worse of both IMHO. (This test requires +- the same amount of work of those who evaluate it as an oral exam would, but without the advantage of feedback and explaining any misunderstandings.)

But I would like to see some of the ideas used in Czech education...

A complete sidenote: what is that radical-like looking notation of division "5) 6/30"? (I would read that rather like maybe (5&radic;30)^6 if you understand what I mean.) I have never seen that! :-)
• No one has yet mentioned that another way to view division is a series of subtractions, just as another way to view multiplication is a series of additions. Using this approach, you can explain that, for example, dividing 4 by 2 is the same as saying, "How many times can I subtract 2 from 4?" If this concept is grasped, you can then ask: "How many times can I subtract 1/4 from 1/2? Drawing a pie divided into quarters and visually relating that two quarters=one half may help.

Glenn Crumpley
• Although I believe it has been mentioned, I don't think it is really useful (but...hmmm, I am a bit older ;-) ), because "How many times can I subtract 1/63 from 1/17?".
• In Scott's reply, he mentions using decimals. Is there a need to use fractions any more? What is wrong with converting everything to decimals? i can't remember ever using fractions outside of my education. the only case i can see using fractions would be for sake of absolute accuracy (22/7 != 3.14), and for higher math classes i guess. Actually now that I think about it, the basic concept of fractions is necessary for higher math so i guess they aren't obsolete. Damn, I thought I was really on to something there ;-)
• One of the fundamental design criteria of the WASL is that the WASL can't be gamed. There is no strategy that can be taught to improve your chances of getting the correct answer.

That's why it's essay based. If the test is multiple choice, then students can attempt to plug the answers into the original problem and come up with the correct answer. But with essay based responses that can't happen.

I actually think that the WASL is an excellent test. I have issues with high stakes tests in general, but as an example of one, the WASL is pretty darned good.
• travis,
How do I represent 1/3 EXACTLY in decimals? How about 1/7?
• 0.3 (with a bar over the 3)
and
0.142857 (with a bar over the 142857)

but yeah, my theory fell apart quite quickly, hehe.
• I liked Richard's idea above. "How many Xs in Y?". It just resonates well in my mind.

Take a piece of paper and cut it into a circle, make that your 'pie'. Cut the circle in half, so you have two halves. Then take one of the halves, and cut that into half, so you have 1/2, 1/4 and 1/4.

Then, ask the students how many of the 1/4s fit into the 1/2. The answer is 2, and they'd be able to see it visually. So then show them that "How many of X fits into Y" is division.
• It is possible to solve fractional division using a number line-like approach. It works best when you use a least common denominator. But this isn't a requirement.

1. Pose your problem. For the sake of simplicity, we'll deal with a specific case instead of a general one - 2/5 divided by 1/5.
2. Draw two parallel and horizontal lines.
3. Divide the bottom line into 5 equal divisions, marking them with their fractional representation (i.e. 1/2, 2/5, ... , 5/5).
4. At the same division points, but on the top line, mark the divisions simply 1, 2, ..., 5.
5. Draw an arc from mark 0/5 to mark 2/5. Connect mark 2/5 using a dotted line with the corresponding point on the top line which happens to be 2.
6. Connect mark 1/5 using a dotted line with the corresponding point on the top line which happens to be 1.
7. The answer is in the form of a fraction. The numerator is the point on the top line for the first dotted line we drew (in step 5) -- 2. The denominator is the point on the top line for the second dotted line we drew (in step 6) -- 1.

So the answer is 2/1 = 2. The nice thing about this, is that the students have a nice visual representation. The arc is equally divided into 2 equal parts sized 1/5 each. The more astute students will probably recognize other subtleties.

Let's try a slightly less trivial example this time around. How about
1/5 divided by 2/5.

The result should be similar to the first one we posed, except the arc has its tail at 0/5 and head at 1/5. The first dotted line is thus at 1. The second dotted line is at the 2/5, 2 mark. So the answer is 1/2. Again, there is a visual representation here. The arc is exactly half of the quantity we are concerned about which is the second dotted line.

If you use this approach for questions like 3/5 divided by 2/5, you will see it yields the correct result. So if the students are shaky with mixed fractions, they have this approach to lean on that they know will give them the right answer.

It is possible to make this work with fractions that don't share a common denominator but it's much more difficult than simply teaching the concept of fractional equivalency and getting the students to find a common denominator.

For fractions whose denominators are different but multiples of each other, you can leave them alone. Just have the students divide the bottom line into N parts where N is the larger denominator. Have them mark each point according to both denominators though. Put another way, if the student is calculating 3/5 divided by 1/10, have them mark the points like: 0, 1/10, {2/10, 1/5}, 3/10, {4/10, 2/5}, ..., {10/10, 5/5, 1}. This will help reinforce the whole concept of fractions.
• I agree with the others above. Division is ALWAYS "how many times can I fit one of *these* into one of *these*"?

The problem here is that we're treating the fractions as numerator and denominator. That's just an symbolic encoding trick. You never think of something as being "one over two of something" - you think of it as a "half". You never think of something as being "one over four of something" - you think of it as a "quarter".

So you have to do it graphically, and ask - instead of "What is 1/2 divided by 1/4?", "how many quarters of an apple are there in half an apple"?

And this way you can do it with a real apple - which, by the way, makes for a wonderful demonstration if you bring in an apple to cut up. Or you can do it with legos if you don't like knives in the classroom :)

The next trick is explaining how to do "1/4 divided by 1/2" - and showing that you're using subtraction to do the real division part.

After that, it should be quite simple to bring in the rule of turning the other fraction upside down and multiplying.

And after that, it's time for the lowest common denominator trick. Or you can even skip that part entirely if you're happy with larger denominators, and multiply each fraction by the other's denominator.

• Ad WASL: I am definitely _not_ for multiple choice (at least not without that magic "none of the previous" choice), in fact multiple-choice tests have appeared only recently in Czech schools (because they are simpler to assign score to). I am just saying that "talking on paper" is generally a bad idea. If the student has perfect understanding of the topic, he/she might be very brief, because the solution seems obvious. Should he/she be given less points just because that? In an oral examination, the teacher would say something like "explain that a little bit more".

Ad division: of course that division is about "how many Xs fit into Y". But do you really think that this helps with those "difficult" fractions like 15/37 divided by 14/31? I am trying to recollect the way I learned fraction math, but I believe that there was no "magic trick" -- we were just shown that simple case "1/4 fits 2 times into 1/2" and after that: "to divide two general fractions, invert and multiply". In fact, I don't think we were even told about that common denominator trick.
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