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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.

• The invert and multiply method need not be an axiom. Simply explain why it's true. There's no better method for dividing fractions -- so learn why it work and trust it. If you understand why it works, it's not merely an axiom. Here's why it works:

a/b / c/d can be "multiplied by 1," in this case d/d.

a/b * d / c/d * d = da/b / c

Now multiply by b/b:

da/b * b / c * b = da/cb

Therefore:

da/cb = a/b * d/c = a/b / c/d

There's no hocus pocus to this. Again, the "invert and multiply" concept need not be an axiom. It's entirely provable.
• You're right Ben, but I'm willing to bet that you're somewhat older than 12 :)

As Valorie pointed out above, these kids are still working their way through GCM/LCD - algebra is beyond their ability (now - this will change).
• Offtopic: Now someone could code a Avalon Pie fraction education sample to demonstrate the power and coolness of Longhorn technologies in coding educational software ;)
• This thread has been interesting to both my husband and myself since we have a 5th grader going through this exact problem and we have done much work with her out of school to try to help her. I guess I feel part of the question is if the child (or anyone) doesn't have a reason for learning then why learn it?

In other words, what benefit now or in the future will this type of math be useful. If they could just get a reason for learning that would be half the battle.

Maybe I use divisional fractions in my life and don't know it but I can't remember the last time I needed to divide fractions of things for life.

• Pam: The simple answer I've come up with for your question:

The next major skill set that your 5th grader will hit in school is algebra. Algebra is dead easy if you know how to manipulate equations. You're going to be taking equations and dividing all the terms on both sides by various values, and sometimes that means that you're going to be dividing by fractions.

So you're right, division of fractions isn't a real life skill. But it IS a requirement to understanding algebra.

And algebra is a skill that you WILL be using during your life. You may not even realize you're using algebra, but you'll be using it (every time you answer "Are we there yet?" or "How long will it take to get there?", you're using algebra).
• 1. Cut an apple in half.
2. Cut one of the halves in half.
3. Show that 1/2 = 2*(1/4).
• Maybe this isn't helping you, but since we are homeschooling our kids I often enjoy trying to figure out how to explain complicated things to them so that they will understand. My kids aren't quite into the dividing fractions stage yet (they are age 2 and 5), I thought I'd think about this one and see what I came up with. Here goes...

I have found so far with math problems that the hardest part is coming up with an easy to understand question that the mathematical equation is trying to express.

For a more complicated example (5/8 / 3/10), you can use something like the following. If you have 5/8 of a pie left in your store and a bunch of people come in wanting slices that are exactly 3/10 in size, how many slices can you make? If your kids know how to get the answer they can calculate that (cross multiply, 50/24 or 2 1/12) and realize that you can get 2 pieces of pie that are 3/10 in size and that there will be 1/12 of a third slice of size 3/10 in size left over. You probably couldn't sell that even at a discount, so maybe you could give it to somebody in need.

If the example was the opposite (3/10 / 5/8), you can use the same thing. If you have 3/10 of a pie left in your store and a bunch of people come in wanting slices that are exactly 5/8 in size, how many slices can you make? Again doing the math (24/50 or 12/25), you can see that you can't make any slices of pie that are 5/8 in size because you don't have enough, but you can offer a slice to one person that is 12/25 of the size that they originally wanted, maybe at half price.

Hope this helps.
• 1/7 in "exact decimal": 0.1<sub>7</sub>

1/3 in "exact decimal": 0.3<sub>9</sub>

Piece of cake :)

Repeated subtraction: You _can_ repeatedly subtract 1/63 from 1/17 either graphically or using LCD. Unfortunately the LCD is pretty awkward (17 is prime and 63 is a prime, 31, times 3 so the LCD is 1071, and there's a remainder. Another misfortune is that graphically you have to have the ability to resolve to 1071ths in order to get the graphical demo to come out right.

I think in carpentry it would be possible to come upon a fraction division situation, especially in English units. For example, "How many 4 5/8 inch items can I cut from this 32 1/4 inch scrap?"

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• Well Pam, what if your child wants to become a doctor? or an engineer? or a teacher? Would you rather they learn fractions now or when they are 20 or 30? Are you teaching them the metric system? What if they go to Canada or Europe? Whenever someone mentions "when am I going to need this in life." rephrase it in this manner, "How stupid do I want to be?"
• I'm not a maths person, but I think I'd find a pie chart the easiest way to explain this.

You know, use geometry to explain fraction things.

Draw a semicircle - "How much of a circle is this?" - it's half.
Bisect the semicircle (is that the right word?) and ask = how many quarters in a half? Two...

It shows the relationship between the objects, and allows Child A to do the *physical* division. You can extend it further with more circles and greater numbers of lines, so that even if the child has a problem with the numeracy aspect (like me, and me), they can still conceptualize a visual way to solve the problem.
• Hi, look at http://www.explorelearning.com/index.cfm?method=cResource.dspResourcesForCourse&CourseID=212

Specifically http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=212

Maybe that will help. Instead of pie slices it uses linear 'lengths.'

I think that using ribbons instead of sliced pies is better because then you can think of fractions as parts of a unit in the sense of cm or inches and not as parts of an object (some kids might ask: how can you give 1/4 of an orange to half a person?). And thus you can work with fractions that are greater than one (i.e. 5/3)

You could bring paper ribbons used in calculators (httphttp://www.rudinfo.com/products/images/swintec/swin-301DPII.jpg) to class and cut 1 foot long pieces of paper. Then you can cut one (paint a red line on it to distinguish it) into 4 pieces (fourths) and another foot of paper (use blue for this one) into 2 pieces (halves). You could then ask: how many pieces of red marked paper fits along one of the blue ones?

Or graphically:

********
******** = 1 intact piece of paper

Then cut one of those into
** ** ** **
** ** ** **
and mark them red,

then cut another on like
**** ****
**** ****
and mark them blue.

Then make them try to align red pieces along one of the blue ribbons (assume that the | character doesn’t take any space, but is used to show you where one red piece ends and the second one starts)
****
****

**|**
**|**

I hope that these ideas are helpful, or at least spark some ideas of your own.
• how do you divide negative fractions?
what is a fraction?
• Well, this year I didn't miss the anniversary of my first blog post.
I still can't quite believe it's...
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