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How do I divide fractions?

How do I divide fractions?

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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.

 

  • hmmmmmm, my mother taught 5th grade for many, many years. I'll ask her how she did it.

    One thing I think I learned first was how to convert fractions so the denominators were the same. Once you convert 1/2 into 2/4 it's easy to see how 1/4 x 2 = 2/4(1/2). Try the same thing with 5/8 and 3/10. (50/80 / 24/80). I've spent so long in a decimal world (chemistry major in college) that fractions are a different world to me. I just automatically convert them to decimals!

    So that's my solution, tell them to try making the bottom numbers the same and then just divide the top numbers. It gets a little more complicated when reducing them down of course. It's been a while since I was in 5th grade, I don't know if that's too hard for them or not.
  • That's a cool idea Scott, I'll pass it on.

    One minor issue is that the class is still struggling with getting Least Common Denominator down, but...
  • I'm of the opinion that learning to phrase mathematical statements in English is an essential step in learning to "do" math. That may seem like an obvious point, but the English language is often not well suited to expressing mathematical statements.

    Pulling two quotes out of the posting,

    The statement "1/2 divided by 1/4 is 2 quarters" is not correct. 1/2 divided by 1/4 is 2. Period. Two quarters is 1/2, which is not the same as 1/2 divided by 1/4. It is a huge mistake to try to explain mathematics to someone by phrasing the explanation in a factually incorrect way.

    "1/4 divided by 1/2 is 1/2 of a half" is equally incorrect. 1/4 divided by 1/2 is 1/2. Period. 1/2 of a half is 1/4, which is not the answer to 1/4 divided by 1/2. I'm not sure what the writer is trying to do with the extra embellishments on the ends of the sentences, but they turn factually correct statements into factually incorrect statements.

    As to how to demonstrate graphically that 5/8 divided by 3/10 = 50/24, my suggestion is to look at the calendar and hope that next year's math teacher has a better grasp of what is a good use of a sixth grader's time and attention span than this year's teacher. My sixth grade math teacher wrote books, designed teaching aids, and lectured around the country on how to teach mathematics using visual manipulable aids. She was quite the authority on the subject, and I can state with confidence that she never tried to inflict nonsense like this on me or any of her other students. She also had a great deal of respect for her students and when I told her "I like your math class but your toys don't help me understand the problems" she let me spend my time working out the answers using the more abstract methods which came naturally to my brain.

    Hoping for another math teacher may not the answer you were looking for, but sometimes that's what out-of-the-box problem solving is all about.
  • Intuitively, if they're both "parts of the whole," you're putting one into the other. For example, 1/4 of the whole fits into 1/2 of the whole 2 times.

    This way, (a/b)/(c/d) is asking "How many times does c/d fit into a/b?" We can see that 1/(c/d) = d/c by drawing it out ( 1/(1/d) = d trivially, and reducing the size of "the whole" by 1/c corresponds to increasing the size of the part by multiplication by c ). From here it's a quick step to (a/b)/(c/d) = (ad)/(bc).
  • I'm of the opinion that learning to phrase mathematical statements in English is an essential step in learning to "do" math. That may seem like an obvious point, but the English language is often not well suited to expressing mathematical statements.

    Pulling two quotes out of the posting,

    The statement "1/2 divided by 1/4 is 2 quarters" is not correct. 1/2 divided by 1/4 is 2. Period. 1/4 divided by 1/2 happens to be 2 quarters, but 1/4 divided by 1/2 and 1/2 divided by 1/4 are completely different problems. It is a huge mistake to try to explain mathematics to someone by phrasing the explanation in a factually incorrect way.

    "1/4 divided by 1/2 is 1/2 of a half" is also incorrect. 1/4 divided by 1/2 is 1/2. Period. 1/2 of a half is 1/4, which is not the answer to 1/4 divided by 1/2. I'm not sure what the writer is trying to do with the extra embellishments on the ends of the sentences, but they turn factually correct statements into factually incorrect statements.

    As to how to demonstrate graphically that 5/8 divided by 3/10 = 50/24, my suggestion is to look at the calendar and hope that next year's math teacher has a better grasp of what is a good use of a sixth grader's time and attention span than this year's teacher. My sixth grade math teacher wrote books, designed teaching aids, and lectured around the country on how to teach mathematics using visual manipulable aids. She was quite the authority on the subject, and I can state with confidence that she never tried to inflict nonsense like this on me or any of her other students. She also had a great deal of respect for her students and when I told her "I like your math class but your toys don't help me understand the problems" she let me spend my time working out the answers using the more abstract methods which came naturally to my brain.

    Hoping for another math teacher may not the answer you were looking for, but sometimes that's what out-of-the-box problem solving is all about.
  • I'd go with the suggestion made by pndmnm. That's the way I explained it to my little sister when her math teacher failed in explaining it to her (her math teacher relied on the "faith" part, which was no good to my sister, who needs to _understand_ things and doesn't take things for granted easily...) It's also very easy to visualize and it will lead to a perfect understanding of what division is all about.
  • I would try to explain like this:

    Begin with 20 divided by four, which yields 5. Then reduce the divisor to one (20/1), which yields 20. So reducing the divisor really increments the result. This behaviour continues when reducing the divisor below zero. When reducing the divisor by 50 percent, result will double. You can easily show this by drawing a graph, keeping the divident constant and changing the divisor. Then it is just a matter of the relationship between divident and divisor. The expression 1/2 could be rewritter to 1/2 divided by one.
  • The rule that got me through school is this - the denominator's denominator's goes to the numerator. So if you (a/b) / (c/d) , now d is the denominator's denominator. So I used to go..ok..d now jumps to the numerator. From then on, it is normal multiplication and division
  • I'm not a mathematician, and this might not be very good but here's my best shot:

    (realizing that the pre-algebraic notation is simply for illustrating the concept for adults)

    Let's say you have an inanimate object that is easily divided into "shares". Pies for instance? Remember, in these samples, the numerator is the number of "shares" alloted while the denominator represents which portion of each share a pie represents.

    When dividing pies in to thirds (x/3), each pie represents three shares of pie. When dividing pies into halves (x/2), each pie represents two shares. Divide a pie into a single share (x/1), each pie represents 1 share.

    In each case, a whole pie represents the number of "shares" indicated by the denominator. The size of each "share" keeps increasing.

    When you reach x/(1/2), each pie represents one half of a "share". Remember that the numerator in this case is the number of whole "shares" that we're interested in. How many pies does it take to make a whole "share"? Two. 1/(1/2) = 2.

    I realize that's the simplest case, but it isn't much different. How many pies do I need to represent 3/5 of a share (3/5) / (1/2)? Well, now that we understand how to figure out what one full "share" is, understanding the axiom is a little more straightforward. The inverse of a given fraction is simply the size of a whole share.

    I think getting the kids to grasp the concept of 1/(x/y) (where x/y is an arbitrary fraction) will help them to grasp what is actually represented by inverting and multiplying.

    That probably only makes sense to me. Ridicule me thoroughly.
  • Scott's advice seems to be the best, though.
    (5/8) / (3/10), when converted to LCD, is (25/40) / (12/40). Once the denominators match, you can basically just do straight division on the numerators, ignoring the common denominator.
  • The way I was taught it was:

    First, 1/4 multiplied by 1 is 1/4.
    1/4 multiplied by 1/2 is 1/8.
    We all know that if a x b = c then a = c / b.
    Therefore 1/8 divided by 1/2 must be 1/4.
    ..and 1/4 divided by 1/2 is 2.

    This kinda killed two birds with one stone.
  • I, too, have been taught it through the inverse of multiplication. “a divided by b is c such that b multiplied by c is a” (although I’m not sure they used this algebraic notation; more likely it went along the lines of “quotient is such a number that, when multiplied by the divisor, yields the dividend” (note different terminology for numerator/denominator and dividend/divisor; it really helps, we even wrote the division as 5/8 : 3/10 (with / actually being a horizontal fraction bar))). Then we were just shown how to invert a fraction and why it works, and then instructed to invert the denominator and multiply.
  • I wonder if we cannot infer the inversion of the divisor by observation and then develop it into a general case? Or at least into a case sufficiently general for the kids to accept!

    Taking a really simple example: 1 divided by 1/2. We shouldn't have any difficulty in pronouncing the answer to be 2. And we can similarly see that 1 divided by 1/3 is three. Notice anything? The answer is the inverse of the divisor. But 2 divided by 1/3 is 6. Which is 3 times 2, or twice the answer to 1 over 1/3. Which is what you would expect.

    Could that approach be developed further? I'm afraid my formal maths training is a looong way behind me and my kids are a few years away from this becoming a personal problem!
  • At my primary school many years ago, we were introduced to the idea of division as a way of sharing. In fact the teacher called it 'sharing' before she started calling it division.
    We began with the simple things that we could share such as 10 apples amongst 5 children. We also were given the idea of the remainder. What happens if you share 11 apples amongst 5 children we get a remainder - 1 for the teacher. This was well before we were introduced to decimals.
    Using the same method we could find out how many children could we give 5 apples to if we had 10 to start with.

    Fractions were introduced in the pictorial form of a pie. When we became familiar with the pictorial ideas of addition and subtraction and multiplication, the idea of division came reasonably easily. The question then boils down to 'if I have half a pie, how many quarter pie slices can I cut?'

    Obviously pictorial methods only take you so far, but the point of maths is that it allows us to abstract to nonpictorial situations.
    So having demonstrated division by fractions in the simple pictorial way for a number of simple cases, the teacher can then proceed to show a general method.



  • Phrasing the question the right way helps. Think of 4 / 2 as the question "how many 2s in 4?". Then 1/2 / 1/4 becomes "how many 1/4s in a 1/2?".

    The inversion rule is just a formal "trick" and should be taught as such - a shortcut and not the definition.

    If you want to get kids to visualize this think of chocolate bars made up of squares, and sharing them by breaking up the squares.
    When you have four squares it's easy to see that each square is a quarter and 2 quarters go into a half.

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