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.

 

Comments (47)

  1. Scott says:

    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.

  2. 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…

  3. Don says:

    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.

  4. pndmnm says:

    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).

  5. Don says:

    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.

  6. Wolfgang says:

    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.

  7. Daniel says:

    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.

  8. 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

  9. 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.

  10. 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.

  11. Mo says:

    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.

  12. Centaur says:

    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.

  13. Mike Woodhouse says:

    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!

  14. John Fannon says:

    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.

  15. 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.

  16. EW says:

    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.

  17. Valorie says:

    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.

    In these days of standardized tests, unless the problem is part of the computational test component, answers to math questions are essay questions. The simplicity of bubbling in the answers on the ITBS or SAT are rapidly coming to an end. Answers in non-computational math parts of the WASL require units and an explanation of how you derived everything. If you choose to write "1/2 divided by 1/4 equals 2," then you are choosing to lose points because you did not use units. If you are not an adept reader or writer, you will likely lose points as well because you cannot explain what you did. At this point in time, if you cannot explain your math in words, your scores on the WASL will say that you are not competent at the skills required for your grade. Kids have to be taught how to answer these types of questions, especially given the ultimately high stakes: you cannot graduate until you pass the WASL. Welcome to the world of "no child left behind". While I think the ideas are good, the current implementations (nation-wide, not just WA state) leave a great deal to be desired.

    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.

  18. Rick Schaut says:

    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 8×5). 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 4×10 rectangle should do, though 5×8 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).

  19. AndrewSeven says:

    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.

  20. Petr Kadlec says:

    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√30)^6 if you understand what I mean.) I have never seen that! :-)

  21. Glenn Crumpley says:

    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

  22. Petr Kadlec says:

    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?".

  23. travis says:

    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 ;-)

  24. 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.

  25. travis,

    How do I represent 1/3 EXACTLY in decimals? How about 1/7?

  26. travis says:

    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.

  27. KC Lemson says:

    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.

  28. Canadian Eh! says:

    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.

  29. 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.

  30. Petr Kadlec says:

    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.

  31. Ben Field says:

    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.

  32. 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).

  33. Joku says:

    Offtopic: Now someone could code a Avalon Pie fraction education sample to demonstrate the power and coolness of Longhorn technologies in coding educational software ;)

  34. Pam McKinnis says:

    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.

  35. 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).

  36. Anonymous says:

    1. Cut an apple in half.

    2. Cut one of the halves in half.

    3. Show that 1/2 = 2*(1/4).

  37. Kirk Munro says:

    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.

  38. Mike says:

    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?"

  39. AYSE (EYE-SH-E) says:

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  40. Scott says:

    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?"

  41. Tristan K says:

    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.

  42. 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.

  43. Eri says:

    how do you divide negative fractions?

    what is a fraction?

  44. Well, this year I didn’t miss the anniversary of my first blog post.

    I still can’t quite believe it’s…