How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspxValorie 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 ofen-USTelligent Evolution Platform Developer Build (Build: 5.6.50428.7875) Larry Osterman s WebLog How do I divide fractions | bird bathshttp://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#9784255Fri, 19 Jun 2009 12:48:09 GMT91d46819-8472-40ad-a661-2c78acb4018c:9784255 Larry Osterman s WebLog How do I divide fractions | bird baths<p>PingBack from <a rel="nofollow" target="_new" href="http://cutebirdbaths.info/story.php?id=1092">http://cutebirdbaths.info/story.php?id=1092</a></p>
<img src="http://blogs.msdn.com/aggbug.aspx?PostID=9784255" width="1" height="1"> Larry Osterman s WebLog How do I divide fractions | Toe Nail Fungushttp://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#9713471Tue, 09 Jun 2009 09:19:08 GMT91d46819-8472-40ad-a661-2c78acb4018c:9713471 Larry Osterman s WebLog How do I divide fractions | Toe Nail Fungus<p>PingBack from <a rel="nofollow" target="_new" href="http://toenailfungusite.info/story.php?id=2726">http://toenailfungusite.info/story.php?id=2726</a></p>
<img src="http://blogs.msdn.com/aggbug.aspx?PostID=9713471" width="1" height="1">why do we change the numbers when dividing fractionshttp://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#8652161Wed, 25 Jun 2008 16:37:29 GMT91d46819-8472-40ad-a661-2c78acb4018c:8652161why do we change the numbers when dividing fractions<p>PingBack from <a rel="nofollow" target="_new" href="http://alejandronewsblog.vwebhosting.com/whydowechangethenumberswhendividingfractions.html">http://alejandronewsblog.vwebhosting.com/whydowechangethenumberswhendividingfractions.html</a></p>
<img src="http://blogs.msdn.com/aggbug.aspx?PostID=8652161" width="1" height="1">Another year, another posthttp://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#552078Wed, 15 Mar 2006 20:21:17 GMT91d46819-8472-40ad-a661-2c78acb4018c:552078Larry Osterman's WebLogWell, this year I didn't miss the anniversary of my first blog post.
<br>I still can't quite believe it's...<img src="http://blogs.msdn.com/aggbug.aspx?PostID=552078" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#139458Sat, 22 May 2004 14:09:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:139458Erihow do you divide negative fractions?
<br>what is a fraction?
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=139458" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#139373Sat, 22 May 2004 08:46:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:139373Daniel KurejwowskiHi, look at <a target="_new" href="http://www.explorelearning.com/index.cfm?method=cResource.dspResourcesForCourse&CourseID=212">http://www.explorelearning.com/index.cfm?method=cResource.dspResourcesForCourse&CourseID=212</a>
<br>
<br>Specifically <a target="_new" href="http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=212">http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=212</a>
<br>
<br>Maybe that will help. Instead of pie slices it uses linear 'lengths.'
<br>
<br>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)
<br>
<br>You could bring paper ribbons used in calculators (http<a target="_new" href="http://www.rudinfo.com/products/images/swintec/swin-301DPII.jpg">http://www.rudinfo.com/products/images/swintec/swin-301DPII.jpg</a>) 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?
<br>
<br>Or graphically:
<br>
<br>********
<br>******** = 1 intact piece of paper
<br>
<br>Then cut one of those into
<br>** ** ** **
<br>** ** ** **
<br>and mark them red,
<br>
<br>then cut another on like
<br>**** ****
<br>**** ****
<br>and mark them blue.
<br>
<br>Then make them try to align red pieces along one of the blue ribbons (assume that the | character doesnâ€™t take any space, but is used to show you where one red piece ends and the second one starts)
<br>****
<br>****
<br>
<br>**|**
<br>**|**
<br>
<br>I hope that these ideas are helpful, or at least spark some ideas of your own.
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=139373" width="1" height="1">I win the google prize!http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#136258Fri, 21 May 2004 01:44:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:136258Larry Osterman's WebLog<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=136258" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#136321Thu, 20 May 2004 23:57:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:136321Tristan KI'm not a maths person, but I think I'd find a pie chart the easiest way to explain this.
<br>
<br>You know, use geometry to explain fraction things.
<br>
<br>Draw a semicircle - "How much of a circle is this?" - it's half.
<br>Bisect the semicircle (is that the right word?) and ask = how many quarters in a half? Two...
<br>
<br>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.
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=136321" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#136294Thu, 20 May 2004 23:24:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:136294ScottWell 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?"<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=136294" width="1" height="1">HOW DO I FIND OUT WHAT FRACTIONS MEAN WHEN THEY HAVE DIFFERENT DENOMINATORS?????????http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#136254Thu, 20 May 2004 22:41:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:136254AYSE (EYE-SH-E)HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP HELP <div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=136254" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135990Thu, 20 May 2004 17:43:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135990Mike1/7 in "exact decimal": 0.1<sub>7</sub>
<br>
<br>1/3 in "exact decimal": 0.3<sub>9</sub>
<br>
<br>Piece of cake :)
<br>
<br>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.
<br>
<br>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?"
<br>
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135990" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135988Thu, 20 May 2004 17:40:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135988Kirk MunroMaybe 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...
<br>
<br>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.
<br>
<br>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.
<br>
<br>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.
<br>
<br>Hope this helps.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135988" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135973Thu, 20 May 2004 17:20:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135973Anonymous1. Cut an apple in half.
<br>2. Cut one of the halves in half.
<br>3. Show that 1/2 = 2*(1/4).<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135973" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135910Thu, 20 May 2004 15:54:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135910Larry OstermanPam: The simple answer I've come up with for your question:
<br>
<br>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.
<br>
<br>So you're right, division of fractions isn't a real life skill. But it IS a requirement to understanding algebra.
<br>
<br>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).
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135910" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135850Thu, 20 May 2004 14:16:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135850Pam McKinnisThis 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?
<br>
<br>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.
<br>
<br>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.
<br>
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135850" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135379Wed, 19 May 2004 21:56:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135379JokuOfftopic: Now someone could code a Avalon Pie fraction education sample to demonstrate the power and coolness of Longhorn technologies in coding educational software ;)<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135379" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135193Wed, 19 May 2004 18:18:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135193Larry OstermanYou're right Ben, but I'm willing to bet that you're somewhat older than 12 :)
<br>
<br>As Valorie pointed out above, these kids are still working their way through GCM/LCD - algebra is beyond their ability (now - this will change).
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135193" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135179Wed, 19 May 2004 18:03:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135179Ben FieldThe 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:
<br>
<br>a/b / c/d can be "multiplied by 1," in this case d/d.
<br>
<br>a/b * d / c/d * d = da/b / c
<br>
<br>Now multiply by b/b:
<br>
<br>da/b * b / c * b = da/cb
<br>
<br>Therefore:
<br>
<br>da/cb = a/b * d/c = a/b / c/d
<br>
<br>There's no hocus pocus to this. Again, the "invert and multiply" concept need not be an axiom. It's entirely provable.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135179" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135171Wed, 19 May 2004 17:58:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135171Petr KadlecAd 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".
<br>
<br>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.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135171" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135137Wed, 19 May 2004 17:26:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135137Simon Cooke [exMSFT]I agree with the others above. Division is ALWAYS "how many times can I fit one of *these* into one of *these*"?
<br>
<br>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".
<br>
<br>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"?
<br>
<br>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 :)
<br>
<br>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.
<br>
<br>After that, it should be quite simple to bring in the rule of turning the other fraction upside down and multiplying.
<br>
<br>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.
<br>
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135137" width="1" height="1">A completely visual approachhttp://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135076Wed, 19 May 2004 16:20:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135076Canadian Eh!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.
<br>
<br>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.
<br>2. Draw two parallel and horizontal lines.
<br>3. Divide the bottom line into 5 equal divisions, marking them with their fractional representation (i.e. 1/2, 2/5, ... , 5/5).
<br>4. At the same division points, but on the top line, mark the divisions simply 1, 2, ..., 5.
<br>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.
<br>6. Connect mark 1/5 using a dotted line with the corresponding point on the top line which happens to be 1.
<br>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.
<br>
<br>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.
<br>
<br>Let's try a slightly less trivial example this time around. How about
<br>1/5 divided by 2/5.
<br>
<br>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.
<br>
<br>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.
<br>
<br>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.
<br>
<br>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.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135076" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135052Wed, 19 May 2004 15:49:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135052KC LemsonI liked Richard's idea above. "How many Xs in Y?". It just resonates well in my mind.
<br>
<br>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.
<br>
<br>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.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135052" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135049Wed, 19 May 2004 15:46:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135049travis0.3 (with a bar over the 3)
<br>and
<br>0.142857 (with a bar over the 142857)
<br>
<br>but yeah, my theory fell apart quite quickly, hehe.<div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135049" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135045Wed, 19 May 2004 15:41:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135045Larry Ostermantravis,
<br> How do I represent 1/3 EXACTLY in decimals? How about 1/7?
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135045" width="1" height="1">re: How do I divide fractions?http://blogs.msdn.com/b/larryosterman/archive/2004/05/18/134697.aspx#135039Wed, 19 May 2004 15:38:00 GMT91d46819-8472-40ad-a661-2c78acb4018c:135039Larry OstermanOne 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.
<br>
<br>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.
<br>
<br>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.
<br><div style="clear:both;"></div><img src="http://blogs.msdn.com/aggbug.aspx?PostID=135039" width="1" height="1">