PDA

View Full Version : Teaching simplifying fractions to a 3rd-grader?



mdowdle
02-09-2011, 04:29 AM
So, my third-grade son is being introduced to simplifying fractions. And he having difficulties (they’re teaching it by drawing blocks within a rectangle, which is not particularly effective when drawing blocks within a rectangle itself requires an understanding of simplifying fractions) (i.e., if you’re asked to draw 10 equal blocks within a rectangle, I’m pretty certain you will bisect the rectangle first than then try to draw 5 equal blocks within each bisection, but this will require that you first understand that 5/10 is equivalent to1/2 in the first place).

So I’m needing to teach simplifying fractions to my son. But he’s much better at deductive reasoning than inductive reasoning. But I can’t figure out if there’s a deductive methodology for identifying simplifiable fractions. Simplifying fraction is really easy when we’re dealing with 6/8. It seems to me to become pretty difficult when it comes to 51/119. But to my son, 6/8 at this point is not much different from 51/119. My question is: is there a relatively simple algorithm for figuring out whether 51/119 can be simplified or not? (Hint: it is.) Or is the algorithm to identify simplified fractions ultimately one of simply randomly trying to see if there is some integer that can divide both the nominator and the denominator into a whole number?

kerzain
02-09-2011, 04:34 AM
How he coming along with his periodic table of elements?

mdowdle
02-09-2011, 04:41 AM
How he coming along with his periodic table of elements?
He is quite clear that the four elements are Reed Richards, Ben Grimm, Susan Storm, and Johnny Storm. So that's not a problem, thank God.

AlanT
02-09-2011, 05:14 AM
Seems a bit complicated for 3rd grade, and 51/119 would be hard for anyone, since not too many people are familiar with the 17 times table. Does he have a really solid grip on multiplication and division?

hong
02-09-2011, 05:29 AM
16/64 = 1/4. Just cancel the 6's on the top and bottom.

Houngan
02-09-2011, 05:37 AM
16/64 = 1/4. Just cancel the 6's on the top and bottom.

Where were you when I was in calculus!?

H.

Tim James
02-09-2011, 05:43 AM
Khan Academy might have something useful for you.

Alistair
02-09-2011, 05:50 AM
I am not an authority (in any way) but I suspect looking for algorithms or processes is not something a young child will do very easily. Memory of what divides by what seems much more likely to help this (and everything else) along...

Matthew Gallant
02-09-2011, 06:12 AM
1. Find the greatest common factor of the numerator and the denominator. http://www.youtube.com/watch?v=3jGAcpK2V7Q

2. Divide each by the GCF.

3. You are done.

The hardest part for him is going to be determining whether or not the larger odd numbers are prime. One handy trick is adding up the digits of the number and if the sum is divisible by three, then the number is also divisible by 3. Of course, telling if a number is divisible by 5 is easy, then you just have to brute force trying to divide by 7, 11 and 13. That should knock down any number into factors that are easier to deal with and will get him up to 289 (17*17).

Which brings me to the next trick: any bigger number like that, take the square root. That sets an upper limit for the primes that you'll have to try. Let's go back to 119. The square root is 10.something. So you only have to try dividing by primes less than 10.

Ryan A
02-09-2011, 06:18 AM
Does he know multiplication and division yet? My third grader came home with multiplication homework at the start of the year and she was struggling with it because they weren't teaching her to memorize basics like the multiplication table. Once we started working on that, her math assignments become trivially easy for her and she stopped stressing out about how many apples or boxes or whatever she was supposed to draw on her math homework.

For obvious reasons, I think mastering multiplication is necessary for kids to be able to learn and master division.... which brings us to fractions. If a kid can divide, a kid can reduce a fraction.

In my opinion, we've screwed over an entire generation of math students by trying to teach them math without ensuring a solid foundation in the fundamental vocabulary and skills that contribute to a student's number sense. It's like trying to teach somebody how to write a novel when they can't read independently.

Brian Seiler
02-09-2011, 07:01 AM
In my opinion, we've screwed over an entire generation of math students by trying to teach them math without ensuring a solid foundation in the fundamental vocabulary and skills that contribute to a student's number sense. It's like trying to teach somebody how to write a novel when they can't read independently.

Welcome to New Math. This is also the crap the schools I graduated from are now shoveling into my nephews, and they can't do multiplication without counting on their fingers and possibly taking off their shoes if they get really stumped. I recognize a lot of the techniques that they're teaching as common Number Sense tricks (as in tricks that you would learn if you were competing in that particular event in the University Interscholastic League), but the eggheads that cooked up this stuff didn't realize that those tricks only work well if you already reflexively know your multiplication tables, just like a Reverse Polish Notation calculator doesn't make a whole lot of sense if you don't really, thoroughly understand the order of operations. They even somehow also managed to completely dick up Geometry, which was probably the most useful, best, goodest overall class I ever had to take in high school because it taught proper reasoning and how to prove a thing, which is the closest thing to a critical thinking class I ever had before Intro to Philosophy in college.


For fractional simplification, the factoring answer is the correct one. Simplifying fractions becomes really easy when you learn to look at 8 and see three twos, and really hard until you get that concept. I'm kind of baffled that factoring exercises like GCD weren't taught before simplifying fractions.

TheTrunkDr
02-09-2011, 07:06 AM
1. Find the greatest common factor of the numerator and the denominator. http://www.youtube.com/watch?v=3jGAcpK2V7Q

2. Divide each by the GCF.

3. You are done.

The hardest part for him is going to be determining whether or not the larger odd numbers are prime. One handy trick is adding up the digits of the number and if the sum is divisible by three, then the number is also divisible by 3. Of course, telling if a number is divisible by 5 is easy, then you just have to brute force trying to divide by 7, 11 and 13. That should knock down any number into factors that are easier to deal with and will get him up to 289 (17*17).

Which brings me to the next trick: any bigger number like that, take the square root. That sets an upper limit for the primes that you'll have to try. Let's go back to 119. The square root is 10.something. So you only have to try dividing by primes less than 10.
I think you've jumped way ahead of 3rd grade math here. The child isn't going to know anything about prime numbers, greatest common denominators, powers or square roots so using them isn't going to help. The work here is likely to be reducing things like 2/4 to 1/2 and such. I doubt any of the values would venture beyond 12 for either the numerator or the denominator.

As others have said it sounds like he's a little weak with the basics of multiplication and division, fractions are generally the first actual use of those principles taught to children. Take a step back and review multiplication and division.

Lizard_King
02-09-2011, 08:06 AM
Your third grader is running headlong into one of the most poorly taught concepts across all subjects in terms of his age group. My training is pretty far from being math or elementary school oriented, but having taught basic statistics and probability to ninth graders along with other economics-related concepts, I can say there are more than a handful of major issues that kids run into when math and specifically fraction teaching, especially operational work, moves too quickly. Even with students that are ahead of their peers (http://www.sci.sdsu.edu/CRMSE/IMAP/pubs/Reflections_on_Fractions.pdf), they are often simply good at memorizing outcomes in the short term rather than really grasping the concepts. That means there's still going to be a gap between them getting good grades and them being genuinely mathematical thinkers that isn't well targeted by present educational pacing (basically, you're waiting for a Eureka moment to come on its own with each kid, which is unnecessary since the gap is usually obvious to anyone that's spends some time with a young student as you have).

On the bright side, that means there's been a lot of work by that handful of people in education that aren't chimps to provide solutions at the macro and micro level to this problem. An early example is the recommendations from projects as far back as 1989 (http://www.cehd.umn.edu/rationalnumberproject/89_1.html), which focus on using manipulatives to develop an understanding of the relationships between fractions (easy fractions, that is) and order/equivalence rather than operations. I would also highlight the recommendation for verbalizing and narrating fractions in a format he already understands based in everyday life or fiction he enjoys, and then bridging back to the numbers and the rectangles. Finally, the first operational step is addition of small fractions using objects if possible or drawings.

Basically, the idea is to move towards emphasis on concepts first rather than procedures or trying to do them simultaneously. With manipulatives, think in terms of what works for them in games: color distinctions, shapes, etc. Good luck, though.

Ezdaar
02-09-2011, 08:50 AM
I'll just leave this (http://www.maa.org/devlin/LockhartsLament.pdf) here again.

Zylon
02-09-2011, 09:04 AM
I'll just leave this (http://www.maa.org/devlin/LockhartsLament.pdf) here again.
Oh great, a reading assignment.

Lizard_King
02-09-2011, 09:12 AM
I'll just leave this (http://www.maa.org/devlin/LockhartsLament.pdf) here again.

That's a great article. Thanks.

Tracy Baker
02-09-2011, 09:35 AM
Khan Academy might have something useful for you.

Always check here first. The guy is a fantastic teacher.

EDIT: And holy crap, Ezdaar. That's one of the greatest things I've ever read. Many thanks for posting it.

Gav
02-09-2011, 10:12 AM
I'm going to go against the tide, and say that I don't think multiplication abilities are the issue.

Simplifying 13/169 needs times tables, but simplifying 6/8 is a matter of understanding how fractions work. I think the easiest way to do it is to draw, say, a cake, and divide it into 4ths, then 8ths, and show how 3 of the former gives you the same amount of cake as 6 of the latter.

My kids got the algorithm down really quickly (their school does drill them on multiplication), but when I started pushing them on what it actually means to simplify a fraction, I realized that they didn't really get it.

I find that concrete demonstrations are always good, which is why things like cake pieces are good. As far as dividing a rectangle into 10 pieces goes, I think that's a great way to go, except that I'd start with 10 small rectangles (say, blocks), then build them up, and show how you can split it into two halves. At the very least, I wouldn't have the kid draw the smaller rectangles -- if he doesn't draw them all the same size, then you can't really show how it splits into half, which ruins the point.

TheTrunkDr
02-09-2011, 11:32 AM
Simplifying 13/169 needs times tables, but simplifying 6/8 is a matter of understanding how fractions work.
These are the same thing, just because we (adults with a firm understanding of the topic) find reducing 6/8ths trivial doesn't change the process or the concept in the slightest.

You still need to find a common denominator of 6 and 8. Just because you (after decades of this stuff) can eyeball it doesn't mean a 9 year old who's just being exposed to the concept can. Again this goes back to multiplication and division. All fractions are is a division and reducing fractions is also just a division.

Gav
02-09-2011, 01:50 PM
These are the same thing, just because we (adults with a firm understanding of the topic) find reducing 6/8ths trivial doesn't change the process or the concept in the slightest.

You still need to find a common denominator of 6 and 8. Just because you (after decades of this stuff) can eyeball it doesn't mean a 9 year old who's just being exposed to the concept can. Again this goes back to multiplication and division. All fractions are is a division and reducing fractions is also just a division.
I disagree. If you cut up a square into 8 parts, it's easy to eyeball it and understand how to reduce from 6/8 -> 3/4, no times tables or division required. It's much harder to eyeball 169 pieces.

I also think that the idea that fractions are a division is too sophisticated for 9 year olds.

Lunch of Kong
02-09-2011, 02:02 PM
I also think that the idea that fractions are a division is too sophisticated for 9 year olds.

I disagree with you. While it may be too sophisticated for a typical American 9-year-old educated solely by a no-child-left-behind curriculum, it is definitely not too sophisticated for 9-year-olds in general.

At age 12, the exam question on our science test wasn't "What is 32F degrees in C?", but "What temperature is identical in both F and C? Show your work."

To get it right, we had to remember the F to C formula we had learned, and apply the algebra we had learned in math the previous year.

F = 9/5C + 32
If F and C are equal:
t = 9/5t + 32
t-9/5t=32
-4t/5=32
5 (-4t/5)=32 * 5
-4t = 160
-t= 40
t = -40

Miramon
02-09-2011, 02:03 PM
Seems to me that the example of cutting an object in half, a similar object in thirds, and then finally another similar object in sixths should illustrate the point.

* "Look, here's one half and one third: there's no way to relate them to one another except by eye: one half plus one third [fit them together] makes some kind of big chunk, but who knows how much it really is."

* "But see here, one half is the same as three sixths, and one third is the same as two sixths. If we convert the fractions into sixths, we can add them together and get five sixths."

* "How should we know to try sixths? Multiply 2 and 3 together, you get 6. We take the 3 from this side to figure out how many sixths a half is, and the 2 from the other side to figure out how many sixths a third is, and there you go. ['Bob's your uncle' omitted]"

Once the concept is grasped visually with the illustration, the actual multiplication to get a common denominator is just a simple mechanical task.

Ryan A
02-09-2011, 04:18 PM
Something to keep in mind: what makes sense to you / how you learn may not necessarily be what makes sense to every kid / how every kid learns.

TheTrunkDr
02-09-2011, 04:57 PM
I disagree. If you cut up a square into 8 parts, it's easy to eyeball it and understand how to reduce from 6/8 -> 3/4, no times tables or division required. It's much harder to eyeball 169 pieces.
The point isn't the specifics it's grasping the concept. Once someone understands the principle the specifics aren't relevant. Yes using simpler fractions to demonstrate the principle is of course going to be easier both to do and to understand.


I also think that the idea that fractions are a division is too sophisticated for 9 year olds.
Fractions are a division, if the student doesn't understand that, they don't understand fractions. I also think you're wrong about it being too sophisticated for the age.

mdowdle
02-09-2011, 06:11 PM
Many thanks for all the advice and perspectives. This is why I <3 QT3.

Regarding to Lockhart's "A Mathematician’s Lament" as recommended by Ezdaar. I think Lockhart's view is part of the problem, to be honest. Not all children have a "natural love of numbers and shapes". My son certainly does not. He will never become a mathematician. An educational curriculum that want to use his "natural love of shapes" as a focus will be completely wasted on him. Of course, it will be very effective for those children who do indeed have a natural love of shapes.

My son's psychological makeup is such that he responds much better to rote learning. I'm not saying that the aesthetically gifted should not be taught to use their mathematical artistry to their advantage. But I do wish that schools would also cater to the needs and particular talents of students who simply do not see an innate beauty in shapes and numbers, in the same way that even after 15 years of advanced musical study, I never understood the appeal of Mozart.

Miramon
02-09-2011, 06:32 PM
Well, not everyone can learn everything well, but still.....

The annoying thing about math is that any given stage, you can learn the necessary operations to solve problems mechanically but then the next thing you have to learn will be very hard indeed. Fractions or long division or the chain rule for derivatives or whatever it is you are having trouble with could be solvable with effort using memorized mechanical procedures, but they would all seem like confounding black box mysteries for which the solution procedure just happens to work, some of the time, for no apparent reason.

The problem is that you can't just go on to the next level in the next grade or the next class with any confidence on the basis of memorizing the solution steps. In math, I believe you really need to be confident in your mastery of the basis for the next thing you are learning. It doesn't matter how smart you are, if you don't really understand basic differential equations in Analysis I, there's just no way you are going to do well with partial differential equations in Analysis II, nor can you go on to ace basic algebra problems in some higher grade (when *do* kids learn algebra these days?) if fractions still make you nervous when you are trying to solve for X.

So for this reason, I think it is a mistake to just assume the kid just has to learn through rote or mechanical learning. Of course, almost everyone just sucks at something or other, and the kid might just suck at mathematical thinking, but I still think it's worthwhile trying to teach math in a way that leads to a more intuitive grasp of the subject, because otherwise he is in for a long series of unhappy mandatory courses for the rest of his school career.

Miramon
02-09-2011, 06:47 PM
Oh, and returning to the initial question about the algorithm for simplification, which one of those links I didn't follow may well address, sure, there is an algorithm, but it's ugly and computationally expensive, which is (well, this is a stretch) why public key cryptography works: factoring is hard.

If you don't have your 17 times table memorized, how long will it take you to figure out that 3x17 is 51? Sure, if you happen to know the adding-up-the-digits rule, it's easy enough to see that 51 is a multiple of 3, but then you'll see that 119 is not a multiple of 3, and you'll just be stuck until you fully factorize 51 and actually try the division of 119/17. I bet there's not a teacher in the entire school who knows how to tell if something is divisible by 7 offhand (http://math.about.com/library/bldivide.htm).

On the other hand, I think 99 out of 100 people won't recognize as 51/119 as simplifiable anyway, including many mathematicians, so what the heck. Anyone who puts that on a 3rd grade exam should be tarred and feathered.

mdowdle
02-09-2011, 07:12 PM
Just to be clear, 51/119 was simply my own way of trying to explain what 6/8 looks like to a 9 year old who just began learning division about 7 months ago.

I understand the argument that rote, process-based learning can lead to cognitive deadends when these processes have to be applied to higher-order tasks. But I'm not clear that that has ever in fact been demonstrated. As I recall, I -- like most of my generation -- learned math by learning rote processing through all of elementary school, and probably into 8th grade geometry. I had no problem progressing all the day to differential equations in college -- although I did fall out -- crash and burn actually -- when I got to "modern algebra". But I'm not certain this has anything to do with rote learning -- it had much more to do with my inability to conceptualize mathematical spaces in which principles of reflexivity and transitivity did not apply. It was precisely because the very objects and diagrams that are now being used to teach math in elementary school ceased to be of use that my cognition failed.

Lizard_King
02-09-2011, 08:49 PM
I'm not going to belabor the point, but I think it's common for students to blame themselves when they burn out just as they are getting to the good stuff in a subject. It's common enough that we should consider questioning the methodology at least as often as we do the commitment and abilities of the student, and I don't think that happens enough.

I've seen rote based teachers that get results, and I've seen inquiry based and other approaches that also get results. Tragically, schools and especially colleges are biased only towards the former in introductory courses, which I think is a big mistake. I know that in my case, the latter would have been far more productive in subjects that I struggled in, and I didn't even know it was an option until I got to advanced courses within my major. Beyond that, I didn't even know it was on the table for basic classes until I became a teacher myself.

There are studies that support all kinds of things, but I think there's a lot to be said for simply differentiating what mathematicians and scientists and historians and everybody else does when they are professionals in their field rather than reprocessing their exciting worlds into a bunch of memorization blocks with fixed gateways.

Chris Nahr
02-10-2011, 01:25 AM
If you don't have your 17 times table memorized, how long will it take you to figure out that 3x17 is 51?

Not very long if you simply try doubling 51 and discover that the difference between that and 119 is 17. :)

Miramon
02-10-2011, 06:26 AM
Not very long if you simply try doubling 51 and discover that the difference between that and 119 is 17. :)

Huh? I don't know this rule.... Suppose I double 51 and find the difference between 102 and 123 is 21. How does that help me simplify 51/123 (17/41)?

Or do I already know that 119 has 17 as a factor -- then it makes sense, but it's a lot easier to figure out 51's factors than 119's anyway.

Chris Nahr
02-10-2011, 06:58 AM
Any common factor of two integers must also be a factor of their difference. If n = ax and m = bx then (n-m) = (a-b)x. So it makes sense to check the difference and see if it's easier to factor.

In the case of 51/119 the difference is directly the number you're looking for. In the case of 51/123 the difference is 21 which has two factors, 3 and 7, and 3 is the one you're looking for.

Miramon
02-10-2011, 07:01 AM
Any common factor of two integers must also be a factor of their difference. If n = ax and m = bx then (n-m) = (a-b)x. So it makes sense to check the difference and see if it's easier to factor.

In the case of 51/119 the difference is directly the number you're looking for. In the case of 51/123 the difference is 21 which has two factors, 3 and 7, and 3 is the one you're looking for.

OK, that makes sense, thanks :)

Houngan
02-10-2011, 07:39 AM
Ah, now I recall (with google.) Squaring two digit numbers:

Move the number down to the next tens digit (if squaring 43, we want 40) and multiply that by the number moved up the same amount (40 * 46) then add the square of the number subtracted/added (3). 40*46 = 1840 + 9. 43 squared is 1849. Takes a bit of arithmatic and practice, but pretty handy.

H.

Ezdaar
02-10-2011, 07:56 AM
Many thanks for all the advice and perspectives. This is why I <3 QT3.

Regarding to Lockhart's "A Mathematician’s Lament" as recommended by Ezdaar. I think Lockhart's view is part of the problem, to be honest. Not all children have a "natural love of numbers and shapes". My son certainly does not. He will never become a mathematician. An educational curriculum that want to use his "natural love of shapes" as a focus will be completely wasted on him. Of course, it will be very effective for those children who do indeed have a natural love of shapes.



Are you sure of this? He's not at all curious why, to take the example from the paper, the area of a triangle is half of the length of the base times the height? Mathematics, as mathematicians see it, is essentially solving puzzles. Unfortunately by third grade your son has probably already made the association math=tedious rote algorithm application. One thing you, or anyone reading this, might look into for their kids is a Math Circle. These are groups that meet every so often with the purpose of teaching actual mathematics to children. Normally they're run by professors, postdocs or grad students in math and they do fun things like play with Möbius strips.

Of course this doesn't help your son to simplify fractions; my link was more to add to the discussion of the teaching of math in schools. I don't agree with everything Lockhart says but I do think that asking why we care that our third graders can simplify fractions or add up 3 digit numbers is a good thing to do.

As for the direction the discussion has taken recently, what you want is the Euclidean Algorithm (http://en.wikipedia.org/wiki/Euclidean_algorithm#Greatest_common_divisor). For things that you or I would be attempting to simplify it works great. You could also write the two numbers as their unique prime factorizations and then cancel out the common factors to get the reduced form.