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?
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?
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…
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.
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.
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.
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.
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, 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, 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.
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.
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.