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## Fractions R Us

 I'm noticing with our novel approach to teaching fractions, that we're discounting the importance of LCM in connection with adding them.  The reason for that is our use of Euclid's Algorithm for the GCD upon instantiation, meaning when we create a fraction object, we automatically convert to the canonical delegate in each equivalence class, namely that for which the numerator and denominator are relatively prime. The background context is "building fractions" (yes, this is constructionism) from the ground up, in some computer language facilitating operator overloading. That's right, we're doing some programming as a part of our standard mathematics curriculum, what of it? TIs have been programmable for years, and we've heard no one fussing ("more features!" has always been the battle cry, no?). As I've posted about numerous times here (math-teach), irrespective of whether programming's involved, it's interesting to see if a curriculum dodges Euclid's Method for finding the GCD, escaping into "factor trees" (which I have nothing against -- very important for teaching the meaning of the Fundamental Theorem of Arithmetic).   I'd call that a kind of litmus test, to see if you proposal meets minimum standards -- let's apply it to the states first and foremost, see which state legislatures are being lobbied intelligently, vs. getting fed a lot of mumbo jumbo.  Good project for a PhD candidate. So how *do* we add fractions then?  Of course the old fashioned way:  a/b + c/d == (a/b)(d/d) + (c/d)(b/b) == (ad + cb)/bd as justified by using multiplicative identity, other axioms (I'm sure Paul could help out here).   Then you get gcd((ad + cb), bd)), call it thegcd (could be 1), so your new (reduced) numerator, denominator (p,q) become p/thegcd, q/thegcd (multiplying by x/x is to multiply by 1, the multiplicative identity -- this came up earlier, in justifying (a/b)==(a/b)(d/d). How does this look in actual Python?  Here's a fraction class (rational number class) developed to the point of enabling its instances (p/q) to multiply and add (p,q integers, q <> 0).  See plaintext view to restore significant indentation. def gcd(a, b):    while b:        a, b = b, a % b    return a class Q:     def __init__(self, p, q):         thegcd = gcd(p, q)         self.p = p/thegcd         self.q = q/thegcd     def __repr__(self):         return "(%s/%s)" % (self.p, self.q)         def __add__(self, other):  # code for adding  (+)         """         (a/b + c/d) == (a*d + c*b)/b*d         """         return Q(self.p*other.q + other.p*self.q, self.q*other.q)             def __mul__(self, other):         return Q(self.p * other.p, self.q * other.q) So used interactively: >>> from whatever import Q >>> a = Q(1,2) >>> b = Q(1,6) >>> a * b (1/12) >>> a + b (2/3) Kirby
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## Re: Fractions R Us

 - --- Kirby Urner <[hidden email]> wrote: > I'm noticing with our novel approach to teaching > fractions, that we're discounting the importance > of LCM in connection with adding them.  The reason > for that is our use of Euclid's Algorithm for the > GCD upon instantiation, meaning when we create a > fraction object, we automatically convert to the > canonical delegate in each equivalence class, > namely that for which the numerator and denominator > are relatively prime. > > The background context is "building fractions" (yes, > this is constructionism) from the ground up, in some > computer language facilitating operator overloading. > That's right, we're doing some programming as a part > of our standard mathematics curriculum, what of it? > TIs have been programmable for years, and we've heard > no one fussing ("more features!" has always been the > battle cry, no?). > > As I've posted about numerous times here (math-teach), > irrespective of whether programming's involved, it's > interesting to see if a curriculum dodges Euclid's > Method for finding the GCD, escaping into "factor trees" > (which I have nothing against -- very important for > teaching the meaning of the Fundamental Theorem of > Arithmetic).   > > I'd call that a kind of litmus test, to see if you > proposal meets minimum standards -- let's apply it to the > states first and foremost, see which state legislatures > are being lobbied intelligently, vs. getting fed a lot > of mumbo jumbo.  Good project for a PhD candidate. > > So how *do* we add fractions then?  Of course the old > fashioned way:  a/b + c/d == (a/b)(d/d) + (c/d)(b/b) > == (ad + cb)/bd as justified by using multiplicative > identity, other axioms (I'm sure Paul could help out > here).   The alternative proof/method I've been pushing is to multiply the whole sum of fractions by 1 in the form of some common denominator divided by itself, and then distribute the common denominator in the numerator through the sum, dividing first on each fraction addend if you're not a computer for easier computation. Here, in your example above where you choose your common denominator to be the product of the denominators, this alternative proof/method is: a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd This alternative to the "old fashioned way" is more streamlined. You do one thing - the exact same thing - to the whole sum every single time, no matter what common denominator you use, least common or otherwise. That is, you never ever have to worry at all about what different forms of 1 you have to use to multiply the different addends by. And by the way, if you would like to see a generalization to contexts where not all elements have multiplicative inverses, we can use this alternative, more streamlined proof/method even in the contexts where we don't have multiplicative inverses on all elements (including those contexts where all elements do not have multiplicative inverses), like the natural numbers. For here, with fraction addition, just "rewrite" what I wrote above as: a/b + c/d == (bd(a/b + c/d))/bd == (ad + bc)/bd That is, where a/b and c/d are defined (and they can be in the contexts I just mentioned), instead of "multiply the whole sum of fractions by 1 in the form of some common denominator divided by itself", we multiply the sum by the chosen common denominator and then divide this product by this common denominator. This application of a cancellative semiring theorem x = (xy)/y enables us to perform the algebra (distributive property, specifically) to get the final form (ad + bc)/bd, not only in the contexts where all elements have multiplicative inverses, but in those contexts where they don't as well (and this includes those contexts where all elements do not have multiplicative inverses). Notice that a beauty of this proof/method I recommend is that it does not require the use of the least common denominator. It's truly a universal method, perhaps the only universal method. > > Then you get gcd((ad + cb), bd)), call it thegcd (could > be 1), so your new (reduced) numerator, denominator (p,q) > become p/thegcd, q/thegcd (multiplying by x/x is to > multiply by 1, the multiplicative identity -- this came > up earlier, in justifying (a/b)==(a/b)(d/d). > > How does this look in actual Python?  Here's a fraction > class (rational number class) developed to the point of > enabling its instances (p/q) to multiply and add (p,q > integers, q <> 0).  See plaintext view to restore > significant indentation. > > def gcd(a, b): >    while b: >        a, b = b, a % b >    return a > > > class Q: > >     def __init__(self, p, q): >         thegcd = gcd(p, q) >         self.p = p/thegcd >         self.q = q/thegcd > >     def __repr__(self): >         return "(%s/%s)" % (self.p, self.q) >     >     def __add__(self, other):  # code for adding  (+) >         """ >         (a/b + c/d) == (a*d + c*b)/b*d >         """ >         return Q(self.p*other.q + other.p*self.q, self.q*other.q) >         >     def __mul__(self, other): >         return Q(self.p * other.p, self.q * other.q) > > So used interactively: > > >>> from whatever import Q > >>> a = Q(1,2) > >>> b = Q(1,6) > >>> a * b > (1/12) > >>> a + b > (2/3) > > Kirby
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 Paul A. Tanner III wrote (in part): http://mathforum.org/kb/message.jspa?messageID=6135572> Here, in your example above where you [[ Kirby ]] > choose your common denominator to be the product > of the denominators, this alternative proof/method is: > > a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd > > This alternative to the "old fashioned way" is > more streamlined. You do one thing - the exact > same thing - to the whole sum every single time, > no matter what common denominator you use, least > common or otherwise. That is, you never ever have > to worry at all about what different forms of 1 > you have to use to multiply the different addends by. I don't know how common this is with other teachers, but it's something I've pointed out in probably every single one of the over 100 classes I've taught since 1983 (high school and college) in which a review of this is sometimes needed (anything at or below the calculus 2 level), not to mention in all the tutoring I also did in the 1970s and 1980s. I no longer teach, but when I did, I called it the "cross-multiplying method", and I mentioned it a few months ago in this group (although I believe you [Paul] may have been talking about it in this group then as well): *********************************************** http://mathforum.org/kb/thread.jspa?messageID=5929430{5} Give several numerical examples. Point out what happens when you multiply the denominators -- this always gives you a common denominator, and it leads you to the sometimes useful "cross multiplying way of adding fractions" a/b + c/d = (ad + bc)/bd. *********************************************** Dave L. Renfro
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 Dave, I believe the "corss-multiplication" is referred to as "The Means-Extremes Property" in the Algebra textbook that I use.  Theresa At 12:40 PM 3/13/2008 EDT, you wrote: >Paul A. Tanner III wrote (in part): > >http://mathforum.org/kb/message.jspa?messageID=6135572> >> Here, in your example above where you [[ Kirby ]] >> choose your common denominator to be the product >> of the denominators, this alternative proof/method is: >> >> a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd >> >> This alternative to the "old fashioned way" is >> more streamlined. You do one thing - the exact >> same thing - to the whole sum every single time, >> no matter what common denominator you use, least >> common or otherwise. That is, you never ever have >> to worry at all about what different forms of 1 >> you have to use to multiply the different addends by. > >I don't know how common this is with other teachers, >but it's something I've pointed out in probably every >single one of the over 100 classes I've taught since >1983 (high school and college) in which a review of >this is sometimes needed (anything at or below the >calculus 2 level), not to mention in all the tutoring >I also did in the 1970s and 1980s. I no longer teach, >but when I did, I called it the "cross-multiplying >method", and I mentioned it a few months ago in >this group (although I believe you [Paul] may have >been talking about it in this group then as well): > >*********************************************** > >http://mathforum.org/kb/thread.jspa?messageID=5929430> >{5} Give several numerical examples. Point out what >happens when you multiply the denominators -- this >always gives you a common denominator, and it leads >you to the sometimes useful "cross multiplying way >of adding fractions" a/b + c/d = (ad + bc)/bd. > >*********************************************** > >Dave L. Renfro
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 I believe the means/extremes cross multiplication refers to a proportion and would involve an equation. However, for addition and subtraction of fractions, I also have used the method Paul and Dave describe in every class to all levels of students for many years. Fractions seem to trouble them at all levels. Each time I have demonstrated this easy procedure, students ask why no one has ever shown them this? And that is a very good question!
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## Re: Fractions R Us

 Why? Well, I'd surmise that most text books would shun this procedure because it is not very transparent to the student new to adding fractions. I'd guess that most books would show that adding fractions with common denominators can be simulated with manipulatives - and then concentrate on creating the common denominator. I'd further guess that teachers often do show your method to students who show difficulty in finding common denominators. The issue is likely further clouded by a common desire to find the lowest common denominator (for ease of arithmetic I suppose) Gary Tupper Terrace BC vlm217 wrote: > I believe the means/extremes cross multiplication refers to a proportion and would involve an equation. However, for addition and subtraction of fractions, I also have used the method Paul and Dave describe in every class to all levels of students for many years. Fractions seem to trouble them at all levels. Each time I have demonstrated this easy procedure, students ask why no one has ever shown them this? And that is a very good question!
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## Re: Fractions R Us

 In reply to this post by Dave L. Renfro - --- "Dave L. Renfro" <[hidden email]> wrote: > Paul A. Tanner III wrote (in part): > > http://mathforum.org/kb/message.jspa?messageID=6135572> > > Here, in your example above where you [[ Kirby ]] > > choose your common denominator to be the product > > of the denominators, this alternative proof/method is: > > > > a/b + c/d == (bd/bd)(a/b + c/d) == (ad + bc)/bd > > > > This alternative to the "old fashioned way" is > > more streamlined. You do one thing - the exact > > same thing - to the whole sum every single time, > > no matter what common denominator you use, least > > common or otherwise. That is, you never ever have > > to worry at all about what different forms of 1 > > you have to use to multiply the different addends by. > > I don't know how common this is with other teachers, > but it's something I've pointed out in probably every > single one of the over 100 classes I've taught since > 1983 (high school and college) in which a review of > this is sometimes needed (anything at or below the > calculus 2 level), not to mention in all the tutoring > I also did in the 1970s and 1980s. I no longer teach, > but when I did, I called it the "cross-multiplying > method", and I mentioned it a few months ago in > this group (although I believe you [Paul] may have > been talking about it in this group then as well): > > *********************************************** > > http://mathforum.org/kb/thread.jspa?messageID=5929430> > {5} Give several numerical examples. Point out what > happens when you multiply the denominators -- this > always gives you a common denominator, and it leads > you to the sometimes useful "cross multiplying way > of adding fractions" a/b + c/d = (ad + bc)/bd. > > *********************************************** > > Dave L. Renfro Hi Dave, Based on what you said and what some others have said in this thread about the theorem a/b + c/d = (ad + bc)/bd, there might be some confusion about what I was referring to when I wrote, "This alternative to the "old fashioned way" is more streamlined. You do one thing - the exact same thing - to the whole sum every single time, no matter what common denominator you use, least common or otherwise." What I was referring to when I said "this alternative" was not the theorem a/b + c/d = (ad + bc)/bd, which I always share with every student I have taught on fraction addition, but the *method* I outlined. This theorem is just a special case of this method, which I've never seen published or used by another teacher. This method is identical on any number of fraction addends using any of the infinitely many possible common denominators. Hence my reference to it being possibly the most general method possible. Here's the method written out abstractly: Where C is any of the infinitely many common denominators (least common denominator, product of the denominators, whatever) of any number of fraction addends, where the method is written out in two alternative ways for the middle step, a/b + . . . + y/z = (C/C)(a/b + . . . + y/z) = [C(a/b) + . . . C(y/z)]/C or a/b + . . . + y/z = [C(a/b + . . . + y/z)]C = [C(a/b) + . . . C(y/z)]/C The theorem a/b + c/d = (ad + bc)/bd is just one special case of this general method (this special case would be where there are only two fraction addends and where C = bd). This general method works much, much better than the old fashioned way (as Kirby calls it) when either there are more than two fraction addends or when we want to use a common denominator other than the product of the denominators. This is especially so if we are adding rational functions, when we would normally want to to use the least common denominator to keep the algebra simpler. In the old fashioned way, we have to multiply each fraction addend by a different number divided by itself. With this method I'm describing, we don't have to worry with that at all. We also can skip the step of writing out a bunch of different fractions all with the same denominator. Example using the least common denominator (again writing the middle step in two different ways): 3/4 + 5/6 + 7/8 + 11/12 = (24/24)(3/4 + 5/6 + 7/8 + 11/12) = (18 + 20 + 21 + 22)/24 or 3/4 + 5/6 + 7/8 + 11/12 = [24(3/4 + 5/6 + 7/8 + 11/12)]/24 = (18 + 20 + 21 + 22)/24 Note: When distributing through the C (in this example C = 24), divide first on each addend for much, much easier computation - or algebra if the context is rational functions. (For instance, with this example on the first addend, do 24/4*3, not 24*3/4.)
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 Paul A. Tanner III wrote (in part): http://mathforum.org/kb/message.jspa?messageID=6136949> Based on what you said and what some others > have said in this thread about the theorem > a/b + c/d = (ad + bc)/bd, there might be some > confusion about what I was referring to when > I wrote, "This alternative to the "old fashioned > way" is more streamlined. You do one thing - the > exact same thing - to the whole sum every single > time, no matter what common denominator you use, > least common or otherwise." Yes, I thought you were advocating this method essentially exclusive of anything else, although my main point was independent of this. My main point was that I don't think this method is as unknown as you said. You later agree that it's very well known and then say: > This theorem is just a special case of this method, > which I've never seen published or used by another > teacher. This method is identical on any number of > fraction addends using any of the infinitely many > possible common denominators. Hence my reference [...] What you described is actually what I did in class, and I'm sure many others do as well (high school and college). I'd point out that any common denominator will serve, and it varies with the situation as to how much effort you want to expend towards getting near to (or actually equal to) the least common denominator. Using the least common denominator involves more work to find, but less work in the subsequent computations ("more" and "less" being their non-strict versions, of course), while using the product of the denominators (or something between the least common denominator and the product) is the reverse. One thing about not using the least common denominator is that you'll *always* wind up having the result not be in lowest terms (proof left to the math-teach reader). On the other hand, it's not hard to come up with examples where you're still not in lowest terms if you use the least common denominator. Two examples that come to mind are trig. function manipulations (identities, etc.) and algebraic evaluation of derivatives (of rational functions and not-too-messy non-rational explict algebraic functions). In the case of manipulations with trig. functions, it's usually worth your while to spend a little extra time finding the LEAST common denominator. In the case of evaluating derivatives using the difference quotient definition of a derivative, it's usually not worth spending a single moment worrying about a least common denominator, since it's almost always the case that the least common denominator is going to be the product of the denominators. To get back to educational matters, fractions are a topic students have a lot of trouble with, and this has been true since at least the early 1800s (any earlier and we'll be comparing apples with oranges in many ways). I also think it's gotten substantially worse in the past 30 some years, especially the past 20 years, due to the lack of the reinforcement of these skills, as compared to what students used to receive in middle school and high school, from the current almost universal use of calculators by students on everything numerical. I once made a  comment in sci.math related to this that is worth repeating. When I was in middle school and in early high school (early 1970s, before anyone had calculators; at least, no one at my school had a calculator before 1975), we always dreaded word problems with decimal numbers, because those were always the hardest to carry out the arithmetic calculations. Naturally, most people hated word problems to begin with, but at least when you had things like 3/8, 2/5, or 22/7, you knew at worst the calculations would be one and two digit multiplications and divisions. But if you had things like 4.52, 0.581, or 3.14, you knew there was going to be some work involved if you had to divide by one of these numbers (or by a number obtained from them). This was especially true in direct variation and inverse variation problems. Now we're not going to get rid of calculators, so the issue comes down to a trade-off of how much additional practice with fractions should we include elsewhere in the curriculum to offset this decreased practice, if any. I don't know for an average high school student, but I will say that I found lots of ways to include this and other skills (rationalizing methods, practice with exponents, etc.) in high school and college precalculus and calculus classes. Instead of lamenting over the fact that students had a lot of trouble with these topics and then review them in probably the same way they were taught, I came up with assignments and worked examples in class that required heavy use of these topics. Here's an example from an old worksheet of mine for calculus 1. Show by hand (i.e. without using a calculator) that (3/2)ln(2) + (1/2)ln(3) + (1/2)ln[(1 + 1/49)/(1 - 1/49)] is equal to ln(5). I certainly allowed the use of calculators (except for certain trig. tests and for certain "calculus gateway" tests), so students were free to check this numerically (or check their work along the way -- in fact, I strongly encouraged them to check their work this way if they were unsure about what they were doing). However, I explained that full credit always required an appropriate amount of work, and if they wanted to argue with me that they did some of it in their heads, they were welcome to come by my office after class and demonstrate this on similar problems. [Very few people took me up on this. In fact, the students that I was most convinced could do a lot of the "appropriate" work in their heads were almost always were the same students who showed the most amount of work on their assignments.] Dave L. Renfro
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## Re: Fractions R Us

 - --- "Dave L. Renfro" <[hidden email]> wrote: > Paul A. Tanner III wrote (in part): > > http://mathforum.org/kb/message.jspa?messageID=6136949> > > Based on what you said and what some others > > have said in this thread about the theorem > > a/b + c/d = (ad + bc)/bd, there might be some > > confusion about what I was referring to when > > I wrote, "This alternative to the "old fashioned > > way" is more streamlined. You do one thing - the > > exact same thing - to the whole sum every single > > time, no matter what common denominator you use, > > least common or otherwise." > > Yes, I thought you were advocating this method > essentially exclusive of anything else, although > my main point was independent of this. My main > point was that I don't think this method is as > unknown as you said. You later agree that it's > very well known and then say: > > > This theorem is just a special case of this method, > > which I've never seen published or used by another > > teacher. This method is identical on any number of > > fraction addends using any of the infinitely many > > possible common denominators. Hence my reference [...] > > What you described is actually what I did in class, > and I'm sure many others do as well (high school > and college). I'd point out that any common denominator > will serve, and it varies with the situation as to > how much effort you want to expend towards getting > near to (or actually equal to) the least common > denominator. I'm still not sure that we are talking about the same thing as to what I was referring to with the phrase "this method." You said, "My main point was that I don't think this method is as unknown as you said. You later agree that it's very well known . . ." But what I referring to with the phrase "this method" was not what was I referring to in terms of "very well known." What I was referring to in terms of "very well known" is the identity a/b + c/d = (ad + bc)/bd itself, not a method of how to get from a/b + c/d to (ad + bc)/bd. I've never seen in print anywhere the method I'm talking about. The only method I've seen used in print to get from from a/b + c/d to (ad + bc)/bd is (a/b)(d/d) + (c/d)(b/b). (This is the old fashioned method Kirby was talking about - it's what he wrote in the original post.) That is, getting from the former to the latter is a proof of the identity a/b + c/d = (ad + bc)/bd, this proof used as a method. A clumsy one, in my view. There is another proof of a/b + c/d = (ad + bc)/bd. One that is one not clumsy. This other proof used as a method is not a clumsy method, in my view. This clumsy method I just talked about involves coming up with a different number for each fraction addend. Here, with the product of the denominators being the chosen common multiple of the denominators, we use two different numbers for the two different fraction addends, these two numbers being d and b. If we have more than two different fraction addends, say n fraction addends, this problem of having to come up with with different numbers for different fraction addends means we have to come up with n different numbers. And on top of all this, this method involves rewriting the sum of n fraction addends with different denominators with n fractions addends with the same number as their denominators. And then we rewrite that as a single fraction. But with the non-clumsy method I'm talking about to get from from a/b + c/d to (ad + bc)/bd is (a/b)(d/d) + (c/d)(b/b), we come up with one and only one number, whatever common multiple of the denominators we choose. Instead of replacing the sum a/b + c/d with (a/b)(d/d) + (c/d)(b/b), we replace it with the equivalent [C(a/b + c/d)]/C or the equivalent (C/C)(a/b + c/d). Then we distribute the top C through the sum, doing the arithmetic. And for n different fraction addends, this means we do not rewrite the sum of n fraction addends with different denominators with n fractions addends with the same number as their denominators. We skip this step.) I've never seen this non-clumsy method I'm talking about in print. If someone can cite a source where it is in print, please do so. I have no doubt that some teachers have realized it's there as an alternative, and maybe you're one of them. So for C being any common multiple of the denominators, are you saying that you taught your students to multiply the whole sum S of fraction addends by 1 in the form of C/C, this written as (C/C)S, or that you taught your students to multiply the whole sum S of fraction addends by C and then divide by C, this written as (CS)C, and then distribute the top C through S, skipping the step of having to write out a new sum of fraction addends all with the same number as their denominators?
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 Dave Renfro:   > Now we're not going to get rid of calculators, > so the issue comes down to a trade-off of how > much additional practice with fractions should > we include elsewhere in the curriculum to offset > this decreased practice, if any. I agree that calculators will continue to be available, including as emulated on computer screens, however some emerging pre-college math curricula displace them with computers, and introduce a modicum of programming as a way to learn the underlying concepts. The advantage of programming over pressing buttons on a calculator is you're focusing on the algorithms.  How do we find the GCD?  Do we build factor trees and circle primes in common, building up a product that way?  That's one method for sure.  Euclid's Method is another, less often taught, but ultimately way more effective, as distilling to primes becomes non-trivial in a hurry, as the numbers get big. So whereas we look at calculators as "black boxes" that "just know" how to add fractions, we might approach the computer as a machine that "needs to be taught". In teaching the computer how to add two fractions, students also teach themselves how it's done. In this day and age, some languages are easy enough, sharable enough, that teaching "adding and multiplying fractions" is actually within the range of a pre-college student.  This approach is not exclusive of using manipulatives, visualizations, other traditional approaches - -- these all become mutually reinforcing, even with a "new kid on the block" (computer programming). Replacing the calculator with an executable math notation (as Kenneth Iverson called APL) is not like just phasing in a more powerful calculator.  The pedagogical implications are potentially huge, deserving of separate threads / conversations vs. the "appropriateness of calculators" threads -- rather old hat and hashed over after this many decades. Students who build "fraction objects", gradually adding capabilities, coming back to their evolving scripts over a period of days, are consistently looking at the algorithms.  Yes, they execute at superhuman speeds, but no, it's not a black box, and comprehension of how and why is the focus. My purpose in this thread was to point out that if you have a way of getting the GCD really easily (have coded Euclid's Method, understand it as well), then it's less important to focus on the LCM, as we can reduce to lowest terms at the moment a "result fraction" is obtained.   In other words, as numerator p and denominator q enter the "birth method" (where a fraction gets instantiated), the gcd algorithm is automatically applied, such that thegcd = gcd(p, q), newp = p/thegcd, newq = q/thegcd. Put another way, fractions automatically reduce to lowest terms: In the interactive session below, a student is creating Q-objects (rational numbers, members of Q). >>> somefraction = Q(10,100) >>> somefraction (1/10) >>> otherfraction = Q(12,144) >>> otherfraction (1/12) If thegcd == 1, no problem, as this means the original p,q were relatively prime, and dividing them each by 1 leaves them as is i.e. we were in lowest terms already. We might pause here to discuss equivalence classes, the fact that p/q with gcd(p,q)==1, is the canonical or normalized representative of an equivalence class in Q. This is usefully connected to a discussion of modulo arithmetic, where we also have equivalent integers modulo N, but a normal delegate < N. Testing for equivalence. Rational number class (coded by students): >>> from whatever import Q >>> Q(1,2) == Q(5,10) True Modulo integer class (coded by students): >>> from tecc_alg2_u5 import M >>> M.modulus = 11 >>> M(10) == M(21) True What's new in this picture is we replace a paper and pencil process with machine execution, without turning the process into a black box.  It remains apparent to students what's happening behind the scenes.  The concepts remain front and center, and yet we're still taking advantage of machine execution speeds, allowing our computations to become interestingly complex without becoming tedious.   Applying a rotation matrix to 12 vector objects, over and over, making an icosahedron rotate in 1 degree increments, isn't something you'd want to do with paper and pencil (nor do current calculators do the job, of actually displaying a polyhedron).   With computers we have the best of both worlds: attractive eye candy, a sense of crafting something interesting, without losing touch with algebraic / algorithmic / symbolic underpinnings. Of course once you understand the concepts, have proved why the algorithms are robust, you'll not be needing to inspect the source code at every turn.  What we're learning, versus what we're simply using, will vary with the lesson.  By they time we're rotating polyhedra, studying matrix multiplication, we're probably not mystified by fraction addition.  But that's just in the nature of mathematics itself, i.e. it's ever been thus. Kirby
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 Dave Renfro wrote (in part): http://mathforum.org/kb/message.jspa?messageID=6137437>> Now we're not going to get rid of calculators, >> so the issue comes down to a trade-off of how >> much additional practice with fractions should >> we include elsewhere in the curriculum to offset >> this decreased practice, if any. Kirby Urner wrote (in part): http://mathforum.org/kb/message.jspa?messageID=6138734> I agree that calculators will continue to be available, > including as emulated on computer screens, however > some emerging pre-college math curricula displace them > with computers, and introduce a modicum of programming > as a way to learn the underlying concepts. I thought about including something like "(or computers, or whatever)", but decided it would distract too much from the flow of the sentence (which is not to say that I succeeded in doing this in other places, such as this sentence). In 40-60 years we'll probably have voice recognition systems that can answer questions or point the way for finding an answer (as appropriate, such as in a classroom) and automatic math features like how MicroSoft Word can automatically (and usually, but not always) correct misspelled words, with programming at the level you're doing going the way of square root algorithms (still useful for certain people, but not the end user). And perhaps in 100-150 years there will be "math chips" and other kinds as well (likely part organic and part non-organinc) that will be implanted in the brains of children at the appropriate time in their development (like what we do with various vaccinations children get today), and much of education at this time will be learning how to use/adapt to these implants. Also, I suspect sometime between 150 and 200 years these implants will more and more allow us to directly link to whatever the internet evolves into. By the way, these things are hardly much different from where we're at today as compared to some of the things you'll find in stories and novels by Greg Egan (for those interested in some really way way out there thinking). Dave L. Renfro
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## Re: Fractions R Us

 In reply to this post by Paul A. Tanner III On Mar 14, 2008, at 7:06 PM, Paul A. Tanner III wrote: >> > > I'm still not sure that we are talking about the same thing as to what > I was referring to with the phrase "this method." You said, "My main > point was that I don't think this method is as unknown as you said.   > You > later agree that it's very well known . . ." But what I referring to > with the phrase "this method" was not what was I referring to in terms > of "very well known." What I was referring to in terms of "very well > known" is the identity a/b + c/d = (ad + bc)/bd itself, not a method   > of > how to get from a/b + c/d to (ad + bc)/bd. > I'VE NEVER SEEN IN PRINT > ANYWHERE THE METHOD I'M TALKING ABOUT. [emphasis added] Perhaps I'm VERY wrong, but there have been many posts like Dave   Renfro's that indicate that contrary to Paul's opinion, his "theorem,"   "method," "technique," "short-cut," or whatever his preferred   appellation might be, is not quite unknown or original to him. I   strongly believe that we need to collectively lie to Paul, tell him   that indeed he's made an earth-shattering discovery for which he   deserves the world's heartfelt appreciation, and suggest that a Fields   Medal may very well be in his future. Otherwise, we're going to continue to be flooded with posts on this   dead-as-a-doornail topic every time someone raises the issue of   fractions and their teaching. That has been the case for years and   years. It's unbelievable that anyone could be so desperate for kudos,   yet considering the individual in question, it's actually completely   obvious and utterly predictable. I'm not sure that offering him   hosannahs and world-wide acclaim would actually suffice to stem the   flow of these periodic dissertations, but I, for one, would be happy   to give it the old college try.
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## Re: Fractions R Us

 In reply to this post by Gary Tupper - --- Gary Tupper <[hidden email]> wrote: > Why? Well, I'd surmise that most text books would shun this procedure > > because it is not very transparent to the student new to adding > fractions. I'd guess that most books would show that > adding fractions with common denominators can be simulated with > manipulatives - and then concentrate > on creating the common denominator. I'd further guess that teachers > often do show your method to > students who show difficulty in finding common denominators. The > issue > is likely further clouded by a common desire to find the lowest > common > denominator (for ease of arithmetic I suppose) > > Gary Tupper > Terrace BC Hi Gary, As I mentioned to Dave, I'm wondering whether we're all talking about the same method, by what vlm217 wrote below where he seemed to be referring to Dave's "cross multiplication method" (the formula a/b + c/d = (ad + bc)/bd) and by what you wrote above, "I'd further guess that teachers often do show your method to students who show difficulty in finding common denominators. The issue is likely further clouded by a common desire to find the lowest common denominator." Problem is, the method I'm talking about involves first finding a common denominator, preferably the least common denominator. I just pointed out to Kirby, when I replied to his original post, that the method I'm talking about to work, it does not require that the common denominator be the least common denominator. In fact, for the method I'm talking about to work, we don't have to skip the step of writing a new sum of fractions all having the same number as their denominators - we just have skipping that step as an option. As to why the method I'm talking about has, as far as I know, never in published materials been given even merely as an alternative when in my experience it makes things easier for those who struggle with adding fractional expressions, including especially adding rational functions, I don't know. > > vlm217 wrote: > > I believe the means/extremes cross multiplication refers to a > proportion and would involve an equation. However, for addition and > subtraction of fractions, I also have used the method Paul and Dave > describe in every class to all levels of students for many years. > Fractions seem to trouble them at all levels. Each time I have > demonstrated this easy procedure, students ask why no one has ever > shown them this? And that is a very good question!
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 Dave L. Renfro wrote (in part): http://mathforum.org/kb/message.jspa?messageID=6139736> In 40-60 years we'll probably have voice recognition > systems that can answer questions or point the way > for finding an answer (as appropriate, such as in > a classroom) and automatic math features like how > MicroSoft Word can automatically (and usually, but not > always) correct misspelled words, with programming at > the level you're doing going the way of square root > algorithms (still useful for certain people, but not > the end user). Thinking about this some more, it occurs to me that this is too awkward. This would probably sound "quaint" to someone at that time. A better guess is that we'll have "smart pads", something whose texture when writing is between that of ordinary pencil/paper and a magic slate (it's processing unit would be at the top, like the top boarder of an 8.5 by 11 notepad with most of the sheets gone), which interprets and corrects what you write -- storing it in digital form that can be reworked in many ways, such as a typed LaTeX document. Dave L. Renfro
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## Re: Fractions R Us

 In reply to this post by Kirby Urner-5 > Dave Renfro wrote (in part): > > And perhaps in 100-150 years there will > be "math chips" and other kinds as well (likely part > organic and part non-organinc) that will be implanted > in the brains of children at the appropriate time in > their development (like what we do with various > vaccinations children get today), and much of > education at this time will be learning how to > use/adapt to these implants. I seriously doubt it.  I think it's a peculiar subculture of science fiction writer that's always thinking of ways to bypass perfectly good I/O (fingers, eyes, ears) and wanting to wire directly to the brain, assuming some major advances in brain science.  Whereas prosthetic devices make sense in some cases, in general there's no point reinventing the wheel and short circuiting what nature has given us. It's also a peculiar subculture that relegates "end users" to the category of Eloi (H.G. Wells), helpless non-technical people who couldn't tell a computer program from a salad, and a small elite of digerati (Morlocks) who know how everything works. One would hope a serious early numeracy curriculum would work against such a distopian vision, but the outgoing generation of math teacher on the whole seemed content to accept it, by leaving programming to programmers, instead of considering it a basic skill.  I am hopeful we won't be repeating that mistake in the future (overspecialization in general has been a major shortcoming of 1900s style thinking). Kirby