# On fractions and the least common multiple

At least in my country, the explanation of the basic operations over rational numbers is done very near to the concept of prime numbers, prime factorization, and the calculation of the least common multiple. In fact, usually l.c.m is explained previous to fractions.

This is a real and typical example of text:

"Addition and subtraction of fractions with same denominator: add or subtract the numerators and keep the denominators [... some examples ...]. To add or subtract fractions of different denominator: convert all fractions to their common denominator, by obtaining the l.c.m. of all denominators [...]".

You can see in the previous text how the concept of common denominator is based on the concept of l.c.m., that easily one of them replaces the other. The drawback of this approach is that students lose the basic concept of the operations in a mix of concepts and a nightmare of l.c.m algorithm.

My questions are:

• Is it done in this way in most other countries ?
• @Number: thanks, for the English correction, fixed. Yes, I know we can use any common multiple :-), this is not the question. In fact, in most of mathematics, the only possibility is to use the product. By example, when denominators are "x" and "y". Another reason to question if l.c.m. is so basic or can be deferred. – pasaba por aqui Jul 8 '18 at 15:57
• Without the technique, students are left with infeasibly large numbers to manipulate in most cases. – Daniel R. Collins Jul 9 '18 at 16:46
• @pasabaporaqui Factorization is not required to compute lcms since ${\rm lcm}(a,b) = ab/\gcd(a,b)$ and gcds can be conputed efficiently by the Euclidean algorithm. – Bill Dubuque Jul 11 '18 at 16:46
• @pasabaporaqui What do you mean by the "big numbers problem"? I do agree that fraction addition should not be defined using lcms. The point of my prior comment was merely to clarify that lcms can be computed simply and efficiently. I have had success teaching gcds and lcms to very young students (primarily for general number theory purposes - not fractions). – Bill Dubuque Jul 11 '18 at 16:55
• In Spain my daughters have been taught (in school) to add fractions in the way that you describe. If one has a lot of practice, and one is adding only fractions with fairly small denominators, then one acquires the ability to quickly know the LCM, and this approach is not then unreasonable. For teaching children it seems wrongheaded, as it gives as a definition what should be the conclusion of a theorem following from a simpler definition. – Dan Fox Jul 12 '18 at 16:02

Hung-Hsi Wu seems to agree with you in Some remarks on the teaching of fractions in elementary school:

The worst case is the rule of adding two fractions. In book after book (with very few exceptions, such as Lang (1988)), $\frac{a}{b}$ + $\frac{c}{d}$ is defined as $(pa + cq)/m$, where $m = \mathrm{lc}m\{b, d\}$ and $m = bp = dq$. Now at least two things are wrong with this definition. First, it turns off many students because they cannot differentiate between lcm and gcd. This definition therefore sets up an entirely unnecessary roadblock in students’ path of learning. Second, from a mathematical point of view, this definition is seriously flawed because it tacitly implies that without the concept of the lcm of two integers, fractions cannot be added.

(Typos present in the PDF from Wu's site have been corrected in the above quote.)

• Excelent reference, thanks – pasaba por aqui Jul 9 '18 at 13:11
• Great discussion by Mr. Wu but as always he criticizes and doesn't give an alternative. – llllllllllllllllllllllllllllll Jul 9 '18 at 15:37
• @inéquation Almost surely Wu does give an alternative - namely use the universal definition $a/b + c/d = (ad+bc)/bd\,$ (as would any professional mathematician). This yields a simpler and more conceptual approach. – Bill Dubuque Jul 9 '18 at 16:49
• @Number accepted your correction (although the error is in the source I copied it from) – user5114 Jul 9 '18 at 18:00
• @inéquation But the question is about this issue, not other "problems". It seems you may not know that Wu has written much on teaching fractions - including extensive introductions, so your claims that he "doesn't give an alternative or solutions" is quite puzzling. He certainly does. – Bill Dubuque Jul 11 '18 at 13:51

In Italy it is standard to teach addition of fractions using l.c.m.. An example from a widely used textbook: "Per addizionare due frazioni aventi denominatori diversi dapprima si riducono le frazioni al minimo comun denominatore e poi si addizionano i rispettivi numeratori" (to sum two fractions having different denominators you first reduce them both to the l.c. denominator and then you had the two respective numerators).

I think the reason is somewhat historical. At some period, a huge stress was placed on all fractions being reduced to minimal terms (do not know if this is the correct terminology: numerator and denominator with no common divisors). I remember as a kid (some 40 years ago) endless exercises on reducing fractions to minimal terms. There no discussion about "equivalence of fractions" defining a single rational number but rather it was implicit that a non-reduced fraction was not a "real" fraction but only something not well defined. If the result of an exercise was a fraction, giving the result in non-reduced form was considered an error. Fractions corresponding to integer numbers are still called apparent fractions (with a rather misleading terminology in my opinion).

I do not see any special advantage in this approach which I find rather clumsy, and I agree with what Wu writes on the topic.

• Thanks for sharing your information. It is surprising, something so common and basic, but done only by tradition, with a lot of answers/papers/comments from experts that are in opposition with the classical method. – pasaba por aqui Jul 11 '18 at 16:08
• @pasabaThe emphasis (or insistence) on reduced / least forms is not only traditional. It also arises from using normal (canonical) form representatives of equivalence classes vs. the classes themselves when constructing quotient structures. Often the normal forms are more useful for computation, but the more flexible classes are more useful for theory. To master algebra on quotient objects one should be proficient with both methods and be able to quickly and effortlessly shift between the two viewpoints. See the links in my comments on Wumpus' s answer for further discussion. – Bill Dubuque Jul 12 '18 at 14:56
• @pasaba In fact some algebraists consider quotient structures to be the pons asinorum of a first course in abstract algebra, e.g. see the quote here from Andy Magid's review of Jacobson's classic textbook Basic Algebra I. – Bill Dubuque Jul 12 '18 at 15:11
• @Number, right. Canonical classes instead of classes themselves, which is of course a less abstract way to think about fractions. But when you say that "often normal forms are more useful for computations", well this is exactly the case in which they are not. If you wish, the point being that the product of canonical representatives is not always in canonical form and therefore an additional effort is required. Not only with fractions it's the case that it was more traditional to use canonical forms and it has become more standard to use the classes. – Nicola Ciccoli Jul 12 '18 at 15:40
• And to add: this point is a very sensible one in teaching fractions. I love Wu's stress on representing fractions on the line to show in which sense "different" fractions are the same fractions. Which is btw the reason why many modern presentations insist on teaching fractions as operators (where you see clearly how different pairs of integers define the same operator). – Nicola Ciccoli Jul 12 '18 at 15:44

This is the only way I've ever seen it taught or read it described in textbooks - except in my classes. I just teach students to multiply the numerator and denominator of the first fraction by the denominator of the second and vice versa. No prime factorizations, least common multiples, etc. are necessary. This is based on the idea that you don't need the least common denominator. You just need a common denominator - any one will do.

You mentioned "roadblocks to learning" and I definitely think is a big part of where people's negative feelings about fractions start. The procedure for finding the least common multiple can be lengthy and involves multiple steps. In my opinion, it's counterproductive to graft this lengthy algorithm onto the beginning of what is already a multi-step process.