Talk:Least common multiple/Student Level

Excellent article in progress! Anyone have any comments to make? I wouldn't be surprised if this goes up for approval quickly. - Greg Martin 23:54, 13 May 2007 (CDT)

Thank you.

I think more should be added before approval. Maybe some stuff about polynomials, some more applications, some things about somewhat abstract number theory... Michael Hardy 13:43, 14 May 2007 (CDT)

I'd be happy to write up a section on finding the least common multiple of polynomials. I'm actually teaching this section right now so it's fresh in my mind. I'll get on it today David Martin 13:17, 15 May 2007 (CDT)

Cancelling before multiplying

Nice, easy-to-read article.

Where it says "Note: Always cancel before multiplying." I would suggest something gentler such as "Note: The arithmetic is simpler if you cancel before multiplying," since if you multiply first and then cancel you will still get the right answer. --Catherine Woodgold 20:35, 14 May 2007 (CDT)

I think the comment should be made stronger. Citing only simplicity of arithmetic understates the point. It is frustrating to keep finding students who are alleged to have passed high-school algebra neglecting this point and acting just as if they've never heard of it. (OK, I'm going to restrain myself from getting carried away about engineering students who can't do arithemetic correctly with or without a calculator.) I'd attend to it instantly but for one fact: The most cogent way of illustrating this point is probably by citing some examples of horrible things that can happen when it is neglected. But it may be cumbersome to think of a simple example that does not get far too far from the present topic. Possibly this should be expanded on in some article on algebra when a link and a stern reminder here. Michael Hardy 16:42, 19 May 2007 (CDT)

OK, maybe the part about doing arithmetic should become a separate article. Students seem FAR too ready to believe uncritically everything they get from a correctly functioning calculator. Often they use calculators as anesthetics rather than for calculation... Michael Hardy 16:45, 19 May 2007 (CDT)

I look forward to eventually finding out what the other reason(s) is/are. --Catherine Woodgold 17:45, 19 May 2007 (CDT)

This is a little divergent but it got me thinking about giving reasons and using real examples in math. Hungary, in the 19th century if I remember correctly, obtained a certain amount of autonomy, in education and in other areas, from the Hapsburg Empire and changed their education significantly. One aspect of this change effected the teaching of math, particularly in the gymnasium in Budapest. They employed real examples and with additional encouragment began to develop world class mathematicians. Placing math within reach and showing how it is needed and why learning math is important has been succesful to an extraordinary degree. Making the article, making the link here ot the subject at hand, pertinent to students who may not understand its signicance, would be a constructive step. --Thomas Simmons 21:06, 20 May 2007 (CDT)

A suggestion

As a high school math teacher, I have to teach this concept all the time. My first suggestion would be to use smaller numbers in the example for the prime factorization method. It's hard to comprehend with such large numbers and so many prime factors. Maybe use 4 and 18. Then, offer a second example using larger numbers. This makes it much more accessible to the common reader.

Also, format the prime factors like so:

${04}={2}\times {2}$

${18}={2}\times$______${3}\times {3}$

     $={2}\times {2}\times {3}\times {3}={36}$

Then you just have to bring down one of each factor per column.

If you were to add a third number to the mix, it'd look like:

${04}={2}\times {2}$

${10}={2}\times$______________$\times {5}$

${18}={2}\times$______${3}\times {3}$

     $={2}\times {2}\times {3}\times {3}\times {5}={180}$

Is this too teacherish for the an article entry? David Martin 13:15, 15 May 2007 (CDT)

I'm going to think about this and some suggestions by Larry Sanger, and probably add some stuff on polynomials. Michael Hardy 22:52, 15 May 2007 (CDT)

As for the question, "Is this too teacherish?" I think that at this level we should be targeting the potential audience which will probably not be graduate students but very young students who are learning this for the first time. Giving numerous simple examples is an advantage here. In addition, this approach will make the entire CZ more accessible for a broader range of readers, in this case teachers as well as students. The ability to explain things simply and provide examples will be well and truly appreciated. --Thomas Simmons 18:37, 19 May 2007 (CDT) +11 hours (EPT)

Supplemental sites for extended teaching

Here are additional sites to extend the article:

• Explanation and some test questions [1]
• Simple explanation and test input by the reader [2]
• Simple explanation and example [3]
• Advanced explanation [4]
• Additional subjects for pre-algebraic functions [5]
• Simple and extended definition with test input for reader [6]
• Lower level card matching game checks reader answers [7]
• Calculates Greatest Common Divisor and Least Common Multiple [8]
• On-line math work sheet generator with answer key [9]

--Thomas Simmons 20:46, 19 May 2007 (CDT)

Article intro

In reading the article intro, I have two points. The first is that it states that the LCM of 9 and 12 is 36 without any explantion. To the common reader, this is not intuitive. It might help to actually list the multiples to show that 36 is the first common multiple (and thus the least).

Also, the application in finding the LCD to add/suntract fractions seems misplaced. It might fit better in a section on applications of the concept. I just feel like a reader will not get the application until after reading the sections on finding it. ...said user:David Martin (talk) (Please sign your talk page posts by simply adding four tildes, ~~~~.)

I think it's OK as it is on this point. Some readers will simply accept the statement that it is the smallest multiple; others will do the rather simple arithmetic to check this for themselves. Just convincing oneself that there don't seem to be any smaller common multiples (or that there certainly aren't) can be done quickly in one's head by many people, I think. --Catherine Woodgold 17:50, 19 May 2007 (CDT)

Actually I think this simply points out the primary assumption--where they are starting. David is correct, it is not intuitive. Ask a class of year 10s in a mid range high school in New Zealand (where they claim to have a good programme)--what are the first five multiples of 5? You'll get blank looks. At which age does the reader know what a multiple is? How often do they even use the term? You may have to jog their memory. It comes back to the question of whom the reader is and what the article is attempting to do.--Thomas Simmons 23:56, 19 May 2007 (CDT)

I don't see why you say it's without explanation. The explanation is there. It says:

Since

36 = 12 × 3, and

36 = 9 × 4,

36 is indeed a multiple of both 9 and 12.

If they don't know what a multiple is, this tells them. Michael Hardy 18:31, 20 May 2007 (CDT)

Additional information

Will the article consider LCMs for polynomials, fractions etc.? --Thomas Simmons 00:53, 20 May 2007 (CDT)

Certainly for polynomials. As for fractions... I've never seen anyone speak of LCMs for fractions, but in Euclidean goemetry, one speaks of the "greatest common measure" of "commensurable" line segments and the like, and that's rather like the gcd, and the lcm is certainly a related concept. Michael Hardy 16:15, 21 May 2007 (CDT)

Finding the least common multiple for the least common denominator.--Thomas Simmons 20:36, 22 May 2007 (CDT)

Which is already there. OK Slipped by the first read. Never mind.--Thomas Simmons 20:39, 22 May 2007 (CDT)

I don't know if it appears in any paper\textbook, but Mathematica extends its GCD and LCM functions to rational numbers, in a natural way. We know that if we write two (whole) numbers as a product of prime powers, the LCM is obtained by taking the maximum of the exponents, and the GCD by taking the minimum. We can do the same when the exponents are allowed to be negative. So for ${\tfrac {2}{3}}=2^{1}3^{-1}$ and ${\tfrac {3}{2}}=2^{-1}3^{1}$, the GCD is ${\tfrac {1}{6}}=2^{-1}3^{-1}$ and the LCM is $6=2^{1}3^{1}$. Meni Rosenfeld 08:21, 11 November 2007 (CST)

"A simple theoretical question and a primitive algorithm"

Is this section really necessary? It seems to me that it uses a lot of words to provide little insight. I think it is clear to a typical reader of this article that the product of two numbers is a common multiple of them. Meni Rosenfeld 15:31, 8 November 2007 (CST)

Try it with fifty undergraduates not specializing in mathematics. Give them a pair of numbers without really obvious common factors, somewhat similar in size, for which the LCM is more than about 8 times the size of either and much smaller than their product. List the first several multiples of each and tell them how the pattern continues: 18, 36, 54, 72, 90, ... etc. Ask them whether those two lists could go on forever without having any numbers in common. Ask those who think the answer is "yes" to raise their hands; similarly with "no". THEN ask whether this section is necessary. Michael Hardy 21:34, 9 November 2007 (CST)

I find myself agreeing with Meni. In Michael's scenario, I would imagine that the problem most of the students have is that NO common multiples immediately spring to mind. Thus, some will guess that maybe there won't be any in common. I would guess that if you tell them from the outset that the product of two numbers is a common multiple of both, then it won't take them long to believe it.Barry R. Smith 16:53, 30 March 2008 (CDT)

I looked at the section again today -- I still think that it is unnecessary for the above reasons, but if it stays, I think one addition is needed. It seems that the section only implies that the first number appearing in both lists will be the lcm, but it should explicitly say this from the outset.Barry R. Smith 22:32, 3 April 2008 (CDT)

Common Denominator Example

Given the intended audience, I think an explanation of the first step of the least common denominator example should be provided -- i.e., where did the extra factors in the numerator and denominator come from, why are you allowed to put them in, and how did adding them accomplish what we needed?

Give more explicit reasons

There is a similar difficulty with the examples on calculating the LCM with and without the prime factorizations. Somewhere in the example with prime factorizations, the reader should be explicitly informed that you always take the largest number of times a given prime appears in any of the numbers and include that power in the LCM. Similarly, in the example without the prime factorizations, they reader should be told that the quotient formed is always the product of the two given numbers divided by the GCD. Also, some mention should be made of the fact that this procedure only works for finding the LCM of two numbers. When you have three or more numbers, the procedure is more involved (but it is still possible to do without the prime factorizations -- you would need to iterate the procedure for finding the LCD of two numbers).Barry R. Smith 22:41, 3 April 2008 (CDT)