How to explain the LCM algorithm to an 11 year old?

Tl;dr: I am trying to teach my sister how the LCM algorithm works but I just can't figure out how to explain it intuitively. The best explanation I can give is that you're trying to construct a number that contains the common factors of each number we are taking LCM of as once, but I'm not sure how to explicitly show the connection of this idea with the steps in the algorithm.

The algorithm is discussed under the section "Using the table-method" in this Wikipedia here

• I think better mathematical development and mathematical insight will result from not initially following an algorithm, but having her work out the LCM's herself by whatever means she can come up with. I have to do this quite a bit with my line of work (see "Not everyone is equally cognizant $\ldots$), and I still do it the way I came up with (and later learned in class) when I was a child, (continued) – Dave L Renfro Dec 2 '20 at 15:34
• which is write the prime factorization of each number and then use the union of the primes appearing each affixed with that prime's maximal-appearing exponent. It is also fairly easy to intuitively see that this gives a common multiple that is the least such. – Dave L Renfro Dec 2 '20 at 15:34
• @DaveLRenfro Not to an 11 year old it's not... – PhysicsMathsLove Dec 2 '20 at 18:40
• @PhysicsMathsLove: FYI, I don't think LCM was mentioned in my school math until 8th grade (age 14-15). Before then it was finding common denominators, and the denominators rarely went past 2-digit integers. Of course, I knew about LCM and GCD from library books, but I don't think I paid much attention to the notions until I had to in school (8th grade), since they didn't have anything to do with stuff I was really interested in (special relativity, trying to learn algebra, higher dimensions, complex numbers, etc.), so I knew about prime factorizations by that point (but not the subtleties). – Dave L Renfro Dec 2 '20 at 19:06
• Don't use a complicated algorithm. Use the definition. What is the first number their lists of multiples have in common? If you try to show her an unnecessarily complicated algorithm, it will teach her the wrong things about what math is. – Sue VanHattum Dec 7 '20 at 4:49

1 Answer

EDIT

Here are the CCSS for grade 6.

http://www.corestandards.org/Math/Content/6/NS/B/4/

At this age, perhaps she is just expected to "find the LCM of two numbers less than or equal to 12". They do not specify a method, but I would still advocate using the definition (as presented below).

I would start by making sure she knows, and can compute with, the definition of the LCM.

For instance, to find the least common multiple of 60 and 28.

First list all the multiples of 60 and all the multiples of 28:

60, 120, 180, 240, 300, 360, 420, 480, ...

28, 56, 84, 112, 140, 168, 196, 224, 252, 280, 208, 336, 364, 392, 420, 448, ...

The least number which is a multiple of both 60 and 28 is 420. Interestingly, if you continue the list, the next common multiple would be 840, and the next one after that would be 1260.

Work with this definition exclusively for a while.

For the small numbers your sister is working with, this method might be sufficient.

However, you might want to develop some ideas which could "speed things up".

For instance, in this example you might notice that since $$60 = 4*3*5$$ and $$28 = 4*7$$, that there is no chance of the multiple $$60*1, 60*2, 60*3, 60*4, 60*5$$, or $$60*6$$ being a common multiple, since none of these have a factor of seven (why?), and ANY multiple of 28 will have a factor of 7. We could have skipped all of those, and the first number we check (in this case) would be 60*7, which actually worked.

Developing this circle of ideas further can lead to the method that Dave L Renfro mentions in the comments. Understand, however, that we are secretly invoking uniqueness of prime factorization when we make such arguments, and that theorem is not obvious.

• and that theorem is not obvious --- Interestingly, especially from a psychological/educational and philosophical viewpoint, it is often not obvious to students why the theorem is not obvious. – Dave L Renfro Dec 2 '20 at 16:32
• @DaveLRenfro I have not read any research on that question, but I would love to better understand why people think it is obvious. Thinking it is obvious must be a mistake, and so I imagine that the reasoning which leads people to believe in it is probably false. There may be no reasoning at all: just a constant bombardment of authority figures declaring it is true, until it is accepted. – Steven Gubkin Dec 2 '20 at 16:35
• I think it seems obvious because the primes are like indivisible multiplicative building blocks, and experience tells us that whenever you factor into numbers, other primes can't enter unless they're factors of one of the numbers you're using, which can't happen for primes. Of course, the problem is that how do we know the product of primes $a,$ $b$, $c, \ldots$ can't equal the product of primes $p,$ $q,$ $r, \ldots$ without all the primes in the first group matching with primes in the second group? Maybe (large prime)*(another large prime) could equal the product of a bunch of little primes. – Dave L Renfro Dec 2 '20 at 16:58
• FWIW, I've got plans sketched out for a talk to an English group on campus about the amount of labor and ink spent on uniqueness proofs, fundamentally to establish whether we should use the article "a" or "the" before a particular object -- which I expect them to find amazing, if not unbelievable. – Daniel R. Collins Dec 3 '20 at 6:11
• @Daniel R. Collins: Misuses of "the" and "a/an" often occur on admissions and scholarship exams in countries where English is not the primary language. Here's an example I've used in an item writing workshop, which was adapted from an actual test item (I was reviewing a certain standardized test's item bank): "A box contains $1$ blue marble and $3$ red marbles. The red marble is randomly chosen …” A test taker might think “red” is a typo and “blue” was intended. Also, this is likely to more strongly impact native English speakers, who are less used to overlooking misuses such as this. – Dave L Renfro Dec 3 '20 at 12:28