Lowest Common Multiple

A Method and an Analogy to Clarify this Topic

During the last few weeks I came up with a way to explain how to calculate the lowest common multiple (LCM) of two or more integers or two or more polynomials.
Many students get confused by this LCM topic. One reason is the simplicity of the fact that, for any two expressions, their product is a common multiple, so, “Why look any further?” many students ask themselves.
They know their teacher told them in general the product is not the lowest common multiple of two expressions, so they know they are going to get marked down if they give that answer, but many do not know how to find the LCM.
Recently I improved my success rate at explaining how to find the LCM when I started using a table format, as follows.
In the head row I write the two or three polynomials or integers for which we are looking their LCM.
On the left margin I make a list (going down) of all prime factors of the expressions involved, without any repetition. Common factors get listed just once regardless of how many expressions they appear in.
Then, having one row per factor, and one column per expression, we fill in the table by writing the exponent each factor appears raised to in each expression, carefully including all exponents (even those with value zero or one).
After all exponents are listed in the table we make another column at the far right, under the heading “Maximum.” There we write the biggest number out of each row.
The next step is to form the LCM as the product of all individual factors listed in the table (in the leftmost column), each raised to its maximum exponent, as listed in the rightmost column (under “Maximum”). This last product is the LCM we were looking for.
Of course this process can be done without the table, but the table makes it explicit, and it helps as a visual aid for the student to see everything that is going on, all at once. It also helps in making very clear that we do not add the exponents, nor do any other operation with them, we only identify and select the biggest one for each factor.
Most students are happy with this process; the table is good enough for them. It gives them a clear method to follow, and it takes away the guessing and the mystery they formerly faced when trying to calculate the LCM. One of them even said: “You just saved my life with that table! Now I know how to do it!”
However, there are always a few students who also want to know why the procedure works, not only how to do it.
For those who ask “Why?” after seeing the table, I have this explanation ready:
“We have to imagine we are watching a movie about spies and intelligence agents, O.K.? Each expression is like a security checkpoint, where our agent has to show the proper clearances to get pass that point. The checkpoints have different sets of requirements. Each requires verification of a certain level of authority for each security category they are checking at that point. The factors of the expressions are the security categories, like “radioactive material,” “fire arms,” “chemical hazards,” and so on. The exponents are the different levels of clearance agents may have in each category. So when determining the LCM we are looking for the bare minimum possible set of clearance levels we need to give an agent for him or her to be able to make it through all the checkpoints, without any extra, unnecessary authority. They don’t lose their credentials when they go through a checkpoint. They only need to show their badges, they do not give them up. That is why we do not need to add exponents; we only need to select the highest from all the expressions for that particular factor.”
I have found this explanation works very well with all students with whom I have used it so far. One of them said: “Oh! I see. The x² from 3x²y is already included in the x³ from 5x³(x+1) because the exponent 3 is higher than 2. We do not need x⁵ or x⁶. Just x³ will be enough.” And I said: “That is exactly how it works!”

Welcome to Number City!

Find your way around. Don’t get lost.

Sometimes students ask me: “How long is going to take for me to pass this test?”
To which I reply: “It all depends on how fast you get to the performance level you need for the score you want.”
The key phrase here is “performance level,” which the tests are supposed to measure.
Sometimes I have to be almost brutally honest by saying: “Look, realistically, as long as you keep hesitating for more than three seconds to come up with the result of a single-digit multiplication, there is no chance you are going to solve a whole problem in less than two minutes. You want to have all those little things down to less than a couple seconds, with no hesitation whatsoever. You have to let go of all those thoughts about not being good at math, or not liking math. If you really want to pass this test, you need to learn how to handle fractions, and all these other things you always hated and have never completely understood so far.”
There is an interesting metaphor I find useful to help students start distancing themselves from their math phobias. I say:
“Think of it this way: Imagine Math is a city you used to visit when you were a child, a city you never liked because you always got lost, or maybe even someone stole your money, or you always got sick when you were there, or something bad like that. I acknowledge it’s only natural for you to harbor bad feelings about that city. Now, because you want to pass this test, it is like now you have to move to that city and live there for a few months. Not only that but, to finally get out of it, you need to work three jobs while you are there, and you need to excel at all of them. You are going to deliver packages during the day, deliver pizzas at night, and drive a taxi cab on the weekends. Do you think you can allow yourself the luxury of being lost again? Are you going to stand there all confused for hours about how to cross the street, or about what avenue takes you downtown? To really do well in those three jobs you want to know all the landmarks, the big buildings, the highways, street names, bus routes, trolley stops, shopping malls, different neighborhoods, and the like, right? So, it’s just like that in math, too. Welcome to Number City. That is why I recommend you to memorize by heart the times tables, square numbers, primes, powers of two, odds, evens, integers, and things like that, so you can easily find your way around and move from place to place as fast as you can without getting lost again. Number sets like “squares” or “primes” are like avenues. Each individual number is like a franchise brand name, with multiple locations around the city. Algebraic operation rules are ways to get fast from place A to place B, like taking the subway or the highway or something like that. You want to set aside your old fears and phobias for a while, and apply yourself to the task of getting to know your way around this city. Then you will pass your test and you will be able to move out and move on with your life. That is what’s needed.”
I find the above analogy helps some students to kind of materialize their math fears and phobias into something external, and objective. They know what is like to familiarize oneself with a new city, so this is a task that looks familiar, doable, and makes sense for them. So they can stop the negative workout on their self-esteem, and focus instead on these concrete and essential memorization steps.

Find and Replace

The substitution method

Most students understand the concept of substitution when the task is to plug in a plain numerical value for a variable in a formula.
A typical example would be to evaluate y = 3x^2 - 5x + 2
when x = -1
Things change dramatically when the task involves plugging in an algebraic expression to replace a variable in another expression, even when the expression we are plugging in is of small complexity.
For example, from y = 3x + 5 plugging in the value 3x + 5 instead of y into the equation 2x - y + 4 = x + 3y - 1.
In the past I often had trouble explaining the process to some students. In my experience, a significant fraction of students taking the tutoring have some difficulties mastering this process. They get the concept in theory, and they are able to apply it in simple examples, but as the replacing expression grows in complexity, they quickly get stuck.
Lately though, I have dramatically increased my success rate for teaching this concept by using the following analogy. I go:
"O.K., time out. I have a question. Are you familiar with the computer program MS Word, the word processor? Have you used it to type some letters?"
They look at me as if I was asking them whether they are from this planet, and they say "Yeah..." Then I continue:
"Have you seen that little binoculars button that says Search and Replace? You know, when you have just finished writing a letter, but you are not very happy with a particular word you used several times, and all of a sudden you think of a better word. Then you click on that Search and Replace button, right? Instead of reading over the whole letter, looking for the word you want to change, and manually typing the new word over and over."
Then their eyes light up and they say: "Oh, yeah! And it gives you the total count for how many replacements were made!"
At this point I say: "Exactly! That is what we are doing here, search and replace. This equation is the letter and this other equation is the Search and Replace button." They silently look at me with a face that yells, "What are you talking about?" Then I proceed:
"Look, this is the letter, O.K.? Our document, from here to here, this equation: 2x - y + 4 = x + 3y - 1 . That is the whole document. And we are the program. This other equation here: y = 3x + 5, that is the Search and Replace button that says:
'Search the document for the letter y and every time you find it, replace it for this other phrase: 3x + 5.' So we perform the instruction, right? We go over the document, symbol by symbol. we copy the 2, we copy the x, we copy the '-' sign, and then we find a 'y.' Well, instead of 'y,' we write this other thing, we write '3x + 5' right? And then we just keep copying the symbols from the original equation until we find another 'y' and we keep doing that search and replace thing until we reach this last '1' here at the end, the last symbol in the original equation."
They totally get it! For confirmation, I ask: "Does that make sense?" They usually say: "Yes, perfect sense! I mean, I get it. Is that all there is to it?" I look them in the eye and I say: "Yeah, that's it" Then they go: "Gosh, let me do the next one!" And they normally get it right in the first try or at most two tries with almost no exception. I am very happy I found this analogy.