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Question

Dear Mitch,

That trick you gave a few weeks ago to help remember different kinds of triangles, well, I love that kind of memory trick. We used to call them mnemonics, but I don't hear that word much anymore. Anyway, would you happen to have any other tricks like that for those of us who have trouble remembering everything we're supposed to?

Sincerely,

I Forget...

Answer

Math mnemonics:

Although it is ALMOST ALWAYS better to understand something than have to memorize it and later rely upon something you don't really understand, everyone is different and some find that they're better off just 'knowing' some things without having to understand why they are true. (After all, the world would probably be a boring place if everyone's brain worked the same way!) So for those who prefer to just plain memorize certain facts or formulas (which is all of us at some point) there are ways to make memorizing easier.

Most of us use memory tricks or techniques whether we realize it or not, but for complicated things that are not natural to us to keep in our heads, most of us do best by knowing/understanding the tricks we're using. Even that understanding helps us to keep the memorized stuff permanent and helps people access it when they need it no matter how long ago it was stuck into a corner of their mind. The memory tricks or helpers that we know we're using when we use them are called 'mnemonics'. There are several different kinds of mnemonics, and sometimes many different mnemonics to remember the same thing, and very often the most effective ones are the ones the user makes up himself. But that can be hard to do and take more time than you have; so here are a few of the 500 or more mnemonics I've shared with people over the years to help make their lives just a little easier than they might be without a few helpful tricks. So without further ado, here are a few that students have found helpful (and in no particular order), so you may have to stumble through some that you'd like to think you don't need; in that case, enjoy the review.

Ever confuse the x-axis with the y-axis? (Mitch Adler Original) Ever label them incorrectly and later feel silly because it's so easy.... Well, do this: look at the two axes drawn the way they always are, like a big plus sign with arrowheads on all four ends. Okay, now think of eggs. Think of a dozen eggs that someone has just given you to take care of, to protect. Now, if you had to leave the eggs with the x- and y-axis for a moment while you ran to the restroom, would you feel safer -- balancing them on the tippy-top of the arrow, or would you feel safer resting them along the nice flat horizontal expanse of the flat axis? I think you'd prefer the flat one, without having to balance anything. So, that flat one that goes across, is the "eggs-axis" spelled exxs-axis (and abbreviated x-axis).

If someone told you it's better to balance the eggs at the top of the arrowhead on the flagpole, you'd probably say "WHY?" ...Y ?

Supplementary or complimentary angles. (Mitch Adler Original) One of these terms means they add up to 180 degrees, the other term means they add up to 90 degrees. Who can recall which is which? Well, after years of reading papers including being an official grader of New York State Regents exams, I've come to the conclusion that just about everyone remembers which is which; the trouble, though, is that they only remember about half the time. The rest of the time they get them wrong. There are two mindless ways (mnemonics) to remember forever, an easy way and a harder way. I'll give you the harder way because it's cool and I thought of it myself. Then I guess I'll give you the easy way, 'cause that'll probably help you more (But first check out how cool I am: complimentary is the 90 degree one, and it begins with a C. Supplementary is the 180 degree one and begins with an S, which is made out and can be chopped up into two C's, one going the right way connected to one going the opposite way, so it's two groups of 90 – and that equals 180 degrees. OKAY, now the easy way: Alphabetical order: The C in complimentary makes it come before the S-word supplementary, so it's earlier in the alphabet and complementary angles is the lower number (90) instead of 180.

Permutations and Combinations. (Mitch Adler Original) Here's one more specifically for older students than the others: For anyone doing the area of mathematics involving permutations and combinations, both of which are usually investigated during or near the study of probability, it seems to be easy to mix up when to use the simple formula for combinations with the even simpler one for permutations. (As both formulas are almost always clearly and repeatedly set out in any textbook that covers the subject, and are only useful when accompanied by clear and appropriate explanation, I would be counterproductive for me to open that door here by giving those two formulas; instead, I can make this most helpful by giving you something that you cannot obtain in any of those books: a memory trick to help you through the very commonly confused selection of when a word problem indicates that it's permutation time and when, instead, it's combination time:

Here's the answer: You use the permutation formula WHEN ORDER MATTERS. You use the COMBINATION formula WHEN ORDER DOES NOT MATTER. That means that if there are a group of three people, John, Jean, and Jan, the three of them standing in that order make up the same combination of people as Jean, Jan, and John standing in a different order, as well as when they stand this way: Jan, John, and Jean. John, Jean, and Jan can stand in any order they like, even on their heads, or stand on each other's heads, and they will STILL be the same combination of three people. HOWEVER, Jan, Jean, and John is a different permutation from John, Jean, and Jan, as well as a different permutation from Jean, Jan, John, etc.

And the big trick to remember when it's perm time and when it's combo time is this:

When a person COMBS his or her hair, it doesn't cost the person anything; he simply takes the comb out of his pocket and runs it through the strands atop his head. BUT when a person gets what's sometimes called a "perm", which involves some type of long sitting in a hairdresser's salon, I am told by my acquaintances who actually have enough hair to either bother combing or perming), and a perm costs money as well as time. (In case you're curious and happen to be as removed from such activities as I am, my understanding is that a 'perm' is when a person has some kind of curls or waves permanently (or as permanently as such things get) ironed into one's hair. Or chemically bent into some unnatural disarray, with the aim, I gather, of adorning a nice, natural, beachy-windswept look.

So, since a perm costs money and a comb is free, to anyone having a perm, the order matters. When one pays money and sits for hours in a hairdressing place, one wants the order to be a certain way; if all the curls one pays for are put in backwards or bunched up one side of the head before the artist runs out of chemicals and rolls and time and patience, the customer may care. But when combing one's hair? If order matters, it certainly pales in comparison to the perm situation. To correct a disarrayed comb-job, one simply plucks the comb from one's pocket again and repeats the process – once again for free and without having to make an appointment.

So: Use the formula for PERMutations when the order of the items in question matters to the answer. And use the formula for COMBinations when the order of the items in question do not matter.

< or > . . . "Less than" or "Greater than"?

Easy, right? Yes, but easy to mix up. So here: look at the first one above, the <. It looks a little like an L that's been bent in a breeze, right? Well, think of it that way: it looks like an L because it stands for LESS THAN. Therefore, of course, the other (>) must be GREATER THAN. And that's that.

Parabolas: It can be tough to tell much about a parabola from an equation (especially when you have to draw one in the middle of a test!). So one helpful hint to help you see if your graph is on the right track, it is worth recalling that if there's a positive coefficient in front of the x-squared term then the parabola will open "up" (like a 'U'). Conversely, if the coefficient is negative, it will open downward (like an upside-down 'U'). So if it's positive, it's a big smile, and if it's negative it's a big frown! Get it? Smile is positive, frown is negative.

So be confident and keep that parabola on your face positive!

See you next week...

--Mitch

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Illustrated volume of 110 mathematical mnemonics by Mitch Adler

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