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This question is motivated by the "shaking hands" method I first saw in this response. I'm constantly learning about little things like this that some students find helpful even if I don't.

I'm curious to know what tricks, mnemonics, etc. you know that might be useful to low level students.

Some other examples:

  1. Singing the quadratic formula to the tune of "Pop Goes the Weasel"
  2. Thinking of the x-y axis as a "ground-wall" axis when graphing.
  3. "low d high, high d low, all over low squared" for the quotient rule.
  4. FOIL
  5. "Cross Multiply"
  6. PEMDAS

A caveat: I don't necessarily agree with the use (abuse) of all these. I cringe when students shout "cross multiply!" at the sight of a fraction without taking the time to think about the problem. But I've learned that some students need help in different ways and some don't abuse these tricks. So perhaps the mention of these would be best left for office hours, but that's a different story. For now I would just like to see what you've used.

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You already said that you don’t necessarily agree with these rules, but let me nevertheless give a voice to all the people who, as I, think that PEMDAS and FOIL do serious damage, even though some might find them helpful. –  k.stm May 8 at 21:06
    
@k.stm - Can you elaborate on why? I view such tricks as simple reminders of the steps involved. Hopefully after a few times, a student wont still be saying 'FOIL' but will know how to do this operation. –  JoeTaxpayer May 10 at 18:07
    
@JoeTaxpayer Firstly and most importantly, by teaching such rules, many students might get the impression that their nature of dictating how to calculate stuff is what algebra is all about, whereas the opposite is true: It’s about relations and identities which can be combined and manipulated to express things in many different ways, giving you many different options how to calculate stuff. Then, these rules seem to be the source of helplessness and confusion when they break down: How do you foil $(√2+1)(√3+2)(√4+3)$? How do you pemdas $2/3·9/4$? –  k.stm May 11 at 1:01
    
@k.stm - I see. Of course, the tricks aren't universal, they are hopefully a steppingstone to understanding the how and why of the bigger picture. And then discarded as one internalizes how to do these problems with no thought of tricks or mnemonics. –  JoeTaxpayer May 11 at 1:50

6 Answers 6

Instead of FOIL, I have had luck with the phrase "every times each."

$(x + 5 + \sqrt2)(x + 2)$ doesn't need to be harder than $(x+5)(x+2)$, but if all the students know is FOIL, then you have to explain the process as something like "FOIIIL" which is comical and terrible.

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I used to call the more general version SUPER FOIL (my made up term) before I heard about (or read about) the handshaking idea. –  Dave L Renfro May 7 at 19:11
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Is there any reason we are not teaching to use a grid, in your example 2x3 would show that 6 terms result, and then just add like terms? I didn't grow up with FOIL, but just knowing that your (x+5)(x+2) gives 4 terms and two have x in the same power, ready to combine. The students who don't just get that are shown a 2x2 set of boxes, etc. –  JoeTaxpayer May 8 at 9:51
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I think my reason to avoid a grid is that I don't want them creating a grid to calculate, for example, $4(x^2 + 3x + 1)$. Since this too is really just the same fact, I want the same implementation. –  Chris Cunningham May 8 at 11:47
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@ChrisCunningham I prefer that as well. In my experience, most troubles are brushed away when they realize that you are just using the distributive (to them that means "multiply everything out".) The grid method is useful as well. I'm glad we are discussing this though. Besides our personal preferences, it's nice to have backup methods just in case someone needs to see it a different way. –  Carlos May 8 at 15:37
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If you're going to do FOIL, show how they've been doing that for years. 23 * 45 = (20 + 3)(40 + 5). They've been foiling vertically since they learned how to do two-digit multiplication. –  thumbtackthief May 9 at 19:46

This was supposed to be a comment, but it became too long.

When you want your students to understand the material, I think it's good practice to avoid remembering procedures (I avoid the term 'trick' because I want to distinguish between two types on tricks: IMO, there is a good and a bad kind) as much as possible. I think they are harmful for understanding of the topic. A much more useful skill is to derive formulas on the spot. I have to admit that there are examples in which it is just not feasible to do this (for example, I would advise to just remember the quotient rule as it's too complex to derive on the spot every time, but I think it's also good to be able to derive it, so that forgetting a rule is no big deal).

So, to recap, I think it's better to teach techniques which enable students to come up with formulas. Remembering a technique (normally, I'd refer to a technique as a trick, rather than a formula or abbreviation) usually works way better than remembering a formula (because there is usually some intuition behind the technique, and our brains just are better at remembering ideas than they are at remembering raw formulas).

I can (hopefully) show what I mean, by an example.

Consider two students that have learned how to find roots of a quadratic equation.

Student A has learned to write $x^2 + bx + c = 0$ in the form $(x-a)^2 = d$, so that you can apply a squareroot on both sides, then solve for x to get: $ x = a \pm \sqrt{d} $. Ideally, he would understand that we translate the graph of the polynomial (by doing a substitution for x) to get rid of the $bx$ term, get an equation that is easy to solve, and then translate/substitute back again.

Student B has learned $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Now suppose they learn about cubic polynomials and their roots. Using some clever geometric tricks, it is possible to solve $x^3 + px + q = 0$.

Student A's knowledge of the technique now enables him to see a similar possibility here. IF you have $x^3 + ux^2 + vx + w = 0$, and you can find an $a$ such that the quadratic term will vanish (just like the first-order term vanished in the quadratic equation), you can write any cubic equation in the simpler form. So student A may realize that the solution to the simpler cubic equations is enough to solve for all cubic equations.

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But this example shows exactly that the formulas are useful. People do NOT derive the formula every time they solve a quadratic equation, just as they do not derive $7\cdot 6$ every time they encounter it. I am really in favour of understanding, but automation of basic calculation frees up important brain real estate to have time to think about the big picture. –  user11235 May 7 at 23:26
    
I agree to some extent: there is obviously a point where you can't derive everything on the spot. But, I the abc-formula is the best example of a formula that is completely unnecessary. 'Completing the square' is just as easy and reveals a technique that can be used in other circumstances as well: translating a function until you have some desirable properties. –  Mon Kee Poo May 8 at 4:29
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Of course, it also matters how often you use something, how hard it is to remember and for what purposes you need it. If you want to impress your friends by giving roots of quadratic equations, it is much easier to write the quadratic formula as $x^2 = ax + b$ and remember that the solutions are $x = \frac{a \pm \sqrt{a^2 + 4b}}{2}$. (As you may have noticed, I have a strong hate against the abc-formula in the form that it is usually teached, haha) –  Mon Kee Poo May 8 at 4:33
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@Mon Kee Poo I like your attacks on the abc-formula -- especially the more-easily-remembered version! But from a practical standpoint I still like the traditional formula. The reason is that I can teach the method "for quadratics, move everything to one side and factor. Every quadratic is factorable (fundamental theorem of algebra), but if the factors are too ugly to find easily, this formula will find them as a last resort." –  Chris Cunningham May 8 at 11:53
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@MonKeePoo I understand where you're coming from and I agree in some sense. However, I think it's important to remember that some students (the one's I'm used to) might not find it so easy to keep track of formulas in their books and the bigger picture AND a deeper understanding of what's going on. Currently, I'm leaning towards user11235's comment about freeing up "brain real estate". Of course this applies only to a particular type of student which is important to keep in mind. –  Carlos May 8 at 15:57

Tina Cardone has been gathering examples of tricks like this in an crowd-sourced online book, available here: http://nixthetricks.com.

I recommend reading her curated book. She has many, many of these tricks described, and the reasons why you should never use any of them.

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In trigonometry, a well-known mnemonic is SOHCAHTOA but this only works for acute angles in right triangles.

For the three basic trigonometric functions of any angle (with an angle $\theta$ in standard position, vertex at the origin, terminal side intersecting the circle centered at the origin with radius $r$ at the point $(x,y)$), I teach my students to relate $x$ with horizontal motion, $y$ with vertical direction; and $r$ with a "road".

Thus, to remember that $\cos\theta=\frac xr$, think of a car (a convertible with the top down) moving over a road. (In the Filipino language, the word that we use for car is kotse, which is quite similar to $\cos$.)

To remember that $\sin\theta=\frac yr$, think of looking up while you're over the road. You will see the sun (which is quite similar to $\sin$).

To remember that $\tan\theta=\frac yx$, think of staying in your car and looking up. You will get a sun tan ($\tan$) because the sun ($\sin$) is over the convertible ($\cos$).

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I know it's not perfect, but it works for me (and I have a weak memory). –  Joel Reyes Noche May 9 at 5:33
    
+1 for SOHCAHTOA, the rest is great for Filipinos, great, non-English mnemonics. –  JoeTaxpayer May 9 at 12:35
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I could never remember "SOHCAHTOA", so when teaching trigonometry I would ask if anyone in the class knew it (usually over half the class knew it) so that I could put it on the board for those it might help. What worked for me (but I realize others learn differently, so I put up SOHCAHTOA and anything else that students suggested) is to associate cosine and sine with the $x$- and $y$-coordinates of points on the unit circle ("c" comes before "s", "x" comes before "y", "a" in adjacent comes before "o" in opposite) as explained here. –  Dave L Renfro May 9 at 13:54
    
@DaveLRenfro, thanks, that's the first time I've heard of that trick. –  Joel Reyes Noche May 9 at 14:16
    
I was taught OH-SAH-COAT instead of SOHCAHTOA with a long rambling story about leaving his coat behind in a restaurant and the waiter running after him with it, and out of breath shouting OH-SAH-COAT (ohs,ahc,oat) –  AndrewC May 9 at 14:44

$X^3 - 1 = (X - 1)(X^2 + X + 1)$

The rule is SOAP - Same, Opposite, Always Positive

$X^3 + 1 = (X + 1)(X^2 - X + 1)$

Let me offer the broader context - the assignment to factor $X^3 - 1$ is first solved via Benjamin's method, as he comments below. Then $X^3 + 1$ is also solved this way. The answers are compared and the general rule is created. When confronted with, say, $X^3 - 8$ in the context of a larger problem to solve or under the time pressure of a standardized test, the student might save valuable time by knowing such tricks. I assume that any tricks the OP would gather would be presented in that spirit.

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Faced with $x^3 - 1$, I would think: This expression is zero when $x = 1$. Then I would factor out an $x-1$, after which I could compute the rest: to get $x^3$, we will need to multiply $x$ by $x^2$; but this also produces a $-x^2$. To get rid of this new term, we add on an $x$, but this produces another $-x$. To get rid of this new term, we add on a $1$, which produces a $-1$ as desired. So we end up with: $x^3 - x^2 + x^2 - x + x - 1 = x^3 - 1$. Similar reasoning works for $x^3 + 1$... –  Benjamin Dickman May 8 at 8:44
    
right, but since the difference of cubes appears to be a common factoring issue, a student told me this acronym/mnemonic. –  JoeTaxpayer May 8 at 9:45
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I'll have to try this. Surprisingly enough, I've had no trouble with students remembering the differences of cubes once I showed them a few examples of higher difference of n powers. Once they see a longer pattern emerge, I think they feel more confident in themselves. –  Carlos May 8 at 15:46

I'm not sure if this qualifies, but there is literature on "tricks" for arithmetic. Many of the, what you call "low level", students that I have encountered could benefit from an additionally multiplication or division trick or two (or more). Two such books that I know of are Rapid Math Tricks and Tips by Edward H. Julius and High-Speed Math by Lester Meyers. The former is almost exclusively arithmetic tricks, whereas the latter contains some applied mathematics as well.

I'm not recommending teaching all of these ideas, but perhaps integrating a few of these types of tricks might help free up some of the working memory of your struggling students.

I'm also linking to this MESE question: Tricks for computing things in your head

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