Trill Talk Wrote:RPurewal Wrote:i'm also a major devotee of flash cards, by the way.
whenever you notice an association or a rephrase - especially in number properties, but in just about any other subject area as well - put it on a flash card that you can use for later.
for instance:
front of flash card: "when n is divided by 2, the remainder is 1"
back of flash card: "n is odd"
front of flash card: "the average of n consecutive integers is an integer"
back of flash card: "n is odd"
front of flash card: "if x is negative, then x^n, where n is an integer, is still negative"
back of flash card: "n is odd"
etc.
obviously you don't want to - indeed, you can't - make all these flash cards at once. instead, every time you notice a new association - which, for number properties, should mean every time you do a problem (at least at first) - make a flash card with that association on it, and watch the pile grow as you study.
Thanks Ron, very helpful. I am still a little confused about what you are supposed to put on the back of the flash card. Can you give a few examples out of OG11 so I can take a look at how you did it? This would be extremely beneficial for me as I would like to start using flash cards.
i'll give you a couple of examples. i don't have my og11 in front of me at the moment, so i'm going from pure memory here.
REPHRASE EXAMPLES:
1
one of the early data suff problems (i think it's #3 in either the og11 or the quant supplement) contains a statement (1) or (2) that says -x|x| > 0.
since this deals only with negative signs and absolute values, it follows that the subject material is positives/negatives. trying the 3 possibilities (pos, neg, 0) reveals that only negatives solve this inequality, so the inequality is equivalent to saying that x is negative.
therefore,
front of card: "-x|x| > 0"
back of card: "x is negative"
2
let's say you see x^2 < x.
if you want, you could solve this the long way: move the x over to make x^2 - x < 0; factor to x(x - 1) < 0; draw a number line with the two zeroes 0 and 1; test the 3 regions into which 0 and 1 divide the number line; ascertain that only the area between 0 and 1 works; deduce that 0 < x < 1 is the solution.
or, you could ASSOCIATE the
comparison of powers with FRACTIONS, which are the numbers that happen to get smaller when you raise them to powers. once you make that association, realize that fractions (numbers between 0 and 1)
do satisfy the inequality, whereas other numbers don't (test cases as appropriate to convince yourself).
no matter how you solve this,
front of card: "x^2 < x"
back of card: "0 < x < 1"
3
front of card: "xy < 0" or "x/y < 0"
back of card: "x and y have opposite signs"
etc.
ASSOCIATION / OPENER EXAMPLE:
let's say you have a system that looks like y + z = j, y - z = k (as in OGPS#245 - please don't post further details of that problem here).
the easiest way to solve this system is to ADD the equations to produce 2y = j + k (whereupon you can divide by 2 to give y = (j + k)/2).
then, you DON'T want to substitute to solve for z; you can just SUBTRACT the original equations to produce 2z = j - k (whereupon you can divide by 2 to give z = (j - k)/2).
therefore,
front of card:
"THIS + THAT = ______; THIS - THAT = ______
best way to solve?"
back of card:
"ADD the equations to isolate THIS; SUBTRACT the equations to isolate THAT"
etc.
basically, whenever you find
any technique that generally works on
anything, you should make a flash card containing that technique.
good luck.