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Question

Dear Mitch,

 

I'm not sure if my internet connection is working properly, but I can't seem to pick up the last couple of days' SAT hints. I was wondering if there was some other place you post them?

Sincerely,

Nicholas B.

 

 

 

 

Answer

Dear Nicholas,

I am not able to check your Internet connection, and there is no other place that
adler-n-subtract.com is posting these tips, though we have been informed that they are being copied on other sites and in that way are making the rounds. For now, if it helps tips get to students, I personally do not feel any immediate need to pursue such small 'piracy'.

But the reason you haven't seen any hints for the last couple of days is that we actually took a break to concentrate on other things, such as getting orders filled. (Apparently, once people began wearing them in various forms of home-adjusted fashions on MYSPACE, our tee shirts have become a bit of a trend. That's certainly nothing we predicted!)

So, to catch up a bit with our original goal of averaging one per day until the PSAT, here goes:

1) For math questions that present variables and ask about the outcome of a given function, it is important to know that there are certain numbers that are ALWAYS worth plugging in to see if you can spot any quick exceptions to the way a particular function appears to behave. This is particularly true for the questions that contain absolutes, such as "always" or "never".

EXAMPLE: Does SQUARING X always give you a larger number than X?

Well, if the domain they give you is the set of all integers, some people initially think: YES. However, 0 is an integer (recall from weeks ago, an integer is any number that appears on a 'big dot' on a number line, but not the numbers that appear on the smaller 'dots' (points) in between, like 1 1/2.

And zero does not get larger when squared.

Next, if the same question gave the domain of all integers except zero, many people would quickly realize that even the negative numbers tend to get larger when squared, because a negative times a negative = a positive, so -6 x -6 = 36. ..

HOWEVER: That domain includes the number 1, which does not get larger when squared. (1 x 1 = 1).

(By the way, do NOT forget to look up the meaning of the math vocabulary words you are supposed to know by test day, such as 'domain' and 'range'. Within the next few days you will find on this site some hints on how to recall which is which!)

Next if the same question were presented this way: "All positive numbers except one...", then it seems pretty good, right? After all, that excludes the 'zero' problem and the 'one' problem...

HOWEVER, NOTICE the little CHANGE: This version replaced the word 'integer' with 'number'. Now you DO have to concern yourself with the quantities that are located on the smaller 'dots' (points), such as 1/2 and 1/4. And, as it happens, both of these numbers get SMALLER when squared! 1/2 x 1/2 = 1/4, and, 1/4 x 1/4 = 1/16.

So, I began by mentioning there are a few 'magic numbers' that are more likely than most to make exceptions pop out to help you.

They are:

0, 1, -1, 1/2 (or anything between zero and one), and -1/2 (or anything between zero and -1).

ALSO, very often there are helpful numbers that are specific to the question, so, for example, if the question were to present something like Q - 13 1/2, as one of its parts, you might want to try plugging in 13 1/2.

Likewise, do not forget about some of the often-missed types, such as this: a fraction that has zero on top (numerator) is always equal to zero, unless it has zero for the numerator (the bottom) – in which case it is always 'undefined', i.e., a problem.

Also, a fraction that always has 1 as its denominator, no matter how complicated the version of 1 is, is ALWAYS equal to the numerator, and a fraction that has the same quantity for its numerator as its denominator is always equal to one, no matter how complicated the form of the number appears – UNLESS that denominator and numerator can be zero; then, once again, you have found an exception to the 'always' question.

If you are confused, you probably have not yet printed it out, read it through seven-to-ten times, circling little bits and underlining other bits, processing each as you go.

Try it. I can assure you that even the exceptions to the exceptions will eventually become so clear that by test-day they will be obvious.

Hope this helps.

HINT #2: (Since we are running a bit behind on our promise):

DO NOT LET THEM FOOL YOU WITH SIMPLE TRICKS THAT MAKE NO SENSE WHEN YOU PAUSE TO CONSIDER THEM. EXAMPLE: If you are asked which of the following numbers is infinite, and your selections include: the set of prime numbers, the set of composite numbers, the set of odd prime numbers, the set of negative numbers, and pi, DO NOT FOR A SECOND THINK THAT pi is infinite! Yes, it goes on for an infinite number of places after the decimal, as far as we can tell, but it is NOT infinite. In fact, if you look at it 'rationally' you will see that it is LESS THAN THE NUMBER FOUR! IS 4 infinite? Probably not?

NOOOOOOOOOOOOO!!!!!!!! IT IS NOT!!!!!!!!!!!!

And I hope that helps!

Mitch