Question
Dear Mitch
I've heard so many people say that there is a "secret language to math" and that there are magic formulas which, once you figure out when to use them (and how to use them, I guess), then they really make a lot of answers pop right out. Is this true? And, to get right to the point, is this true on the SAT test or the PSAT?
Sincerely,
Scott M
Allentown, PA
Answer
Dear Scott,
Well, the 'secret' part of the phrase secret language probably comes from the fact that anything seems like a secret to those who are unfamiliar with it. This is especially true of languages, such as math and written score for music. And occasionally, when I'm visiting a foreign country whose language I do not know, I have to occasionally remind myself that NOT everyone in the elevator (or bus or whatever) is talking about me! (At least I hope they're not!)
As far as "magic formulas", I have found that the more I work with certain formulas and try to come up with my own new ones, all formulas are both magical and ordinary at the same time. What I mean by that is yes, it is remarkable how an arrangement of variables can make sense of a pattern that is difficult to otherwise see and help the user come up with a solution to a problem that might otherwise take a very, very long time, but on the other hand, as one comes to understand how each one is derived they become a drop less mysterious and 'magical' – though often they become more impressive and (in that way) miraculous.
Now, for the math sections of the SAT and the PSAT, I have traditionally made a point of persuading people that a lot of memorization is really not the most effective way to assure yourself a high score. In fact, the opposite: there is very little knowledge required beyond the beginnings of high school math for the SAT exam and the PSAT exam (I am not referring to the SAT subject exams, which are different and come later and are 'achievement' based rather than 'aptitude'-based). On the top of the test is a box in which they print practically every formula you could possibly wish to have during the exam, such as the area of a triangle (1/2bh), area and circumference of a circle, a couple of volume formulas, and the ratios of two "Special Right Triangles".
However, now that I've stated that position, which is still one I believe should keep you in good stead during the test, I will present a very different (and equally valid) view, since you did ask:
YES, there are some formulas that seem to work wonders for students who prefer to have such things packed into their head, and although they are NOT necessary or expected on the exams you mention, they can occasionally be used. Now, just as most formulas appear the first time one sees them, they might be scary, or at least exotic. However, like anything easily memorize-able, they can very quickly become mindlessly easy to apply – just make sure you are doing so at the right time and in the right way.
O.K., TODAY'S SAT/PSAT HINT #1:
As mentioned above, the standard way to figure out the area of a triangle is to multiply the base by the height, and then multiply that result by 1/2. However, sometimes you are not given the height, or have to do a number of steps to figure it out. There is a formula which is very old and very famous and much admired in the world of geometry, and it was developed by a Greek mathematician whose name was 'Heron' (though, for some reason, many people refer to his formula as 'hero's formula'. No, it is 'Heron's formula', and it gives you the area of any triangle whose base you do not know, but whose sides you do know. And I have seen SAT questions on which one can readily use HERON's FORMULA to arrive at the correct answer – though this method is by no means expected or necessary.
Anyway, here's what you do:
FIRST: You add up the lengths of the three sides.
SECOND: You divide that sum by 2. (In other words, take half of it). This is usually referred to as s, for semi-perimeter, or 'half of the perimeter').
THIRD: You take each of the triangle's three sides, a, b, and c, and subtract them one at a time from that semi-perimeter number. So you get three new numbers: s-a, s-b, and s-c.
FOURTH: You multiply these three numbers together, and then multiply that result by your s (the semi-perimeter).
FINALLY: you take the square root of your result.
AND THAT IS THE AREA OF THAT TRIANGLE.
EXAMPLE: A TRIANGLE WITH SIDES MEASURING 3, 4, 5.
S= (3+4+5)/2
S = 12/2
S=6
The sq-rt of (S (6-3)(6-4)(6-5)) = the area of the triangle
Sq-rt of 6(3)(2)(1)=
Sq-rt of 36 = 6
So a triangle with sides 3,4,5 has an area of 6.
(And it makes sense: consider a right triangle with one leg = 3, one leg = 4, and the hypotenuse = 5 (a very famous right triangle, actually). Well, if it were the full rectangle of 3x4, the area would be 12, and since it is half cut by the diagonal of 5, indeed the area is 6.)
IT WORKS!
SAT HINT # 2 for today: I do not know why, but most students tend to overlook the two 'special triangles' presented in the box of information at the top of the exam booklet.
DO NOT OVERLOOK THEM. ALMOST ALWAYS, ONE OF THEM WILL LEAD YOU RIGHT TO THE ANSWER OF ONE QUESTION AND THE OTHER WILL LEAD YOU RIGHT TO THE ANSWER OF ANOTHER QUESTION. THEY ARE THERE FOR A REASON. GET FAMILIAR WITH THEM. AND USE THEM. THEY WORK.
Hope this helps,
Mitch