Math Symbols

The SAT likes to include a math few problems with unusual symbols, such as ◊, or ♠. For some exam takers, this is a signal to panic, since they have no idea what ◊ or ♠ mean, but really, these are simple replacement questions, similar to the f(x) questions I addressed in an earlier post.

Really, no one knows what what ◊ or ♠ mean.  The College Board (the writers of the exam) make up a meaning for the symbols as they write the question.  In each case they give you the definition of the symbol, then ask you to apply that definition.

For example:

a ♣ b = a + (b x 2)

3 ♣ 7 = ?

The first line defines the symbol ♣.  We'll use that definition as a template for answering the question on the second line.

Note that in the definition, a appears before the ♣. And in the question, 3 appears before the ♣. The test makers have replaced a with 3 in the left half of the equation; we just need to make the same replacements in the right side of the equation:

a ♣ b = a + (b x 2)

3 ♣ 7 = 3 + (b x 2)

We still have to account for b, though. In the left side of the equation, the test maker replaced b with 7, so now we need to make the same replacement on the right:

a ♣ b = a + (b x 2)

3 ♣ 7 = 3 + (7 x 2)

At this point, we have an equation that we can easily solve. Following the usual order of operations (PEMDAS), we get:

3 ♣ 7 = 3 + (7 x 2)
= 3 + 14
= 17

Essays: keep your eyes on the prize

NOTE: The MCAT no longer includes an essay.

I once had an SAT student who wrote a brilliant essay in which she evaluated both sides of an issue before deciding that there was merit to both sides, and that we could determine which side to go with based on a particular criterion, which she listed and discussed. As I said, it was brilliant, and it was better than many of the essays I receive from MCAT students. Unfortunately, she failed to adequately address the question that had been asked of her, so I couldn't give her a good grade.

When you're asked to write an essay, the first step is to be sure that you understand what you're assignment is. the second step is to be sure that you actually write to address that assignment. Both steps are important.

The first step doesn't take much: you just have to take the time to read the assignment and make sure that you understand it. Try to paraphrase the assignment question (put it into your own words) to be sure that you understand it. If you taking an exam where you're allowed to do so, and you're uncertain about the assignment, ask your instructor.

The second can be trickier, since it can be tempting to use the assignment as a jumping-off point for an essay that ultimately charges off into other territories, or to only answer part of a more complex assignment. Once we're writing, our thoughts may focus on the what's in front of us - is this fact correct, is my grammar ok - and we can lose sight of where we're actually supposed to be going.

The solution to keeping on track is to plan out the essay before writing it. Take a few minutes to sketch out the points you want to make, with their examples and/or reasoning. Look at the completed sketch to make sure that it actually matches the assignment. And then write the essay, keeping to the sketched-out plan: if a new example comes to us as we write, don't add it unless there is the time to go back and rework the original plan to include it (in other words, it's okay to add examples to a take-home assignment, but not for an in-class exam).

If f (x) = 3x + 4

Function problems, like f (x), g (x), h (x), etc, leave a lot of students confused. This is unfortunate, because function problems are nothing more than substitution problems. I'll give you an example:

f (x) = 3x + 4
f (2) = ?

We note that in left half of the question, 2 has been plugged in where x was. To solve the problem, then, all we need to do is substitute (plug in) 2 for x in the right half of the equation:

f (2) = 3(2) + 4 = 6 + 4 = 10

Similarly,

f (3) = 3(3) + 4 = 9 + 4 = 13
f (4) = 3(4) + 4 = 12 + 4 = 16
f (5) = 3(5) + 4 = 15 + 4 = 19
f (6) = 3(6) + 4 = 18 + 4 = 22
f (7) = 3(7) + 4 = 21 + 4 = 25
f (8) = 3(8) + 4 = 24 + 4 = 28
f (9) = 3(9) + 4 = 27 + 4 = 31
f (10) = 3(10) + 4 = 30 + 4 = 34
f (y) = 3(y) + 4 = 3y + 4
f (z) = 3(z) + 4 = 3z + 4
etc.

Sometimes, we might find multiple functions used together. When this is the case, we just follow the usual rules of math to untangle the question:

if f (x) = 3x + 4, and g(x) = 5x - 7
f (4) - g (2) = ?

As before, we merely substitute in. Let's work with each function separately, then put them together, being sure to keep straight that 4 was given to us for the f function and 2 was given to us for the g function:

f (4) = 3(4) + 4 = 12 + 4 = 16
g (2) = 5(2) - 7 = 10 - 7 = 3

Taking the original equation and then substituting in these values, we have:

f (4) - g (2) = ?
16 - 3 = ?

and of course that equals 13.

We may also find cases where functions are nested within parentheses:

if f (x) = 3x + 4, and g(x) = 5x - 7
f (g (10)) = ?

Note that I've defined a new function for g. These are solved in the same way: by following the usual rules of math. g (10) is found inside a set of parentheses, so we start with that:

g (10) = 5(10) - 7 = 50 - 7 = 43

We then substitute this value in for g (10):

f (g (10)) = f (43)

And then we solve f (43):

f (43) = 3(43) + 4 = 129 + 4 = 134

PS: If you had to reach for your calculator to do any of that math, then you're relying on your calculator too much.