Ever see those weird symbol questions on the SAT? Something to the tune of “if xΩ = 2x-1, then what is the value of 3Ω?” You freeze and think to yourself—we haven’t covered horseshoes yet in precalc!

1) The “weird symbols” questions are really functions in disguise. Whenever you see a weird symbol, its corresponding relationship (here, 2x-1) will ALWAYS be defined immediately after. So don’t worry if you haven’t covered horseshoes yet. In fact, most of the time you can substitute f(x) for the weird symbol. The question above could be rewritten as, “if f(x)=2x-1, what is f(3)?” So take these questions and treat them exactly like functions. Just the same as a function could be called “f(x), g(x) or h(x),” why couldn’t it be called xΩ?

2) The horseshoe, spade, weird brackets, smiley face, triangle and whatever other strange symbols that show up on the exam have no actual defined mathematical meaning. No theoretical mathematician knows what xΩ is either! That’s why it will always be defined for you.

3) Sometimes the SAT is testing your ability to draw visual parallels . For instance, take the following question:

Because there is no triangle in real mathematics, only in SAT land, the SAT wants to see if you reason—“hmm, in the example, the lower left-hand number (a) was multiplied by the top number (b) and then the lower right-hand number (c) gets subtracted from that product. Maybe I should do the same thing to the numbers in the question! In which case, (7(3)-2) = 21-2 = 19.

QUICK TIP: When you see strange symbols, don’t freak out, treat them as functions, and be on the lookout for visual parallels in how the variables are spatially laid out.

Some SAT Algebra questions will not only have one variable, but two or three for you to keep in mind. On occasion you’ll be given a set of equations, a “system” of equations, and asked to solve for one or all of the many variables in that system. You will also encounter questions like these that will give you a fifth option, choice E: This question cannot be solved with the information given. Remember this general rule of thumb: you must have the SAME NUMBER of equations as unknowns. If there are three variables and only two equations provided, chances are you will not be able to solve the problem. If there are two variables AND two equations given, chances are you will be able to solve it.

However, the SAT has a few tricks up its sleeve—here are two of the most common ones:

2x-y=14, and 4x-2y=28.

Two equations, two variables—should be simple, right? Let’s try our first tactic, substitution, and see what happens. Solve the first equation for y, and we get 2x-14=y. Fantastic—plug that expression in for y in the second equation. 4x-2(2x-14)=28. Distribute: 4x-4x+28=28. Reverse PEMDAS, and we’re left with… 0=0.

Well, while 0=0 is certainly true, it isn’t very helpful! Where was the trick? 2x-y=14 and 4x-2y=28 are the same equation. The second is simply the first multiplied by two. None of the relationships have changed between the variables and thus no new information is provided. This is really a two-variable, one-equation problem—it can’t be solved with the information provided.
Here’s a second trick, one that may make you give up before you even examine the numbers.

2m+z+5p=10, and 2m-z+3p=6. Find the value of p.

Shoot, only two equations, but three variables; it must be unsolvable… Let’s try another technique before we write this one off and move on. This problem requires a little bit of heavy lifting in the world of systems of equations: linear combination (combining multiple equations by either addition, subtraction, multiplication or division in order to cancel out variables). In this case, subtract the second equation from the first (setting up the problem vertically):

2m+z+5p=10
+ 2m-z+3p=6
0 + 0 +8p = 16 becomes 8p=16. Divide both sides by 8, and p=2.

Using linear combination, you were able to cancel out two variables, leaving you with 8p=16, a one-variable, one-equation problem!

QUICK TIP: Use the general rule of thumb that you must have the same number of equations as unknowns, but keep in mind the two ways the SAT can disguise these problems!

There is a subset of word problems on the SAT I: those involving tables, charts and graphs.  These provide a great chance for the SAT to try to trip you up by asking for information that does not readily appear to be on the graph!

The main difficulty students have with line graphs is keeping track of what the axes represent.  To avoid this difficulty, you simply need to read the question with this in mind, then look at the graphs to confirm where this information can be found.

Here’s a question with a line graph:

According to the graph, which of the following is closest to the decrease per year in the number of books sold between 1929 and 1936?

(A)    2
(B)    4
(C)    13
(D)    30
(E)    52

So our two axes represent both the year that the books were sold, and the total amount sold that year.  The axes DO NOT include information about the decrease each year, the information you’ll need to answer the question.   We’ll have to dig for this.

In 1929, approximately 73 books were sold.  In 1936 approximately 43 books were sold.  That’s a decrease of 30 books.  Seven years passed in that time, so per year the decrease was:

The correct answer choice is B.

QUICK TIP: This advice goes for all bar and line graphs, pie charts, tables and scatter plots.  Make sure to take a few seconds to examine the figure before diving right in.  Misinterpretation of the graph (not the question!) is the number source of error on these types of problems!

The SAT will frequently trip you up with what may seem the most simple numbers of all—0, 1 and 2.  In the context of Arithmetic Questions on the SAT, these numbers are deceptively simple, but all have specific special properties.  Memorize these few facts about 0, 1 and 2 and be prepared for their tricks.

Facts About 0

1. Any number multiplied by 0 equals 0.

example –>  6 x 0 = 0

2. You may NOT divide a number by 0.

example –> 6/0 is undefined

*Similarly, 1/(x-3) is undefined when x=3.  Therefore, x=3 could not be a solution to a question containing this expression.

3. 0 to any power is 0.

example –> 0^2 = 0,  0^1.5 = 0, 0^78 = 0, etc.

4. Any number to the power of 0 is 1.

example –> 1^0 = 1,  147^0 = 1,  1,000,000^0 = 1, etc.

Facts About 1

1. Any number multiplied by 1 is that number.

example –> 147 x 1 =147,  132.667 x 1 = 132.667, (a/b) x 1 = 0

*Be on the lookout for ones in disguise, usually in fraction form, i.e. 3/3, (x-2)/(x-2) or b/b.

2. One to any power is still 1.

example –> 1^3 = 1, 1^2.3 = 1, 1^-3 = 1, etc.

3. Any number to the power of 1 is that number.

example –> 147^1 = 147, 0^1 = 0, (x-b)^1 = x-b

4. 1 is NOT a prime number.

A prime must have only two distinct factors.  While 1 has only two factors (1 and 1), they are not distinct (different) from each other; therefore, 1 is NOT prime.

Facts About 2

1. Any integer multiplied by 2 will be even.

example –> 3×2=6, 1×2=2, 0×2=0 (0 is even)

2. 2 is a prime number.

It is the lowest prime number and the only even prime number.

When preparing for Algebra questions on the SAT, remember that ETS hates negative powers.  Whenever you see something like x^-2 in an expression, know that you must get rid of the negative exponent in order to solve the problem.  However, you can’t just delete sections of a question to suit your fancy!  In order to “get rid” of something in an expression, you must either simplify, or change the format of that piece.  Remember, negative exponents can be re-written as positive exponents by putting them in the denominator.  X^-2 becomes 1/x^2.  Voila, we have turned our negative power into a positive one.  This rule goes for all negative exponents: x^-n = 1/x^n.  We haven’t really changed the overall quantitative value of the expression, just its format.

Converting negative exponents will quickly set you up to start simplifying.  For example:

(x^2)(x^-4)=
(A)    1/x^8
(B)    1/x^2
(C)    X^8
(D)    –(x^2)
(E)    -4x^2

Let’s try a combination of converting negative powers and simplifying:  (x^2) (x^-4) =  (x^2) x (1/x^4), becoming (x^2)/(x^4).  Two exponents dividing each other with the same base but different powers?  That’s your clue to subtract the powers and simplify the expression to x^-2.  Go one more round of converting negative powers, and we’re left with 1/x^2, choice B.

Alternatively, you can also recognize that the initial expression (x^2)(x^-4) has two exponents being multiplied with the same base and different powers—in this case you add the powers, getting x^-2.  Triumphant, you look at the answer choices, only to find no options with a negative power!  Again, convert it to a positive one, 1/x^2, and choice B becomes apparent.

QUICK TIP: Always convert negative exponents to positive ones.

Using a calculator effectively on the test can really help you on the SAT Math Section. Here’s a few tips to make the most of using a calculator on the SAT.

-    Have the quadratic formula programmed into your calculator.

-    Have the distance formula in both 2- and 3-dimensions programmed into your calculator.

-    Don’t let the proctor wipe the memory of your calculator or take your graphing calculator away.  The TI-83, TI-84, TI-89 and TI-90 are all allowed on the test and may have programs stored in their memory.

-    You may NOT have your calculator out during the Writing or Critical Reading portions of the test, so don’t count on storing a dictionary or grammar rules in your calculator.

-    If your graphing calculator is new, practice with it before the exam.  Test day is not the time to be searching for the cube root function!

-    Recognize that every problem on the SAT I can be completed completely without a calculator.  Choose when to use your calculator—don’t reach for it on every problem.  When you’re required to multiple 3-digit numbers, by all means go ahead.  But using your calculator to find “8^2” or “49 – 30” can just suck up your time.  Make sure your calculator is a help, not a hindrance.

-    Most importantly, remember that your calculator doesn’t have any common sense—it won’t tell you that the average of 3 and 26 can’t possibly be 27, or that 212 can’t be someone’s age.  It won’t remind you that the question asked for a negative root, when your screen showed only positive 4.

For those of you who hate math, never fear.  Read on and let us take the mystique out of the SAT Math section.  SAT Math is broken down into three sections, with 44 multiple-choice questions and 10 grid-in (or “open response”) questions.

The SAT does not test beyond traditional high school sophomore math.

Believe it.

There is no trigonometry and no calculus.  There are no proofs and no imaginary numbers. There will be questions that concern arithmetic, algebra, geometry, probability & statistics, sequences, ratios, and the interpretation of figures.  It’s all, or at least mostly, stuff you’ve seen before.  It’s just asked about in strange ways.

Remember that even kids taking college Calc II in high school are not scoring perfect 800s on SAT Math.  This is because the SAT phrases questions in weird ways and attempts to trick you by asking things you’d never be asked in a classroom setting.  For instance, a problem may require you to solve for 6w.  I guarantee your high school math teacher has never asked you for 6w.  No, he’d ask you to solve for w.  In addition to understanding the material, the SAT can be an exercise in understanding their tricks and reading directions and problems very, very carefully.

However, there is a wonderful boon on SAT Math.  You’ve been given 44 out of 54 answers!  They’re all there in the test booklet—in the form of multiple-choice.  When it comes to the multiple choice questions, you want to use the principle that one of those five answer choices must be right.  This little fact is really wonderful.  There is an uncountable infinity of numbers out there in the world, but only 5 of them will be answer choices on a given problem.

In order to further increase your chances of getting a problem correct, you must get into the habit of eliminating incorrect answers just as much as looking for right answers.  You should guess on the SAT whenever you can eliminate at least one answer choice as wrong.  That might turn out to be a lot of guessing, but even with the small penalty for guessing wrong, you’ll do better if you guess when you can eliminate at least one answer choice.

We’ll talk about more of these tricks and traps to watch out for in future posts.  For now, rest assured that you’ve already learned everything you need to know in math class to be prepared academically for the SAT.  Now a little review and a few tips should have you fine-tuned in no time!

Word problems never cease to be intimidating, whether in math class or on standardized tests.  You probably find yourself thinking, “If this is math class, then why are there all these words in the question?  If they want me to do the math, why don’t they write out the problem properly?!”

While that is a question best left to pedagogical studies, we can give you a way to decipher those problems on the SAT and “translate” them into the mathematical expressions you find so familiar.

Luckily, just like an encrypted code, each key word in a word problem represents a symbol or operation in a number expression.  Take a look at the table below.

English equivalents of math symbols

Now, let’s translate almost word for word the problem below, strictly adhering to the chart!

EXAMPLE:
If 25 percent of 60 percent of a positive number is equal to 30 percent of p percent of the same number, what is the value of p?

TRANSLATION:
If 25 percent (25 over 100) of (multiply) 60 percent (60 over 100) of (multiply) a positive number (positive x) is equal to (equals sign) 30 percent (30 over 100) of (multiply) p percent (p over 100) of the same number (positive x)…

Without the words, that would look something like this–>

Wow!  This is much more manageable than the word problem we were given a moment ago.

QUICK TIP:  Practice “translating” word problems according to the chart above.  You’ll be surprised at how easily you can break down the problems.  Don’t forget to look for that little word “of,” in particular!

Sometimes remembering the definitions of math terms is all you need to answer a question on the Arithmetic portions of the SAT Math sections.

Here’s a quick review of basic (but important!) math terms:

WHOLE NUMBERS: a whole number can’t be a fraction of a number, a percentage, or have a decimal. Whole numbers are always positive, and are all also called the “counting numbers,” because they include the first numbers we ever learn to count as children (1, 2, 3… 147… a million…). 0 is a whole number.

INTEGERS: all whole numbers and their negatives, from negative infinity to positive infinity (-3, -2, -1, 0, 1, 2…).

RATIONAL NUMBERS: includes all integers, as well as terminating or repeating decimals, such as 1.125, or 2.7777777… These numbers could all be expressed as a fraction.

IRRATIONAL NUMBERS: decimals that never end and never repeat, such as pi or e.  These numbers cannot be written as fractions or integers.

REAL NUMBERS: includes all rational numbers as well as irrational numbers.  Excludes i.

QUICK TIP for SAT Math Arithmetic: In a Venn diagram, whole numbers are the smallest group, completely inside integers.  Integers and fractions are inside rational numbers.  Rational and irrational numbers are inside real numbers. Imaginary numbers are separate from everything.

Remember these definitions and you’ll be ready to learn more strategies for approaching Arithmetic questions on the SAT Math sections.

Algebra is a core subject area of the SAT math section. Master the basics and you will be well positioned for the test.  The first rule of algebra is to ALWAYS BALANCE the equation, meaning “whatever-you-do-to-one-side,” do to the other!  It is important though to not only perform the same operation to both sides of an equation, but to every term as well.

Here’s an example:  3x-7=15.  Simple, right?

Just divide both sides by 3 in order to isolate the variable (get the x on its own), and you get x-7=5.  Perform reverse PEMDAS, add 7 and we find that x=12. Ta-da, done.

But we forgot that we must divide every single TERM by 3, not just both sides of the equation.  So we really end up with x- (7/3) = 5.  Now we’ve made ourselves a fraction, something even more complicated than before, and answer is 7 1/3, or 22/3 –something very different from x=12.

In this particular example, it’d be simplest to start with reverse PEMDAS and add 7, getting 3x=22, and then divide by 3.  But remember the rule of thumb, if you’re manipulating an equation, change every term, not just one term on both sides of the equals sign.

QUICK TIP: One way to avoid careless errors and keep track of your work is to work vertically, and mark ALL YOUR WORK.  Re-copy the entire equation at each step.