# SAT Math Tips Post

## The Top 10 SAT Math Formulas You Need to Know for the New SAT and PSAT...and the rest of them too.

: I am a Harvard grad, SAT/ACT perfect scorer and full-time private tutor in San Diego, California, with 18 years and 18,000 hours of teaching and tutoring experience.  For more helpful information, check out my my SAT Action Plan as well as my free e-book,
Master the SAT by Brian McElroy.

Despite what many high school students believe, you need to know relatively few formulas for the New SAT Math section.

The reason why there are so few formulas necessary for SAT Math is that the SAT is meant to test your reasoning skills more than your ability to memorize (though in some cases, of course, memorization is necessary). There are always multiple avenues to the solution of a problem, and I teach my students how to take a consistent, accurate approach that utilizes a minimum of formulas and takes the path of least resistance to each answer. Usually, this involves solving the problem differently than you would in math class, stressing technique and common sense over pure memorization.

Take, for example, the distance formula. It’s a big, complicated mess of roots and pluses and minuses, and it’s easy to make a small mistake and screw the whole thing up. Well, no worries, because the distance formula is completely useless on the SAT--and it's just a rearranged Pythagorean theorem anyway. You’re better off just plotting the points on a grid, forming a right triangle and using the Pythagorean theorem. “But wait,” you say, “don’t I still have to memorize the Pythagorean theorem?” Nope. It’s provided for you at the beginning of each math section (though any student of geometry and trigonometry should know it anyway). The Pythagorean theorem is easier, more basic, and less prone to mistakes than the distance formula. So unless you are a whiz at the distance formula and never make careless mistakes on math questions, I would stick with the advice of Mr. Pythagoras.

That being said, there are still a few things you must know by heart on test day.

HERE ARE THE FORMULAS YOU MUST MEMORIZE FOR THE SAT:

1) Percentage and Percent Change ( (Part/Whole) and (Difference/Original) x 100)
2) The Circle Proportionality Formula (Slice/Area = Arc/Circumference = Measure of Inner Angle/360)
3) The Formula for a Line (standard y=mx+b format as well as point-slope format: y-y1 = m(x-x1), and the slope equation (y2-y1) / (x2-x1) ).
4) All 3 Quadratic Identities (unfactored to factored form)

(x2-y2)=(x+y)(x-y)
x2+2xy+y2=(x+y)2
x2-2xy+y2=(x-y)2

5) The Third Side Rule for Triangles (a-b) < c < (a+b) if c represents the “third side” and b and a represent the lengths of the other two sides.
6) Direct and Indirect Proportion (a1/b1)=(a2/b2) and (a1a2 = b1b2), respectively
7) Average = (Total / Number of things)
8) Probability = (Desired Possibilities / Total Possibilities).
9) Surface Area of a Cube = 6s2
10) Distance = Rate x Time (#38 C Test 5, #9 C Test 3)

These are the only formulas you needed to know for the old SAT, but there are some additional formulas and concepts that you will need for the new SAT and PSAT.  On the new SAT (starting March 2016) and new PSAT (starting October 2015) you must also be familiar with the following:

-----
11) The Quadratic Equation (#14 NC Test 3, #15 NC Test 4).  Also know what the discriminant is.  If the discriminant is POSITIVE, then there are 2 real roots ("roots" is another word for "solutions" when equations are written in ax^2+by+c = 0 form).  If the discriminant is ZERO, then there is 1 real root.  If the discriminant is NEGATIVE, then there are no real roots.  (#13 C Test 6)
12) Understanding (not calculating!) Standard Deviation (#23 C Test 4)
13) Binomial and Synthetic Division
14) Weighted Averages (#19 NC Test 5)
15) Simultaneous Equations / Substitution (#19 C Test 1)
16) Functions
17) Imaginary numbers (i) and the iterations of i.  Binomial addition involving constants and i by combining like terms (adding and subtracting complex numbers)
18) Multiplying by the conjugate of the denominator with complex numbers (#11 Test 2)
19) Completing the square
20) Sin x = Cos (90-x) (#19 NC Test 1)
21) Concept: the vertex of a parabola is located at the midpoint of its x-intercepts (#12 NC Test 3)
22) The vertex (h,k) form of a parabola: a(x-h)^2 + k
23) Area of a triangle = 1/2 ab sin C
24) Concept: when an upward projectile reaches its highest point, its velocity is zero.
25) Concept: when an upward projectile lands, its height is zero.
26) Concept: the sides of similar triangles all have the same respective proportions. (#17 NC Test 1, #18 NC Test 2)
27) Concept: in a system of linear equations, there is no solution if the slopes of the two lines are the same (parallel) and the y-intercept is different. (see #9 Test 3)  Conversely, there are infinitely many solutions is the slopes of the two lines are the same and the y-intercept is also the same (#20 NC Test 2)
28) Concept: to find the intersections of two lines, set them equal to one another (#13 test 4)
29) Concept: the “zeroes” or "roots" of a function are the x-coordinates where it crosses the x-axis (and where the y value outputs zero).
30) Concept: the arc measure formed by an angle with its vertex on a circle is double the measure of the angle. (#36 C Test 5)
31) Concept: the value of a function is undefined when the denominator is equal to zero (#36 C Test 1)
32) Concept: the proportion of distance that you travel along the hypotenuse of a right triangle is equal to the proportion of distance that you travel along both legs. (#16 NC Test 4)
33) Concept: a polynomial of Nth degree has at most N-1 changes in direction.
34) The equation of a circle with center (h,k) and radius r:  (x-h)2 + (y-k)2= r2 (#24 C Test 1)
35) Polynomial Remainder Theorem (#29 C Test 1) (#7 NC Test 3)
36) Domain and Range
37) Manipulating Absolute Value Inequalities
38) Negative and Fractional Exponents (#3 NC Test 3)
39) Rules of Exponents: "Same Root" Tricks (multiplication = add the exponents, division = subtract the exponents, taking to a power = multiply the exponents).  "Same Exponent" Trick (perform the operation on the base and keep the exponent the same for multiplication and division operations)
40) Parallel Lines and Transversals (#36 C Test 1)
41) Positive and Negative Associations in Graphs (#5 C Test #1)
42) π radians = 180 degrees (#19 NC Test 2)
43) Box and whisker plots (showed up on March 2018 SAT)
-----

That’s all you need to know as far as formulas and concepts!

YOU SHOULD ALSO KNOW THE DEFINITIONS OF THE FOLLOWING TERMS:

-PEMDAS AND THE ORDER OF OPERATIONS. If you don’t know what I’m talking about here, talk to your math teacher, pronto! Just a reminder…Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Also remember that a TI-83 (perfectly legal on this test) automatically performs PEMDAS so long as you enter the expression correctly.

- MEAN, MEDIAN, MODE. Mean is the same as average. Median is the number in the middle after rearranging from low to high. In the case that the list has no true middle because it has an even number of terms, find the average of the middle two. So the median of the list { 1 1 5 5 } is (1+5)/2 which equals 3. MODE is quite simply the number that appears the MOST.  Multiple modes are possible if there is a tie for greatest frequency: the example I just listed, for example, has two modes, 1 and 5.

-INTEGERS. Integers are whole numbers, including zero and negative whole numbers. Think of them as hash marks on the number line. (For those who don’t know what hash marks are, picture the while yardage markings on the grass of a football field.) Don’t forget that zero is an integer and that negative whole numbers are integers too. Remember that -3 is less than -2, not the other way around (sounds simple but is a common mistake. If I fooled you initially with that one, think of “greater than” as “further to the right” on a number line, and “less than” as “further to the left.”

-PRIME NUMBERS. Prime numbers are positive integers that are only divisible by themselves and the number 1. Be able to list all the primes you between 1 and 50…remember that 1 is not a prime and there are no negative primes. By the way, 51 is not prime…that question actually showed up on a recent SAT. 17 x 3 = 51. What, you forgot your times tables for 17? ;)

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, etc…

Also, be able to use a factor tree and find all the factors of a number and perform a “prime factorization” of a number (this means you find a series of prime numbers that multiplies together to equal that number). The prime factorization of 18, for example, is 3 x 3 x 2.

-PYTHAGOREAN TRIPLES. These are particular types of Right Triangles that just happen to have exact integers as sides. The SAT loves to use them, so know them by heart and save yourself the trouble of calculating all those roots. Here are the ones they use:

3/4/5, 5/12/13, 6/8/10, 7/24/25, 8/15/17

Please note that Pythagorean Triples are not the same as 45/45/90 and 30/60/90 trianges, which are provided for you at the beginning of each Math section.)

-“Y LESS THAN X”
(for example, “x-7” is the correct mathematic translation of “7 less than x.” Be careful because many students will write this as “7-x”, which is incorrect.)

-THE WORD “OF.” (“of” always means multiply.)

-DIGITS. Digits are to numbers what letters are to words. There are only 10 possible digits, 0 through 9.

-MULTIPLES. The MULTIPLES of x are the ANSWERS I get when I MULTIPLY x by another INTEGER. For example the multiples of 5 are 5,10,15,20 etc. as well as 0 (a multiple of everything because anything times zero is zero) as well as -5, -10, -15 and other NEGATIVE MULTIPLES.

-FACTORS. The factors of x are the answers I get when I divide x by another integer. For example the factors of 60 are 30, 20,15,12,10,6,5,4,3,2,1, as well as -5,-6,-10 etc.

-REMAINDER. Remainder is the whole number that’s left over after division. For example 8/3 equals 2 remainder 2. Remainder is particularly helpful on pattern and sequence problems.

-CONSECUTIVE INTEGERS. Consecutive integers are integers in order from least to greatest, for example 1,2,3. The SAT may also ask for consecutive even or odd integers. For example -6,-4,-2, 0, 2, 4 etc (yes zero is even) or 1, 3, 5 etc.

-SUM. Sum means the result of addition. The sum of 3 and 5 is 8. I know, duh, but you’d be surprised how many students will say “15” if they are not paying close attention.

-DIFFERENCE. Difference is the result of subtraction.

-PRODUCT. The result of multiplication. Do not confuse with sum!

-ODD AND EVEN NUMBERS. Even numbers are all the integers divisible by 2, and odd numbers are all the other integers.

-POSITIVE and NEGATIVE NUMBERS. Be aware that if the problem asks for “a negative number,” that does not necessarily mean a negative INTEGER. -1.5 will do just fine. Zero is neither negative nor positive. Be aware of strange tricks with negatives, and that negatives taken to EVEN powers are positive and that negatives taken to ODD powers are negative.

-POSITIVE AND NEGATIVE ROOTS.  Although you might think that the root of 9 is "positive or negative" 3, the rules of math say that it's actually positive 3 only.  Here's how to remember it:  if you see the root symbol, then you want the positive answer only.  However, if the question says x2 = 9, then the answer could be either positive or negative 3.   Strange, I know, but that's the rule. Beware: this concept appeared on both the October and November 2018 exams!

In addition, you’re going to have to remember basic geometrical concepts (vertical angles are congruent, perpendicular lines have slopes that are negative reciprocals of each other, etc.), and how to re-write expressions with negative or fractional powers.  The fewer formulas you need to remember, the more you can focus on technique, and good technique is the true key to an excellent SAT score. I don’t teach my students unnecessary formulas because I can teach them to find the answers using a more logical approach to the problem.

“So why did I spend all those years in math class, memorizing formulas,” you might ask, “when most of these formulas are unnecessary for the SAT?” Well, as I mentioned earlier, formulas are de-emphasized on the SAT because the SAT is meant to be a test of logic more than a test of raw facts. All those formulas you learned in math class are fine to know, and yes, the new SAT requires that you memorize more formulas and equations than ever before, but if you respond to all the SAT Math problems in exactly the same way your math teacher taught you, you’re probably going to run out of time, and you’re most likely not going to get a very good score.

This isn’t math class, where you have to show your work or use “proper” technique. This is the SAT, where the only thing that matters is that you get the correct answer as quickly as possible. So you can get away with shortcuts galore. This is why the best SAT math tutors focus on problem recognition, technique and logic more than they focus on pure memorization.

-Brian

• SAT Tutoring

• ACT Tutoring

• LSAT Tutoring

• GRE Tutoring

• GMAT Tutoring