SAT Math Tips Post
The 10 Essential Math Formulas You Need to Know for the SATPosted by Brian R. McElroy at 2015-10-13 00:00:00 Please note: I am a Harvard grad (class of '02), SAT perfect scorer and full-time private tutor in San Diego, California. Since 1998, I have amassed over 15 years and 15,000 hours of private tutoring experience. For more helpful information, check out my Test-Day Tips for the current SAT and/or new PSAT, as well as my SAT Action Plan. Update 10/10/15: What additional math formulas do I need to know for the new PSAT (beginning October '15) and SAT (beginning March '16)? There are some additional formulas and concepts you will need for the new PSAT. I suggest you try the free videos on Khan Academy if you are unfamiliar with any of the following: The Quadratic Equation (#14 test 3, #15 test #4) Understanding (not calculating!) Standard Deviation (#23 Test 4) Synthetic Division Imaginary numbers (i) and the iterations of i. Adding and subtracting complex numbers Multiplying by the conjugate of the denominator with complex numbers (#11 Test 2) Completing the square Sin x = Cos (90-x) Concept: when an upward projectile reaches its highest point, its velocity is zero. Concept: when an upward projectile lands, its height is zero. Concept: the sides of similar triangles all have the same respective proportions. Concept: in a system of linear equations, there is no solution if the slopes of the two lines are the same (parallel). (see #9 Test 3) Concept: to find the intersections of two lines, set them equal to one another (#13 test 4) Concept: the “zeroes” of a function are the x-coordinates where it crosses the x-axis. The equation of a circle with center (h,k) and radius r: (x-h)^2 - (y-k)^2 = r^2 Polynomial Remainder Theorem Domain and Range / Functions are Undefined when divided by zero. Manipulating Absolute Value Inequalities—- That’s all you need to know as far as formulas! 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. Ties don’t count—the example I just listed, for example, has no mode. -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. 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. You WON’T need the distance formula, the Quadratic Equation, Trigonometric Identities, the equations for permutations/combinations, the equation of a circle, fancy programs for the calculator, or even the arithmetic/geometric sequence formulas. 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, 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 much more than they focus on pure memorization. -Brian
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