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SAT Math - Khan Academy - Tutorials & Exercises by
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# SAT Math - Khan Academy - Tutorials & Exercises

SAT Math - Khan Academy Here are some of the math subjects you may see tested on your upcoming SAT. - Watch the tutorials - Do the exercises

## Absolute Value

Finding Absolute Value

## Angles

Understanding Angles

## Area & Perimeter

Find Area of Trapezoid

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## Even & Odd Numbers (see notes)

Even and Odd Numbers Even numbers are numbers that can be divided evenly by 2. Even numbers can be shown as a set like this:   { … -4, -2, 0, 2, 4, … } Odd numbers are numbers that cannot be divided evenly by 2. Odd numbers can be shown as a set like this: { … -5, -3, -1, 1, 3, 5, … } Zero is considered an even number. Is It Even or Odd? To tell whether a number is even or odd, look at the number in the ones place. That single number will tell you whether the entire number is odd or even. An even number ends in 0, 2, 4, 6, or 8. An odd number ends in 1, 3, 5, 7, or 9. Consider the number 3,842,917. It ends in 7, an odd number. Therefore, 3,842,917 is an odd number. Likewise, 8,322 is an even number because it ends in 2. Adding Even and Odd Numbers even + even = even 4 + 2 = 6 even + odd = odd 4 + 3 = 7 odd + odd = even 5 + 3 = 8 Subtracting Even and Odd Numbers even - even = even 4 - 2 = 2 even - odd = odd 4 - 3 = 1 odd - odd = even 5 - 3 = 2 Multiplying Even and Odd Numbers even x even = even 4 x 2 = 8 even x odd = even 4 x 3 = 12 odd x odd = odd 5 x 3 = 15 Division, or The Fraction Problem As you can see, there are rules that tell what happens when you add, subtract, or multiply even and odd numbers. In any of these operations, you will always get a particular kind of whole number. But when you divide numbers, something tricky can happen—you might be left with a fraction. Fractions are not even numbers or odd numbers, because they are not whole numbers. They are only parts of numbers, and can be written in different ways. For example, you can't say that the fraction 1/3 is odd because the denominator is an odd number. You could just as well write that same fraction as 2/6, in which the denominator is an even number. The terms “even number” and “odd number” are only used for whole numbers and their opposites (additive inverses).

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## Integers (see notes)

Whole Numbers and Integers Whole Numbers Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, … (and so on) No Fractions! Counting Numbers Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero. So they are 1, 2, 3, 4, 5, … (and so on). Natural Numbers "Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject. Integers Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed! So, integers can be negative {-1, -2,-3, -4, -5, … }, positive {1, 2, 3, 4, 5, … }, or zero {0} We can put that all together like this: Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } Example, these are all integers: -16, -3, 0, 1, 198 (But numbers like ½, 1.1 and 3.5 are not integers) Confusing Just to be confusing, some people say that whole numbers can also be negative, so that would make them exactly the same as integers. And sometimes people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing! My Standard I must admit that sometimes I say "negative whole number", but usually I stick to: Name Numbers Examples Whole Numbers { 0, 1, 2, 3, 4, 5, … } 0, 27, 398, 2345 Counting Numbers { 1, 2, 3, 4, 5, … } 1, 18, 27, 2061 Integers { ... -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … } -15, 0, 27, 1102 But nobody disagrees on the definition of an integer, so when in doubt say "integer", and if you only want positive integers, say "positive integers". It is not only accurate, it makes you sound intelligent. Like this (note: zero is neither positive nor negative): Integers = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... } Negative Integers = { ..., -5, -4, -3, -2, -1 } Positive Integers = { 1, 2, 3, 4, 5, ... } Non-Negative Integers = { 0, 1, 2, 3, 4, 5, ... } (includes zero, see?)

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## Linear pairs of angles

Linear Pair of Angles A pair of adjacent angles formed by intersecting lines = 180 degrees or a straight line. Angles 1 and 2 below are a linear pair. Angles 2 and 4,angles 3 and 4, and angles 1 and 3. Linear pairs of angles are supplementary.

## Made up Functions (see notes)

Sometimes functions will use made-up symbols. In these questions, the SAT test makers choose a strange symbol—something you might not have seen before—and gives it a specific definition. This type of question might seem scary, but they really require nothing more than simple substitution. Let’s look at some questions. x ∑ y is defined by the equation x ∑ y = x2 – 1/y. What is the value of 4 ∑ 2? Here, all you have to do is substitute  a 4 for x and 2 for y and then solve: 42 – 1/2 As you can see, we’re just doing simple substitution. Solving, we have: 16 – 1/2 15.5 Simple right?  One way the SAT can make this kind of question harder is to “chain” symbols like (4 ∑ 2) ∑ 3.  If you run into one of these, all you have to do is do the operation in the parenthesis first and then repeat the process again. 2. Let the function ⏀x be defined as ⏀x = (x + 3)(x – 3). If ⏀v = v + 3, what is one possible value for v? (A)  9 (B)  6 (C)  4 (D)  3 (E)  0 If you are a little confused about where to start, remember that you can always use our TAC strategy—Test Answer Choices. Plug in the answer choices into the function to see which one will yield “v + 3.” If our answer is too large, we will be able to eliminate (A) and (B). If it’s too small, we’ll eliminate (D) and (E). (C)  ⏀4 = (4 + 3)(4 – 3) ⏀4 = (7)(1) = 7 7 does equal 4 + 3, so (C) must be the correct answer. Let’s look at another question where we can use TAC: 3. Let the function f⋆g be defined as f⋆g = 2f-g. If h⋆12 = h, then h = (A) 2 (B) 4 (C) 8 (D) 16 (E) 76 We’ll start by following the pattern of the symbols. If h⋆12 = h, then 2h-12 = h. Let’s start with choice (C): If h = 8, then 28-12 = 8. 2-4 = 8, which isn’t true, so (C) cannot be correct. We need a LARGER number since our exponent is negative, so let’s move to choice (D). If h = 16, then 216-12 = 16. 24 = 16 16 = 16 (D) is correct.

Divisiblility

## Parabolas

Parabolas   In total, there are only six things to know about a, b, c. If you can learn the “Parabola Six” you can answer most, many and hopefully all parabola questions on your SAT. The Parabola Six: a is negative - parabola points down a is positive - parabola points up b is negative - parabola moves to the right b is positive - parabola moves to the left c is negative - parabola moves down (y intercept <> c is positive - parabola moves up (y intercept > 0) Parabolas Are Symmetrical This is the most important thing to remember about parabolas, because this is the key that unlocks most of the SAT's most difficult parabola questions. The awesome thing is that you probably already knew this. The not awesome thing is that the SAT still finds ways to make you miss these questions. Let's look at an example: The graph above represents the parabolic function f(x). If the function's minimum is at f(-3), and f(0) = 0, which of the following is also equal to 0? (A) f(3) (B) f(-1) (C) f(-4) (D) f(-5) (E) f(-6) Right. So let's translate this into English first.  What they're saying here is that the line of symmetry for this parabola is at x = -3, and that the function goes right through the origin: f(0) = 0 means that this graph contains the point (0,0). They're basically asking us to find the other x-intercept. Here's where they whole symmetry thing really comes in. If we know the line of symmetry, and we know one of the x-intercepts, it's CAKE to find the other. Put very simply, they have to be the exact same distance from the line of symmetry as each other. Since (0,0) is a distance of 3 from the line of symmetry at x = -3, our other x-intercept has to be a distance of 3 away as well! So we're looking for the point (-6,0). Choice (E) is the one that does that for us: f(-6) = 0. Can they find ways to make symmetry questions difficult? You bet. Will you be ready for them? The equation of a parabola: y = ax2 + bx + c Again, let me say that there are many more equations of parabolas. If you go on to do advanced math in college, you'll need to learn some of them. If you're doing three-dimensional math now maybe you know some of them. But you won't need any of them for the SAT. Know this one (and what the coefficients signify), and you're good to go. In this equation: a tells you whether the parabola opens up or down. If a is positive, it's a smiley face. If a is negative, it's a frowny face. Easy to remember, no? b is pretty useless for you, as far as the SAT is concerned. c is your y-intercept. If there is no c, that means your parabola has a y-intercept of 0 (which is to say, it goes through the origin).

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## Rational & Irrational Numbers

Imaginary Numbers

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