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

## How many arrangements

## Absolute Value

## Angles

## Area & Perimeter

## Area: Finding by breaking up

## Area of a Circle

## Circle & Inscribed Angle

## Angles of a Polygon

## Convex Polygons

## Difference of Perfect Squares

## Distributive Law

## Even & Odd Numbers (see notes)

## Answer Grid (see notes)

## Exponents

## Foil

## Fractions & Percent

## Fractions

## Equations vs Functions: the Difference

## Graphing Functions

## Integers (see notes)

## Proportionality constant for direct
variation

## Linear pairs of angles

## Made up Functions (see notes)

## Mean, Mode, Median (see notes)

## Multiples and Factors

## Parabolas

## Quadrilaterals

## Percent

## Percent Problems

## Perpendicular Line Slope

## Plug in a Number (see notes)

## Positive and Negative Numbers

## Prime Numbers

## Probablility

## Ratios

## Rational & Irrational Numbers

## Word Problems

## Slope

## Substitution Method

## Slope Example

## Speical Right Triangles

## Types of Triangles

## Treating Units algebraically
dimensional analysis

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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

How Many - Additional

Finding Absolute Value

Understanding Angles

Find Area of Trapezoid

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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).

Information provided by Kaplan Student-Produced Response questions, or Grid-ins, is a question type that simulates a more natural math test-taking experience by offering questions outside the multiple-choice format. On Grid-ins, you're supposed to come up with your own response. The catch is that you have to fill your answer into a funny-looking grid. There are 10 Grid-ins on the quantitative section of the SAT, so Grid-Ins account for one-sixth of your quantitative score. The math tested on Grid-ins is the same as that tested on Regular Math questions: junior and senior high school level arithmetic, algebra and geometry. To answer a Grid-in question, you must fill out a grid with four boxes and a column of ovals beneath each. It's recommended that you first write your numerical answer in the boxes above and then shade in the corresponding ovals below. You must fill out the grid properly to get credit for a correct answer. So take some time to learn the directions for Grid-ins. Directions: For each of the following questions, solve the problem and indicate your answer by darkening the ovals on the special grid. For example: If the correct answer is 1 1/4, then grid-in the number as 1.25 or 5/4, like so: Write answer in boxes Grid in the result It is recommended, through not required, that you write your answer in the boxes at the top of the columns. However, you will receive credit only for darkening in ovals correctly. Grid only one answer to a question, even though some problems may have more than one correct answer. Darken no more than one oval per column. No answers are negative. Mixed numbers cannot be gridded. For example: the number must be gridded as 1.25 or 5/4 (if it is gridded as 11/4, it will be interpreted as , not ). Decimal accuracy: Decimal answers must be entered as accurately as possible. For example, if you obtain an answer such as 0.1666. . ., you should record the result as .166 or .167. Less accurate values such as .16 or .17 are not acceptable. Acceptable ways to grid 1 / 6 .166 .167 What all this means to you: Your answer must have at most 4 characters, including the decimal point or fraction bar. The grid cannot accommodate negative numbers, mixed numbers, or numbers greater than 9,999. A fractional number with 4 digits won't fit. Mixed numbers must be changed to decimals or fractions before you grid. Decimals must be as complete as possible (for long or repeating decimals, use as many columns as possible) but do not have to be rounded up for accuracy's sake. Many questions may have more than one correct answer, and there may be many ways to fill in the grid correctly (just choose a single safe one). You will only get credit for filling in the ovals correctly—if you fill in two ovals in the same column, the computer reads this as an omission.

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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 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.

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.

Mean/Median/Mode All three of these terms are types of averages, but obviously they're asking for very different things. Mean is usually what we mean when we say 'average,' and the one that any student who has tried to frantically figure out his GPA before report cards comes out knows how to do. We'll come back to how that will be tested in a moment. Median Median is the middle number when numbers are arranged least to greatest. A median is also that cement strip in the middle of the road – median=middle. The easiest way to find a median is to simply rearrange the numbers given from least to greatest, and then eliminate the first and last number in the list until you're left with only one number in the center. If the original list contains an even number of numbers, you may be left with two: Find Joe's median test score if he scored 85, 84, 85, 82, 83, 72, 96, 81, and 93 on this semester's tests. Rearrange: 72, 81, 82, 83, 84, 85, 85, 93, 96 -SAT Math Hint: Count the number of terms in the original list and count the number of terms in you're rearranged list to make sure you got everything. Then eliminate first and last: And continue until you're left with one or a pair in the center: His median test score is 84. If you're left with two in the center, add the two numbers together and divide by two to get the median. -SAT Math Hint: Your answer in this case will most likely NOT be one of the numbers on the original list. That's okay. Sara buys coffee at the local coffee store at least once a day. On days when she's particularly busy, she visits multiple times. She made one visit on Monday, two on Tuesday, one on Wednesday, one on Thursday, three on Friday, and two on Saturday. What was her median number of visits for the week? First, write down your numbers: 1, 2, 1, 1, 3, 2 Then rearrange: 1, 1, 1, 2, 2, 3 Then cross out first and last until you get to the center: You're left with 1 and 2, so add them together and divide by 2: She had a median number of 1.5 trips for the week. -SAT Math Tip: A median is always found by first arranging the numbers least to greatest. Do NOT start slashing away at pairs until you've rewritten the whole string of numbers first, including every single one. So if there are 3 14s in your list, you better have 3 14s written down in your rearrangement. Mode Mode means "most." That's it. When they're asking you for mode, they're asking you which number in the list appears most often. What is the mode in each of the following sets? Set W: {4,3,3,5,2,1} Set X: {2,7,2,9,6,7,2} Set Y: {2,6,2,8,9,6,3} Set Z: {1,5,7,4,8} In Set W, every other number appears once, but 3 appears twice, so 3 is the mode. In Set X, both 2 and 7 appear more than once, but 2 appears more often than 7 so 2 is the mode. In Set Y, both 2 and 6 appear an equal number of times, both more often than any other number. Set Y has two modes: 2 and 6. In Set Z, no number appears more often, so there is no mode. -SAT Math Tip: Remember, a mode must appear more often than any other number, not just be repeated. There can be no mode; there can also be more than one mode if those numbers appear an equal number of times but more than any other number. Mean Let's talk about mean, or averages. The formula for average is "where the total is all the numbers you were given added up and the number of things is the number of things you were given. We can shorthand the formula to .

Divisiblility

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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The Top SAT Math Strategy If I only had time to teach one thing about taking the SAT, the strategy of plugging in numbers would be it. The SAT loves to be tricky, loves problems with shortcuts, loves to test if you can find the quickest, easiest way to a solution. One of the tricks it uses all the time is to find abstract, vague, convoluted ways of asking a question to see if you can cut through all the junk and figure out the answer. And the easiest way to cut through the junk is to make the problem more concrete. Allow me to be more concrete. Here’s an algebra-based question where this strategy would be helpful: Example 1* If the average (arithmetic mean) of x and y is k, which of the following is the average of x, y, and z? (a) (2k + z)/3 (b) (2k + z)/2 (c) (k + z)/3 (d) (k + z)/2 (e) (2k + 2z)/3 I could use algebra to solve this problem, if I could figure out how, but that would take a while and there would be several steps to it. Instead, I’m just going to replace the variables with actual, concrete numbers. I can do this because variables can represent any number (with certain limits). That’s the point of variables, after all. Plugging in Numbers Here, then, is how to solve a problem by plugging in numbers: 1) Choose a number that fits the constraints – if it says X is a positive integer, I can’t choose -8. 2) Choose a number that’s easy to work with – don’t use 456 when you could use 2. 3) Plug that number in for the variable in the question and find the answer. 4) Plug that same number in to the answer choices and find the one that comes out with the same answer as in step #3 5) If more than one answer choice works, don’t panic, just cross of the answers that didn’t work, go back to step #1 and repeat. Sometimes you just happen to choose a number that works for more than one answer choice, but it won’t happen twice. In this problem the average of x and y is k. There are no other constraints, so I’m going to say x = 1, and y = 3, because those are easy to work with. It is crucial that you keep track of what you make the variables, so write them down. If x = 1 and y = 3, then k = 2 (since that’s the average of 1 and 3; this is easy to figure out because I did step #2). Now, I need a z. I’ll say z = 5, because I don’t want to repeat a number and get confused. The question asks what the average of x, y, and z is. x + y + z = 1 + 3 + 5 = 9, 9/3 = 3. So the answer to the problem, using my plugged in numbers is 3. When I plug in my same numbers (z = 5, k = 2) into the answers, I’m looking for the one that gives me 3. (a) (2*2 + 5)/3 = 9/3 = 3. Keep this one around. (b) (2*2 + 5)/2 = 9/2 = 4.5. Cross this one out. (c) (2 + 5)/3 = 7/3 = 2.33. Cross it out. (d) (2 + 5)/2 = 7/2 = 3.5. Cross it out (e) (2*2 + 2*5)/3 = 14/3 = 4.67 Cross it out. My answer is (a), and I didn’t have to do anything more than know what an average is and do some simple arithmetic.** There will be multiple questions like this on every section in the SAT. Maybe you’ll know how to do the algebra behind it, maybe you won’t, but either way it will be faster to plug in a number and make the problem concrete than it will be to work through the steps in the algebra. Plus, when the problem is concrete, it’s easier to check to make sure you were right. Example 2 – Constraints Here’s another problem where the constraints are important If x and y are positive consecutive odd integers, where y > x, which of the following is equal to y2 – x2? (a) 2x (b) 4x (c) 2x + 2 (d) 2x + 4 (e) 4x + 4 Again, there’s a way to do this with algebra, if you can see it. Most students won’t, and even if they did it would be harder and take longer than plugging in numbers.*** It’s important to choose numbers for x and y that fit with the constraints. I can’t choose 24.5 and -12, because I need positive consecutive odd integers. So…1 and 3. They’re positive, they’re odd, they’re consecutive, and most importantly they’re easy to work with. When x = 1 and y = 3 y2 – x2 = 9 – 1 = 8. Now I use x = 1 in the answers to see which one gives me 8, and I get 2, 4, 4, 6, and 8. The answer is (e), and I barely had to think to get there. Example 3 – Percents Another great time to use this strategy is with questions using percents. It’s easy to get confused when a question asks what is 20% of 35% of 30% of 29% of X. But, if you just make X an actual number, like, say 100 (remember #2 ab When to Plug-in Numbers Here is astrategy of plugging in a number whenever an SAT Math question mentions a number or integer. This number plug-in strategy works equally as well for questions with equations in the answer choices – questions that are often among the hardest on the entire SAT Math section. Whenever you see an SAT Math question with equations in the answer choices, plug in a number. Pick a number and plug it into the question to get a value. Then plug the number into each answer choice to see which one produces the same value. Remember: When plugging in numbers, be sure to pick EASY numbers and ALWAYS plug in for ALL answer choices. Example 1: The number plug-in strategy also works great for word problems with equations in the answer choices. Example 2:

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Imaginary Numbers

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