# Translating English to Math

If you have trouble studying, bookmark this handy list of study tips that can make your study time more effective.

## Mathematics: Standard Multiple Choice

A florist buys roses at $0.50 apiece and sells them for$1.00 apiece. If there are no other expenses, how many roses must be sold in order to make a profit of $300? First, note the bottom line. p =$300, r = ?

Assess your options. You know how much the roses cost and how much they sell for, so you could calculate the amount of profit earned with each flower. You could also translate the problem into a formula to determine how many flowers will yield $300 in profit. The first solution is more efficient, so begin by finding the amount of profit each flower brings in. Since they cost fifty cents and sell for a dollar, each flower earns fifty cents in profit.$1.00 - $0.50 =$0.50

Next, how many flowers are needed to add up to $300 total? .5r = 300 $r=\frac{300}{.5}$ r = 600 Loop back to the bottom line to check whether you answered the question correctly. You were looking for the number of roses that would yield$300 in profit. Since that is what you found, look at the answer choices to see which one matches.

A) 100

B) 150

C) 200

D) 300

E) 600

On sat.collegeboard.org, 74% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Algebra Equations: Translation

Today, I returned to History.com's "This Day in History" resource to see what kinds of things happened on the May 18ths of the past. On May 18, 1896, the Supreme Court issued an opinion on the landmark case Plessy v. Ferguson and dealt a major victory to supporters of racial segregation by supporting the legal standard of "separate but equal." Another quick online search brought me to this page from PBS with more details of the case, including the fact that Homer Plessy, a very light-complected man who was considered black because of his heritage, deliberately sat in the white car on a train in order to challenge Louisiana's Separate Car Act.

## 5/18 Algebra Equations: Translation

Read the following SAT test question and then select the correct answer.

Always follow the Knowsys Method for math problems. It will save you time not only on the SAT, but also on math tests and quizzes in your school classes. Read the question carefully and identify the bottom line. Assess your options. Ask what you could do and then what you should do, and solve the problem quickly and efficiently once you have decided on a strategy. Finally, loop back to double check that you answered the question correctly.

First, 3 is subtracted from x and the square root of the difference is taken. Then, 5 is added to the result, giving a final result of 9. What is the value of x?

Wow! All those math terms close together are very effective at making this problem look scary. It is not really complicated, but when you come across problems like this (or ones like this) it is important not to let the test writers intimidate you. First, as always, look for the bottom line. At least that's easy to find:

x = ?

Next, assess your options. How could you approach this problem? On the test, of course, you could choose to skip this problem entirely, but I would not recommend it. Instead, you could try translating the problem from English sentences to a mathematical equation.

First, 3 is subtracted from x    So, $x-3$

and the square root of the difference is taken.    So, $\sqrt{x-3}$

Then, 5 is added to the result    So, $\sqrt{x-3}+5$

giving a final result of 9.    So,$\sqrt{x-3}+5=9$

Now you have a simple algebraic equation, and all you have to do is solve for x!

Start with    $\sqrt{x-3}+5=9$

Subtract 5 from both sides.    $\sqrt{x-3}=4$

Square both sides to remove the radical.    $x-3=16$

Add 3 to both sides.    $x=19$

Now, loop back to your bottom line. You were looking for the value of x, and you found x = 19, so that is the correct answer to the question! Check the answer choices.

A) 3

B) 4

C) 5

D) 16

E) 19

On sat.collegeboard.org, 58% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Slope

Write it Down! This infographic linked today from www.coolsiteoftheday.com discusses the importance of taking notes, a few different methods, and the potential benefits and drawbacks of taking notes digitally or the old--fashioned way. Did you know that your brain actually processes information differently while you're taking notes? This is a good resource to bookmark and revisit when you notice that your class notes are less than helpful--it might be time to try out a different method.

## 5/15 Slope

Read the following SAT test question and then select the correct answer.

Remember to read carefully, identify the bottom line, assess your options, attack the problem, and loop back. When you use this method, you will get more problems right and you will move faster through the test.

In the xy-plane, line l passes through the points (a, 0) and (0, 2a), where a > 1. What is the slope of line l?

First, read carefully. You have two points on a line, which means you can visualize that line if you wish. Picking a number for a might make that easier if the variable trips you up. Next, identify the bottom line. The question asks for the slope of line l, so at the top of your scratch work write "slope = ?"

Now assess your options. Since you need to find the slope of the line, a good place to start is with the formula for slope: rise over run. There are two choices here; you can use a as a variable or you can pick a number for a. Using a directly involves fewer steps because you don't need to plug in the value, but manipulating the variable can be confusing for some and can cost time. Which tool you choose to solve the problem is up to your personal preference.

Either way, the first step in the problem is to set up your formula. Since a must by greater than 1, I'll use 2.

$\frac{rise}{run}=\frac{2a-0}{0-a}$                                                             $\frac{rise}{run}=\frac{2(2)-0}{0-(2)}$

$\frac{rise}{run}=\frac{2a}{-a}$                                                                    $\frac{rise}{run}=\frac{4}{-2}$

$\frac{rise}{run}=-2$                                                                    $\frac{rise}{run}=-2$

Now loop back to make sure that you answered the right question. Your bottom line asks for the slope, so you found the change in y-coordinates (rise) and the change in x-coordinates (run), divided one by the other and reduced. That is the slope, so -2 is the answer you need.

A) -2

B) $\frac{-1}{2}$

C) 2

D) -2a

E) 2a

On sat.collegeboard.org, 42% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Ratios, Rates, and Proportions

George Mason University's History News Network is an unusual news site that puts current events in a broad historical context. Normal news stories focus only on what has happened recently, but HNN strives to connect current events to the history that created them.

## 5/12 Ratios, Rates, and Proportions

The c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks?

First, note the bottom line.

5 cars 2 weeks = ?

Next, assess your options. Since the problem gives so much information about the cars using words rather than numbers, a good place to start is to translate its question into mathematical terms.

c = total number of cars

g = total gallons of gas per week

The gas used in two weeks is easy to find: 2g. The tricky part involves determining how much gas is used by only 5 cars. It is tricky rather than difficult because if you know the trick, this problem is easy. Simply find the gas used by one car over the course of a week and multiply that by 5 cars.

$\frac{g}{c}$ = gas per week for 1 car

$\frac{5g}{c}$ = gas per week for 5 cars

Since the question asks how much gas will be used in 2 weeks, multiply this term by 2. This incorporates the 2g you identified earlier.

$\frac{10g}{c}$ = gas for 5 cars for 2 weeks

Now look at the answer choices.

A) $10cg$

B) $\frac{2g}{5c}$

C) $\frac{5g}{2c}$

D) $\frac{g}{10c}$

E) $\frac{10g}{c}$

On sat.collegeboard.org, 32% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Triangles

History.com's This Day in History is a great place to look for interesting historical events that might otherwise be overlooked. Common examples like Martin Luther King Jr. or the Holocaust will not make your essay stand out, but the fact that on May 9th, 1950, L. Ron Hubbard published Dianetics or that in 2001, soccer fans were trampled in Ghana will make your essay stronger.

## 5/9 Geometry: Triangles

Remember to always follow the Knowsys Method for math problems. The method will save you time and errors not only on the SAT but also in your regular math classes and problems. First, read the question carefully and identify the bottom line. Once you know what the problem is asking, assess your options by asking "What could I do?" "What should I do?" Select the most efficient method, attack the problem, and loop back to make sure that you answered the question correctly.

If triangle ABC above is congruent to triangle DEF (not shown), which of the following must be the length of one side of triangle DEF?

First, at the top of your scratch work, write one side of DEF = ?

Next, assess your options. How can you find the side lengths of a triangle that is not shown? The problem mentions that ABC and DEF are congruent, which means all their side lengths and angle measurements are the same. That means that you can simply change the labels on ABC to DEF. To find the answer, though, you will need to figure out the side lengths. You could try to use the Pythagorean Theorem here, but it would be very difficult. Instead, you should notice that the triangle is one of the special right triangles that you have memorized. You can use that information to find the side lengths.

Triangle DEF is a 30-60-90, which means the side lengths are $x-x\sqrt{3}-2x$. The hypotenuse is 12, so 12 = 2s and 6 = s. The hypotenuse of the triangle is 12, the short leg is 6, and the other leg is $6\sqrt{3}$.

Loop back to the bottom line. You are looking for any side of Triangle DEF, so now that you have all three, you only need to look at the answer choices and find one that matches any of these three numbers.

A) 18

B) 24

C) $3\sqrt{6}$

D) $6\sqrt{3}$

E) It cannot be determined from the information given.

On sat.collegeboard.org, 36% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Graphing Equations

Since I shamefully allowed both Star Wars Day and Cinco de Mayo to pass without comment, I'll make up for it with two links. First, a brief history of Cinco de Mayo explains what the holiday is as well as who celebrates it, why, and how. Second, this article about Star Wars as a religious allegory reveals that no one is sure which (if any) religion is portrayed in Lucas' famous series. Happy Cinco de Mayo, and May the Fourth be with you!

## 5/6 Graphing Equations

Always follow the Knowsys Method: Read carefully, getting information not only from the question itself but also from any charts, graphs, or figures included. Make careful note of labels and scales on all images. Identify the bottom line and copy it at the top of your scratch work. Assess your options, select the fastest way to solve the problem, and attack. Loop back to make sure you answered the question correctly.

The function y=f(x), defined for $-1.5\leq x\leq 1.5$, is graphed above. For how many different values of x is f(x) = 0.2

First, write at the top of your scratch work # times f(x) = 0.2 = ?

Next, assess your options. You could try plugging in 0.2, but you have nothing to plug it into. You could guess, but (while that is always an option on the test) that choice will not help you during practice. Your only option is to look at the graph and count.

Make sure you pay attention to the labels on the graph. You're looking for points where f(x) = 0.2, so first draw a line at y = 0.2. Since the scale on this graph is 0.5, you need to draw a horizontal line a little less than halfway up to the first tick mark above the x-axis. Next, simply count how many times the two lines intersect.

A) None

B) One

C) Two

D) Three

E) Four

On sat.collegeboard.org, 39% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Remainders

"Many students discover the need to develop or hone their time management skills when they arrive at college. Unlike high school where teachers frequently structured your assignments and classes filled your day, in college, you will have less in-class time, more outside of class work, and a great deal of freedom and flexibility." Check out these pages for advice on how to manage your time in college.

## 5/3 Remainders

First, always remember to read the question carefully and identify the bottom line. If you don't know what the question is asking, you cannot answer it correctly! Next, assess your options. Based on what you need to find and on what the problem gives you, what COULD you do? What SHOULD you do? Select the most efficient method and attack the problem. Solve it as quickly as you can without making mistakes, then loop back up to the bottom line to check you answer against the question. Finally, once you have found the correct answer, select it from among the answer choices.

If it is now 4:00 p.m. Saturday, in 253 hours from now, what time and day will it be? (Assume no daylight saving time changes in the period.)

First, read carefully. The fact that daylight saving time (commonly mispronounced "daylight savings time") doesn't come or go is helpful to know; that will make the problem simpler. We start at 4 p.m. on Saturday and move forward 253 hours. 253 is a large number, and not one that is easy to work with, so trying to count the days and hours would be a huge waste of time. Instead, divide 253 by 24 to get the number of days that passed during that time.

$253\div 24=??$

Since 253 doesn't divide evenly by 24, what should you do with the left-overs? Keep them in the form of a remainder.

$253\div 24=10r13$

10 days and a remainder of 13 hours passed. If this time started at 4p.m. on Saturday, what day and time did it end? Exactly ten days from 4 p.m. on Saturday would be 4 p.m. on Tuesday. The remaining 13 hours move you forward into Wednesday, landing at 5 a.m. Now look at the answer choices.

A) 5:00 a.m. Saturday

B) 1:00 a.m. Sunday

C) 5:00 p.m. Tuesday

D) 1:00 a.m. Wednesday

E) 5:00 a.m. Wednesday

On sat.collegeboard.org, 50% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Functions

How to manage time and adapt to college life is a major concern for many students. A site full of College Study Tips could be a great help! This website has "college study tips, college study skills, study guides and tricks to help you manage your time, take better notes, study more effectively, improve memory, take tests, and handle the stresses of college life."

## 4/30 Functions

Always try to solve the problem as if the answer choices weren't there. This has two benefits: it makes you more efficient at multiple-choice questions, and it makes you more confident on grid-in questions. First, read carefully and look for important information. Identify the bottom line, then assess your options and select the most efficient way to solve the problem. Attack the problem, solve it quickly, and loop back to ensure that the answer you found matches the bottom line. Finally, check the answer choices and select the correct answer.

If the function f is defined by $f(x)=\frac{(x-a)(x-b)}{(x-c)}$, where 0 < a < b < c, for which of the following values of x is f undefined?

I. a
II. b
III. c

Read carefully and focus on the bottom line.

f undefined when x = ?

The most important clue here is the word "undefined." What does "undefined" mean in a math problem? It means that you have attempted to do something impossible, such as dividing by zero. In order for this function to divide by zero, x - c must equal zero because it is the denominator. Given that, there is only one possible value for x. Any number minus itself equals zero, so if x = c, then x - c = 0. Any other answer choice would yield a positive or negative number for the denominator, and the function would not be undefined. Only choice III makes the function undefined.

Now look at the answer choices:

A) None

B) I only

C) III only

D) I and II only

E) I, II, and III

On sat.collegeboard.org, 53% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Ratios

Private consulting groups like College Funding Solutions, Inc, and GetCollegeFunding can help you find ways to pay for college. How do you get the best financial aid? Scholarships? Grants? Tuition discounts?Which forms do you fill out, and how? Where should you put your money to make the best impression on government funding groups? Their services may seem expensive, but they can save you enough money in college that you come out far ahead.

## 4/27 Ratios

Always remember to follow the Knowsys Math Method. This may take longer than simply solving the problem at first, as you learn the method, but it will save you time once you begin to use it consistently. Reading carefully will help you make sure you don't miss anything. Identifying the bottom line makes it clear exactly what you are looking for. Stopping to assess your options will help you select the most efficient way to solve every problem and keep you from losing time by spending too much time on a problem. Finally, looping back will ensure that the answer you found matches the question that was asked; if you found the value of m, but the question asked for m + 3, you might get that problem wrong even after doing all the math correctly.

A jar contains only red marbles and green marbles. If a marble is selected at random from the jar, the probability that a red marble will be selected is $\frac{2}{3}$. If there are 36 green marbles in the jar, how many red marbles are in the jar?

When reading carefully, take note of facts that could help you solve the problem. For example, the fact that the jar only has red and green marbles means that this problem will involve only two variables, probably r and g. Later, the value of g is given, and the problem asks how many red marbles there are. The marbles are selected at random; that's good because it means you can rely on the probability given. If you reached into the jar looking for a red marble, the odds of finding one would be extremely high, no matter what the ratio of red marbles to green marbles is.

Next, identify the bottom line. The question asks "how many red marbles are in the jar?" That can be summarized as

r = ?

Now, assess your options. You could try plugging in the answers until you find one that works, but that could take a while. Or you could try setting up a proportion with the red and green marbles to calculate the number of red marbles in the jar. Conveniently, a ratio is already provided! You're halfway done already! So if there are two red marbles for every... Oh wait.

This is an example of why reading carefully is important. The ratio you need to find to solve the problem is r:g, but the ratio the problem gives you is r:a, or the ratio or red marbles to all the marbles in the jar. So, if 2 out of every 3 marbles are red, the remaining 1 must be green. Now you can set up a proportion.

$\frac{red}{green}=\frac{2}{1}=\frac{x}{36}$

It is essential that you always label your scratch work so that it is clear not just what you are doing, but what you did. When you reach the end of a section and begin to work backwards, double-checking problems you're not sure about, labels are invaluable because they show what you did to solve the problem. Now that the proportion is set up, you can solve it easily.

$36(\frac{2}{1})=36(\frac{x}{36})$

36(2) = x

x = 72

Now look at the answer choices:

A) 18

B) 24

C) 54

D) 72

E) 108

On sat.collegeboard.org, 46% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Fractions

College can seem intimidating, and all the more so because it practically involves a different language. Learn the terms around college admission, and you will be able to navigate the process much more easily.

## 4/24 Fractions

Remember to always follow the Knowsys Method when approaching math problems: First, read carefully. Then identify the bottom line and assess your options. Next, choose the most efficient method and attack the problem! Finally, loop back to make sure you have the correct answer.

If f(x) = x + ax, and $a=\frac{7}{2}$, what is $f(\frac{3}{2})$?

First, as always, identify the bottom line.

$f(\frac{3}{2})=?$

Next, assess your options. You have variables in the question and numbers in the answer choices, so you could work backward from the answer choices. However, there are also numbers in the question, so you could plug in those numbers to see what that gives you. The second method will be far more efficient.

First, plug in the numbers. $a=\frac{7}{2}$ and $x=\frac{3}{2}$, so plug each of those in where appropriate.

$f(\frac{3}{2})=\frac{3}{2}+\frac{7}{2}(\frac{3}{2})$

Next, following the Order of Operations (PEMDAS), multiply the last two numbers together. When multiplying fractions, simply multiply the numerators and then the denominators.

$f(\frac{3}{2})=\frac{3}{2}+\frac{21}{4}$

Obviously, the next thing to do is to add the fractions together. Remember that when adding fractions, the denominators must match, and you can ONLY add the numerators. You'll need to do something to make the denominators the same.

$f(\frac{3}{2})=\frac{3}{2}(\frac{2}{2})+\frac{21}{4}$

$f(\frac{3}{2})=\frac{6}{4}+\frac{21}{4}$

$f(\frac{3}{2})=\frac{27}{4}$

Now loop back. Is $f(\frac{3}{2})$ what you actually needed to find? Yes, it is! Now look at the answer choices.

$A)\frac{1}{3}$

$B)\frac{3}{2}$

$C)\frac{7}{2}$

$D)\frac{21}{4}$

$E)\frac{27}{4}$

On sat.collegeboard.org, 57% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Digits

http://www.collegeview.com/collegesearch/index.jsp is another college-finding search engine. Start by selecting the criterion most important to YOU--then look for colleges that fit what you want.

## 4/21 Digits

Remember to always follow the Knowsys Method to avoid selecting the wrong answer. Read carefully and identify the bottom line, assess your options, attack the problem, and loop back to ensure that what you found is actually what the problem asked for.

The sum of the digits of a three-digit number is 12. If the hundreds digit is 3 times the tens digit and the tens digit is $\frac{1}{2}$ the units digit, what is the tens digit of the number?

First, as always, identify the bottom line.

tens = ?

Next, consider your options. What could you do to solve this problem? What should you do? The best strategy here is to assign a variable to the problem. Say the tens digit is x, and put the rest of the problem in terms of x. You know that the hundreds digit is 3 times the tens digit, so it becomes 3x. The tens digit is also half of the units (ones) digit, so the units digit can be translated 2x.

Finally, the problem says that the sum of the three numbers is 12. Therefore, 3x + x + 2x = 12. Now you just have to solve for x and you will have the value of the tens digit.

3x + x + 2x = 12

6x = 12

x = 2

A) 2

B) 3

C) 4

D) 6

E) 9

On sat.collegeboard.org, 58% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Rates

College.gov has your starting place for financial aid. One of the major concerns for prospective college students is money; college is expensive, and rising tuition costs combined with less government funding show no end in sight. However, there is money available to help students who need it, and the FAFSA is the best place to start.

## 4/15 Rates

Always remember to follow the Knowsys Method--even in your math classes. Thinking strategically and logically will help you be more efficient far beyond the SAT. First, read carefully to see what the question is actually asking. Then assess your options and select the best one. Attack the problem efficiently, then loop back to make sure that the answer you found matches the question that was originally asked.

A train traveling 60 miles per hour for 1 hour covers the same distance as a train traveling 30 miles per hour for how many hours?

First, note the bottom line.

train 2 hours= ?

Next, look back at the question to determine how you could solve it. You could determine the total distance traveled by train 1 and then calculate the time it would take train 2 to travel the same distance. You could also calculate the times relative to one another. That might sounds odd, but it is actually more efficient than the first method.

The first step is to set up a ratio of  the two rates given. Put the "new" rate, that of train 2, on top because it is the variable you are trying to find. Always reduce ratios to lowest terms.

$\frac{train 2}{train1}=\frac{30}{60}=\frac{1}{2}$

You now have a ratio of the two rates. Here's the cool part: simply flip it over to find a ratio of the two times.

If a car or train travels at twice the planned speed, the trip will take half as long as projected. If it travels at half the planned speed, the trip will take twice as long. This rule applies when traveling 2/3, 5/4, or any fraction of the original rate; the ratio of the times will be the reciprocal of that fraction.

$\frac{train2}{train1}=\frac{2}{1}$

This means that train 2 spent twice as much time as train 1 covering the same amount of ground.

train2 = 2(train1)

Since train 1 traveled for 1 hour, train 2 traveled for 2 hours. Loop back to make sure you answered the right question.

train 2 hours = 2

Good job! Now look at the answer choices.

A) 3

B) 2

C) 1

D) $\frac{1}{2}$

E) $\frac{1}{3}$

On sat.collegeboard.org, 81% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Functions

Remember that your goal is to get as many questions right as possible, not to answer as many as you can. In some cases, leaving a question blank can be better for your score than guessing, but best of all is to solve the problems efficiently and confidently enough that you can answer every question right. Solve each problem as if it did not have answer choices by reading carefully, noting the bottom line, assessing your options, attacking the problem, and looping back to double-check your answer.

In the xy - plane, line l is perpendicular to the graph of the function f(x) = 5x - 2. Line l could be the graph of which of the following functions?

Your bottom line in this case is "Which of the following?" In WOTF questions, always check E first and work your way back up to A. WOTF questions are much more likely than normal questions to have D or E coded as the correct answer. (The exception to this is Roman Numeral questions, in which the actual answer choices will say things like A) I and II, B) II only, etc. These are random.)

Next, assess your options. You could guess, and since this is a "which of the following" question, you would be relatively safe in guessing D or E. Or you could work from what you know to make a prediction about the correct answer.

You should know that the slope of any line is the negative reciprocal of the line perpendicular to it. That is, if the first line has a slope of m, the second line has a slope of $-\frac{1}{m}$. You should also know that slop-intercept the formula for a line is y = mx + b. Since the equation in the problem is f(x) = 5x - 2, the slope of the given line is 5. Therefore, the slope of the perpendicular line is $-\frac{1}{5}$.

Now look at the answer choices:

A) g(x) = -5x

B) $g(x)=-\frac{1}{5}x$

C) g(x) = x - 2

D) $g(x)=\frac{1}{5}x$

E) g(x) = 5x

The only option that has the correct slope is B. That is the correct answer. Remember that rules of thumb, like starting with E on WOTF questions, are not always reliable.

On sat.collegeboard.org, 62% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Inequalities

This page lists some of the great mathematicians of the ages, including Newton, Archimedes, Euclid, and others. Using any of them in an essay will help you stand out and earn a higher score.

## 4/9 Inequalities

Be sure to read carefully--reading carelessly will cost you points. Mark the bottom line and assess your options, then choose the fastest route to the answer. Remember: The long way is the wrong way! After you find an answer, check it against your bottom line before looking among the answer choices for the number you found.

If a number is chosen at random from the set {-10, -5, 0, 5, 10}, what is the probability that it is a member of the solution set of both 3x - 2 < 10 and x + 2 > -8?

First, look for your bottom line. It is in two parts here, so one thing you could do is try to combine them to simplify the problem. You could also find the numbers that satisfy each half of the bottom line and then combine them. The long way is the wrong way, so you will need to combine the inequalities. First, isolate x.

p (both inequalities) = ?

3x - 2 < 10                                                               x + 2 > -8

3x < 12                                                                    x > -10

x < 4

Next, you can combine the two inequalities into one compound inequality.

-10 < x < 4

Now look for the values of the given set that satisfy the inequality. There are only two: -5 and 0. Since the set has 5 terms, the probability that any random term is one of the two you found is $\frac{2}{5}$.

A) 0

B) $\frac{1}{5}$

C) $\frac{2}{5}$

D) $\frac{3}{5}$

E) $\frac{4}{5}$

Alternate Method: If rearranging the inequalities doesn't occur to you, don't sweat it. If it takes longer to remember one tool than it does to use a different one, then that isn't the right tool. Instead, after isolating x in each inequality, you can check to see which numbers fit each of the two inequalities you have.

x < 4                  -10, -5, 0

x < -10               -5, 0, 5, 10

Only two numbers fit both inequalities: 0 and -5. Again, these are two numbers out of five, so your probability and your answer are the same.

On sat.collegeboard.org, 52% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Graphs

Graphs are actually a fairly new form of mathematical analysis, first systematically investigated in the 1930s. This link has a short and, hopefully, informative article on the history of graphs.

## 4/3 Graphs

Always take the time to carefully read the question, identify the bottom line, and assess your options. Then attack the problem and loop back to check your answer. Only after that should you look at the answer choices to select the correct one.

In the figure, the slope of the line through points P and Q is $\frac{3}{2}$. What is the value of k?

First, note the bottom line at the top of your scratch work.

k = ?

Next, assess your options. Getting from a slope to one part of a coordinate pair may take a few steps, so in this case the best way to start is to simply begin with what you know and move toward what you want.

You know that slope is rise over run, and you can calculate the total rise since you have both of the y coordinates for the two points. Using this information, you can set up a proportion.

7 - 1 = 6

$\frac{rise}{run}=\frac{3}{2}=\frac{6}{x}$

From here, you can either cross-multiply and solve for x or use multiples to determine its value (2*3=6, so 3*3=x). Either way, x = 4.

Now that you have solved for x, loop back to the bottom line to see if you are finished.

k = ?

You found the difference between the two x-values, but not the value of k. There is still one step left! Since x, the difference between the x coordinates of P and Q, is 4, simply add the value of the first x coordinate to find the value of k.

4 + 1 = 5

Now look at the answer choices

A) 4

B) 5

C) 6

D) 7

E) 8

On sat.collegeboard.org, 49% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Algebra

If you want a fun way to build your math skills, check out Math Playground. The games here deal with simple math, so tackling them without a calculator is a great way to exercise your brain.

## 3/31 Algebra

Always follow the Knowsys Method and read carefully before you do anything else. Then identify the bottom line. Stop to assess your options: What could I do? What should I do? Attack the problem. After you have an answer, loop back to the bottom line to ensure that your answer is the correct one. Finally, select your answer from among the answer choices.

If  $5x-3=2a$, then $\frac{5x-3}{2}=$

First, at the top of your scratch work, note the bottom line.

$\frac{5x-3}{2}=?$

Next, assess your options. You could try to solve for x or for here, or you could pick one of the answer choices and work backwards to determine whether it is the right answer. However, the fastest and easiest way to solve this problem is simply to solve for the bottom line. The problem provides 5x - 3, so it only takes one step to find the answer:

$\frac{5x-3}{2}=\frac{2a}{2}$

$\frac{5x-3}{2}=a$

Now look at the answer choices.

$A)\frac{a}{4}$

$B)\frac{a}{2}$

$C)a$

$D) 2a$

$E)4a$

On sat.collegeboard.org, 81% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Integers

Pythagoras, best known to high school students for his Pythagorean Theorem, actually discovered much more than that one formula. Even if you are not mathematically inclined, the beginning of this paper has some interesting notes on how the Pythagoreans--the followers of Pythagoras--lived.

## 3/28 Integers

Always attempt to solve the problem before looking at the answer choices. Read carefully, then identify the bottom line--what the question is actually asking--and mark it at the top of your scratch work. Assess you options by asking "What could I do?" to open your toolbox, then "What should I do?" to select the best way to solve the problem. Attack the problem fearlessly, then loop back to the bottom line to check whether what you found is the correct answer.

If p is an odd integer, which of the following is an even integer?

At the top of your scratch work, write even = ?

Next, ask "What could I do?" You could think through each answer choice abstractly, determining that if p is odd then... but that is difficult and gets confusing quickly. You could pick a number for p, then use that number to find a value for each answer choice. The smallest odd number is the best for this. Pick one. Since this question includes the phrase "which of the following," the answer is very likely to be D or E. Start at the bottom and work your way up.

E) $p^{2}-p$
If p = 1, then $1^{2}-1=0$. 0 is neither positive nor negative, but neutral; however, it is still even. This distinction confuses some students, so make sure you know it. Now loop back to the bottom line. $p^{2}-p=0$, so it is even, so it is the answer. On the SAT, you could continue on from this point or check the other answers.

D) $(p-2)^{2}$
$(1-2)^{2}=(-1)^{2}=1$ is odd.

C) $p^{2}-2$
$1^{2}-2=1-2=-1$ is odd.

B) $p^{2}$
$1^{2}=1$ is odd.

A) $p-2$
$1-2=-1$ is odd.

On sat.collegeboard.org, 62% of responses were correct.

For more help with math, visit www.myknowsys.com.

# Equation of a Line

The BBC is a great source for international news or simply a different perspective on American news. Look at this site or other news sites in the last week or two before you take the SAT to find your five current events examples. Even if you don't normally keep up with the news, looking like you do can increase your score!

## 3/25 Equation of a Line

Remember to follow the Knowsys Method and note the bottom line, assess your options, attack the problem, and loop back to check the question before you select your answer.

A line segment containing the points (0,0) and (12,8) will also contain the point

The bottom line here is which answer choice lies on the given line.

There are multiple ways to solve this problem. You could find the equation of the line, or, since the line goes through the origin, you could use ratios to find a point that has the same relationship between its x and y coordinates.

First, reduce the coordinates to lowest terms. You can arrange them in a ratio format if you wish; whether you prefer $\frac{x}{y}$ or $\frac{y}{x}$ does not matter. You might use $\frac{y}{x}=\frac{8}{12}$ since "rise over run" is also the formula for slope. Reduce this to its lowest terms and then check the answer choices for multiples.

$\frac{y}{x}=\frac{8}{12}=\frac{2}{3}$

Now convert it back to (x,y) format. Make sure the x and y go in the right places.

(3,2)

A) (2,3)

B) (2,4)

C) (3,2)

D) (3,4)

E) (4,2)

On sat.collegeboard.org, 58% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Functions

history.com The History Channel's website is full of fascinating articles and videos. Watch their current series, "Full Metal Jousting," and learn some math facts about St. Patrick's Day.

## 3/22 Functions

Remember to follow the Knowsys Method: Read carefully, identify the bottom line, and assess your options. After you ask, "What could I do?" and "What should I do?" attack the problem, then loop back to check whether you answered the question correctly. Finally, take a look at the answer choices and select the correct one.

The graph of y=f(x) is shown above. If $0\leq t\leq 5$, and if (t,v) is on the graph of f, which of the following must be true?

A glance at the answer choices shows that they all have v, so v=? will do for an imperfect bottom line. Now look at the question again. If t is somewhere from 0 to 5, what does that say about v? Note that t is the x value, and v is the y value, so you need to focus on the part of the graph from x=0 to x=5. In that range, the smallest y value is 5 and the greatest is 10. Now look at the answer choices.

$A) -10\leq v\leq -5$

$B)-5\leq v\leq 0$

$C) 0\leq v\leq 5$

$D) 5\leq v\leq 10$

$E)10\leq v\leq 15$

On sat.collegeboard.org, 56% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Exponents

Remember to follow the Knowsys Method: read carefully, identify the bottom line, assess your options, attack the problem, and loop back to ensure that you answered the question correctly.

If $x^{\frac{1}{3}}=y^{2}$, which of the following must be equivalent to x?

After reading, find the bottom line and note it at the top of your scratch work.

x=?

Next, assess your options. There are two courses of action apparent here: you could pick numbers and plug them into x and y, or you could apply the exponent rules to solve for x. Which would be faster and easier? The exponent rules.

$x^{\frac{1}{3}}=y^{2}$

What can you do here? Since you need to isolate x, pay attention to its exponent. Normally, a fraction in an exponent indicates that you need to take a root--in this case, a cube root--but since you cannot take the root of a variable, do the opposite. Cube both sides.

$x^{\frac{1}{3}^{3}}=y^{2^{3}}$

The rule for "power to a power" situations, when an exponent is itself the base of another exponent, is to simply multiply the powers together.

$x=y^{6}$

Loop back to your bottom line. You were looking for x, and you found that  $x=y^{6}$. Now look at the answer choices.

$A)y^{\frac{1}{6}}$

$B)y^{\frac{2}{3}}$

$C)y^{\frac{3}{2}}$

$D)y^{3}$

$E)y^{6}$