The SAT Prep Black Book
The Recommended Math Path
Now that we’ve explained the rules and patterns that you’ll find on the SAT Math section, we can look at the process that I recommend for those questions. I call it the “Math Path,” mostly because that rhymes.
The Math Path is a set of guidelines to help us figure out how to attack tough questions. You won’t need to use it on every question, and you can modify it as you keep practicing. I’m teaching it to you because it’s a good way to keep all of the elements of SAT Math questions in mind. If you practice with these ideas, you’ll find that they become second nature.
1. Read the question carefully.
This might sound kind of strange, but if you asked me to pick the single mistake that costs people the most points on the Math part of the SAT, I’d say it’s the mistake of not reading the questions and the answer choices carefully.
In fact, we should really think of the entire SAT, including the Math portion, as an extended test of reading skills. Most of what we do on the Math portion of the test will depend on our ability to notice key phrases and details in each question.
By the way, because of the way SAT Math works, if you know the meanings of every word in a particular question, then you know enough math to be able to answer that question. Trust me on this—we’ll see proof of it as we continue.
2. Read the answer choices if there are any.
Most students ignore the answer choices in a Math question until they’re basically done with the question. They typically read a question, try to figure out the answer on their own, and look for that answer (or a similar answer) in the answer choices. Now, if you could successfully do that for the course of an entire test without making a single mistake, it’s true that you wouldn’t miss any questions. But so many questions become so much easier to answer when we consider the answer choices as part of the question from the very beginning.
Remember that the College Board likes to play little games in the answer choices of the SAT Math section. We talked about some of those “games” a few pages ago when we covered the hidden patterns of these questions. Sometimes wrong answers will include elements of the right answer, for instance, or sometimes they’ll form a series, and so on. Sometimes, simply noting that all the answer choices are one-digit numbers can be enough to help you realize how to approach a question.
So after you read the main part of the question, look over the answer choices and see what kind of options the test is giving you. Try to figure out why the test is presenting the answer choices that way—look at the values in the choices, but also look at the relationships between those values, and try to think about how those relationships might be important to the question.
I said earlier that it’s important to remember that every SAT Math question can be answered in less than 30 seconds each if we’re really on our game. Often, a large part of finding these super-fast solutions involves thinking about how the answer choices relate to the question from the very beginning.
So remember to think of the entire question, including the answer choices, as one big system of ideas. We’ll see several examples of how this works when we look at some real SAT questions from the College Board’s blue book in a few pages.
3. Consider diagrams, if there are any.
As we discussed earlier, there are two questions you should always ask yourself when a math question includes a diagram:
1. Is this drawn to scale?
2. Are any dimensions of the diagram left out of the diagram itself but included in the text underneath it?
If a diagram is drawn to scale, we can often (but not always) find the answer to the question just by looking at the diagram itself.
If the diagram is not to scale, the reason is almost always that drawing it to scale would have given away the correct answer. So if we re-draw the diagram to scale, we’ll often be able to figure out how to answer the question pretty easily.
Any dimensions that are left out of the diagram itself but included in the text of the question will probably be the basis for the first step in the solution to the question.
4. Think about what areas of math might be involved.
Now that you’ve read the question and the answer choices, and considered the diagram if there is one, you should have a pretty good idea of which specific math terms and concepts are mentioned in the question.
Many test-takers overlook the fact that the solution to a question can only involve concepts that are immediately related to the concepts in the question. It sounds kind of obvious once it’s pointed out, but everything in math proceeds in a step-by-step fashion, with each step building on the previous one.
When most people get stumped on the SAT Math section, they panic and try to call to mind every single math concept they know in the hope that one of those concepts will miraculously reveal the answer. Instead, we want to narrow our focus and confine our thought process to the concepts in the question and the concepts that are related to them.
For example, if an SAT Math question involves words like “degrees” and “radius” and “center,” then it must be a question about circles, and there are only a few circle-related concepts that the SAT is allowed to ask us about (look back at the toolbox if you don’t remember what they are). That means that the solution to the question must somehow involve those circle-related concepts, so we should focus our attention on them.
5. Look for a 30-second solution.
In this step, we try to use everything we’ve already figured out (the clues from the diagrams, the clues in the answer choices, and the relevant mathematical concepts) to help us string together the right basic math ideas that will let us connect the prompt to the correct answer choice. And don’t forget—the best solutions will take you less than 30 seconds to work out.
Of course, as I said before, you can still get the question right even if you can’t find a solution in under 30 seconds. But it’s a good idea to get in the habit of looking for fast, simple solutions, because the majority of the difficulty that people have on the SAT Math section comes from not catching small details in a question, and wasting a lot of time and effort as a result.
6. Solve the problem.
After you have read the question, considered the answer choices, considered the diagram, considered the likely areas of math to be involved, and decided on a straightforward solution, you’ve finally earned the right to go ahead and solve the problem. IF YOU TRY TO SOLVE THE PROBLEM WITHOUT GOING THROUGH THE EARLIER STEPS, YOU’LL PROBABLY JUST WASTE YOUR TIME.
This is one more way that SAT Math questions differ from the math questions you encounter in school. In school, the questions on a math test are basically just like the questions you’ve been doing for homework and the questions your teacher has been doing in lectures, so you build up a kind of instinctive, automatic approach to doing math, in which you memorize formulas and then automatically apply a certain formula in a certain situation.
But that won’t work on the SAT Math section, where questions seem to be specifically written so that formulas are of little help. If you read a math question on the SAT and dive right into it without thinking about it first, you’re probably doing something wrong. Don’t try to solve the problem until you’ve read it and thought about how it fits the SAT’s patterns and rules.
7. Re-check the answer choices.
One of the best ways to double-check your work is to look at all the choices you think are wrong and see if you can figure out why some of them are included. In other words, if you can figure out the mistakes that the College Board wanted you to make for some of the wrong choices, then there’s a pretty good chance that you’ve handled the question correctly. But if you look back over the wrong answers and you don’t have any idea why any of them are there, that’s typically a sign that you misunderstood the question. Be on the lookout for hidden patterns like the ones we talked about.
If you’re fully satisfied that you know why your answer is right and why at least a few of the other answer choices are present as wrong answers (for multiple-choice questions), mark your answer and move on to the next question. AS ALWAYS, IF YOU’RE NOT COMPLETELY SURE THAT YOU HAVE THE RIGHT ANSWER, SKIP THE QUESTION. DON’T GUESS! If you don’t remember why you shouldn’t guess, go back and look at our earlier discussion of the problems with guessing.
SAT Math Path Conclusion
The important thing about SAT Math questions is that you shouldn’t try to solve them without reading them carefully and setting them up first. Taking a few seconds to get your bearings will make answering the question a lot easier. Remember to keep the solution to every problem as simple as possible.
It may seem like this process is pretty long or complicated, especially for questions that seem obvious when you first look at them. But it’s important to remember that you don’t have to use this process on every question—only on the ones that you can’t figure out at first. And you can modify it as you see fit, depending on the question and your own preferences.
The important thing is to be aware of all the elements involved in the Math Path (careful reading, considering the answer choices, evaluating diagrams, identifying relevant concepts, trying to find the simplest possible solution, and catching your mistakes). Try to implement them in your practice sessions, so they can become second nature when you see challenging questions on the test.
What About Grid-In Questions?
Many students wonder if the Student-Produced Response questions (or “grid-in” questions) require a different approach from the multiple-choice questions. For the most part, the Math Path process that we just discussed is the process I would follow for the grid-in questions (with the obvious exception that we won’t have any answer choices to consider in deciding how to attack the question.
There are a few special considerations we should keep in mind for the grid-in questions, though.
If The Question Refers To The Possibility Of Multiple Solutions, Make Sure You Understand Why
The grid-in format allows the College Board to ask more open-ended math questions, which is one of the reasons it exists on the SAT in the first place. So be aware that you might see questions that offer more than one valid solution. Such a question will often use a phrase like “one possible value,” as in, “If X has a value between 3.9 and 4, what is one possible value of X?”
If you realize that you’re dealing with a question that refers to the possibility of more than one valid response, make sure you can figure out why. In other words, if you can only think of one possible answer for such a question, then you’ve probably misunderstood it in some fundamental way, and there’s a very good chance that the answer you’re thinking of is wrong.
I’m not saying that you actually need to work out more than one solution in order to know that you’ve got the question right; I’m just saying that you need to understand where other solutions might come from. For instance, in the hypothetical question I just mentioned, you might say, “I know that 3.91 is between 3.9 and 4, and I know they used the phrase ‘one possible value’ because there’s an infinite number of numbers between any two numbers.” You wouldn’t necessarily have to work out that 3.92 and 3.93 are also valid solutions in order to be certain you understood the question correctly.
Don’t Be Afraid To Guess, But Don’t Expect Much To Come Of It
You aren’t penalized for missing a grid-in question, so you should never leave a grid-in question blank. On the other hand, since there are thousands of possible ways to fill out the answer grid for each question, the chance of guessing right is extremely small—which is why they don’t penalize you in the first place.
If you do decide to guess on a grid-in question, make that decision as quickly as you can so that you don’t waste any more time on the question than necessary. I would recommend guessing either 0 or 1, if those answers seem like they have any chance of being correct, just because I feel like I see those answers appearing more frequently than any other individual number. But the advantage of that, if there even is one, is extremely slight, and your chance of being correct on a random guess is basically zero anyway.
The Last 2 or 3 Questions Might Seem Extra Weird
We’ll often find that the strangest questions on the whole SAT Math section appear as questions 16, 17, and/or 18 on the grid-in section. We won’t always find exceptionally weird questions in these positions, but if there are going to be exceptionally weird math questions on a particular test, this is probably where they’ll appear.
These questions must still follow the same rules and patterns that the rest of the test follows—they still can’t involve trigonometry, for instance, and they must still be answerable in under 30 seconds and without a calculator. It’s just that sometimes they seem noticeably weirder than other questions on the SAT Math section.
Unfortunately, simply knowing that this might happen doesn’t do much to help us answer these questions. I just wanted to mention that we sometimes see extremely strange questions in these positions so that you know to look out for them, and so that you don’t think the test is suddenly changing dramatically and begin to doubt yourself.
We’ve just discussed a few special considerations for the grid-in questions, but remember that we want to approach them in basically the same way we would approach any other SAT Math question: by reading carefully, paying attention to details, thinking about which areas of math might be involved, looking for the simplest possible solution, and so on.
Sometimes You Can’t Solve For Every Variable
We’ve seen several examples of the way the College Board likes to make questions more challenging by handling things differently from the way they’re handled in school. One of the best examples of this is the way the test sometimes asks questions that involve multiple variables.
In school, if you have a question that involves four variables, you’re usually supposed to figure out the individual values of each variable. But the SAT often asks questions in which it’s unnecessary—or even impossible—to work out values for individual variables.
This doesn’t mean that questions with multiple variables are impossible to figure out on the SAT. It’s just that sometimes it’s possible to know the value of an expression that involves several variables even if you can’t know the values of the variables themselves.
For instance, if I tell you that ab2 - 7 = 53, and then ask you for the value of ab2, you can still figure out that ab2 is 60, even if it’s impossible to know the values of a or b individually.
One of the test’s main signals that it might not be possible to solve for every variable in an expression is when they ask you for the value of an expression that contains multiple variables, rather than asking for the value of an individual variable within that expression. In the ab2example I just gave, I couldn’t ask for a or b individually because there wasn’t enough information about them; that’s why I had to ask for the entire expression ab2.
So if you see a question that asks for the value of x + y or some other expression with more than one variable, don’t make the mistake of assuming that you have to find x and y individually. It may be more expedient to try to solve for the entire x + y expression all at once.
One example of a question that involves multiple variables that can’t be approached individually is question 16 from page 919 in the second edition of the College Board’s Blue Book. My explanation for that question appears later in this book, when I go through a selection of challenging SAT Math questions.
It’s Not School Math—Your Work Doesn’t Matter
Lots of test-takers experience significant difficulty on the SAT Math section for a reason that might seem strange to a lot of people: they try to approach each question in a formalized way that would satisfy a math teacher.
But by now we know that the SAT doesn’t reward the same things that school rewards. And the SAT Math section is no different.
The bottom line is that the SAT doesn’t care what kind of work you do to arrive at the answer that you choose. The SAT only cares if the answer that you choose is correct. That’s it.
This fact has two very important implications for us as test-takers. First, it means that we can, and should, get in the habit of looking for the fastest, most direct route to the answer, even if that route doesn’t involve solving a formal equation (or even writing anything down at all). Second, it means that we have to make sure we don’t make any small mistakes in our solution that might lead us to mark the wrong answer even if our overall approach is formally sound, because the College Board will never know what our approach was. For the College Board, a wrong answer is a wrong answer no matter how solid the approach to the question was, and a right answer is a right answer no matter what you did to arrive there.
So remember that the SAT Math section often features questions where formulaic solutions are literally impossible to use.
In the parts of this book where I provide solutions to real SAT Math questions from the College Board’s Blue Book, you’ll often see that the approach I recommend wouldn’t be acceptable to most math teachers, because it’s not formal. This isn’t because I’m not good at math; it’s because I’m very, very good at SAT Math, and in SAT Math there’s no value in approaching things formulaically. In fact, there’s usually a lot more value in abandoning formulaic math whenever possible.
So try to get in the habit of finding the most direct approach to a question that you possibly can, and remember that the only thing that matters to the College Board is that you mark the correct answer!
Avoid Decimal Expressions Unless A Question Uses Them
In high school and college math classes, we’re often encouraged to use decimal values rather than fractions. For instance, we might write “0.8” instead of “4/5.”
On the SAT, it’s usually a bad idea to express things in terms of decimals, unless the answer choices are also in decimal form. When we work with decimals, we often miss opportunities to simplify and reduce expressions that are much easier to see when we keep everything in fraction form.
For instance, if a question involves multiplying 4/5 by 5/6, then using fractions might help me see right away that the 5’s cancel out and I’m left with 4/6, which is the same thing as 2/3. If I had to enter that into a calculator, I’d probably lose time, and I’d also run the risk of hitting the wrong key or something. In general, we’ll have a much better chance of finding the shortest possible solution if we get in the habit of avoiding decimal expressions on the SAT.
The only real exception to this comes when a question involves decimal expressions in its answer choices. When the College Board sets a question up like that, they’re usually trying to get us to realize that we can simply approximate the answer, often by using the scale of a diagram or some other clue in the question, and that the correct answer choice will be the only one that’s close to our approximation. In these situations, it can be very helpful to use the decimal approximation to solve the question—but in just about every other situation, it’ll be smarter to stick with fractions.
Ignore The So-Called “Order Of Difficulty”
Test-takers are often encouraged to believe that the questions on the SAT Math section get harder as the section goes on. But I would recommend ignoring that idea.
It’s true that the College Board assigns difficulty-level rankings of 1 through 5 to each question, and that we’ll typically see that the earliest questions in an SAT Math section are ranked 1 or 2, while the last questions are 4’s and 5’s.
But that shouldn’t mean anything to us, as trained test-takers. Let me explain why.
The difficulty ranking is simply an indication of the percentage of the test-taking population who miss a particular question. In other words, if a question has a difficulty ranking of 1, then the vast majority of people who see it will get it right; if it has a difficulty ranking of 5, then the vast majority of people who see it will answer it incorrectly.
The difficulty ranking has nothing to do with the number or type of math concepts that appear in a particular question. Sometimes a question that’s based on the definition of the word “even” might be ranked as a 1, and sometimes it’s ranked as a 5. The deciding factor is simply the percentage of test-takers who get it right or wrong.
And here’s the important thing to keep in mind: the vast majority of SAT-takers have no idea at all of how to take the test. They do almost everything wrong. If you’re familiar with the concepts in this Black Book, you’ll be looking at SAT Math questions the right way, and the percentage of regular test-takers who get a question right has nothing to do with whether you, yourself, will get that question right.
We’ll generally find that questions near the end of an SAT Math section tend to be more abstract or conceptual, while questions towards the beginning of a section tend to be a little more similar to the kinds of things you might be asked about in school. But that doesn’t mean the later questions are always harder. In fact, we’ll often find that questions towards the end of a section are more likely to be things you can solve without actually doing any calculations (because they’ll rely on things like properties and definitions of terms). This can often mean that we can answer later questions much more quickly than earlier questions, if we’re approaching them correctly.
But this cuts both ways. Just as we shouldn’t be intimidated by later questions, we also shouldn’t take earlier questions for granted. Remember that doing well on SAT Math is a question of reading carefully and thinking carefully, and that wrong answer choices are often designed to attract students who make small mistakes. This means that we have to be alert to possible mistakes at all times, even when a question might seem extremely easy.
In fact, when I work with students who are trying to score a perfect 800 on the SAT Math section but who might be missing one or two questions per test, I find that they nearly always miss questions in the first half of the section because of small mistakes—usually because they’ve been incorrectly taught that those questions are always “easy” and that they don’t have to pay full attention to them.
So here’s the bottom line: you should ignore the idea of an order of difficulty, because it serves no purpose. Instead, treat each question as a separate event. Don’t be afraid of any questions, and don’t take them for granted either.