## The SAT Prep Black Book

**SAT Math**

**Hidden Test Design Patterns of SAT Math Questions**

Most of the hidden patterns on the SAT math section have to do with using the answer choices to help you check your answers, where that’s possible. Looking at the answer choices can reassure you that you have the right answer (or help show you your mistakes so you can correct them).

Many students are surprised to find out that these patterns appear reliably and consistently in real SAT Math questions from the College Board, because they really have very little to do with actual math rules—they’re purely related to the test-design principles required by the standardization of the test. So before we get into these patterns, I’d like to take a moment to remind you why they’re a part of the SAT.

As we discussed at the beginning of this book, it’s important to remember that the SAT is not a normal test. It has a very specific purpose and must, therefore, follow very specific rules to make sure that questions are designed to test the same skills in the same underlying ways, without actually repeating the questions. It’s also important to remember that the SAT is predominantly a multiple-choice test, and that the multiple-choice format only requires a test-taker to separate right answers from wrong answers, rather than requiring you to provide correct answers on your own. This means that the relationships among right answers and wrong answers must remain constant for all real tests, because changing those relationships would involve changing the nature of the multiple-choice questions and breaking the standardization rules of the test.

Keep these things in mind as we discuss the patterns in the SAT Math section, and as you encounter real SAT Math questions in the future.

Hidden Pattern 1: Halves and Doubles

Very often, one of the wrong answer choices will be twice as much as the right answer choice, or half as much as it. This is especially true when the problem involves multiplying or dividing an amount by 2. If you solve a problem and get an answer like 18, a wrong answer choice like 36 might reassure you that you’re right.

Remember that this pattern is an indication that you’re probably right, not a confirmation that you’re definitely right. Also, it’s important not to get it backwards—in the same hypothetical example, the right answer might be 36, and the wrong answer might be 18! Be very aware of this useful pattern, but don’t rely on it exclusively.

Hidden Pattern 2: Right Answer, Wrong Time

One of the ways that the SAT will try to confuse you is by giving you a problem that involves two or three steps. When it does that, one of the wrong answers will often be the number that you would get if you stopped after one of the earlier steps. For example, a problem might ask you to find the price for pens by giving you the prices for different combinations of pencils and erasers. The problem might require you to figure out the price of pencils in order to figure out the price for erasers, and one of the wrong answer choices would be the price of pencils. Because this wrong answer is actually a number that you found in the process of solving the problem, seeing it in as a wrong answer can reassure you that you’re on the right track.

Hidden Pattern 3: Substitution

As I said before, the SAT likes to give you problems that look complicated but have very simple solutions. The test often does this by showing you a rather complicated expression that you can simplify by substituting one thing for another. If you start looking for substitution opportunities, you’ll find them all over the test, and they’ll make your life easier. For example, the SAT will often reward you for substituting a difference of squares like *x*^{2} – *y*^{2} with the expression (*x – y*)(*x* + *y*). On the other hand, it might also reward you for going in the other direction, and realizing that (*x – y*)(*x* + *y*) can be restated as *x*^{2} – *y*^{2}.

Don’t worry if this sounds a little vague right now. We’ll see several examples of it when we go through real SAT questions from the College Board in a few pages.

Hidden Pattern 4: Try To Avoid Firsts And Lasts In A Series

Sometimes some or all of the answer choices in a math question will form a series. These series might be pretty easy to recognize in some cases, like if the answer choices are 7, 8, 9, 10, and 11. In other cases, the series might be less obvious, and it might be related to a concept in the question: if the question is talking about dividing some quantity by 4, then the answer choices might contain the series 3, 12, 48, because each number in the series is one fourth of the next number in the series.

The College Board seems to include series in the answer choices when it hopes that you’ll make a mistake and repeat a step in the solution one time too many or too few, ending up with one of the wrong answers in the series. In other words, if a question involves finding the perimeter of a triangle with sides of 5 units each, the answer choices might include the series 10, 15, 20, because the College Board is hoping you’ll either add 5 one time too few (ending up with 10) or one time too many (ending up with 20).

For this reason, when a series is involved in the answer choices, we’ll typically find that the correct answer isn’t the first or last number in the series. The College Board seems to like to put the correct answer near the middle of the series in order to allow you to make a mistake in either direction and still find a wrong answer that reflects your mistake.

Remember, as with the other patterns in this section, that this isn’t an unbreakable rule. So I’m not saying that we’ll never, ever find the right answer at the beginning or the end of a series; sometimes we will. I’m just saying that it’s more common to find it in one of the middle positions of a series, and that it helps to be aware of that.

Hidden Pattern 5: Wrong Answers Try To Imitate Right Answers

The College Board likes to create wrong answers that incorporate elements of correct answers, probably in an attempt to make it harder for you to eliminate answer choices on the basis of a partial solution. In other words, if you’re working on a question where the answer choices are all algebraic expressions and you figure out that the correct answer should include the expression 2*r* along with some other stuff, then you’ll often find that a majority of the answer choices include 2*r*. This way, the College Board can try to force you to figure out the rest of the question in order to identify the correct answer.

While this can be an annoying thing for the College Board to do, you’ll find that you can actually use it to your advantage in many cases: after you think you’ve solved a question, if you see that the wrong answers seem to include a lot of the elements in common with the choice that you like, you can often take that as a good sign that you’ve thought about the question correctly.

In other words, if the wrong answers seem to be imitating parts of the right answer, that’s typically a sign that you’ve understood the question correctly. (Notice I said, “typically,” and not “always.”)

We’ll see several examples of this, and of the other SAT Math patterns, when we go through some questions from the Blue Book in a few pages.