ACT Math Questions, and What to Do About It

How You'll Get Stuck in SAT/ACT Math Questions, and What to Do About It SAT / ACT Prep Online Guides and Tips So you’ve been staring at one math problem for what feels like forever, or maybe you’ve gone through your solve and none of the answer choices match what you found. Or maybe you just feel like somewhere along the way you made a huge mistake†¦. Well, never fear! Right now you might be stuck on a math problem, but we’ve all been there and there is always a way to recover. We’ll walk you through both how to recognize when you’re stuck (it’s not always obvious until too late) and what to do about it when it happens. Overall Test Structure The SAT math section and the ACT math section are designed and structured differently. Though many principles of how to solve a math problem (and how to get out of a math bind) will hold true for both tests, there are some unique pitfalls and benefits hidden in each test. So let’s look at a little background and structure for both the SAT and the ACT to give a better foundation for the kinds of ways you can get stuck on a math problem and how to get out of it. About the SAT Math Section The SAT math section requires that you put your mathematical knowledge to work in ways that may seem strange or unfamiliar to you. Most SAT math questions won't look like the kinds of questions you see on your math tests in school and this may be a challenge to deal with at first. But while the questions may be unusual, each and every math topic used on the test is one that you are likely familiar with and have studied for a number of years. If you’re not used to the structure of the SAT, there are two main ways you can find yourself trapped into the no-man’s-land of selecting wrong answers: by overestimating the difficulty of each question or by underestimating the difficulty of each question. (Yikes!) Some students get so turned around by the twisting and unusual presentation of the questions that they forget the fact that they will be familiar with most, if not every topic on the SAT math section. This kind of student tends to panic and either make wild guesses or leave many questions blank that she likely could have solved. Other students, however, go in the complete opposite direction and underestimate how tricky the questions on the SAT can be. Many questions (though not all) are set up in ways that will steer anyone who isn’t paying close attention off the right track. And all the answer choices on the multiple choice section are most often generated by common student errors, so it can be easy to fall for the many â€Å"bait† answer choices available. In the next section, we’ll show you how to tell when you’re falling into one of these traps and how to get out of it. About the ACT Math Section The ACT math section will occasionally give you a question that is set up in a strange or unfamiliar way, but for the most part, ACT math questions will be fairly â€Å"straightforward.† Do you know how to set up and solve for an average? Do you know what a rational number is? A significant portion of your ACT math questions will look just like the ones you've seen in school. Though many ACT math questions are far from easy, the presentation of the questions is not specifically designed to trick or trap you. ACT math questions are challenging mostly for two reasons: you will be on a much stricter time crunch than you would be on the SAT (see our article on the differences between ACT math and SAT math for more detail on this) and the topics in question may be more obscure to you. For instance, most students will be far less familiar with trigonometry or sequences than they are with single variable equations, but these are all topics that will be on the ACT. Because the ACT math section must be completed quickly, it's crucial that you realize when you’re going down the wrong mathematical path. So let’s take a look at how to spot it when it happens. Time is a precious resource on both tests, so it's important to change course quickly if something's going wrong. How to Tell When You’re Stuck (or Going Down the Wrong Path) Almost every math question on both the ACT and the SAT can be solved in 30 seconds or less. If you’re familiar with the material, you can absolutely solve almost every single math question with plenty of time to spare. In addition, each and every question on the test is designed in such a way that a student can solve it without a calculator. Though you should use a calculator on the test to boost your speed and accuracy, always keep in mind that you don’t technically need one. How does this knowledge help you? Well, if you find you’re going down the rabbit hole and need 30 steps to solve a problem, or if you start getting bizarre strings of decimals or weird root systems, then stop! If the problem is taking two or three minutes to solve or you think it would literally be impossible (or just really, really difficult) to solve without a calculator, then something went wrong and you should stop and go back to the beginning. Let’s look at a few examples of ACT and SAT problems and some common student errors to see this in action. Let’s say you got to this problem and were going so quickly through the test that you misread the word â€Å"area† and thought you were supposed to solve for the perimeter instead. This may seem like an obvious mistake right now, but never underestimate the effects of adrenaline and fatigue on your ability to think logically. It’s all too easy (and all too common) to make a mistake like this on the test and pay the price in lost time if you do. But if you read â€Å"area† as â€Å"perimeter,† then you would see that the â€Å"perimeter† of triangle DFH was 10, which would mean that each side was $10/3 = 3.33$. Hmm. This is already mighty suspicious. Remember- you should be able to solve any question without a calculator, so if you have decimal points, they will almost always be 0.5 or, more rarely, 0.25. To get a decimal of 0.33 is suspect. But if you kept going, you would find that each side of triangle DFH is made up of two sides of the smaller triangles. This would mean that the side of each smaller triangle was: $3.33/2 = 1.67$ Now a red alert should be sounding in your head! You’ve gotten two â€Å"weird† decimal values in a row and you still haven’t found your answer yet. It’s a good idea to pause and see if you’ve made a mistake somewhere before you waste time going any further. [Note: the way to actually solve the problem is to either find the area of each small triangle- $10/4 = 2.5$- and multiply by the total number of small triangles, or to set up a proportion of the area of DFH to AFK. Either way, your final answer will be E, 62.5] Now let’s take a look at another example, Last time, we said that perhaps you simply misread the problem. This time, maybe you thought you knew enough about the topic to solve the problem, but ended up stuck halfway through. For instance, let’s say you were familiar enough with quadratic equations to recognize that this was one, but didn’t know a whole lot else about them. Well it says that the value of $x$ is -3, so that must mean we replace $x$ with -3 and solve from there, right? $x^2 + mx + n = 0$ $-3^2 + 3m + n = 0$ $9 + 3m + n = 0$ Hmm. Well this doesn’t look promising. We could try to isolate $m$ to find it’s value, in which case we would say: $3m + n = -9$ $3m + -9 - n$ $m = -3 - {n/3}$ But we still don’t know the value of $n$, so that doesn’t help us much. We could try to factor it, but without knowing the value of $n$, we still can’t find $m$. We’re well and truly stuck. [Note: the way to solve the problem is by understanding how factoring quadratic equations works at its core. If -3 is the only value for $x$, then it must mean that our quadratic factors out to be a square. Our factors must therefore look like: (x + 3)(x + 3) for $x$ to equal -3 since $x + 3 = 0$ = $x = -3$. If we properly distribute $(x + 3)^2$, we will get $x^2 + 6x + 9 = 0$, which means the answer is C, $m = 6$.] Sometimes, you’ll be able to see pretty quickly (as with our first example) when you’re going wrong, and sometimes it will take a few steps before you hit a wall (as with our second example). But, as you get more practice solving SAT and ACT math questions, you’ll start to get instincts to feel when you’ve taken a left turn somewhere, and this is the point at which you must take a step back. But what do you do once you realize you’ve run into a problem? Let’s take a look. We can save this, not to worry. Well...probably. Pretty sure. (Gross) What to Do When You’re Stuck As you go through your test, answer your questions by following these steps: Step 1: Always Answer the Easiest Questions First As a general rule of thumb when taking the either the SAT or the ACT, always root out and answer the questions you can complete the fastest and with the most confidence. Remember- each and every question is worth one point, so it doesn’t matter if it was the easiest problem or the hardest. No one is judging how you complete your test, so don’t feel you have to ignore the simpler problems for the sake of the more difficult ones. Get your points where you can and keep moving forward. The ACT math section is arranged in ascending order of difficulty, so the earliest questions will also be the "easiest." The SAT is a little more complicated, since the math portion of the test is divided into three sections- two entirely multiple choice and one combination multiple choice and grid-in. SAT multiple choice questions are also arranged in order of ascending difficulty (so the early questions will also be "easiest"), but the difficulty level resets for the grid-in section. This means that the first question on the grid-in portion will be much easier than the last multiple choice question in that math section. Be especially careful in the combination multiple choice and grid-in SAT math section for this reason. If any question starts to give you trouble or seems to be taking a long time, mark it and come back to it only after you’ve completed all of your easy questions. Once you’ve identified these potential â€Å"problem questions,† approach them based on why they were problematic in the first place. Step 2: If You Can See Your Error, Redo the Problem Once you’ve identified that you’re going down the wrong track, stop working and read the question again. Did you try to find the wrong variable? For instance, did you solve for $a$ instead of $2a$, the perimeter instead of the area? Did you write down the wrong values for your givens? Or did you realize you simply don’t know enough about, for instance, functions to solve the problem? If you can spot your error and correct it in a few seconds, go ahead and start over and solve the problem again the right way. If you really think you can solve the problem this time, then it’s definitely worth your time to work through it again. If you can’t spot why or how you went wrong (just that you did), move on to the next step. Step 3: If You Can't See the Error, Find an Alternative Solving Method Sometimes you might know enough about the topic (or are just familiar enough with the test) to see that you don't know how to solve a particular problem in the traditional way. Now is a good time to turn to one of your alternate solving methods, namely plugging in answers or plugging in your own numbers. Let’s say that you went up against this question: Maybe you didn’t know where to begin, or maybe you attempted the problem and felt that you started getting trapped in the algebra. Well luckily, there is almost always another way to solve any ACT or SAT question! In this case, we have variables in both the problem and in the answer choices, which is a key feature for a PIN question. With that in mind, we can pick our own value for $x$ and find the answer choice that corresponds to this value. Let’s say that we said $x = 2$. (Why 2? Why not!) Now we find the value of our given function. ${x + 1}/{x^3 - x}$ ${2 + 1}/{2^3 - 2}$ $3/6$ $1/2$ Now we need to find which answer choice is equivalent to $1/2$, when we use the same value of $x$ ($x = 2$). Answer choice F gives us: $1/{x^2} - 1/{x^3}$ $1/4 - 1/8$ $1/8$ This does not match our found value of $1/2$, so we can eliminate answer choice F. Answer choice G gives us: $1/{x^3} - 1/x$ $1/8 - 1/2$ $-{3/8}$ This does not match our found value of $1/2$, so we can eliminate answer choice G. Answer choice H gives us: $1/{x^2 - 1}$ $1/{4 - 1}$ $1/3$ This does not match our found value of $1/2$, so we can eliminate answer choice H. Answer choice J gives us: $1/{x^2 - x}$ $1/{4 - 2}$ $1/2$ This does match our found answer of $1/2$, so we can keep answer J in the running. We should still test answer choice K, however, just to make sure we do not have any duplicate correct answers. Answer choice K gives us: $1/{x^3}$ $1/8$ This does not match our found answer of $1/2$. We can eliminate answer choice K. This leaves us with only one answer that matches what we found as our given when $x = 2$. This means that answer choice J (and only answer choice J) must be correct. Our final answer is J, $1/{x^2 - x}$ Step 4: If Your Alternate Solve Doesn't Work, Move On Sometimes taking a standardized test means learning to let go of a question. Your time is precious and limited, so if you can solve two questions in the time it takes you to solve one, it’s always best to solve the two. If you find yourself trapped on a question and can’t find any way to solve it, let it go (for now). If you’re taking the ACT, bubble in your best guess answer, but do so lightly enough that you can come back and change it later, time permitting. If you’re taking the SAT, simply skip the question for now (don’t bubble in a guess!) and come back later. If you have time after you’re done with the rest of your questions, come back to any that you had to skip or bubble a guess on. Sometimes all you need is fresh eyes, and then the method to solve the problem will become apparent. If you still can’t find the correct answer in any of the â€Å"traditional† ways, you can try to strategically eliminate answer choices until you’ve at least narrowed down your odds. For more information on how to do this, check out our guides for how to best guess on the SAT math section and how to best guess on the ACT math section. Otherwise, simply let it stay blank (if taking the SAT) or stick with your first guess answer (if taking the ACT). Learning to let go of a problem can be just as important as learning when to stick with a problem and try an alternate solving method. The more you practice, the better you'll get at balancing these techniques. The Take-Aways The more practice you have with taking standardized tests, the more skilled you’ll become at spotting your mistakes and heading them off at the pass. So don’t despair if it seems like an impossible task right now- you’ll get better at it. Be sure to practice on quality ACT and SAT study materials and stop if you feel any misgivings while solving any particular question. Remember- if it takes more than 30 seconds, or you think you would absolutely need a calculator, you’re probably on the wrong track. Take a breath, back up, and see where and how you can resolve the problem. And don’t be afraid to let a question go if you need to. One question won’t matter too much in the long run, but getting stuck for five minutes will absolutely take away time from other questions and have a detrimental impact on your final score. What’s Next? Worried about your math formulas? Learn which formulas you'll need to memorize for the ACT, which formulas you'll need to know for the SAT, and how to put your formulas to their greatest effect on both the ACT and the SAT. Running out of time on your math sections? Learn how to beat the clock on both the ACT and the SAT so you can reach your greatest potential in the time allowed. Trying not to procrastinate? Our guide will help you beat those procrastination urges and get you back on track. Want to get a perfect math score? Check out how to get a perfect 36 on the ACT math section or a perfect 800 on the SAT math section. Want to improve your SAT score by 160 points? Check out our best-in-class online SAT prep program. We guarantee your money back if you don't improve your SAT score by 160 points or more. Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math strategy guide, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next. 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