ACT Math For Dummies (2011)
Part I. Getting into the Act: An Overview of ACT Math Basics
Chapter 2. Boosting Your Test-Taking Skills
In This Chapter
Taking the test in two passes
Knowing when to guess an answer
Understanding multiple-choice questions
Using your calculator to your advantage
Some folks say that the only thing tests measure is how well you take a test. Such news would be great, because then you could skip all the math stuff in this book and head directly to tips such as “always guess Choice (D).” But guess what? That tactic isn’t gonna help!
So before you take any test, you need to know a few strategies that relate specifically to that test. In this chapter, I focus on some basic facts about ACT math and show you how best to approach the test to take advantage of these facts.
To begin, I discuss the strategy of tackling the math section in two passes — the first to handle the easy questions and the second to focus on the more difficult ones. I also discuss a few strategies about guessing on the ACT and provide you with a few ways to approach multiple-choice math questions. Finally, I end the chapter with some advice on selecting a calculator, including some tips for using the one you select.
Two-Timing the Test: Taking Two Passes to Answer Questions
On the ACT math test, you have 60 minutes to answer 60 questions. So you have roughly 1 minute per question. Every question you answer correctly is worth 1 point toward your raw score.
But not all ACT math questions are created equal. Generally speaking, the questions increase in difficulty as you proceed from Question 1 to Question 60. Here’s the general breakdown of difficulty:
Easy: Questions 1 through 20
Medium: Questions 21 through 40
Hard: Questions 41 through 60
You should obviously start the test at Question 1 and proceed in order as much as you can. However, every student is different, so you may find some questions along the way that are more difficult for you than later questions. So if you read a question and don’t have a clue how to answer it, feel free to jump over it. The next question may be easier for you to answer.
I suggest you use the tried-and-true strategy of taking two passes over the ACT:
Pass #1: Start with Question 1 and work your way forward, answering questions that look relatively quick and easy and jumping over those that look difficult or time-consuming.
Pass #2: After you’ve answered all the quick and easy questions, circle back to the first question you skipped over and work your way forward to the end again.
This strategy maximizes the number of questions you can answer with confidence. It also helps you save time for the tough questions, which usually take more than 1 minute to solve.
In my humble opinion, every ACT math section includes a few questions that are practically begging for you to skip over them. For example, you may consider passing over questions that
Are very long and wordy
Seem purposely confusing and don’t make a lot of sense even the second time you read them
Have large or complicated numbers that will involve long or difficult calculations
Of course, not every problem with the preceding characteristics is as difficult as it looks. But as you run across problems like this, feel free to jump over them — even on Pass #2. If you have time at the end of the test, you can always try to pick off a few of these questions. But if you’re going to skip questions, you may as well skip these hairy beasts.
No penalty exists for guessing on the ACT, so be sure to answer all 60 questions before your time runs out. I discuss this point further in the next section, “To Guess or Not to Guess.”
To Guess or Not to Guess
On the ACT, you don’t lose points from your raw score when you fill in a wrong answer. So strategically you should fill in every answer, even if you have to make a wild guess.
Of course, you don’t want to guess on questions that you may be able to answer correctly — especially among the earlier questions, which tend to be easier. And keep in mind that an educated guess is always better than a wild guess. So whenever possible, rule out answers that you know are wrong. Keep track of these wrong answers by crossing them out in your test booklet.
Don’t guess at any answers while you’re still on the first pass (see the previous section, “Two-Timing the Test: Taking Two Passes to Answer Questions,” where I discuss tackling the test in two separate passes). Instead, begin guessing on your second pass. At this point, if you can confidently rule out a couple of answers but don’t know how to proceed with a question, you can save time by guessing at the answer and moving on to the next question.
Whenever possible, keep track of the questions that you guess on. If you have time at the end of the test — or if you have an unexpected brainstorm — you can revisit these questions.
Finally, keep close track of your time. When your 60 minutes are almost up, take a moment to guess at all the remaining answers — don’t leave any blank. With a bit of luck, you may pick up a few additional points on some of these questions.
After your time is up on any part of the test, the ACT rules state that you may not return to that part, even to fill in guesses. If you get caught doing this, you could be expelled from the test with no score and no refund. ’Nuff said!
ACT versus SAT: When do I guess?
On the ACT, you don’t lose points when you fill in a wrong answer. This rule is different on the SAT, where a quarter of a point is taken off for each wrong answer. This small distinction makes a big difference in your strategy if you’re planning to take both tests.
When you take the ACT, be sure to fill in every answer, even if you have to guess. To do this, keep a sharp eye on the time. Then when you’re down to the last minute or two, make sure that every answer is filled in. In contrast, if you also take the SAT, don’t fill in any answer for which you can’t make an educated guess. That is, don’t answer unless you can confidently rule out at least one choice as wrong.
Answering Multiple-Choice Questions
The math section of the ACT comprises 60 multiple-choice questions. Each question provides five possible answers. You likely have been taking standardized tests for most of your life as a student, so you’re probably already familiar with this type of question. However, math teachers generally don’t use multiple-choice questions when assigning homework or testing. So in this section, I provide a few strategies for approaching multiple-choice math questions.
Considering the five answer choices
Every multiple-choice question gives you a little extra information, because you know the correct answer must be one of the five choices given. Always take a moment to notice these answer choices, because they may guide you as you work on solving the problem. The following example shows you how you can rely on answer choices to correctly solve a problem.
If j2 – 14j + 48 = 0, which of the following shows all of the possible values of j?
(C) 6, 8
(D) –6, 8
(E) –6, –8
You can solve the equation j2 – 14j + 48 = 0 by factoring. (I show you the details of factoring in Chapter 5 and explain how to apply this technique to solving quadratic equations in Chapter 7.) In this case, every value in each of the five answers includes either 6 or 8 (give or take a minus sign), so you have a head start on the factoring:
j2 – 14j + 48 = 0
(j – 6)(j – 8) = 0
At this point, you only need to fill in the signs (+ or –) inside the parentheses. Because 48 in the original equation is positive, the two signs must be the same (either both + or both –). And because –14 is negative, at least one of the signs is negative. Therefore, both signs are negative:
(j – 6)(j – 8) = 0
Now you can solve this equation by breaking it into two separate equations:
j – 6 = 0 j – 8 = 0
j = 6 j = 8
Thus, the correct answer is Choice (C).
Plugging and playing
Multiple-choice questions give you an opportunity to arrive at the correct answer by plugging in the answer choices and solving. Please note that plugging in answers can be a little time-consuming, so if you can find a better way to solve the problem, go for it. But when you get stuck, this tactic gives you a chance at answering questions that you really aren’t sure how to solve. Consider the following example.
If , then x =
You may or may not know how to solve this type of equation (flip to Chapter 5 to see how it’s done). And in any case, solving it may be time-consuming. So you can try to plug in each possible answer for x to see which one works. Start with Choice (A) and plug in 4 for x:
This answer choice is obviously wrong, because 21 isn’t a square number. Therefore, is irrational and doesn’t equal 3. In fact, this wrong answer choice may suggest a way to save even more time: The reason this answer is wrong is that the value of evaluates to an irrational number, which messes up the equation. So has to be a rational number, which means 5x + 1 must be a square number. Try testing Choices (B) through (E) in this way, keeping in mind that you’re looking for a value of x that makes 5x + 1 a square number:
(B): 5(5) + 1 = 25 + 1 = 26 not a square number
(C): 5(6) + 1 = 30 + 1 = 31 not a square number
(D): 5(7) + 1 = 35 + 1 = 36 a square number!
(E): 5(8) + 1 = 40 + 1 = 41 not a square number
Only one value produces a square number, so the correct answer is Choice (D). You can verify this by plugging in 7 for x:
Some questions ask you for the greatest or least number that has a certain property. These questions provide a great opportunity to test answers individually until you find the correct one. Consider the following strategies:
When looking for the lowest or least value, begin with the lowest number and work your way up.
When looking for the greatest or highest value, begin with the greatest number and work your way down.
The following example illustrates this strategy.
What is the least common denominator when adding three fractions with denominators of 6, 9, and 16?
Because you’re looking for the least common denominator, you can find the correct answer by testing numbers and ruling out wrong answers, starting with the lowest number.
Begin by testing to see whether 60 is divisible by 6, 9, and 16:
60 ÷ 6 = 10
60 ÷ 9 = 6 r 6 not divisible
So Choice (F) is wrong. Now test 120:
120 ÷ 6 = 20
120 ÷ 9 = 13 r 3 not divisible
So Choice (G) also is wrong. Next, test 144:
144 ÷ 6 = 24
144 ÷ 9 = 16
144 ÷ 16 = 9
So Choice (H) is the correct answer. By the way, before moving on, notice that 288 is also divisible by all three denominators. However, Choice (K) is wrong because the question asks for the least common denominator, which is why you started plugging in the lowest numbers first.
Calculating Your Way to Success: Calculators and the ACT
If you hate doing long division as much as I do, you’re probably glad to hear that calculators are allowed on the ACT. In this section, I answer a few basic questions about calculators.
When should I use a calculator?
A calculator is a great tool for solving problems more quickly than you can either in your head or using a pencil and scratch paper. At the same time, however, you want to avoid overusing it for calculations that you can easily and accurately do in your head.
When you take a practice test, notice how you use your calculator. Do you almost forget about it? If so, here are a few tips to train yourself to use the calculator to your advantage:
Look over the practice test and see whether you performed any long calculations that a calculator could have saved you time with.
Be sure you know how to use your calculator. If you aren’t familiar with it, you’re less likely to use it when appropriate. Check out the tips for calculator use later in this chapter.
Consider upgrading to a calculator that can handle things like square roots, powers, fractions, or other types of math that you need help with on the ACT. I provide a few ideas later in this section.
On the other hand, do you use your calculator for just about every problem? If so, consider these tips for backing off a bit:
Notice a few places (especially at the beginning of the test) where you may have done an easy calculation more quickly without your calculator.
If you overuse your calculator for simple arithmetic, such as for multiplying 6 × 8 or dividing 14 ÷ 2, you probably can benefit from beefing up your basic math skills. Check out Chapter 3 for a list of must-know math skills for the ACT.
If you second-guess yourself too much — that is, you do basic math correctly without a calculator but then doubt the answer and check it to be sure — you probably need to convince yourself that you’re on track. Start to notice how often you check simple math and find that your answer was already correct. If you’re almost always correct, you’re spending time using your calculator that could be used in a better way.
What kind of calculators can I use?
The calculators that are allowed on the ACT are divided into these categories:
Basic calculators: You can buy this type of calculator for less than $10 in almost any store that sells stationery. Basic calculators are perfect for balancing your checkbook or keeping a tally of your groceries, but they’re simply not adequate for the ACT. If you’re currently using this kind of a calculator, I strongly suggest that you consider upgrading to a scientific or graphing calculator.
Scientific calculators: This type of calculator typically costs more than $10, but you get a lot of functionality not found on a basic calculator. Depending on the model, a scientific calculator usually includes exponents, square roots, logarithms, trig functions, a reciprocal function, and lots of other stuff that may come in handy on the ACT.
Graphing calculators: A graphing calculator has all the bells and whistles of a scientific calculator, plus a larger screen for visual display of graphs and tables. If you’re thinking of upgrading to a graphing calculator from either of the other two types, consider this: The main advantage you gain is directly related to your proficiency with these visual elements. So plan to spend at least four or five hours practicing with your new toy, creating input-output tables for functions, graphing lines and parabolas, and exploring other related visual options. If you’re not convinced you’re really going to practice, you may as well save your money. Stick with a scientific calculator, which should serve you well enough.
What kind of calculators can’t I use?
The elders of the ACT weren’t born yesterday, which is how they got to be elders! So don’t try to pull a fast one on them. Even though you’re allowed to use a calculator on the test, you may not use a calculator that includes any of the following features:
Texting and Internet access: Sure, your iPhone (or iPad or laptop) may have a calculator function, but this function doesn’t make it a calculator. It also has lots of other fancy capabilities that aren’t allowed on the ACT. Obviously the elders don’t want you texting your genius Uncle Roy at MIT or looking up answers in Wikipedia if you get stuck on a question.
Talking or other weird noises: If your calculator makes noise and disturbs people, the monitors may separate you from it for the duration of the test. Of course, that separation wouldn’t be good for your test score.
Electrical access: I can’t guarantee that your testing site will have a place to plug in a calculator that requires power. Even though you may get lucky, your best bet is to bring a battery-powered calculator (along with a fresh set of batteries).
The point here is that the ACT elders are traditionalists. So if you stray much beyond the old-fashioned scientific and graphing calculators, you may run into problems.
How do I use my calculator?
After you purchase your calculator, don’t let it sit in its impenetrable plastic packaging until the night before the ACT. Use it for at least one practice test so you can get the feel of where the important keys are. At a minimum, make sure you know how to enter the following:
Negative numbers: Scientific calculators almost always have a special key to enter a negative number. This key usually is distinct from the minus sign used for subtraction. Find this key and test it by calculating a few things, such as –1 – 5 = –6
Parentheses: When you enter complicated calculations, you may need parentheses.
The most common example of this is when you enter a fraction such as . Enter
this fraction as (2 + 7)/(4 – 1) and make sure you get the answer 3.
Pi (π): On most scientific calculators, the π function is simple to use, requiring only one or two key strokes. Locate it and calculate 10π — the answer should be about 31.4.
Square roots: On many popular scientific calculators, the square root function doesn’t have its own key. Instead, it’s often the 2nd function on the x2 key used for squaring a number. This manipulation isn’t complicated, but you want to know it cold before you take your ACT. Make sure you can calculate some square roots, such as .
Trig functions: Truthfully, you may never need to use the sin, cos, or tan keys. Even so, be prepared. Locate them and calculate cos 0 = 1.
Also very important: Scientific calculators accept trig inputs in either degrees or radians, depending on which mode you choose. (Here’s a quick test: Enter sin π and see what you get. If the answer is 0, you’re in radian mode; if it’s something weird, you’re in degree mode.) Both modes work equally well, so decide whether you like working with degrees or radians best and make sure your calculator is set for this mode.
If you have a fancy graphing calculator, here are some of the useful features that are worth checking out:
Input-output tables: This feature allows you to enter a function such as y = 2x – 3, and then the calculator builds a table showing the resulting y-values (given x = 1, x = 2, and so on).
Solving equations: You won’t want to miss this time-saving feature. Be sure to enter equations carefully, using parentheses as needed, especially for complicated fractions.
Graphs: This feature allows you to enter a function and view the resulting graph. More advanced features allow you to solve equations — including quadratic equations — using a graph.
For more tips on using a graphing calculator, check out TI-83 Plus Graphing Calculator For Dummies (Wiley) and TI-89 Graphing Calculator For Dummies (Wiley), which are both written by C. C. Edwards. Finally — and I know that in the fun department, this ranks somewhere between comparison shopping for snow tires and cleaning the lint filter on your dryer — you can learn a lot about your calculator from that little manual that comes along with it. Don’t just toss it aside!