ACT Math

The Power of Elimination - Mastering Multiple Choice

It’s one of the oldest tricks in the book -- but is a very powerful, underrated strategy!

“So, what can you eliminate?”

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My students will tell you that I must say this phrase at least 100 times each lesson, but there is a method to my madness. In almost every question on the ACT Math section, there is usually one answer that is completely out of left field. As soon as you see that answer, crossing it off immediately will ensure that you won’t accidentally choose it if it comes down to guessing. It will also help lead your brain to the right answer by narrowing your focus to the other answer choices. In the case of “plug and chug” questions, having to plug in 3 numbers versus 5 can save you precious seconds in the race against the clock.

Eliminating will not only help you increase your chances of selecting the correct answer, but it will exercise your intuition and confidence. Knowing the wrong answer can be just as useful as knowing the right one!

6 Math Formulas to Know Before Taking the ACT

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For many students, the ACT math section is the final frontier on the journey to their dream score. While the section can sometimes feel daunting — those word problems can go on forever — there’s some easy prep that can save you some time and earn you some major points. By far, the one thing that makes the biggest difference for my students is getting familiar with the most common formulas. Because the ACT math section is relatively short (just about a minute per question) and you don’t get a formula sheet, knowing these formulas can be the difference between feeling like a champ after your test and leaving the test center scratching your head. Here are my top six formulas to know before the ACT:

1. Special Right Triangles

One of the first things I ask my students to memorize. For some people, the meaning of life is happiness, or success; for the ACT, it’s special right triangles. It feels like half of the geometry problems are really just triangle problems in disguise, so knowing the sets of side lengths (or angles) that always make perfect right triangles definitely comes in handy.


2. Area of a Trapezoid

This one might seem a bit random, but there’s always at least one trapezoid problem on the ACT, and it’s an easy way to guarantee yourself a point. It’s also one of the easiest to memorize, since it’s so close to the triangle area formula.

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Watch out here- sometimes you’ll be need to find the height, where the Pythagorean Theorem (or your knowledge of special right triangles) will be a big help.

3. Distance and Midpoint

Two very popular questions in coordinate geometry:

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...are easily solved when you have the equations for distance and midpoint between two points. (there’s also a nifty program, Points, that can do this for you- know the formulas, but don’t be afraid to take advantage of the technology!).

4. Slope of a Line

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Slope. Gradient. Rise over run. A slope by any other name works just as well- as long as you remember that your change in Y always goes above your change in X.

5. Slope-Intercept Form of a Line

Speaking of slopes, remembering how to find the slope-intercept form of a line is a must. While the ACT doesn’t play as many tricks as other, similar tests (see: SAT), one thing the test writers love to do is hand you an equation that looks like this:

...and ask you for the slope. Proceed with caution! Most students pull the coefficient off the X (in this case, that would give us C), but this only works when your line is in slope-intercept form:

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2x + 3y + 6 = 0

3y = -6 - 2x

y = -6 -2/3x

Here, it’s evident that the answer is actually D. My advice? Anytime you get an equation that looks like this, rearrange it so it’s in slope intercept. You’ll still be able to plug and chug if you need to, and you’ll save yourself one of those.

6. SOHCAHTOA

Not really a formula as much as it is a mnemonic device, but an essential one, especially on the second half of the test. Most of the right triangle trig questions on the test are pretty straightforward— just remember to double check which angle you’re using when you’re figuring out your opposite vs. adjacent sides.

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P.S. don’t forget- tan can be rewritten a fraction (sin/cos)!

These formulas are a great start for anyone starting their ACT math prep, or a good refresher for anyone looking to bulk up on some math knowledge mid-program. Learn these, and you’ll be flying through the math section in no time!

Using Eyeballing to Solve Equations

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When I began teaching ACT Math four years ago, I primarily emphasized teaching students the “mathematical” ways of solving questions — that is, solving questions in ways that their math teachers would be proud of. As a math major, I felt — and still do — that it’s important for students to understand the concepts underlying the math that they’re doing. While I still encourage a conceptual understanding, I’ve also learned that -- on standardized tests like the ACT and the SAT -- students benefit most from having multiple ways to solve any given question. In fact, these tests routinely reward creative solving-problem. To that end, one surprisingly powerful technique that test-takers can use on the ACT Math — one that typical math teachers would probably not approve of — is eyeballing.

Take a look at this math question, taken from a real ACT, as featured in The Real ACT book, test 2:

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This question can be solved in multiple ways:

The standard “math” way to solve it is to recognize that the two angles on the line are supplementary and so must add up to 180 degrees. We then set up the equation (4x + 6) + 2x = 180, and solve to get that x = 29. Since the question is asking for the measure of the smaller angle, which is 2x, we then double this to get (D) 58 degrees.

Another approach to the question is to approximate the measure of the smaller angle by simply eyeballing it: it looks to be slightly more than 45 degrees. We then go to the answer choices. The figures on the ACT Math are drawn roughly to scale, so what answer choices can we eliminate? Well, (A), (B), and (C) are all far too small. We can also eliminate (E) because we know that it’s possible to set up an equation to solve for the smaller angle.

Interestingly, for this particular question, eyeballing the figure to arrive at the answer is actually faster than solving the question algebraically. In addition, eyeballing avoids a common mistake students make when solving this question algebraically. That is, many students set up the equation and correctly solve for x, finding that x = 29. They forget, though, that the question is asking for the measure of the smaller angle (which is 2x), and they choose (C). (Note: Solving this question algebraically is still great as a primary strategy and can be done very quickly if you’re comfortable with the algebra.)

While eyeballing can be helpful, it should be thought of more as an extra tool rather than as a primary problem solving-strategy. The technique is only relevant for questions with figures, and, even on such questions, it often can’t be used by itself to narrow down to one answer. However, it 1) can be the most efficient way of solving certain questions, 2) will often allow you to eliminate at least two answers on many other questions if you need to make an educated guess, and 3) provides a way to double-check your work if you solve the question using a more standard math approach. For example, in the question above, if a student decides to solve the question algebraically, he or she can then quickly glance at the figure to see whether the answer makes sense given the scale.

As a final example, consider this question, again taken from a real ACT, as featured in The Real ACT book, test 4:

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This question appears among the last 10 questions of an ACT Math section. At this stage of the Math section, many students are pressed for time and/or are unsure of how to solve certain questions using standard math approaches. In situations like this, when a student needs to make an educated guess, eyeballing can come in handy.

The question asks for the height of the building. Based on the given length of the shadow, which is 24 yards, and given the scale, we know that the height of the building is slightly more than 24 yards. We can use this to eliminate (F), (G), and (H). In a situation when we might need to guess, (the correct answer is (K)), we’ve very simply and quickly increased our chances.

Let us know if you ever use the eyeballing strategy on the ACT!

The ACT Math Shortlist

The Math section of the ACT covers a wide range of topics, from basic arithmetic and pre-algebra to logarithms, imaginary numbers, and advanced Algebra II. Preparing for such an exam can seem like an overwhelming task, and it is easy to feel paralyzed by the sheer quantity of material covered in so many years of math courses. Fortunately, not all math topics are created equal, and the test tends to favor certain topics over others. Contained here is a shortlist of topics to start with for those of you who don’t quite know where to start!

Linear Equations

Linear functions are of paramount importance in mathematics and thus appear frequently on the ACT. The test will require you to find slopes of lines, given the coordinates of two points (remember: “rise ÷ run”), and to use these slopes to find the equations of the lines. Once you have the slope, you should use slope-intercept form if a y-intercept is provided, and point-slope form if the coordinates of a point on the line are provided.

 

Systems of Linear Equations

Linear functions also appear frequently on the ACT in systems of linear equations. These types of problems usually contain two equations with two variables each – often x and y, but not always – and have no exponents. The most common way to solve these equations is through the substitution method. Keep in mind that there are precisely three possible outcomes: one solution for each variable (i.e. the two lines intersect at exactly one point), zero solutions for each variable (i.e. the two lines are parallel), and infinite solutions for each variable (i.e. the two equations are actually one and the same line).

 

Quadratic Equations

Next come quadratic equations, which graph to parabolas. The first step to solving quadratics is to get everything on one side of the equation and zero on the other. Only once this has been done can the equation be solved! The three primary methods of solving are then:

  1. Factoring the expression and setting each factor equal to zero
  2. Graphing the function on your calculator and using the ZERO function under the CALC menu
  3. Plugging in the constants of the function (a, b, and c) into the quadratic formula.

Triangles

Triangles appear on the ACT in a wide variety of contexts, but there are a number of basic things you must always keep in mind. All triangles contain angles that sum up to 180°, and the area of any triangle equals base x height ÷ two. For right triangles, in particular, the Pythagorean Theorem can be used to solve for the length of a third side if given the lengths of the other two. Right-triangle trigonometry will make an appearance as well, so at some point, you will likely be asked to set up (and possibly solve) a trigonometric equation using SOH-CAH-TOA.

 

Area, Perimeter, and Volume

In addition to the area of a triangle, you should also have memorized the equations for the areas of a circle, rectangle, and trapezoid. For circles, you should also know how to calculate the perimeter, known as the circumference, of the circle. 3-D solids appear less frequently than 2-D shapes, but you should also be prepared to calculate the volumes of a rectangular prism and cylinder.

 

This is by no means an exhaustive list of topics on the ACT Math, but it should serve as a starting point for any student preparing for the exam. If you can master these topics, you should be able to answer a majority of the first 30-40 questions of the test, and you’ll be off to a great start. Good luck!

How To Handle Overwhelming ACT Math Questions

Some ACT math questions are straightforward. Some are complex. But there is a third category of math question, one that I call “overwhelming.” These questions might not be all that hard, per se, but it is mighty difficult to figure out what exactly you’re supposed to do with them. These questions are frequently word problems that throw a ton of information at you all at once, and it’s not immediately clear what sequence of steps will lead you to the answer. But don’t worry -- here’s how to handle those overwhelming questions.

 

Process the given information “chunk by chunk” 


If an overwhelming question hurls four sentences of information at you in a row, don’t freak out. Stop after each “chunk” of given information, and process it fully. For instance, if a geometry question tells you there’s a pair of parallel lines, STOP before reading on and mark that information on the figure. Then, take it one step further - if there is a pair of parallel lines, what are the alternate interior angles? Mark them. How about the corresponding angles? Mark those, too. Only once you’ve fully processed a “chunk” of information should you read on to the next.

Follow the invisible path


Once you’ve processed all the given information, you may still be unclear on how exactly to get to an answer. But even though you may not be able to see every piece of the puzzle, there is often an invisible path through the question. If you take a first step, the second becomes clear. And once you’ve taken the second, the third falls into place. And then you’re off to the races! The key is to see that these overwhelming ACT questions often guide you to take a particular first step, and if you’ve correctly processed all the given information, that first step is usually unveiled to you. So trust in the test and take that first step, and watch the path unfold in front of you.

 

By following these simple steps, you can wrangle overwhelming questions and make them much more manageable. All of a sudden, a whole host of questions that seem overwhelming on a first pass become, well, just whelming. Good luck!
 

How to Stay Sharp Over the Summer

As the school year winds down, students are understandably looking ahead eagerly to the summer break. Regardless of where you are in your high school career, keeping your mind sharp over the summer is essential.

For freshmen, your sophomore year will offer a more challenging course load, sometimes featuring your first AP classes. It is crucial to build on the success you established your first year or turn the page and start anew if you struggled.

For sophomores, junior year will be seen as the doorway to college acceptance since you will likely be taking ACTs and SATs for the first time, along with juggling your busiest course load of high school.

Juniors who have finished with the SAT or ACT still have the rigors of college applications and a challenging fall semester to look forward to, while those who have not finished with the tests will have to gear up again for the fall exams.

And even for the seniors who are already accepted into college, I would remind you that the level of comprehension necessary for college courses far surpasses that of high school classes.

With all this in mind, I recommend that students do their best to avoid the trap of summer complacency, which can make starting the new school year all that much more painful. You should all pursue intriguing, unique experiences over the summer, but there are simple steps you can take to keep your mind functioning at a high level.

For most students, the most important step is to read consistently at a high level. Some of you are ambitious enough readers to tackle full novels, in which case you should check out our summer reading list here.  For those of you who feel overwhelmed by the commitment of a novel, challenge yourself to read one article each day in a publication such as The Atlantic, Popular Science, Scientific American, The New York Times, or The Wall Street Journal. These publications are written at a level similar to or above that which you will usually find on the SAT or ACT.

For those of you trying to get ahead on your SAT or ACT prep or make a strong final push for the fall exams, I would strongly recommend a consistent review of the English rules and math equations, as well as steady practice with your past mistakes. Even 10 to 15 minutes of work each day can make a significant impact on your readiness at the start of the next school year.

So challenge yourself to stay sharp and keep yourself ahead of the curve this summer!

 

-Jamie K, Test Prep Advisor & Instructor