College Prep

Conquering Trigonometry on the ACT

vivian kerr September 2, 2010

Trig is some of hardest math on the ACT (it isn’t tested at all on the SAT) but luckily the Trig tested on the ACT requires the use of only a very special formulas! Some of these you may have seen before while some may be entirely new, but you’ll need to memorize all of them to adequately prepare for the test.

Sine = Opposite / Hypotenuse

Cosine = Adjacent / Hypotenuse

Tangent = Opposite / Adjacent

These are the three most basic trig identities. In plain English, they mean that if you are looking for the “sine” of a certain angle for example, you would divide the length of side opposite that angle by the length of the hypotenuse of the triangle. It’s important to remember that the “opposite” and “adjacent” sides change depending on which angle you are using, so always think of it from the point of view of the angle. The easiest way to remember these basic identities is the acronym SOHCAHTOA.

You will definitely encounter questions that require you to use SOHCAHTOA, and you may encounter questions that ask about reciprocal trig identities. Each of the three basic trig identities has a corresponding reciprocal trig identity:

Cosecant = Hypotenuse / Opposite

Secant = Hypotenuse / Adjacent

Cotangent = Adjacent / Opposite

Notice how Sine and Cosecant are the same except the numerator and denominator is flipped. That’s what we mean by reciprocal. It’s easy to remember that “tangent” and “cotangent” are reciprocals since they sound so much alike, but how what about the other two? I once had a math teacher who used, “Co-co no go” as a mnemonic device to help my high school class remember. What he meant was that your brain may think that “cosine” and “cosecant” are reciprocals since they both begin with the prefix “co-“ but that isn’t true. “Sine” goes with “cosecant” and “cosine” goes with “secant.”

Finally, there are two more trig equations that appear from time to time in ACT questions:

Sin²θ + Cos²θ = 1

The second equation is referred to as the law of sines (where a, b, and c are the sides of the triangle and A, B and C are the opposite angles).

Let’s try a few practice problems!

For right triangle XYZ, what is cos X?

A. x/y

B. z/y

C. x/z

D. y/x

E. y/z

We remember from SOHCAHTOA that cosine = adjacent / hypotenuse. From the point of view of angle X, y is the adjacent side and z is the hypotenuse. Therefore the answer must be E.

If cos theta = ⁴/₅ and ³^π/₂ < theta < 2π, then sin theta =?

A -³/₄

B -³/₅

C ³/₅

D ⁴/₅

E ⁵/₄

For this question we have two ways to solve: draw and label the triangle, or use the formula Sin²θ + Cos²θ = 1. For both, we first need to review our understanding of radians.

A radian is simply another way of measuring an angle. We are used to measuring and expressing angles in degrees. Some harder problems on the ACT will use radians instead of degrees.

There are 2π radians in one circle. Each point on a circle corresponds to a certain number of radians.

By telling us that angle theta is between 3π/2 and 2π, we know that the angle must be in the 4^th quadrant of the circle.

We know cosine = adjacent / hypotenuse, so we can label those two sides 4 and 5. Since we’ve been studying our Pythagorean triplets we know that the third side must be 3! But because it’s in the 4^th quadrant, we can see that it will be a -3.

Sin theta = opposite / hypotenuse = -3/5. The correct answer is B.

If we had used the equation Sin²θ + Cos²θ = 1, we would have squared the cosine 4/5 to become cosine 16/25 and then solved for sine. It would still have required us to know, however, that the 3 is negative since it’s in the 4th quadrant.

Using New Function Definitions

vivian kerr August 31, 2010

There are several different types of functions tested on the SAT: linear, quadratic, as well as function questions which ask about domain and range (the ones that look like f(x)). Today we’re going to look at symbol functions. In symbol functions, the SAT test makers choose a new symbol, something that you probably haven’t seen before or at least not in a mathematical equation, and give it a new definition.

Many students get nervous when they see symbol functions because they think they should recognize the new symbol in the same way they do common math symbols like +, -, x and ÷. Remember that the question will tell you exactly how the new symbol functions. While these problems may seem intimidating at first, simple plugging in is usually all that is called for to get the right answer.

Example: Let z € y be defined by the equation z € y = z² – 1/y. What is the value of 3 € 1?

Here we plug in 3 for z and 1 for y to solve.

3² – 1/1

9 – 1

Now let’s take a look at a more challenging question you may have seen on Grockit!

Example: Let the function $x be defined as $x = (x + 3)(x – 3). If $a = a + 3, what is one possible value for a?

A 9

B 6

C 4

D 3

E 0

If you are a little confused about where to start, remember that you can always plug in the answer choices into the function to see which one will yield a + 3. Since the answer choices are listed numerically, let’s start with answer choice C. If our answer is too large, we will be able to eliminate A and B as well. If it’s too small, we’ll eliminate D and E.

C $4 = (4 + 3)(4 – 3)

$4 = (7)(1) = 7

7 does equal 4 + 3, so C must be the correct answer!

Look out for more opportunities to use the answer choices to your advantage on symbol function questions! Plugging in and working backwards is one SAT strategy that really pays off!

Angles, Parallel Lines, and Perpendicular Lines

jordan schonig August 27, 2010

Lines and angles are the bread and butter of ACT geometry; though you might have learned your triangles and squares first, all polygons are essentially made up lines and angles. Here are the basics of lines and angles so you can nail those basic geometry questions.

Angles

An angle is formed by the union of two lines that share an endpoint (the vertex of an angle). The angle measurement corresponds to how far you have to rotate one of the lines to reach the other line.

Angles are measured in degrees, symbolized by the symbol º. No, that’s not an exponent of 0, but it looks pretty close. A complete rotation has 360 degrees, so it makes sense that a circle has 360 degrees, and the four angles produced by two intersecting lines (seen below) add up to 360 degrees.

While a full revolution is 360 degrees, a half revolution (a.k.a. a straight angle) has 180 degrees, and a quarter revolution (a.k.a. right angle) has 90 degrees.

Obtuse and Acute

You may remember that “acute” angles are less than 90 degrees while obtuse angles are more than 90 degrees but less than 180.

Complementary and Supplementary

The terms complementary and supplementary refer to special pairs of angles. Complementary angles add up to 90 degrees ad supplementary angles add up to 180 degrees.

Vertical Angles

When two lines intersect, we have two pairs of equal angles that are opposite each other.

In the diagram, angles 1 and 3 are equal and angles 4 and 2 are equal.

Parallel Lines

Lines that never intersect are called parallel lines. You may see this symbolized on the test as | |. Think of train tracks as parallel lines–they always run along each other and never converge.

Traversals

You will almost always run into at least problem that presents two parallel lines intersected by a third straight line known as a traversal. When this happens, eight angles are formed with special relationships to each other. Essentially, you can figure out all eight angles when given only one angle.

In the diagram above, angles A, D, E, and H are equal to each other while angles B, C, F and G and equal to each other. The sum of any two adjacent angles, like A and B or F and H, is always 180 degrees (since they are supplementary). For example, if angle A was 110 degrees, and I asked you to find the rest of the angles, you would immediately know that D, E, and H are 110 degrees while B, C, F, and G each has 70 degrees (180 – 110= 70).

There are special names for these related angles in the diagram. In the diagram, angle pairs like A and H are alternate exterior angles, angle pairs like C and F are alternate interior angles, and angle pairs like A and E are corresponding angles.

Perpendicular Lines

Two intersecting lines that form 90 degrees (a.k.a. a right angle) are called perpendicular lines. Simply put, when two lines form a cross or a “T,” they are perpendicular.

Angles and lines are used in diagrams throughout the test. Knowing these basics will help you immensely with even the most complicated geometric diagrams.

Sentence Completions

jordan schonig August 25, 2010

In one-blank sentence completion problems, we are presented with a sentence that has a missing word. It is our job to figure out the best word for that sentence. If you’ve never seen one of these problems before, you might be thinking “How should I know what word to use? I didn’t write the sentence.” Each sentence completion problem, though, offers clues to help you figure out the answer, and each question is 100% answerable.

If I had a rough idea for a sentence completion problem, and it looked like this “The student felt _____,” you might come up with some conflicting possibilities for fitting words; that’s because this sentence lacks clues. What if I changed it “After receiving an ‘A’ on the test, the student felt _____.” Now, we can expect that the fitting word is positive, perhaps “happy,” “relieved,” or “proud.” If I said “After putting in hours of valuable study time, the student felt ____,” you might expect the answer to be something like “prepared.” And, if I only slightly alter that sentence to “Although he put in hours of valuable study time, the student still felt ____,” you might conclude that now the answer is something like “unprepared.” This is how sentence completions work: clues in the sentence always tell us what kind of word fits the blank.

Let’s start with an example:

Building a new space shuttle was expected to take several years; thus, officials were surprised when the Plutonic Lander was built with ——–.

A.honor

B.caution

C.trepidation

D.celerity

E.deliberation

Here’s how to approach the one-blank sentence completion, step by step.

1. Identify Clue Words: Clue words are those words in the sentence that will help you predict what kind of word best fits in the blank. Clue words help you figure out what logical direction the sentence is going.

Words that signal a continuing thought include: and, also, consequently, as a result, thus, hence, so, for example.

Words that signal a reversal in thought include: but, yet, although, on the other hand, in contrast, however, nevertheless.

In addition to these words that show logical direction, we must also identify those key words that help us figure out what exactly is going on in a sentence. Here is the question reproduced with the clue words in bold:

Building a new space shuttle was expected to take several years; thus, officials were surprised when the Plutonic Lander was built with ——–.

Our first main idea is that the space shuttle regularly takes several years to be built; evidently, it is a long, involved process. The word thus tells us that the second clause agrees with the assertions made in the first clause. The most important word in the sentence is surprised; this word tells us that something about the construction of the Plutonic Lander was unexpected. We can infer that, if officials were surprised with its construction, then it must have been built quickly, i.e. much shorter than several years.

2. Predict: After you identify the logical flow of the sentence, you can predict what kind of word will go in the blank. We’ve figured out that this lander was built much more quickly than were its predecessors. So, we can say that this lander was “built with quickness or rapidity.”

3. Eliminate: Now that we have a prediction–quickness–we can eliminate those words we know do not fit our prediction. Here is the list of answer choices:

A.honor: Honor has nothing to do with quickness. It’s gone.

B.caution: If anything, more caution will result in a longer construction time. Get rid of it.

C.trepidation: Trepidation means fear. That’s not quickness. Say bye-bye.

D.celerity: Celerity actually means quickness. Sounds good to me.

E.deliberation: Deliberation has a few definitions. It can mean thoughtfulness, careful consideration, thorough discussion, or unhurriedness. If anything, these definitions connote–and in one case, denote–a slow carefulness. It’s not our word.

The answer is, in fact, D. Provided that you know the definitions of all the answer choices, one-blank sentence completions are pretty simple. Of course, this will not always be the case. What if you didn’t know the definitions of trepidation and celerity, arguably the most difficult words in the answer choices? The good news is that, by predicting your answer, you eliminated A, B and E. At that point, you have a fifty percent chance of answering correctly. Another good technique is to try to guess the positive or negative charge of a word. More often than not, students can correctly guess the charge of a word even when they are unfamiliar with it. Maybe it’s just me, but “trepidation” just sounds negative to me, and it certainly is. That negative charge would rule it out, and we’d be left with “celerity,” a fancy word for quickness.

Geometry part 3

jake becker August 23, 2010

One of the key things to remember with circles is that once you know one piece of information, you know everything about the circle itself. Additional angles and lengths inside are not always so simple, but it is possible to convert circumferences to areas, to radii and diameters without intermediate steps.

Arc Lengths and Sector Areas

Arc Lengths (portions of perimeters) and Sectors (pie slices) seem more complicated than they really are. Both relate directly to the internal angle at the circle’s center, represented by θ in this diagram. Here are the equations:

Sector Area = A = θ/360 * πr²

Arc Length = L = θ/360 * 2πr

θ in these formulas refer to the angle measure in degrees, not radians. If you’re using your calculator, make sure that your calculator is in the right mode.

Literally, all we are doing is finding the fraction of the circle and applying it to either the area or circumference formula, respectively. The pie is always a certain fraction of the circle, meaning the area of the pie and the arc length of the pie is also the same fraction of the circle’s area and circumference respectively.

In the above diagram, a square is inscribed in a circle, which is inscribed in another square. If the area of the larger square is 64 and the area of the green region is 4, which is larger, the green region or the yellow region?

The information we have is:

A(large square) = 64
s(large square) = 8
diameter = 8
r = 4

First, recognize that the figure is symmetrical. So while we may not explicitly be given an angle to find the sector area (not drawn) in which the yellow region resides, we do know its measure. The diagonals of a square intersect at a right angle, so we can deduce that the sector including the yellow region is 1/4 of the area of the circle. Since we know the radius…

A(large sector) = 1/4 * 16π = 4π

To find the yellow region itself, we must subtract the imaginary (not drawn) triangle from 4π. (Note that this imaginary triangle will be twice the green triangle.)

A(imaginary triangle) = 1/2 * r * r = 1/2 * 4 * 4 = 8

A(Yellow Region) = 4π – 8

The area of the green region can be found in two ways. Either we can see that it’s simply one-half of the 8 we just found, OR we can find both sides of the green triangle with the common 45-45-90 1:1:√2 formula. With a hypotenuse of 4, we derive 2√2 for each side, which yields an area of 4.

One More Problem

A circular loop of wire is attached to the two straight wires of a dipole antenna at points X and Y. Point Z is the base of the antenna where the two straight wires meet. The perimeter of the circular loop of wire is 3 feet. The center of the circular loop of wire is 2 feet from point Z. How far is point X from the center of the circular loop of wire?

The key point to remember here is that all radii are equal. That is, the distance from the center of the circle (C, not drawn) to both X and Y is the same.

Before we mentioned any if you have any one piece of information about the circle, then you have it all. This is perfect example. If we know the circumference (perimeter) of the wire, we know the radius, regardless of the presence of Z in the diagram. The information about Z is simply thrown in there to confuse you. The distance to X is actually the radius which is circumference/pi = 3/pi.

Here are just a couple examples, but keep working hard. Givens in geometry provide a series of information, not just what’s stated. Make sure to keep that in mind for Data Sufficiency questions.

Good luck!

Geometry Series part 2

jake becker August 20, 2010

To start off, let’s quickly review the essentials. These are formulas/concepts you must know:

a² + b² = c², but only when a right triangle. If you don’t know it’s a right triangle, Pythagorean theorem does not apply!
Common special right triangles include 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples.)
45-45-90 triangles are ALWAYS in the ratio 1:1:√2
30-60-90 triangle are ALWAYS in the ratio 1:√3:2
Angles and opposite sides are in the same relative size order, but are NOT proportional.

Let’s continue with a standard diagram in which we have an equilateral triangle inscribed in a circle, which is inscribed in a square.

The center point of all three figures (triangle, circle, square) are all the same, but this is ONLY true if the triangle is equilateral (all the sides are the same length). Therefore, if given ANY piece of information about the circle, square or triangle, we can derive the rest. We draw a perpendicular line from the center to the side of the triangle.

Note that the hypotenuses of the smaller triangles are equal to the radius of the circle. We also know that the smaller triangles are each 30-60-90 because you are taking the 120-degree internal angle from the circle’s center and cutting it in two. Here are your basic conversions:

r = ½d = ½s, where s is the side of the square.
The sides of the 30-60-90 triangles become ½r : (r√3)/2 : r respectively
The side of the equilateral triangle becomes 2*(r√3)/2 = r√3

If given the area of the square, we should be able to derive essentially any other information.

Area of an Equilateral Triangle

The area of an equilateral triangle equals (s²√3)/4. Memorize this. It will save you the time of drawing a 30-60-90 triangle, solving for the base, finding the height, multiplying and dividing by 2. That was long to write, imagine how long it takes to do!

If the area of the square = 64 and we needed to find the area of the triangle, we just use the conversions above:

d = 8
r = 4
side of triangle = 4√3

Area of triangle = [(4√3)²√3]/4 = 16*3*√3 / 4 = 4*3*√3 = 12√3

Angle Relationships

Another important rule is that the interior angle created from of two radii extending to the outside of the circle is exactly twice the measure of any angle on the circle extending to those same points. In the image above, 2b = a.

Let’s take a look at this practice question:

In the figure above, a circle is inscribed in a square. If the perimeter of the square is 32 and x=35, what is the area of the shaded region?

We can determine r by knowing that the length of the square’s side. If s = 32/4 = 8, then d = 8 and r = 4. But we still need to know the interior angle. Since x=35, we know that the interior angle of the shaded region is 2(35) = 70. So A(shaded) = (2x/360) * πr² = 70/360 * 16π = 28π/9

There are infinite variations of these concepts. Be flexible in your reasoning, and practice makes perfect!

Good luck!

Geometry Series Part 1: Circles inscribed in Squares

jake becker August 19, 2010

In this series, we will cover many types of geometric scenarios encountered on the SAT and ACT. A basic knowledge of simple formulas (area, perimeter, etc) is essential, but there are numerous shortcuts to geometry questions that will save you time. Today, we’ll explore circles inscribed squares.

Some things to remember

The center of the square is the same point as the center of the circle
Draw lines! Depending on what the question asks for, draw in lines that create simple shapes. (Squares can be turned into triangles, for example).
Shared angles will normally not be explicitly stated, unless necessary.
Trust the pictures, but not too much. Inferences must be drawn from fact. Just because it looks like 90-degrees doesn’t mean it is! (Many of these common inferences will be detailed in this series)
Lengths cannot be negative.

For circles:

d=2r and all lines from the center to the exterior equal r.
C = 2πr = πd
A = πr²
Tangent lines create right angles with the radius that meets that tangent.
If you know r, you know everything about the circle!
Use π = 22/7 with caution. Remember 22/7 > π.
The diagonal equals (length of a side * √2), since it creates 45-degree angles.
The intersection of the diagonals creates a right angle.
When a circle is inscribed inside a square, the side equals the diameter. (Inscribed means that the circle fits perfectly inside the square with its edges touching the side of the square like in the diagram)

For squares:

Usually, you will be provided with one bit of information that tells you a whole lot, if not everything. If given the length of the side of the square in the above image, we can actually find the length of the hypotenuse of the internal triangle (s = d = 2r, so the hypotenuse = (s√2)/2).

Shaded Areas

Find the large area and subtract the small area from it. When dealing with circles along with other figures, eliminate answer choices that ONLY have π’s in them or don’t have any at all. Typically, your answer will look like x + yπ because it involves a circular shape and a square/rectangle.

The picture above depicts a circle perfectly inscribed in a square. What is the area of the shaded region?

By knowing the area of the large square we also know the lengths of its sides. (Note that 64 is a perfect square, which should be a clue.) If the side is 8, then so is the diameter, which means the radius equals 4. In the image, we can see that the “larger figure” is the top-right square bordered by two radii and the outer border. How do we know that it’s a square? Two pieces of information: All sides are equal to 4, and the radius meets the large square at a right angle because it is a tangent. In this instance, the area of the smaller square equals 16. Since the interior angle is 90-degrees (360/4), the area of the sector of the circle can be represented by A = πr²/4. So that A = 16π/4 = 4π.

A(shaded) = A(small square) – A(sector) = 16 – 4π

SAT Reading – Tone and Style

vivian kerr August 16, 2010

You’ve probably seen an SAT question that looked like this:
The author’s tone in the passage can best be described as:

Questions that ask about tone and style may not be as common as Detail or Inference questions, but they often come up on the SAT. The first step to tackle them, is to make sure you did your note-taking on your first reading of the passage (you can check out an article I wrote on note-taking for passages here: How To Deal with One Long Passage).

In that article, I discussed the importance of paying attention to the author’s point of view and to note the places in the passage where the author reveals his/her opinion. After all that work, now is the time for the payoff!

Unlike detail questions, there are no line numbers to help you find the answer for tone/style questions. Only by paying attention to the author’s voice and style as you read will you be able to get these questions right.

Now let’s talk strategy. What to do if you encounter a tone/style question:

1. Refer back to your passage notes. Ask yourself, what does the author like and what does he dislike? It’s important to note that while the author will have opinions, they may not be obvious. The passages are often scholarly and balanced in tone, so you must look carefully at the adjectives and adverbs (and the descriptive phrases) to find the places where the author reveals his opinion. Think of yourself like a detective looking for clues. They may be subtle, but they are definitely there.

2. Make a prediction. Don’t even think about reading those answer choices until you come up with your own prediction. If you’re tempted, cover up the choices with your hand. The SAT Reading section is testing your ability to think critically, and you must remember that the answer choices are not there to help you. Once you read them, you’ll never get them out of your head. Use the descriptive words of the passage as your prediction, or even a simple positive (+) or negative (-) sign.

3. Eliminate answers that don’t match your prediction. Trust that you’ve done your homework and that you know what the answer should be. Got more than one answer left after eliminating? Here is when you get into the nitty-gritty of SAT Passage-based Reading. You may encounter two words with very similar meanings, for example “dislike” and “despise.” How do they differ? Is one of them overly emotional, informal, or extreme? Unless it’s truly appropriate to the passage, go with the more “middle-of-the-road” word. In this case it would be “dislike.” The tone of most SAT passages is academic and technical, not emotional.

Tackling Word Problems

vivian kerr August 12, 2010

Even the strongest Math student can be troubled by the occasional tough word problem. It’s important not to rush when you read these types of questions. Make sure to read methodically and be confident you understand each part of the problem before you move on. Many students find themselves setting up equations and solving algebraically before they’ve even understood what the question is really asking!

Make sure to circle the question at the end of the word problem. You typically want to define whatever the question is asking for as x. It is also good practice to write down what you defined x to be. Sometimes, it might be easier to let something else be x. In those situations, always write down what the final answer you are looking for is, to remind yourself not to stop at solving for x. For example, if x is the length of a side of a square and the question asks for the perimeter, the final answer would be given by 4x.

One of the ways you can quickly sharpen your word problem skills is to practice translating English into Math. Certain words and phrases commonly occur in word problems and knowing the Math processes they represent will help you gain confidence. Here are a few examples:

Addition: increased by, more than, combined, together, total of, sum, added to

Subtraction: decreased by, minus, less, difference between/of, less than, fewer than

Multiplication: of, times, multiplied by, product of, increased/decreased by a factor of

Division: per, out of, ratio of, quotient of, percent (divide by 100)

Equals: is, are, was, were, will be, gives, yields, sold for

Make sure that you name variables after what they stand for so you can easily remember them. For example, if a problem says “Julie’s age is four years less than Sarah’s,” it is easier to write it as “J = S – 4” than as “x = y – 4”. You can always choose variables for unknown quantities.

Remember that with Subtraction and Division the order matters so read carefully so you know what is being subtracted or divided from what! For example, a common mistake to make with the above example would be to translate it as “J = 4 – S”. The order the words appear in English in a sentence is not necessarily the order in which they should appear in Math.

If translation and setting up algebraic equations is not your strong suit, you may also want to consider using one of two Math strategies: Picking Numbers or Backsolving.

Picking Numbers is a great strategy when there are variables in the question stem (like, a, x, n, etc.) and in the answer choices. Instead of setting up an equation, you simply pick nice, easy low numbers for the variables (such as 2, 3, 4, etc.) and plug them in, finding a solution. Once you have a solution, you plug the SAME numbers into the answer choices. Whatever matches your answer choice must be correct!
Backsolving is a great strategy when there are numbers in the answer choices. Instead of picking numbers for variables, you simply work backwards by plugging in each answer choice as if it’s correct. It’s a little bit like checking your work as you go! When backsolving, make sure to start with answer choice (C), typically the middle value, because the answer choices are usually listed in ascending order. That way, if you find the number is too small, you can then try (D) or (E). If (C) is too big, then try (A) or (B).

When practicing these pesky word problems, look for opportunities to Pick Numbers and Backsolve. It may seem awkward at first, but having multiple ways of solving can help on Test Day when you get stuck.

To get started on some practice, check out Grockit’s live games now!

Identifying ACT Verb Errors

vivian kerr August 9, 2010

The most common way verbs are tested on the ACT is in subject-verb agreement; however, sometimes the questions in the English section will contain other verb errors. While it’s important to make sure that verbs always agree with their subject in number, it is also important to check to make sure that the verb tense makes logical sense in the context of the sentence.

Verb tense errors have to do with when the sentence takes place. If the action of the sentence is happening in the present, a verb in present tense is required. If the sentence describes something that has not yet taken place, it requires the future tense. There are six verb tenses you should be familiar with for the ACT. These three you probably already know:

Present Tense: I clean.

Past Tense: I cleaned.

Future Tense: I will clean.

The other three are part of what is called the perfect tense. To express this tense, we write the word “have” or “had” before the conjugated verb. The conjugated verb is called the “past participle” in this tense.

Present Perfect: I have cleaned.

The present perfect is used to describe either something that started in the past but is still going on in the present or something that occurred in the past at some unspecified time.

Past Perfect: I had cleaned.

The past perfect tense is used in a sentence that already contains a verb in the past tense. If the sentence describes something that took place before the verb in past tense you would use the past perfect tense. Example: I cleaned the bathroom after I had cleaned the living room. The past perfect tense makes it clear that the living room was cleaned first.

Future Perfect: I will have cleaned.

The future perfect tense is used to describe something that will occur after another future event. However, unlike past perfect, many sentences using future perfect will not require the future tense as well. Example: At the end of my diet, I will have lost ten pounds. A phrase like “at the end of my diet” implies the future tense so a second verb is not required.

As you notice with the above examples, to go from present tense to past tense usually involves simply adding –ed to the verb. We call these “regular” verbs. For the perfect tenses, simply adding “have” or “had” to the past participle is all that is required.

Some verbs are “irregular”, meaning they have very different forms in the present and past tense and require a unique past participle to form the perfect tenses. These are especially tough for non-native speakers to remember. Here are some of the most common “irregular” verbs- you can find more by searching online. Make sure to write down any you don’t know and study them until you can recognize the irregular forms.

Verb: to be

Present: I am

Past: I was

Perfect Tenses: I have/had been

Verb: to begin

Present: I begin

Past: I began

Perfect Tenses: I have/had begun

Verb: to drive

Present: I drive

Past: I drove

Perfect Tenses: I have/had driven

Verb: to speak

Present: I speak

Past: I spoke

Perfect Tenses: I have/had spoken

Verb: to grow

Present: I grow

Past: I grew

Perfect Tenses: I have/had grown

Don’t worry if you can’t remember the names of the tenses themselves. The ACT won’t even use phrases like “past participle” or “perfect tense.” You’ll only need to know which tense is correct in context; as an English speaker, you already possess this skill! To spot these verb tense errors, remember to ask yourself: when does this sentence, or this part of the sentence, take place?