Grockit SAT ACT Prep

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Archive for August, 2010

Using New Function Definitions on the SAT

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.

Test your SAT math skills with this SAT multiple choice practice question!

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Angles, Parallel Lines, and Perpendicular Lines

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.

Geometry part 3

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.

Good luck!

Geometry Series part 2

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

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

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)

Test your SAT math skills with this SAT multiple choice practice question.

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SAT Reading – Tone and Style

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 on the ACT

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.

Test your skills with this ACT math word problem practice questionRead more »

Identifying ACT Verb Errors

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?

Inferences

On the SAT Reading, “inference” questions ask you to make a reasoned judgment about the passage that goes beyond the material on the page. Authors will often imply information but not state it directly; inference questions test your ability to spot the author’s implications without straying too far from the text.

We make inferences all the time during everyday conversations, but our casual inferences are often riddled with unsubstantiated leaps in logic. For example, if I told you that my lawn was wet, you might infer that it rained last night. This seems like a fair inference, but there are other logical reasons for my lawn being wet–I could have watered the lawn or perhaps a neighbor was washing his car and sprinkled my lawn. More information surrounding my original sentence might lead me to conclude that it rained last night, but from the simple statement “my lawn is wet,” I can only infer that my lawn is not dry.

The SAT inference questions will often offer you some tempting answers that infer too much information. Beware of these choices–they are the most common pitfalls for students.

What do inference questions look like?

Before diving into an example, let’s make sure you know how to spot an inference question. Most inference questions are characterized by the words suggest, infer, or imply. They might look something like this:

What might be inferred by the final paragraph?

The author implies that the frontiersmen quickly packed because…

By revealing the results of the scientific study, the author suggests…

Example of an inference question

Let’s look at an inference question from Grockit.

This passage concerns the speaker’s feelings about buying art and how they reflect upon her notions of adulthood. In the first paragraph, she describes how she chose not to buy art in her teens and twenties because each artwork would tend to ossify her then mutable conception of self. Here is the paragraph that follows:

As I grew older, though, I began to set down roots, as
(20)    people generally do. Moving every year or so became a
hassle, not an adventure, and meeting new people was a
diversion but not a necessity, since I had collected friends
and loved ones along the road to being a grown-up. I
became more willing to settle on a particular version of
(25)    myself: not a transitory, tacked-up-above-the-futon
version that could be bought for under $5.00 and
disposed of without regret, but a framed and matted,
carefully considered version that might not be
permanent, but was certainly going to arrive in the
(30)    expectation of a long tenure.

The author suggests that the new “version” of herself mentioned in line 24 will be:

  1. More dedicated to cultivating a collection of art
  2. More financially stable than she was in her youth
  3. Sticking around for a while
  4. Spending more time with friends
  5. Focused on her career

Our first step, as always, is to locate the line and read a little before and after it. According to those lines, the speaker has now grown older and more secure in her stable identity. She has become more willing to “settle” on a version of herself that is “not…transitory,” a version that, while not totally permanent, will expect a “long tenure.”   Using the context clues, we can logically infer that if her new version of herself is not “transitory” or temporary, then it must be more permanent or at least lasting a long time. Notice that C is the best articulation of the idea of permanence. Let’s look at some other choices to see why they’re incorrect.

Answer choice “A” diverts us from the important matter at hand; in the passage, the speaker’s decision to purchase art stands for her increasingly stable sense of self. To say that this new version of self is only more dedicated to collecting art is entirely missing the point; in this case, the wrong answer choice is overly literal.. “B” is the kind of inference that lacks any referent in the text. Although someone might infer that the speaker becomes more financially stable with age, such an inference lacks a textual basis–it is purely speculative. “D” simply locates a minor detail in the passage and overemphasizes its significance. “E,” like “B,” preys upon our biased expectations about adulthood–though we may associate adulthood with career ambitions, there’s no textual evidence for such an inference.

To avoid these trap answers and nail the right one, remember that inferences are objective and logical deductions based in fact. Leave your assumptions and biases at the door, and stick to what’s on the page!

Sentence Completions: Cause and Effect

“Cause and effect” sentence completions can get a bad rap because students often find them less obvious than blanks that give definitions or provide an outright contrast. While it’s true that it can sometimes be tough to predict an answer for the blank in “cause and effect” SCs, they provide a great opportunity to use logic to help you eliminate answer choices and get the correct answer!

If you’ve been studying for the SAT or ACT for any length of time, you already know all about how identifying key words and making predictions can help you get the correct answer on sentence completions. Let’s try an easier “cause and effect” question to warm up!

Now that the board members have reached consensus on several issues, their meetings are not as ——– as they were.

A  cooperative

B  instructive

C  contentious

D  unfounded

E  legitimate

Here we can identify the keyword “consensus” as something positive. If they are “now” something positive, then in the past, they were probably something negative. Let’s predict something easy like “bad” for the blank. The only two words that we can’t eliminate are C and D. Between “contentious” and “unfounded”, “contentious” is the more negative word. The “cause and effect” idea in this one had to do with the change over time. You’ll often see time play a factor in sentence completions. Try to find out what changed and if it was for the better or worse.

Try this SAT sentence completion practice question and find out why Grockit is the ideal environment for improving your SAT score.

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