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Cheat Sheet / Updated 06-28-2022

The American College Testing exam (ACT) tests your knowledge of grammar, reading, science, and math. In addition, the ACT includes an optional writing test. Many colleges require or recommend and entrance exam, such as the ACT, as a component of your application for admission.

View Cheat SheetCheat Sheet / Updated 02-23-2022

Making sure you have a handle on algebra, geometry, and trigonometry is a solid start for success on the math section of the ACT. However, to boost your confidence — and your score — even higher, you should master some helpful test-taking strategies, as well as make sure you know how to translate word problems into equations and use sketches to figure out what a tricky-sounding question is really asking.

View Cheat SheetArticle / Updated 05-22-2020

Angle problems make up a big part of the ACT geometry test. Fortunately, understanding angles is easy when you memorize a few basic concepts. After all, you don’t have to do any proofs on the test. Finding an angle is usually a matter of simple addition or subtraction. Here are a few things you need to know about angles to succeed on the ACT: Angles that are greater than 0 but less than 90 degrees are called acute angles. Think of an acute angle as being a cute little angle. Angles that are equal to 90 degrees are called right angles. They’re formed by perpendicular lines and indicated by a box in the corner of the two intersecting lines. Don’t automatically assume that angles that look like right angles are right angles. Without calculating the degree of the angle, you can’t know for certain that an angle is a right angle unless one of the following is true: The problem directly tells you, “This is a right angle.” You see the perpendicular symbol, indicating that the lines form a 90-degree angle. You see a box in the angle, like the one in the following figure. Angles that are greater than 90 degrees but less than 180 degrees are called obtuse angles. Think of obtuse as obese; an obese (or fat) angle is an obtuse angle. Angles that measure exactly 180 degrees are called straight angles. Angles that total 90 degrees are called complementary angles. Think of C for corner (the lines form a 90-degree corner angle) and C for complementary. Angles that total 180 degrees are called supplementary angles. Think of S for supplementary (or straight) angles. Be careful not to confuse complementary angles with supplementary angles. If you’re likely to get them confused, just think alphabetically: C comes before S in the alphabet; 90 comes before 180 when you count. Angles that are greater than 180 degrees but less than 360 degrees are called reflex angles. Angles around a point total 360 degrees. The exterior angles of any figure are supplementary to the two opposite interior angles and always total 360 degrees. Angles that are opposite each other have equal measures and are called vertical angles. Just remember that vertical angles are across from each other, whether they’re up and down (vertical) or side by side (horizontal). (The following figure shows two sets of vertical angles.) Angles in the same position around two parallel lines and a transversal are called corresponding angles and have equal measures. (The following figure shows two sets of corresponding angles.) When you see two parallel lines and a transversal (that’s the line going across the parallel lines), number the angles. Start in the upper-right corner with 1 and go clockwise. For the second batch of angles, start in the upper-right corner with 5 and go clockwise. Note that in the preceding figure, all odd-numbered angles are equal and all even-numbered angles are equal. Be careful not to zigzag back and forth when numbering. If you zig when you should have zagged, you can no longer use the tip that all even-numbered angles are equal to one another and all odd-numbered angles are equal to one another.

View ArticleArticle / Updated 02-18-2020

Many of the geometry problems on the ACT require you to know a lot about triangles. Remember the facts and rules about triangles given here, and you’re on your way to acing geometry questions. Classifying triangles Triangles are classified based on the measurements of their sides and angles. Here are the types of triangles you may need to know for the ACT: Equilateral: A triangle with three equal sides and three equal angles. Isosceles: A triangle with two equal sides and two equal angles. The angles opposite equal sides in an isosceles triangle are also equal. Scalene: A triangle with no equal sides and no equal angles. Sizing up triangles When you’re figuring out ACT questions that deal with triangles, you need to know these rules about the measurements of their sides and angles: In any triangle, the largest angle is opposite the longest side. In any triangle, the sum of the lengths of two sides must be greater than the length of the third side. In any type of triangle, the sum of the interior angles is 180 degrees. The measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. Zeroing in on similar triangles Several ACT math questions require you to compare similar triangles. Similar triangles look alike but are different sizes. Here’s what you need to know about similar triangles: Similar triangles have the same angle measures. If you can determine that two triangles contain angles that measure the same degrees, you know the triangles are similar. The sides of similar triangles are in proportion. For example, if the heights of two similar triangles are in a ratio of 2:3, then the bases of those triangles are also in a ratio of 2:3. Don’t assume that triangles are similar on the ACT just because they look similar to you. The only way you know two triangles are similar is if the test tells you they are or you can determine that their angle measures are the same.

View ArticleArticle / Updated 02-04-2020

As a parent, you may wonder what you can do to help your student study for the ACT. Well, wonder no longer! Here are ten specific steps for helping your child do his or her best. Give him awesome test-prep materials If you bought a study prep guide for your child, you did him a huge favor. Taking full-length practice tests give your child an edge over other juniors and seniors who haven’t prepared. Nicely done! Encourage her to study Help your child work out a study schedule and give her incentives to stick to it, such as picking out the family’s dinner menu for one week or allotting her a larger share of the family’s talk and text minutes. Supply him with a good study environment Make sure your student has a quiet study area where he can concentrate without being disturbed by siblings, pets, friends, TV, the computer, or his cellphone. Quality study time is time spent without distractions. Take practice tests with her You’ll be better able to discuss the questions and answers with your child if you take the practice tests, too. Pretend you’re a test proctor and be the official timer for your student when she takes the full-length practice tests. After she’s done, read through the answer explanation with your whiz kid and help her discover which question types she may need to improve on. Then look up those particular topics for a refresher on the rules that govern them. Model good grammar for him Help your child recognize mistakes in English usage questions by speaking properly with him and gently correcting his grammar mistakes in your conversations. Before you know it, he’ll be correcting you! Help her memorize math formulas The online Cheat Sheet has a list of tips your student needs to know for the test; check it out at Dummies.com and search for ACT Cheat Sheet. Quiz her to make sure she remembers them. Encourage him to read One of the best ways to improve reading scores is to actually read. Go figure! Incorporate reading into your family’s schedule and set up times to read short passages together and discuss their meanings. Explore colleges with her Your child’s ACT score becomes more important to her when she realizes what’s at stake. Taking her to college fairs and campus visits can foster her enthusiasm for college and make taking the ACT more relevant. Get him to the test site on time If the test site is unfamiliar to you, take a test drive before the exam date to make sure you don’t get lost or encounter unexpected roadwork on the morning of the test. That day, make sure your kid’s alarm is set properly so he rises with plenty of time to get dressed, eat a healthy breakfast, and confirm he has the items he needs to take with him to the exam. Help her keep a proper perspective Remind your student that although the ACT is important, it isn’t more important than her schoolwork or being good to her family. Her exam score isn’t a reflection of her worth (or your parenting skills). It’s just one of many tools that colleges use to assess students’ skills and determine whether they’re a proper fit for their freshman classes.

View ArticleArticle / Updated 02-03-2020

Every great essay (whether it's written for the ACT or not) is organized like a big, juicy hamburger. Yes, you heard right. Ever taken a bite of a big, juicy hamburger from a fast-food restaurant? Well, okay, we don’t blame you for not wanting to see what’s really lurking between the buns (even though it tastes darn good). But if you’re feeling adventurous (and want to ace the essay part of the ACT), you may want to follow along as we dissect the classic fast-food burger and match each ingredient with a specific part of your essay. No matter your prompt’s topic, the ACT graders want to see a specific format to your writing. In other words, they don’t want all the ingredients thrown in any old way. By following the organization we outline in the next few sections, you can give the test graders a supersized essay worthy of a supersized score. The top bun: Introduction The top bun includes the funnel of information that leads to your thesis. We show you how to write it in the previous sections. Now you can move on to the essay’s body paragraphs. The three meats: Example paragraphs Think of your supporting arguments in terms of three different kinds of meat — perhaps two beef patties and some bacon or a chicken club with turkey and bacon. Each meat represents a separate paragraph in your essay, the purpose of which is to add specific examples that help prove the position that you state in your top bun. (Are you getting hungry yet?) Each meaty paragraph needs to include the following elements: Three to five sentences A solid topic sentence that relates directly to your position (remember, you already wrote your ideas in the top bun — your thesis) A variety of reasons, details, and examples that illustrate that specific topic In the thesis, we wrote about the uniform prompt, and we said that clothing can be distracting (see the section, “Throwing a Good First Punch: The Hook,” for more on this sample thesis). You can use that thought as the topic sentence for your first meat paragraph. For example, you may open your first body paragraph with something like this: Uniforms should be required because a variety of clothing choices can be very distracting in the learning environment. Now you have to write a few sentences that prove that clothing can be distracting. Make sure that you use specific and clear examples from a variety of sources, including personal experience, history, culture, and literature. Don’t stray off topic, or in this case, begin writing about anything other than the fact that clothing can be distracting. In other words, don’t get distracted when writing about distraction! Here’s a sample meat paragraph that you (and the graders) can really sink your teeth into: Uniforms should be required because a variety of clothing choices can be very distracting in the learning environment. Social media and advertisements flash images of young girls wearing practically nothing, for example, a fashion that most teenagers try to emulate (culture reference). However, wearing skimpy clothes and showing body parts can make some people look and react, which may interrupt an important part of class. That can be quite distracting when you’re trying to learn the Pythagorean theorem (personal experience reference). Furthermore, paying attention to the teacher is difficult when you hear people discussing another student’s $150 Dolce and Gabbana jeans (cultural reference). A uniform does away with these distractions by enforcing a more conservative style of clothing, allowing the focus in the classroom to remain on education rather than fashion. Sounds good, right? Well, your essay isn’t full, yet, even after a meaty paragraph like this one. You still have two more meats to gobble down! Lucky for you, you’ve already decided which topics you’re going to discuss in the next two meaty paragraphs: You mentioned distractions, school violence, and fitting in as part of your essay’s introduction (see the section, “Throwing a Good First Punch: The Hook,” for details). You just wrote about distractions in the first meat paragraph, so your second meat is about school violence and your third is about fitting in. Don’t get so caught up in your own argument that you forget the task at hand, though, and that includes careful consideration of the other perspectives provided. Pepper your paragraphs with nods to the opposition. Maybe you’re refuting the points made by others, or maybe you’re agreeing — but you do need to acknowledge them and consider their merits, if any. To make things easier, structure the second and third examples by including the following elements: A solid topic sentence that defends your position A few sentences in which you give reasons, details, and examples that support the topic of this paragraph or refute a counterargument. A variety of examples taken from different areas, such as literature, culture, personal experience, and history You’ll want to acknowledge arguments you don’t agree with (the three perspectives provide examples, but you can come up with your own, too), and then show why they’re not strong enough to change your position. For example, you could point out that the clothes you wear aren’t the only form of personal expression and that the lack of distractions created by uniforms may actually make it easier to express yourself in other areas, such as art, music, and writing. The lettuce, tomato, and special sauce: Transitions Like the sandwich, your essay needs to taste good (that is, read well) as a whole. Transitions serve as the special sauce and other burger fixin’s that help smooth out the differences between your paragraphs. You must include transitions between your first and second, and second and third meat paragraphs. The most obvious way to do so is by using transitional words, such as secondly, finally, another idea, another example, furthermore, and in addition, just to name a few. Using these obvious transitions will be good enough to earn a score of 5, but to achieve the perfect 6, your transitions will need to be subtler. For example, you may transition from one paragraph to another by alluding in the second paragraph to a concept mentioned in the first one. The bottom bun: Conclusion No matter how full of this essay you are by the time you add your three meaty paragraphs and all the saucy transitions, you need to consume the bottom bun before you’re done. Ideally, the bottom bun or conclusion of your essay should include the following two elements: A restatement of your position An expansion of your position that looks to the future You can address both elements in three to four sentences. Just make sure you include your position, references to your meat topics, and one sentence that pulls everything together. Here’s an example: Implementing a uniform policy would be beneficial (restatement of your position). Requiring uniforms has the potential to limit distractions in the classroom, reduce school-related violence, and help students find more creative ways to fit in (references to your meat topics). School uniforms would direct the appropriate focus back on education rather than keep it fixated on an adolescent fashion show (looking toward the future).

View ArticleArticle / Updated 02-03-2020

To create a great ACT essay, you must use specific examples, reasons, and details that prove your position on the prompt, and help refute counterarguments made by others. The ACT folks are looking for two things here, which we discuss in the following sections: Specific examples Variety of examples Use specific examples To get a handle on how specific your examples should be, consider the last time your parents questioned you about your Saturday night activities. We’ll bet their questions included all the old stand-bys: Where did you go? Who was there? Why are you home so late? Who drove? How long has he had his license? You know that vague answers never cut it. This skill that you’ve been practicing for years is going to come in handy when you take your ACT Writing Test, because you’re already great at giving the specifics (or making them up). Really good examples discuss extremely specific details, events, dates, and occurrences. Your goal is to write in detail and to try not to be too broad and loose. For example, say that you’re trying to find examples to support uniforms. You can conclude that allowing students to wear whatever they want leads to distraction among the students. Great, but you need to be more specific. You need to give an example from your life when you witnessed this distraction, or site a relevant article you’ve read. In other words, give dates, mention people, rat on your friends! Just choose examples that you know a lot about so that you can get down to the nitty-gritty and be extremely specific. Mix things up with a variety of examples Over the past few years, you may have had to come up with a variety of excuses for breaking curfew — the car broke down, traffic was horrendous, the movie ran late, you forgot the time, you fell asleep … you know the routine. Again, thank your parents for helping you with yet another skill you can apply to the ACT Writing Test. Coming up with specific examples about how you feel about uniforms just from your personal life is easy, but it’s also boring. Use a broad range of examples from different areas, such as literature, cultural experiences, your personal life, current events, business, or history. If you spend just a few moments thinking about the topic, you can come up with great examples from varied areas. So, to answer the question, “Should schools require students to wear uniforms?” you may strengthen your own perspective by using examples like these: Personal life: A scenario where you saw a girl wearing a short skirt and teeny top and noticed how it interfered with other students’ ability to concentrate Current events: An example from a magazine article you read about a high school shooting that explains how the boys who fired guns in their school were trying to hurt the kids who looked and dressed like jocks Cultural experience: The concern regarding wearing gang-related colors and logos and the potential implications doing so may have regarding violence in the schools A nice variety of examples like these definitely gets the attention of the ACT folks and helps you sound like the smart writer that you are. Form logical arguments The ACT Writing Test provides you with an issue and three perspectives and expects you to examine the whole to create a logical thesis. Accomplishing this task is easier when you know a little about how to form arguments. A logical argument consists of premises and a conclusion. The premises give the supporting evidence that you can draw a conclusion from. You can usually find the conclusion in the argument because it’s the statement that you can preface with “therefore.” The conclusion is often, but not always, the argument’s last sentence. For example, take a look at this simple deduction: All gazelles are fast. That animal is a gazelle. Therefore, that animal is fast. The premises in the argument are “All gazelles are fast” and “that animal is a gazelle.” You know this because they provide the supporting evidence for the conclusion that that animal is fast. The perspectives in the Writing Test prompt are unlikely to be so obvious as to include a conclusion designated by a "therefore," but you can form your own "therefore" statement to determine the conclusion. In deductive reasoning, you draw a specific conclusion from general premises as we did for the earlier gazelle argument. With inductive reasoning, you do just the opposite; you develop a general conclusion from specific premises. Consider this example of an inductive argument: Grace is a high school student and likes spaghetti. (Specific premise) Javi is a high school student and like spaghetti. (Specific premise) Gidget is a high school student and like spaghetti. (Specific premise) Manny is a high school and likes spaghetti. (Specific premise) Therefore, it is likely that all high school students like spaghetti. (General conclusion) Because an inductive argument derives general conclusions from specific examples, you can’t come up with a statement that “must be true.” The best you can say, even if all the premises are true, is that the conclusion can be or is likely to be true. The perspectives you see in the Writing Test will be based on inductive reasoning. Inductive reasoning often relies on three main methods. Knowing these ways of reaching a conclusion can help you analyze perspectives and effectively draw your own conclusions: Cause-and-effect arguments: This argument concludes that one event is the result of another. These types of arguments are strongest when the premises prove that an event’s alleged cause is the most likely one and that there are no other probable causes. For example, after years of football watching, you may conclude the following: “Every time I wear my lucky shirt, my favorite team wins; therefore, wearing my lucky shirt causes the team to win.” This example is weak because it doesn’t take into consideration other, more probable reasons (like the team’s talent) for the wins. Analogy arguments: This argument tries to show that two or more concepts are similar so that what holds true for one is true for the other. The argument’s strength depends on the degree of similarity between the persons, objects, or ideas being compared. For example, in drawing a conclusion about Beth’s likes, you may compare her to Alex: “Alex is a student, and he likes rap music. Beth is also a student, so she probably likes rap music, too.” Your argument would be stronger if you could show that Alex and Beth have other similar interests that apply to rap music, like hip-hop dancing or wearing bling. If, on the other hand, you show that Alex likes to go to dance clubs while Beth prefers practicing her violin at home, your original conclusion may be less likely. Statistical arguments: These arguments rely on numbers to reach a conclusion. These types of arguments claim that what’s true for the statistical majority is also true for the individual (or, alternately, that what’s true of a member or members of a group also holds true for the larger group). But because these are inductive reasoning arguments, you can’t prove that the conclusions are absolutely true. When you analyze statistical arguments, focus on how well the given statistics apply to the conclusion’s circumstances. For instance, if you wanted people to buy clothing through your website, you may make this argument: “In a recent study of consumers’ preferences, 80 percent of shoppers surveyed said they prefer to shop online; therefore, you’ll probably prefer to buy clothes online.” You’d support your conclusion if you could show that what's true for the majority is also true for an individual.

View ArticleArticle / Updated 04-26-2017

There's no getting around it: you'll probably have to solve some questions on the ACT Math exam that deal with circles. To solve the following practice questions, you'll need to know the formulas for the area of a circle and the general equation for a circle. Practice questions The figure shows a portion of the graph of y = (1.5)x and a circle with center (3, 1). The two meet at the point indicated in the graph. Which is nearest to the area of the circle in square units? Which of the following represents the equation of a circle in the standard xy-coordinate plane that is tangent to the x-axis at 3 units and to the y-axis at 3 units? A. x2 + y2 = 9 B. (x + 3)2 + (y – 3)2 = 9 C. (x – 3)2 + (y – 3)2 = 9 D. (x – 3)2 + (y – 3)2 = 6 E. (x + 3)2 – (y + 3)2 = 6 Answers and explanations The correct answer is Choice (A). Don't let the equation with the x exponent throw you. Just use a couple of familiar formulas. The formula for the area of a circle is So find the radius of the circle, apply it to the formula, and you're done. The radius is the distance from (3, 1) to the point on the circle that intersects with the graph. Find the coordinates of that point, and you can use the handy dandy distance formula to discover the length of the radius. The point's x-coordinate is obvious. The dotted line on the figure indicates that it's 2. The y-coordinate, then, is what you get when you plug 2 in for x in the equation of the curve: The y-coordinate is 2.25. So the coordinates of the second point are (2, 2.25). With the coordinates of the two points, you can use the distance formula to find the radius. The radius of the circle is Plug that value into the area formula (Don't bother to find the square root of 2.5625 because you just square it again in the area formula): The correct answer is Choice (C). For this problem, you need to know the general equation for a circle: (x – h)2 + (y – k)2 =r2, where h and k are the x- and y-coordinates of the center of the circle and r is its radius. Because the circle is tangent to the x and y at 3 units, the radius of the circle is 3. Eliminate Choices (D) and (E) because they don't have 32 on the right side of the equation. Choice (A) is wrong because it's the equation for a circle with a center on the origin (0, 0). There's no way this circle's center is on the origin if it's touching both axes. Choice (B) adds rather than subtracts within the first parentheses, which would be true only if the center were (–3, 3). Choice (C) is the only equation in proper format for a circle with a center point of (3, 3).

View ArticleArticle / Updated 04-26-2017

In geometry, polygons cover a lot of ground, so you can bet that some questions on the ACT Math exam will involve polygons—specifically, finding the interior angles of a polygon. Fortunately, as you'll see in the following practice questions, there's a handy formula that you can use to find a missing interior angle in a polygon, whether it's a square, a hexagon, or whatever. Practice questions In the figure, the following is true about the value of the degree measurement of angles a and b: 70 < a + b < 150. Which of the following describes all possible values in degrees of c + d? A. 210 < c + d < 290 B. 30 < c + d < 110 C. 120 < c + d < 200 D. 390 < c + d < 470 E. 570 < c + d < 650 The regular polygon shown here has 6 congruent sides and 6 congruent interior angles. Two of the sides are extended until they meet at point A. What is the measure of angle A? A. 160 degrees B. 120 degrees C. 72 degrees D. 60 degrees E. 35 degrees Answers and explanations The correct answer is Choice (A). To find the sum of the interior angles of a polygon, you use this formula: where n is the number of sides in the polygon. If you can't remember that formula, simply divide the shape into triangles. The sum of the interior angles in each triangle measures 180 degrees, so for each triangle add 180 degrees and you get the sum of all the angles in the polygon. The polygon in this problem has 4 sides, so you know its interior angles add up to 360 degrees. The problem tells you that the sum of angles a and b is more than 70 degrees. The lowest possible value for a + b is 71 degrees. If at its lowest, then at its highest. That means that The answer that states c + d < 290 is Choice (A). Double-check the rest of the information to make sure Choice (A) is the right answer. The problem says that a + b < 150. If a + b < 150, then its highest value is 149 degrees; the lowest the sum of c and d can be is 360 – 149 = 211. Choice (A) works. The correct answer is Choice (D). The formula for finding the interior angle measures of a regular polygon is as follows, where n represents the number of sides in the polygon: Apply the formula to the figure in the question to find the measure of each interior angle in the polygon: So each angle in the hexagon measures 120 degrees. The angles in the triangle that's formed by the two extended sides must each be 60 degrees because two of those angles form a straight line with the two 120-degree angles in the hexagon and 180 – 120 = 60. If two angles in a triangle each measure 60 degrees, the third angle, angle A, must also measure 60 degrees.

View ArticleArticle / Updated 04-26-2017

A big part of geometry involves working with angles, so it shouldn't be a surprise that the ACT Math exam contains a number of questions involving them. You may want to brush up on the properties of angles before you take on the following practice questions (and definitely before you tackle the ACT!) Practice questions In the figure, A, B, and C are collinear. The measure of angle ABD is 4 times that of angle DBC. What is the measure of angle ABD? A. 36 degrees B. 45 degrees C. 72 degrees D. 108 degrees E. 144 degrees The next figure shows three straight lines that intersect at point M. If angle EMF measures 47 degrees and angle AMB measures 29 degrees, what is the degree measure of angle CME? A. 72 degrees B. 76 degrees C. 104 degrees D. 133 degrees E. 151 degrees Answers and explanations The correct answer is Choice (E). Because the value of angle ABD measures 4 times that of angle DBC, angle ABD = 4x and angle DBC = x. Because A, B, and C are collinear, the sum of angle DBC and angle ABD is 180 degrees. To find the measure of angle ABD, set up an equation and solve for x: Don't stop there and pick Choice (A), though. The value of x is the measure of angle DBC. Multiply 36 by 4 to get 144 degrees, which is 4x and the measure of angle ABD. The correct answer is Choice (D). Scan the figure to determine which angles are equal. Angle AMB and angle DME are vertical angles, so they're equal. Angle DME also measures 29 degrees. The same goes for angle EMF and angle BMC. They're equal, so angle BMC also measures 47 degrees. The remaining two angles, angle CMD and angle AMF, are also vertical angles and equal. The degree measures of the 6 angles in the figure total to 360 degrees because they circle around the center point M. Create an equation to solve for the degree measure of the remaining two angles: Hang on, though. This is the degree measure of angle CMD, but the question asks for the measure of angle CME. You need to add 29 degrees to 104 degrees for a degree measure of angle CME, which equals 133 degrees.

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