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Notes from Crash Course for the ACT 4th editionMath test—My notes are specifically for the math portion of the ACT.The Math test is made up of the following questions: 14 pre-algebra, 10 elementary algebra, 9 intermediate algebra, 9 coordinate geometry, 14 plane geometry, and 4 trigonometry questions. The bulk of the questions are in geometry (23) and algebra (33)—stuff you probably finished studying in tenth grade. There is a GENERAL trend of increasing difficulty throughout the Math test.Reading about test taking techniques is not enough. Follow up with practice—timed practice. Score your test. Rework what you missed. Figure out why you missed what you missed. Then practice more! The better you know the ACT, the better you are going to perform.Mistake: trying to do the work in your head. Write it down! Keep nothing in your head. The ACT is designed to take advantage of your inclinations. The questions on the ACT are full of partial answers, misleading answers, and distracting answers that are there to trip you up. As the writers of the test are composing the answer choices, they build the wrong answers by working the problem and making the sorts of mistakes that a careless (or rushing) student might make. If you’re doing work in your head, you’re going to fall into one of their traps. Write! For the Math test, set up the problems on paper and use your calculator to do the crunch work. If you don’t set it up on paper, you risk making careless errors that will cost you points.For every question, pay attention to the answer choices. You’ll find that you can eliminate obvious wrong answers in every section of the test. Process of Elimination (POE) is a great tool on the ACT.Use a technique called Ballparking. Ballparking is the name for the process of reading a question, figuring out roughly what that right answer will be (without actually working the problem), and then crossing out any answers that are too big or too small. Ballparking will rarely eliminate all four wrong answers, but it will frequently eliminate two or more wrong answers, and that’s a real help. Keep in mind that you can’t use Ballparking on any questions that have variables in the answer choices, only on questions that have numbers in the answers.When the ACT offers you some version of “It cannot be determined,” you’re often dealing with a trap. Be very suspicious of the answer choice. It’s almost never right (just the opposite of OMIT on the English test). Keep that in mind as you finish the question.Every question on the test can be worked without a calculator. However, you should have a calculator to do the math problems on the ACT. Not all calculators are permitted for use on the ACT. Check for specifics.Don’t let yourself get stuck on the Math test. The worst thing you can do is spend a long time on a hard question when there are still easy questions left to be done. The way to avoid this is to use a two-pass system. You want to take the test this way so that you don’t miss any of the easy points.Do the Math test in two passes. On your first pass through, deal with two types of questions: those you are sure you can do, and those you know are impossible for you. Also, circle and skip problems you think you can do but that will take some work. Once you’ve worked your way through the entire test in this way (finishing your first pass), you’re going to come back and work on the problems you circled and skipped (pass two). Do NOT leave any blank answers!There are three major changes taking place in the Math test of the ACT. First, there are more long word questions with elaborate setups. Second, ACT is starting to put more useless information in the questions. Finally, ACT is asking more theoretical questions on recent exams. All of these changes force you to do more thinking about the mathematical concepts behind the problems and prevent you from just tossing numbers into your calculator on every question.There are plenty of percentage problems on the ACT. The 10 Percent Rule is a quick way to estimate percentages. To get 10% of any number, take the decimal point and move it one place to the left. To get 1% of any number, move the decimal point two places to the left.Translating English to math: Set up the question in English and then translate into math.% means to divide by 100Is, are, will be means =Of means multiplyWhat, some amount means a variableWhat percent means x/100On every ACT there are a number of questions that test your knowledge of mathematical vocabulary terms—not your calculator prowess. Some terms that are sure to be tested include integer, reciprocal, positive, negative, product, sum, difference, quotient, rational, irrational, bisect, square, greatest common factor, variable, coefficient, absolute value, midpoint, congruent, regular, circumference, perimeter, isosceles, equilateral, diameter, area.Absolute Value: If there is a variable inside the absolute value bars, there are going to be two solutions to the equation. Get the absolute value on one side by itself. Then, set up two equations—one with a positive answer and one with a negative answer—and solve them both. (If this were an inequality, you’d need to flip the inequality sign on the negative answer.) What comes out of an absolute value operation can never be negative.Recently the writers of the ACT have been asking more and more questions that involve conceptualization instead of calculation. It sometimes helps to sketch out little diagrams. Draw as much as you can and label. Go to the answer choices to see which conform to what the problem tells you. Use POE to get rid of the bad choices. The way to get better at visualizing these kinds of questions is to practice.Plugging In Technique: This technique should be used whenever you have variables in the answers, whether it is an algebra question or a geometry question. Don’t break any algebraic or geometric rules! Pick numbers (easy numbers that make life easier) for all the variables. (Any question that has variables being raised to powers is going to be a great candidate for plugging in 0 or 1.) Write them down above the variables in the question so you keep track of what’s what. Solve the question using your numbers. Circle your answer. Put your numbers into the answer choices and solve them. Find the one that matches your circled answer. Always check every answer choice. If you get more than one that works, just use a different set of numbers and try again. (Be careful choosing 0 or 1.) Whenever you have multiple variables, you want to plug in for the variable that’s buried in the middle of stuff first and work your way out to the other ones.PITA: Plugging in the Answers. Pita is what you use when the ACT tells you a little story and then asks you how many or how much. When using PITA, you want to start with the middle answer choice. Because the answer choices are in numerical answer, you will know which direction you need to go to find the correct answer.Manipulating equations, or solving for x, is a skill you can’t live without on this test. There are two basic types of solving for x questions you will find on the ACT.All the values for the variables are given and you have to work out the math—carefully (PEMDAS). Beware of the negative! (Don’t let the devil get you!) Plug and chug. Remember to set up the problem (plugging in the value) on paper before you begin punching at your calculator.You are given an equation and have to solve for the variables on your own. The key to success is remembering that everything you do on one side of the equation must be done on the other side of the equation. You must keep the equation balanced.Got fractions? The first thing you always want to do is to get rid of any fractions. The way to get rid of fractions is to multiply every term in the equation by the LCD.The next step is to get the x terms on one side and the numbers on the other side.Finally, get rid of the number “stuck to” x. Divide both sides by that number.Got inequalities (< or >)? Work it the same way (steps a, b, c). The difference: When you multiply or divide by a negative number, you must flip the inequality sign.Number lines. Open circle means < or >. Closed circle means < or >. Knowing how this works is a great POE tool.Quadratic equationsQuadratic formula x =[- b + sq rt of (b2 – 4ac)] / 2aFactoringAx2 + Bx + C bottoms up method; factors are ( x + something) ( x + something); same signAx2 – Bx + C bottoms up method; factors are ( x – something) ( x – something); same signAx2 + Bx – C bottoms up method; factors are (x + something) (x – something); different signsAx2 – Bx – C bottoms up method; factors are (x – something) ( x + something); different signs.Check using FOIL.Or use POE with the answer choices.Proportions (fraction equal to another fraction) allow you to project a relationship (direct variation) between two things onto a different scale.Always keep the same units on the top of each fraction and the same units on the bottom of each fraction. Write this down.Go fishing to solve the proportion.Make sure your answer is in the unit of measure the test question asks for. You may need to convert. If necessary, you may convert at the beginning or in the end.Twenty-three questions (more than 1/3) of the Math test are geometry questions of some type. These questions are based on fairly basic geometry concepts. The vast majority of the geometry questions fall into these four categories: angles, perimeters, areas, and coordinate geometry. ACT doesn’t give you any of the formulas, so you need to have these memorized.On any geometry question, the first thing you want to do is label everything in the figure (i.e., WRITE). The figures on the test are almost always drawn to scale, so you want to be sure to ballpark whenever you can. This will eliminate lots of answer choices that don’t make any sense.Parallel lines cut by a third line called a transversal. There are two kinds of angles—Big and Little. All the Big angles are congruent (the same). All the Little angles are congruent. One Big angle + one Little angle = 180o.Parallelograms have parallel lines cut by transversals so remember #1.Triangle: angle 1 + angle 2 + angle 3 = 180o.Isosceles triangle: two equal sides AND two equal angles.Pythagorean Theorem: lengths of the sides of a right triangle a2 + b2 = c2Special right triangles30o-60o-90o: ratio of the side lengths is 1: sq rt of 3: 245o-45o-90o: ratio of the side lengths is 1: 1: sq rt of 2Quadrilateral (ANY 4-sided figure): angle 1 + angle 2 + angle 3 + angle 4 = 360o.Perimeter = the lengths of all the sides added together; distance around a figure.Circle: 360o.Circumference of a circle (“perimeter” of a circle) = 2?rOn circle questions, ACT nearly always asks about the radius, diameter, circumference, or area. As a rule of thumb, it’s a good idea to determine the radius before you start attacking a circle question.Use a piece of paper to measure and do some Ballparking.The basic area formulas you will need to knowSquare or rectangle l x wCircle ?r2Triangle ? bhDo you need to make unit conversions? Do it before finding the area.Graphing linear equationsThe x-axis goes left and right. The y-axis goes up and down.Slope = rise over run = (y2 – y1) / (x2 – x1)Slope-intercept form y = mx + bParallel lines have the same slope.Perpendicular lines have slopes that are opposite reciprocals of each other.Distance between two points. Sketch the line segment connecting the two points. Build a right triangle. Use the Pythagorean Theorem to find the length of the segment (which is the hypotenuse).There are only 4 trigonometry questions on the ACT. Two of those four trig questions are always based on SOHCAHTOA. The other two questions deal with things such as trig graphing. Remember that the trig questions are frequently based on the right triangle relationships of the trig functions sine, cosine, and tangent.Focus before bubble: Focus on a set of questions. Then bubble in the answers. Going back and forth hurts your focus. Never leave a question blank. Guessing is not penalized. Keep track of the time. If there are only two minutes left for a test section, go ahead and bubble in an answer for the remaining questions. Bubble in the same answer for all of them.When you take the ACT, use every minute they give you to work the questions. Check your work in that “spare time.” The answer is always there on the page in those sections, and the only way you’ll find it is by spending more time looking for it. More time means a higher score.General advice: One problem a lot of people have on the ACT is having enough energy to finish. Get a good night’s rest before the test. Eat a good breakfast. Bring a snack to eat at the break. Stay focused.Know where you are going for the ACT. Drive there before hand. Arrive at the testing location no later than 7:45. Dress in layers. Get your stuff together the day before so you will be ready the morning of the test. You will need the following: admission ticket, sharpened #2 pencils (not mechanical), calculator with fresh batteries, backup calculator, watch, snack, photo ID. ................
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