ACT Math Lesson 1: Introduction to the Math Section Key Skills and Learning Objectives: Introducing students to the different topics of the ACT Math exam Understanding common problems for each topic Identifying strategies for the ACT Math section
Pre-Algebra (23%) • The following pre-algebra topics: Number Problems; Ratios; Proportions; Percentages; Multiples, Factors, and Primes; Absolute Value; Least Greatest Problems, One Variable Linear Equations; Simple Probability; Charts, Tables, and Graphs; Simple Statistics; Exponents; Square Roots; Exponents and Roots; Sequences • These are the math basics. You probably learned them so long ago that you might have forgotten the rules to deal with things like integers, exponents, absolute value, and negative numbers. Watch out for: • 1. Simple computation errors. Because the problems appear so simple, you're going to be tempted to make computation errors in this section. For example, make sure you read problems like this carefully: • -2(10 – (7 × 8 + 5)) = ? • See all the negative numbers and parentheses? This is testing order of operations, or the rules for combining numbers when several operators are present (addition, subtraction, multiplication, and division). If you're going to use your calculator to solve problems like these, input these kinds of problems with caution. You should start within the smallest parentheses and work your way out. • 2. Negative numbers.Least greatest problems are also areas ripe for silly mistakes with negative numbers. These are problems that make you order things from least to greatest—but difficult kinds of ordering. Don't be fooled. • 3. Multiples. A mistake that the test makers are just waiting for you to make is not picking the least common multiple out of a list of multiples. Questions testing the LCM concept will often include true multiples of the numbers involved. When going through these problems quickly, you might be tempted to choose the first multiple you see. It might be the shiniest, but it isn't always the right one.
Elementary Algebra (17%) • The following elementary algebra topics: Variables to Express Relationships; Simplifying Expressions; Substitution; Polynomial Basics and Factoring; Integer Exponents and Square Roots; Linear Equations and Inequalities; Simple Quadratic Equations • The most important concept in this area is the variable. Variables are letters that represent unknown quantities. You will need to know how to relate the same variable of different orders (x and x2) to find roots, simplify equations, or solve simple linear equations and inequalities. • Watch out for: • 1. Simplifying errors. You will need to simplify polynomials that will have x terms of several orders (x, x2, maybe even x3). Don't add polynomials of different orders to simplify them. • 2. Using the quadratic formula when factoring will do. When you're asked to find roots or zeros for a simple quadratic, or the values of x when the whole equation is set to zero, usually factoring will do the trick. The ACT test makers don't want to involve you in intensive computation. It's more trouble for them to create computationally complicated problems that trip you up. • 3. Flipping the inequality when dividing by a negative number. Practice this until you can do it in your sleep. On that note, make sure you get enough sleep before the exam.
Intermediate Algebra (15%) • The following intermediate algebra topics: Binomials, Quadratic and Fractional Equations, Radical and Rational Expressions, Inequalities and Absolute Value, Quadratic Inequalities. For more complex operations, there are usually shortcuts. For example, use Pascal's triangle to avoid calculating factorials for the binomial coefficient. Memorize the formulas for arithmetic and geometric sequences, or sequences of numbers involving additive and multiplicative changes. The ACT does not provide formulas on the test. • Watch out for: • 1. Recognizing systems of equations in word problems. Systems of equations often appear masked in word problems. Systems are sets of equations that relate two different variables to each other in two different ways. It's up to you to recognize when two variables are being flagged in the problem as x and y. For example: • Janie and Matt have fifteen books between the two of them. Janie has two times as many books as Matt. How many books does Matt have? • x = Matt, y = Janie • x + y =152x = y • 2. Sequences. If you get desperate on the test, you could be persuaded to write out entire sequences. Don't do it. The formulas are always the way to go, if you forget the formulas, you should make a best guess and move onto other problems rather than writing out a bunch of numbers. • 3. Testing intervals with absolute value inequalities. Solving an absolute value inequality, such as |x + 4| > 7, is not done once you plug in an equals sign and solve for x. You then have to test all the intervals and figure out if the solution is an "and" solution (4 ≤ x ≤ 18), or an "or" solution (x ≤ 4 or x ≥ 18). • 4. Polynomials and LCDs. If you need to add fractions with polynomials as denominators, sometimes it's hard to see the lowest common denominator (LCD), or the most simplified expression common to both denominators,rightaway. Factoring is always the way to get around this roadblock.
Coordinate Geometry (15%) • This is algebra's graphical friend. If you are solid on linear and quadratic equations, you should be able to tackle the core of the coordinate geometry problems. The key is to be able to recognize what the graph will look like from an equation before you even pull out your pencils, or graphing calculators. To do this for polynomials, you need to know how the sign, power, and roots of a polynomial affect its graphical shape. For lines, the slope and y-intercept are the two most quantities things to know for a graph. For circles, the radius is included in the equation and will determine the size of the circle. Transformations of these graphs depend on numbers added, multiplied, and subtracted to the key terms in the equation. • Watch out for: • 1. Slope-intercept form. Whenever a question asks you to visualize where a line would fall on the coordinate plane, you should always rewrite the equation in slope-intercept form. Trying to visualize a line on a coordinate plane from an equation like 5x + y = 12 is just too intense. • 2. Being afraid to draw. Not everyone is an artist. Not even some artists. But if you don't draw out problems that involve shapes on a coordinate plane, you will be confused by problems like this: • A triangle has a base of 5 in and a height of 3 in. The end of the base is at (2, 3) on the coordinate plane with units in inches. Which of the following points could also be an end of the triangle? The answer could be anything, until you draw it out and see for yourself. • The rules of transformations. If you instantly know what happens to a graph when a constant is added, subtracted, or multiplied to a term in the equation, these problems will be no-brainers. If you forget, then you will waste time with your graphing calculators trying to figure it out. And it won't be quality time. Those graphing calculators can be nasty.
Plane Geometry (23%) • This is the meat of the geometry portion of the test. Unlike coordinate geometry, which shares a lot with algebra, plane geometry has its own rules. You should spend a significant amount of study time memorizing key facts about polygons and angles. Just knowing formulas for the sum of angles in a polygon, the relationship between the sides and angles of triangles (these will help with proofs for congruent triangles), and how to calculate perimeter and area for 2D shapes, and volume and surface area for 3D shapes, will go a long way. It might seem like a lot to take in, but polygons have a lot in common with each other—including their tendency to show up together on tests. • Watch out for: • 1. The polygon within the polygon.Some plane geometry problems might present you with a square inside a circle, or a triangle inside a square. Failing to recognize, for example, that the side of a square can give you the radius of a circle can make all the difference. For these circle-square problems especially, knowing the types of special right triangles (45-45-90, 30-60-90) will be crucial. • 2. The Pythagorean Theorem. This is a test maker favorite. Any right triangle you see on the test should immediately call to mind a portrait of Pythagoras and his formula a2 + b2 = c2. Okay, the portrait is optional, but the formula is not. • 3. Break large shapes up into smaller shapes. Sometimes plane geometry questions can look scary and complicated. You should know that every polygon can be one or more triangles, and triangles have all sorts of great rules that will be helpful in finding things like area and perimeter. • 4. Shaded regions. Shaded regions mean only the shaded regions, not including the rest of the shape. Test makers expect you to overlook this when working quickly, and it will lead to silly mistakes on these types of problems. • 5. Scale. Don't rely on the pictures to give you hints about the measurements of angles or sides. Figures are not drawn to scale.
Trigonometry (7%) • Despite its mouthful of a name, trigonometry is less of a big deal than geometry on the ACT because it makes up fewer questions. Problems in this area will primarily ask you to recall the trig functions sine, cosine, and tangent, and you should use the pneumonic SOH ) CAH ) TOA ) to remember how to calculate them. • It's not exactly a witch chant, but might work some magic if you're lucky. • You will need these functions to find sides and angles of right triangles. Memorizing a few of the trig identities won't hurt either (but the most important is sin(θ)2 + cos(θ)2 = 1). As for modeling and graphing trig functions, you should know the amplitude, or maximum, and periodicity, or how long it takes to complete a cycle, for each trig function. This will help you apply trig functions to word problems and to understand trig graphs. • Watch out for: • 1. Calculating sine, cosine, and tangent. • Use the information provided. This sounds like an obvious one, but when a question provides a trig identity or an unfamiliar formula, use it. It's like when your teacher drops hints about a pop quiz. Similarly, when the answers in a multiple choice question involve certain trig functions and not others, it's a sign (sine?) that you should be using those and not go off the rails with other trig formulas. For trig problems especially, paying close attention to the information provided in the questions is a big deal.
Test Strategies • 1. Calculator. If you're going to use a calculator, use the same calculator that you know and love from math class during the exam. No need to introduce another unknown into the equation. • 2. Calculations. These shouldn't get too lengthy or complicated. The test is designed to be doable without a calculator. • 3. Drawing. Can't stress this enough. Draw all over the test to help yourself solve problems, especially on geometry problems involving angles. And we don't mean doodles. • 4. Answer everything. Even if you don't know the answer, you should guess. Wrong answers are not penalized. • 5. Smart elimination. If guessing is the way you're going to go, then focus on eliminating answers that don't seem probable. For example, the very largest or very smallest numbers out of all the options for a least common multiple. • 6. Steady pace. If you can knock off some pre-algebra problems in under a minute, great. But don't spend more than a minute on any one problem. Guess first on every problem you're stuck on and then circle them to go back to if you have time later. • 7. Check answers. Speaking of going back, if you have time at the end, check your work on problems you were unsure about.