ACT Math Review

Key Concepts and Formulas

• Understand the properties of real numbers (commutative, associative, distributive)
  • Commutative property: $a + b = b + a$ and $a \times b = b \times a$
  • Associative property: $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$
  • Distributive property: $a(b + c) = ab + ac$
• Know how to solve linear equations and inequalities
  • To solve linear equations, isolate the variable on one side of the equation by performing inverse operations
  • To solve linear inequalities, follow the same steps as solving equations, but reverse the inequality sign when multiplying or dividing by a negative number
• Understand the properties of exponents and radicals
  • Multiplying powers with the same base: $a^m \times a^n = a^{m+n}$
  • Dividing powers with the same base: $\frac{a^m}{a^n} = a^{m-n}$
  • Negative exponents: $a^{-n} = \frac{1}{a^n}$
• Recognize and apply the Pythagorean theorem: $a^2 + b^2 = c^2$
• Familiarize yourself with the basic trigonometric ratios (sine, cosine, tangent)
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Understand the concepts of mean, median, and mode for data analysis
• Know how to calculate the probability of simple events
  • Probability of an event $A$: $P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$

Number Theory and Operations

• Understand the properties of integers (whole numbers, their opposites, and zero)
  • Integers are closed under addition, subtraction, and multiplication, but not division
• Know how to perform operations with fractions and decimals
  • To add or subtract fractions, find a common denominator and add or subtract the numerators
  • To multiply fractions, multiply the numerators and denominators separately
  • To divide fractions, multiply the first fraction by the reciprocal of the second fraction
• Recognize prime numbers (numbers divisible only by 1 and themselves) and composite numbers (numbers with factors other than 1 and themselves)
• Understand the concept of absolute value (the distance of a number from zero on the number line)
  • $|a| = a$ if $a \geq 0$
  • $|a| = -a$ if $a < 0$
• Know how to find the greatest common factor (GCF) and least common multiple (LCM) of two or more numbers
  • The GCF is the largest factor that divides all the given numbers
  • The LCM is the smallest multiple that is divisible by all the given numbers
• Understand the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right)
• Recognize and apply the properties of rational and irrational numbers
  • Rational numbers can be expressed as the ratio of two integers (fractions or terminating/repeating decimals)
  • Irrational numbers cannot be expressed as the ratio of two integers (non-terminating, non-repeating decimals)

Algebra and Functions

• Understand the concept of variables and how to manipulate algebraic expressions
  • Variables represent unknown quantities in expressions and equations
  • To simplify algebraic expressions, combine like terms and apply the distributive property
• Know how to solve systems of linear equations
  • Solve systems of equations using substitution, elimination, or graphing methods
• Recognize and graph linear functions in the form $y = mx + b$
  • $m$ represents the slope (rate of change) and $b$ represents the y-intercept (where the line crosses the y-axis)
• Understand the properties of quadratic functions and how to solve quadratic equations
  • Quadratic functions have the general form $y = ax^2 + bx + c$, where $a \neq 0$
  • Solve quadratic equations by factoring, using the quadratic formula, or completing the square
• Know how to factor polynomials and solve polynomial equations
  • Factor polynomials by finding the greatest common factor (GCF) or using special factoring techniques (difference of squares, perfect square trinomials)
• Recognize and graph exponential and logarithmic functions
  • Exponential functions have the general form $y = a^x$, where $a > 0$ and $a \neq 1$
  • Logarithmic functions have the general form $y = \log_a x$, where $a > 0$, $a \neq 1$, and $x > 0$
• Understand the properties of rational functions and how to solve rational equations
  • Rational functions are the ratio of two polynomial functions
  • To solve rational equations, find the least common denominator (LCD) and multiply both sides of the equation by the LCD to clear the denominators

Geometry and Measurement

• Understand the properties of parallel and perpendicular lines
  • Parallel lines have the same slope and never intersect
  • Perpendicular lines have slopes that are negative reciprocals of each other
• Know how to calculate the area and perimeter of basic shapes (rectangles, triangles, circles)
  • Area of a rectangle: $A = lw$
  • Perimeter of a rectangle: $P = 2l + 2w$
  • Area of a triangle: $A = \frac{1}{2}bh$
  • Perimeter of a triangle: $P = a + b + c$
  • Area of a circle: $A = \pi r^2$
  • Circumference of a circle: $C = 2\pi r$
• Understand the properties of angles (complementary, supplementary, vertical, and adjacent angles)
  • Complementary angles add up to 90°
  • Supplementary angles add up to 180°
  • Vertical angles are congruent (equal in measure)
  • Adjacent angles share a common vertex and side
• Recognize and apply the properties of congruent and similar triangles
  • Congruent triangles have the same size and shape (SSS, SAS, ASA, AAS)
  • Similar triangles have the same shape but different sizes (AA, SAS, SSS)
• Know how to find the volume and surface area of basic 3D shapes (cubes, cylinders, spheres)
  • Volume of a cube: $V = s^3$
  • Surface area of a cube: $SA = 6s^2$
  • Volume of a cylinder: $V = \pi r^2 h$
  • Surface area of a cylinder: $SA = 2\pi r h + 2\pi r^2$
  • Volume of a sphere: $V = \frac{4}{3}\pi r^3$
  • Surface area of a sphere: $SA = 4\pi r^2$
• Understand the concept of coordinate geometry and how to find the distance between two points
  • Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Trigonometry Basics

• Understand the definitions of sine, cosine, and tangent in right triangles
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Know the values of trigonometric functions for special angles (30°, 45°, 60°)
  • $\sin 30° = \frac{1}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$, $\tan 30° = \frac{\sqrt{3}}{3}$
  • $\sin 45° = \frac{\sqrt{2}}{2}$, $\cos 45° = \frac{\sqrt{2}}{2}$, $\tan 45° = 1$
  • $\sin 60° = \frac{\sqrt{3}}{2}$, $\cos 60° = \frac{1}{2}$, $\tan 60° = \sqrt{3}$
• Understand the relationship between the trigonometric functions (reciprocal, quotient, and Pythagorean identities)
  • Reciprocal identities: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$
  • Quotient identities: $\tan \theta = \frac{\sin \theta}{\cos \theta}$, $\cot \theta = \frac{\cos \theta}{\sin \theta}$
  • Pythagorean identities: $\sin^2 \theta + \cos^2 \theta = 1$, $1 + \tan^2 \theta = \sec^2 \theta$, $1 + \cot^2 \theta = \csc^2 \theta$
• Know how to solve right triangles using trigonometric ratios
  • Use the appropriate trigonometric ratio (sine, cosine, or tangent) to find the missing side or angle in a right triangle
• Recognize and apply the law of sines and the law of cosines for non-right triangles
  • Law of sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
  • Law of cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

Data Analysis and Statistics

• Understand how to interpret and create various types of graphs (bar graphs, line graphs, pie charts, scatter plots)
  • Bar graphs compare categories using rectangular bars
  • Line graphs show trends or changes over time
  • Pie charts represent data as slices of a circle
  • Scatter plots display the relationship between two variables
• Know how to calculate and interpret measures of central tendency (mean, median, mode)
  • Mean is the average of a set of numbers
  • Median is the middle value when the data is arranged in order
  • Mode is the most frequently occurring value
• Understand the concept of standard deviation and how it relates to the spread of data
  • Standard deviation measures the dispersion of data from the mean
  • A larger standard deviation indicates that the data is more spread out
• Recognize and interpret the correlation between two variables in a scatter plot
  • Positive correlation: as one variable increases, the other variable also increases
  • Negative correlation: as one variable increases, the other variable decreases
  • No correlation: there is no apparent relationship between the variables
• Know how to calculate and interpret the probability of simple events
  • Probability of an event $A$: $P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$
  • Probability of the complement of an event $A$: $P(\text{not } A) = 1 - P(A)$
• Understand the concepts of permutations and combinations
  • Permutations: the number of ways to arrange $n$ objects in a specific order
  • Combinations: the number of ways to select $r$ objects from a set of $n$ objects, where the order does not matter

Problem-Solving Strategies

• Read the problem carefully and identify the given information and the question being asked
  • Underline or highlight key information and the question to stay focused
• Draw a diagram or picture to visualize the problem, if applicable
  • Sketching a diagram can help you better understand the relationships between the given information
• Break down complex problems into smaller, manageable steps
  • Solve the problem step by step, focusing on one aspect at a time
• Look for patterns or relationships between the given information
  • Identifying patterns can help you develop a strategy to solve the problem
• Use the process of elimination to narrow down answer choices
  • Eliminate answer choices that are clearly incorrect or do not make sense based on the given information
• Plug in numbers or use examples to test your understanding of the problem
  • Substituting simple numbers for variables can help you better understand the problem and check your work
• Check your answer to ensure it makes sense and answers the original question
  • Review your solution and make sure it is reasonable and answers the question being asked

Common Mistakes and How to Avoid Them

• Misreading or misinterpreting the problem
  • Read the problem carefully and make sure you understand what is being asked before attempting to solve it
• Making calculation errors
  • Double-check your calculations and use a calculator when necessary to avoid simple arithmetic mistakes
• Forgetting to use the appropriate formula or concept
  • Identify the key concepts or formulas needed to solve the problem and make sure to apply them correctly
• Rushing through the problem without fully understanding it
  • Take your time to read and understand the problem before attempting to solve it
• Not checking your answer or ensuring it makes sense
  • Always review your solution and make sure it is reasonable and answers the original question
• Misinterpreting graphs or diagrams
  • Pay close attention to the labels, scales, and units on graphs and diagrams to avoid misinterpreting the information
• Not using the given information effectively
  • Make sure to use all the relevant information provided in the problem to solve it accurately
• Relying too heavily on memorized formulas without understanding the underlying concepts
  • Focus on understanding the concepts behind the formulas to apply them more effectively in different situations