✏️PSAT Math

Key Concepts and Definitions

• Understand the difference between an equation (mathematical statement with an equals sign) and an expression (combination of numbers and operations without an equals sign)
  • Equations can be solved to find the value of an unknown variable
  • Expressions can be simplified by combining like terms and performing operations
• Familiarize yourself with the order of operations (PEMDAS) which dictates the sequence in which operations are performed
  • Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Recognize the properties of real numbers such as the commutative property (order of addition or multiplication does not affect the result), associative property (grouping of addition or multiplication does not affect the result), and distributive property (multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products)
• Understand the concept of a function as a relation between an input (independent variable) and an output (dependent variable)
  • Functions can be represented using equations, graphs, or tables
• Know the difference between linear equations (constant rate of change, straight-line graph) and nonlinear equations (variable rate of change, curved graph)
• Comprehend the meaning of slope as the steepness of a line, calculated by the change in y divided by the change in x
• Recognize the equation of a line in slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept

Fundamental Formulas

• Memorize the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for solving quadratic equations in the form $ax^2 + bx + c = 0$
• Know the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for finding the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$
• Understand the Pythagorean theorem $a^2 + b^2 = c^2$ for right triangles, where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse
• Familiarize yourself with the area formulas for common shapes such as rectangles ($A = lw$), triangles ($A = \frac{1}{2}bh$), and circles ($A = \pi r^2$)
  • $l$ is length, $w$ is width, $b$ is base, $h$ is height, and $r$ is radius
• Know the volume formulas for basic three-dimensional shapes like rectangular prisms ($V = lwh$), cylinders ($V = \pi r^2 h$), and spheres ($V = \frac{4}{3}\pi r^3$)
• Understand the formula for the sum of an arithmetic sequence $S_n = \frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term
• Recognize the compound interest formula $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the number of years

Problem-Solving Strategies

• Read the question carefully and identify the given information and what you are asked to find
• Draw a diagram or picture if helpful to visualize the problem
  • Label the diagram with the given information
• Break down complex problems into smaller, manageable steps
• Look for patterns or relationships between the given information
• Consider using algebraic equations to represent the problem and solve for the unknown variable
• Estimate the answer first to check the reasonableness of your final answer
  • Use rounding and mental math to quickly approximate the solution
• Work backwards from the answer choices, plugging them into the problem to see which one satisfies the conditions
• Eliminate answer choices that are clearly incorrect based on the given information or your estimated answer

Common Question Types

• Solving linear equations and inequalities
  • Isolate the variable by performing the same operation on both sides of the equation or inequality
• Graphing linear equations and inequalities
  • Identify the slope and y-intercept from the equation in slope-intercept form
  • Plot the y-intercept and use the slope to find additional points
• Solving systems of linear equations
  • Use substitution or elimination methods to solve for the variables
• Factoring quadratic expressions
  • Find the common factor, difference of squares, or use grouping to factor the expression
• Solving quadratic equations
  • Factor the equation, use the quadratic formula, or complete the square
• Analyzing functions
  • Determine the domain and range, find the zeros (x-intercepts), and identify the maximum or minimum points
• Geometry problems involving angles, triangles, and circles
  • Use properties of parallel lines, triangle congruence and similarity, and circle theorems
• Word problems involving rates, ratios, and proportions
  • Set up a proportion or equation based on the given information and solve for the unknown quantity

Practice Techniques

• Work through practice problems regularly to reinforce concepts and improve problem-solving skills
  • Focus on understanding the problem-solving process rather than just getting the correct answer
• Use official PSAT practice tests to familiarize yourself with the format and types of questions
  • Analyze your mistakes and identify areas for improvement
• Create flashcards for key formulas, definitions, and concepts
  • Review the flashcards frequently to commit the information to memory
• Explain concepts and problem-solving strategies to others
  • Teaching others helps deepen your own understanding
• Collaborate with study partners to discuss difficult problems and share different approaches
• Break study sessions into manageable chunks to maintain focus and avoid burnout
  • Take short breaks and engage in physical activity to refresh your mind
• Seek help from teachers, tutors, or online resources when you encounter challenges or need further explanations

Time Management Tips

• Pace yourself during the test by dividing the total time by the number of questions
  • Allocate approximately 1-2 minutes per question, depending on the difficulty level
• Answer the easy questions first to build confidence and ensure you don't miss any points
  • Skip the difficult questions and come back to them if time allows
• Read each question carefully, but avoid spending too much time on any single question
• Show your work on your scratch paper to avoid making careless errors and to help you review your solutions
• Bubble in your answers on the answer sheet after completing a section to save time
  • Make sure to bubble carefully and avoid stray marks
• Use any remaining time to check your answers and make sure you have bubbled in all your responses
• Practice with timed tests to improve your speed and accuracy
  • Gradually work on reducing the time per question while maintaining accuracy

Common Mistakes to Avoid

• Misreading the question or overlooking important information
  • Read each question carefully and underline or circle key words and numbers
• Making careless errors in calculations or bubbling
  • Double-check your work and make sure your answer makes sense in the context of the problem
• Spending too much time on a single difficult question
  • If you are stuck, make an educated guess and move on to avoid running out of time
• Forgetting to use the given information or making assumptions
  • Use all the information provided in the question and avoid making assumptions that are not supported by the given facts
• Rushing through the test and making mistakes due to time pressure
  • Work at a steady pace and focus on accuracy rather than speed
• Leaving questions blank
  • There is no penalty for guessing on the PSAT, so eliminate the clearly incorrect choices and make an educated guess
• Relying too heavily on memorized formulas without understanding the concepts
  • Make sure you understand the underlying concepts and can apply the formulas in different contexts

Test-Taking Strategies

• Get a good night's sleep before the test and eat a healthy breakfast to fuel your brain
• Arrive at the testing center early to avoid last-minute stress and to give yourself time to relax
• Bring all the necessary materials, including several sharpened pencils, an eraser, and an approved calculator
• Read the instructions carefully and make sure you understand the format and timing of each section
• Use the process of elimination to narrow down the answer choices
  • Cross out the choices that are clearly incorrect and focus on the remaining options
• Make educated guesses if you are unsure of the answer
  • Eliminate the choices that are clearly wrong and choose the best answer from the remaining options
• Use your scratch paper to work out problems, organize your thoughts, and keep track of your progress
• Manage your stress by taking deep breaths, stretching, and maintaining a positive attitude
  • Remember that one question or section does not determine your entire score
• Stay focused and avoid distractions
  • Concentrate on one question at a time and do not let your mind wander to other sections or topics
• Trust your instincts and don't second-guess yourself
  • If you have studied and prepared well, your first answer is often the correct one