🧮

SAT Math Concepts Overview

Sep 12, 2025

View transcript

Overview

This lecture covers all essential SAT Math concepts, strategies, and formulas, organized by increasing difficulty, from basic algebra to advanced geometry and statistics.

Algebra, Functions, and Linear Equations

  • Isolate variables and apply the order of operations (PEMDAS/BIDMAS).
  • Linear functions have the form y = mx + b, where m is the slope and b is the y-intercept.
  • Slope is the rate of change: (change in y) / (change in x).
  • Use Desmos to solve single-variable equations quickly.
  • Given two points, find the slope and use one point to solve for the y-intercept.

Variables, Systems, and Exponents

  • Variables change; constants do not; coefficients multiply variables.
  • Systems of equations are sets of constraints—solutions are points where equations intersect.
  • Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes.
  • Translate word problems into equations (e.g., "is" means "=", "of" means multiply).
  • Memorize exponent rules for operations with powers.

Geometry Basics

  • Right angles sum to 90°, straight lines to 180°, circles to 360°.
  • Vertical angles and alternate interior angles are congruent.
  • Triangle angles sum to 180°; quadrilaterals to 360°.
  • Pythagorean theorem for missing side lengths in right triangles.
  • Know density = mass/volume.
  • Isosceles triangles have two equal sides; equilateral have three.
  • Perimeter = sum of sides; area and volume formulas are given on the SAT.

Statistics Fundamentals

  • Mean = sum of values / count; median = middle value in a set.
  • Range = maximum - minimum.
  • Mean is affected by outliers; median is not.
  • Use the mean to find the total by rearranging the formula.

Intermediate Concepts: Integers, Percentages, and Circles

  • Integers include positive, negative whole numbers, and zero.
  • Translating/reflecting functions: horizontal and vertical shifts.
  • 20% of x is 0.2x; increasing x by 20% is 1.2x; decreasing is 0.8x.
  • Circle equation: (x - h)² + (y - k)² = r².
  • Arc length/sector area: find the fraction of the circle corresponding to the angle.
  • Inscribed angle is half the central angle.
  • Convert degrees to radians: 180° = π radians.

Triangles, Probability, and Data Interpretation

  • Congruent triangles: identical in form; similar triangles: same angles, proportional sides.
  • Prove triangle similarity using AA, SSS, or SAS (not SSA).
  • Conditional probability considers subsets based on given information.
  • Box plots show minimum, 25th, median, 75th, and maximum percentiles.
  • For unbiased sampling, use random, representative samples.
  • Quadratic functions: use Desmos for solutions and vertex.

Quadratics, Polynomials, and Exponentials

  • Standard quadratic form: ax² + bx + c; vertex form: a(x-h)² + k.
  • Vertex x-coordinate: h = -b/(2a).
  • The sum of roots: -b/a; product: c/a.
  • Exponential growth: a * r^t for repeated multiplication.
  • In polynomials, odd-powered roots cross the x-axis; even-powered roots bounce.

Advanced Geometry and Statistics

  • Sine of one angle equals cosine of its complement.
  • When converting area or volume units, raise the conversion factor to the appropriate power (e.g., square for area).
  • Doubling side length quadruples area, octuples volume.
  • Surface area problems may require setting up custom equations.

Higher-Level Topics and Strategies

  • The altitude from a right angle in a triangle divides it into two similar triangles.
  • For problems with unknown constants, compare to quadratic standard form and solve for relationships.
  • For pyramid surface area, distinguish between slant height and vertical height—use the Pythagorean theorem.
  • Memorize the quadratic formula and discriminant.
  • Triangle inequality: sum of any two sides > third side.
  • Practice hardest problem types and review mistakes for improvement.

Key Terms & Definitions

  • Variable — A symbol representing a changeable value.
  • Constant — A fixed value in an equation.
  • Coefficient — A number multiplying a variable.
  • Mean — Average of a set.
  • Median — Middle value of a data set.
  • Range — Difference between max and min.
  • System of equations — Set of simultaneous equations.
  • Vertex — Turning point of a parabola or highest/lowest point.
  • Congruent triangles — Triangles identical in shape and size.
  • Similar triangles — Triangles with proportional sides and equal angles.
  • Exponent — The power to which a number is raised.

Action Items / Next Steps

  • Complete algebra, geometry, and exponent drills (referenced in the video description).
  • Practice systems of equations, translation, and similarity proofs.
  • Memorize essential formulas (circle equation, vertex formula, exponent rules).
  • Use Desmos for checking and practicing graph-based problems.
  • Review the most challenging problems in each topic and revisit previous errors.

Full transcript