SAT Math Core Concepts

Overview

This lecture covers all core SAT Math concepts, strategies, and formulas, organized from foundational to advanced topics, including algebra, geometry, functions, statistics, and problem-solving techniques.

Basic Algebra & Functions

• Know how to isolate variables, add fractions, and use correct order of operations (PEMDAS/BIDMAS).
• Use Desmos to solve single-variable equations quickly.
• A function takes an input (x) and gives an output (y), often graphed as a straight line for linear functions.
• To find a line through two points, calculate slope (rate of y change over x change) and solve for the y-intercept.

Variables, Systems, & Exponents

• Variables change, constants stay the same; a coefficient multiplies a variable.
• Systems of equations represent two constraints; their solution is the intersection point(s).
• Parallel lines have equal slopes, perpendicular lines have negative reciprocal slopes.
• Translate word problems: "is" means "=", "of" means multiplication.
• Memorize exponent rules (product, quotient, power, etc.).

Geometry Essentials

• Right, straight, and full angles sum to 90°, 180°, 360° respectively.
• Triangle angles sum to 180°, quadrilaterals sum to 360°; n-gon sum is 180(n-2).
• Use the Pythagorean Theorem for missing side lengths in right triangles.
• Know the difference between isosceles (2 equal sides) and equilateral (3 equal sides) triangles.
• Perimeter is total side length; area and volume formulas are on the SAT formula sheet.

Statistics Basics

• Mean = sum of values ÷ number of values; median = middle value; range = max - min.
• Mean is affected by outliers more than median.
• To find total from mean: mean × count = sum.

Intermediate Concepts: Numbers, Circles, & Probability

• An integer is a whole number (positive, negative, or zero).
• Percentage increase: x × (1 + rate); percentage decrease: x × (1 - rate).
• General circle equation: (x - h)² + (y - k)² = r².
• Arc length and sector area are fractions of circumference and area, found by central angle/360.
• Inscribed angle = ½ central angle; 180° = π radians for conversions.
• Similar triangles have proportional sides and identical angles; congruent triangles are exact copies.
• Conditional probability calculates chance given certain criteria.

Data Representation & Quadratics

• Box plots show min, 25th, median, 75th, and max; scatter plots help with trend lines.
• For unbiased results, use random samples.
• Quadratics: Standard and vertex form; vertex x = -b/2a; sum of solutions = -b/a; product = c/a.
• Use Desmos to find quadratic solutions or vertex; memorize key formulas.

Advanced Algebra, Exponentials, & Geometry

• For exponential functions: repeated multiplication is expressed as a^t.
• For right triangles, sin(angle) = cos(complement angle).
• When converting square and cubic units, square or cube the unit conversion factor.
• Doubling side length multiplies area by 4, volume by 8.
• Surface area may require decomposing irregular shapes.

Advanced Statistics

• If data is symmetric, mean = median.
• Standard deviation measures how spread out data is.
• Margin of error gives a confidence interval for sample-based estimates; larger samples reduce margin.

Expert-Level Topics

• The altitude from the right angle of a triangle creates two similar triangles.
• For polynomial equations with variable constants, set up all equations and analyze divisibility or integer constraints.
• Pyramid surface area = base area + 4 × triangle area (with slant height).
• When many constants are present, memorize quadratic formula, discriminant, and how to complete the square.
• Slope in standard form Ax + By = C is -A/B.
• Arc length = radius × (angle in radians).
• 3-4-5 and 5-12-13 triangles are common.
• Triangle inequality: sum of two sides > third side.
• Review hardest problem types in every category and address weaknesses.

Key Terms & Definitions

• Function — A process that turns an input into an output (y = f(x)).
• Coefficient — Number multiplying a variable in an equation.
• Vertex (of a parabola) — The highest or lowest point; x = -b/2a.
• Median — The middle value in a data set.
• Standard Deviation — A measure of data dispersion.
• Margin of Error — Range indicating confidence in an estimate.

Action Items / Next Steps

• Practice from linked drills and sample problems for each topic.
• Memorize main formulas and rules highlighted above.
• Use Desmos for graphing and calculations where permitted.
• Identify weak topics and practice more challenging questions in those areas.