Overview
This lecture systematically reviews all major SAT Math topics, focusing on key concepts, formulas, problem types, and strategies across algebra, geometry, statistics, functions, and advanced problem-solving.
Algebra & Linear Functions
- Basic algebra involves isolating variables, adding fractions, and following order of operations.
- Use Desmos to solve single-variable equations quickly by finding intersection points.
- A linear function is written as y = mx + b, where m is the slope, and b is the y-intercept.
- Slope measures the rate of change; calculate with (change in y) / (change in x).
- For two points, find the slope and plug into y = mx + b to solve for the line's equation.
Variables, Systems & Exponents
- Variables change; constants remain the same; coefficients multiply variables.
- Systems of equations represent constraints and can be solved graphically or with Desmos.
- Parallel lines have the same slope; perpendicular lines have negative reciprocal slopes.
- Translate English to math: "is" means "=", "of" means multiplication.
- Memorize key exponent rules (product, quotient, power, negative, zero, and fractional).
Geometry Essentials
- Angles in a right angle sum to 90°; semicircle, 180°; full circle, 360°.
- Triangle angles sum to 180°; quadrilateral, 360°; n-gon, 180(n-2)°.
- Pythagorean Theorem: a² + b² = c² in right triangles.
- Perimeter is the sum of all side lengths; area/volume formulas are provided on the SAT sheet.
- Know the difference between isosceles (2 equal sides) and equilateral triangles (3 equal sides).
Statistics & Data
- Mean = sum of values / number of values; median = middle value.
- Mean can be skewed by outliers; median is not.
- Range = max - min.
- Box plots show min, 25th, median, 75th, and max percentiles.
- Standard deviation measures data spread; margin of error describes confidence in estimates.
Intermediate Concepts
- Integers are all whole numbers, including zero.
- Percentage problems: increase by n% = 1.n × value; decrease by n% = (1-n) × value.
- Equation of a circle: (x-h)² + (y-k)² = r²; directly derived from the Pythagorean theorem.
- Similar triangles have proportional sides and equal angles; congruent triangles are identical.
- Conditional probability: probability given a specific condition, focus only on the relevant subset.
Quadratics & Polynomials
- Standard form: ax² + bx + c; vertex form: a(x-h)² + k; vertex at x = -b/2a.
- Sum of solutions = -b/a; product of solutions = c/a.
- For parabolas, vertex is midpoint of roots/solutions.
- Graphs bounce at even-powered roots, pass through at odd-powered roots.
Exponential Functions & Advanced Algebra
- Exponential growth formula: initial × (1 + rate)^t.
- Exponential functions repeat multiplication, not addition.
- Function transformations shift graphs horizontally/vertically.
- For complex algebraic identities, plug in values to test answer choices.
Advanced Geometry & Miscellaneous
- Altitude in triangles is perpendicular from a vertex; divides triangle into similar triangles.
- Surface area of a pyramid = base + four triangles; slant height differs from perpendicular height.
- Unit conversion: square units require squaring conversion factor.
- Triangle inequality: sum of any two sides must exceed the third side.
Key Terms & Definitions
- Coefficient — Number multiplying a variable.
- Exponent — Power to which a number/variable is raised.
- Mean — Average of a data set.
- Median — Middle value in ordered data.
- Vertex (quadratic) — Highest/lowest point of a parabola.
- Standard deviation — Measure of data spread.
- Margin of error — Confidence range for survey/statistical result.
Action Items / Next Steps
- Practice algebra and linear functions using provided drills.
- Review exponent rules and geometry basics.
- Memorize circle and quadratic formulas.
- Use Desmos for graphing and solutions.
- Focus on weak categories and review missed problems for harder variants.
- Try practice problems for each topic, especially those involving circles, quadratics, and statistics.