Overview
This lecture provides a comprehensive overview of essential SAT Math concepts, strategies, and formulas, organized by increasing difficulty, to help students master all tested topics.
Level 1: Fundamentals
- Basic algebra includes isolating variables, adding fractions, and understanding order of operations (PEMDAS/BIDMAS).
- Use Desmos for solving single-variable equations quickly and visually.
- Linear functions: y = mx + b, where m is the slope (rate of change), and b is the y-intercept.
- Slope calculation: change in y divided by change in x.
- To find a line through two points, calculate slope then solve for the y-intercept.
Level 2: Intermediate Concepts
- Variables can change; constants and coefficients stay the same in equations.
- Systems of equations involve finding variable values that satisfy two constraints; solutions are intersection points on a graph.
- Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
- Translate English to math: "is" means "=", "of" means multiplication.
- Memorize six exponent rules for simplifying expressions.
- Key geometry facts: right angles (sum 90°), straight lines (180°), full circle (360°), triangle angles (180°), quadrilateral angles (360°).
- Use Pythagorean theorem for missing triangle sides.
- SohCahToa helps solve right triangles: Sine (opposite/hypotenuse), Cosine (adjacent/hypotenuse), Tangent (opposite/adjacent).
- Density = mass/volume; know perimeter (sum of sides), area, and volume (formulas provided on test).
- Mean is average; median is the middle value; range is max - min.
Level 3: Advanced Essentials
- Integers are positive/negative whole numbers and zero.
- Translate function graphs horizontally or vertically; Desmos can help visualize.
- Percentages: 20% of x is 0.2x; increasing/decreasing by a percentage multiples x by 1±rate.
- Circle equation: (x–h)² + (y–k)² = r², where (h,k) is the center and r is the radius.
- Tangent line to a circle is perpendicular to radius at the point of tangency.
- Arc length/sector area is proportional to the central angle's fraction of 360°.
- Inscribed angle is half the central angle it subtends.
- Convert between degrees/radians: 180° = π radians.
- Congruent triangles are identical; similar triangles have proportional sides and identical angles.
- Prove triangle similarity with AA, SSS, or SAS (not SSA).
- Conditional probability: P(A|B) = P(A and B)/P(B).
- Box plots show min, quartiles, median, and max; scatter plots show relationships and trends.
- Quadratics: factor to solve, or use Desmos for roots and vertex.
Level 4: Pre-Expert Skills
- Standard quadratic form: ax² + bx + c; vertex form: a(x–h)² + k; vertex at x = –b/2a.
- Vertex is midpoint between roots; sum of roots = –b/a; product of roots = c/a.
- Know the behaviors of polynomial functions at roots (odd power: crosses axis; even power: bounces).
- Exponential functions: repeated multiplication, modeled as y = initial × (growth factor)^t.
- Key trig identity: sin(angle) = cos(complementary angle).
- Unit conversions: square unit conversion scales by square of length conversion; for volume, by cube.
- Surface area problems require understanding shape composition, not memorizing formulas.
- Standard deviation measures spread; symmetric data implies mean = median.
- Margin of error gives a plausible range, not all possible values; larger samples shrink margin.
Level 5: Mastery Topics
- Triangle altitude from right angle to hypotenuse forms three similar triangles with proportional sides.
- For polynomial factor matching problems, set up and compare coefficients to deduce integer conditions.
- Square pyramid surface area includes base and four triangles; use slant height and right triangle logic as needed.
- With many variables/constants, expect to solve by hand (memorize quadratic formula, discriminant, and completing the square).
- Slope of a line Ax + By = C is –A/B; arc length = radius × central angle (in radians).
- Know special triangles: 3-4-5, 5-12-13; triangle inequality: two sides' sum > third side.
- Practice hardest question types: mean with integers, exponential with fractional exponents, and advanced quadratics.
Key Terms & Definitions
- Variable — a symbol (like x or y) that can change value.
- Constant — a fixed value in an equation.
- Coefficient — a number multiplying a variable.
- Slope — rate of change; rise/run between two points on a line.
- Y-intercept — value of y when x = 0.
- System of Equations — set of equations with common variables.
- Parallel Lines — lines with equal slopes, never intersect.
- Perpendicular Lines — lines with slopes that are negative reciprocals.
- Mean — average of a data set.
- Median — middle value of a data set in order.
- Range — difference between maximum and minimum values.
- Standard Deviation — measure of data spread.
- Quadratic Equation — polynomial of degree two, ax² + bx + c = 0.
- Vertex — highest/lowest point of a parabola.
- Exponential Function — repeated multiplication, y = ab^x.
- Arc Length — distance along a circle’s edge between two points.
- Sector Area — area of a "slice" of a circle.
- Box Plot — a diagram showing the five-number summary of a data set.
Action Items / Next Steps
- Memorize exponent rules and key formulas (circle, quadratic, area, and volume).
- Drill algebra, systems, and geometry problems, using Desmos where allowed.
- Review sample problems for each topic, focusing on missed or weak areas.
- Practice translating word problems into equations.
- Prepare by solving hardest question types in each category.
- Study statistical concepts: box plots, standard deviation, and margin of error.
- Complete assigned practice sets or drills as provided in course materials.