SAT Math Section Lecture Notes

Overview

• Focus on six lessons covering key SAT math topics:

  1. Algebra (Solving equations, evaluating functions, composite functions, multi-variable functions)
  2. Converting sentences into equations (Solving word problems)
  3. Ratios, proportions, probability
  4. Averages, fractions, percentages
  5. Graphs of linear, quadratic, and absolute value functions (Slope, arithmetic and geometric sequences)
  6. Geometry review

• Each lesson includes a review of important topics, concepts, and multiple choice problems.

• Emphasis on learning through problem-solving.

Lesson 1: Algebra

Review of Key Concepts

Exponents

• Multiplying common bases: Add the exponents
  • Example: x³ * x⁴ = x⁷
• Raising an exponent to another exponent: Multiply the exponents
  • Example: (x³)⁴ = x¹²
• Dividing common bases: Subtract the exponents
  • Example: x⁹ / x² = x⁷
• Radical to fractional exponent conversion
  • Example: ∛(x⁵) = x^(5/3)

Evaluating Exponents and Roots

• Example: 4³ = 4 * 4 * 4 = 64
• Example: ∛(64) = 4
• Example: ∜(16) = 2
• Simplifying fractional exponents
  • Example: 8^(5/3) = (8^(1/3))^5 = 2^5 = 32
  • Example: 16^(5/4) = (∜(16))^5 = 2^5 = 32
• Solving equations with exponents
  • Example: x^(2/3) = 16, solve for x: (16^(3/2)) = 64

Absolute Value

• |a| = a if a ≥ 0
• |a| = -a if a < 0
• Solving absolute value equations involves creating two separate equations
  • Example: |x+1| = 3 → x+1 = 3 or x+1 = -3 → x = 2 or x = -4

Fractions

• Adding fractions: common denominator
  • Example: 5 + 2/3 = 17/3
• Multiplying fractions: multiply numerators and denominators
  • Example: (3/2) * (5/6) = 5/4
• Dividing fractions: keep-change-flip
  • Example: (8/4) ÷ (3/5) = (8/4) * (5/3) = 6/5

Factoring

• Trinomials (ax² + bx + c)
  • Example: x² + 8x + 15 = (x+3)(x+5)
• Leading coefficient ≠ 1: Split middle term
  • Example: 2x² - 3x - 2 = (2x+1)(x-2)
• Difference of squares: a² - b² = (a+b)(a-b)
  • Example: x² - 25 = (x+5)(x-5)
• Perfect square trinomials: a² ± 2ab + b² = (a ± b)²

Functions

• Evaluating functions: plug in value of x
  • Example: f(x) = 3x + 5, find f(2) → f(2) = 11
• Composite functions: f(g(x)), solve inside out top-down
  • Example: f(g(2)), if g(x) = x², and f(x) = 2x + 3

Multiple Choice Problems

• Practice each concept with example problems for deeper understanding and implementation.

Lesson 2: Converting Sentences into Equations

Translating Sentences to Equations

• Break down sentences into algebraic expressions
  • Example: