Overview
This lecture covers foundational pre-algebra topics, including integer operations, order of operations, simplifying algebraic expressions, solving equations, exponents, factoring, fractions, percentages, and similar triangles.
Adding and Subtracting Integers
- Use a number line: move right to add, left to subtract.
- Adding two negatives results in a more negative number; subtracting a negative is equivalent to adding.
Multiplication and Division of Integers
- Multiplication is repeated addition; negative × positive = negative; negative × negative = positive.
- Division is the inverse of multiplication; negative ÷ positive = negative; negative ÷ negative = positive.
Order of Operations (PEMDAS)
- Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
- Always perform operations inside parentheses first.
Evaluating and Simplifying Algebraic Expressions
- Substitute given values for variables before simplifying.
- Use the distributive property: a(b + c) = ab + ac.
- Combine like terms (same variable and exponent).
Solving Linear Equations
- Isolate the variable using inverse operations (addition/subtraction, multiplication/division).
- For equations with fractions, multiply by the common denominator to clear denominators.
- Cross-multiply to solve proportions.
Exponents and Negative Exponents
- Exponent indicates repeated multiplication; a^n = a × a × ... × a (n times).
- Negative exponent: a^(-n) = 1 / a^n.
Factoring Monomials and GCF
- Factor numbers and variables into primes and lowest terms.
- GCF is the highest product of common prime factors in all terms.
Simplifying Fractions and Monomials
- Cancel common factors in numerator and denominator.
- Subtract exponents when dividing terms with the same base.
Multiplying and Dividing Monomials
- When multiplying, add exponents of like bases.
- When dividing, subtract exponents of like bases.
Percentages
- Percent means "per hundred;" convert percent to decimal by dividing by 100.
- To find n% of a number, multiply the number by n/100.
Similar Triangles and Proportions
- Setup proportions using corresponding sides: side1/side2 = side3/side4.
- Solve for the unknown by cross-multiplying and isolating the variable.
Key Terms & Definitions
- Integer — A whole number (positive, negative, or zero).
- PEMDAS — Order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
- Distributive Property — a(b + c) = ab + ac.
- Greatest Common Factor (GCF) — The largest factor shared by two or more numbers.
- Exponent — Indicates how many times a number is multiplied by itself.
- Monomial — An algebraic expression with one term.
- Proportion — An equation stating two ratios are equal.
Action Items / Next Steps
- Practice problems on integer operations, order of operations, and solving equations.
- Review percent calculations and proportions with similar triangles.
- Study and memorize multiplication tables and basic exponent rules.