Algebra 1 Unit 1: Expressions & Operations

Essential Question for the Unit:

"How can algebraic expressions be used to represent and evaluate real-world situations?"

Weekly Pacing at a Glance

  • Day 1: Translate verbal quantitative situations into algebraic expressions.
  • Day 2: Evaluate algebraic expressions using substitution and the order of operations.
  • Day 3: Derive and apply the laws of exponents (product, quotient, power rules).
  • Day 4: Simplify multivariable and monomial expressions using all laws of exponents.

1. Translating Verbal to Algebraic Expressions

Translating words into mathematical symbols is a crucial first step in solving problems.

Verbal Phrase Operation Example
Sum, plus, more than, increased by Addition (+) The sum of a number and 5 → n + 5
Difference, minus, less than, decreased by Subtraction (-) 7 less than a number → x - 7 (Order matters!)
Product, times, of Multiplication (⋅, (), x) The product of 4 and y → 4y
Quotient, divided by, ratio of Division (/, ÷) The quotient of z and 10 → z / 10
Squared, to the power of 2 Exponent (²) A number squared → b²

2. Evaluating Expressions & Order of Operations

To evaluate an expression, substitute the given numbers for the variables and simplify using the Order of Operations.

Order of Operations (PEMDAS)

  1. Parentheses (or any grouping symbols like | |, √ )
  2. Exponents
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

Worked Example:

Evaluate 3x² - |y - 10| when x = -2 and y = 4.

  • Substitute: 3(-2)² - |4 - 10|
  • Parentheses/Grouping: 3(-2)² - |-6| → 3(-2)² - 6
  • Exponents: 3(4) - 6
  • Multiplication: 12 - 6
  • Subtraction: 6

3. Laws of Exponents

These rules help simplify expressions with exponents. Assume a, b ≠ 0.

Law Rule Example
Product of Powers aᵐ ⋅ aⁿ = aᵐ⁺ⁿ x⁵ ⋅ x² = x⁵⁺² = x⁷
Quotient of Powers aᵐ / aⁿ = aᵐ⁻ⁿ y⁸ / y³ = y⁸⁻³ = y⁵
Power of a Power (aᵐ)ⁿ = aᵐⁿ (b⁴)² = b⁴*² = b⁸
Power of a Product (ab)ᵐ = aᵐbᵐ (2z)³ = 2³z³ = 8z³
Power of a Quotient (a/b)ᵐ = aᵐ/bᵐ (c/5)² = c²/5² = c²/25
Zero Exponent a⁰ = 1 (-5)⁰ = 1
Negative Exponent a⁻ᵐ = 1/aᵐ d⁻³ = 1/d³

Key Vocabulary

  • Expression: A mix of numbers, variables, and operations (no equal sign).
  • Variable: A symbol (like x) that represents an unknown number.
  • Coefficient: The number multiplied in front of a variable.
  • Term: Parts of an expression separated by + or -.
  • Evaluate: To find the numerical value of an expression.
  • Base: The number being multiplied in an exponential expression.
  • Exponent: Shows how many times to multiply the base by itself.
  • Absolute Value: A number's distance from zero.