1. Why Do We Need Variables?

The world is full of quantities we do not know yet — variables let us reason about them.

• How much should I charge for something to make a profit?
• How long will it take to drive there at this speed?
• If I save x dollars per week, when will I have enough for a new phone?
• How does temperature affect a chemical reaction?
• Why does a basketball arc the way it does?

2. What Is Pre-Algebra?

Pre-algebra bridges arithmetic and algebra by introducing variables, expressions, equations, and abstract reasoning.

• Variable: a letter representing an unknown or changing quantity
• Expression: a math phrase with numbers and variables (3x + 5)
• Equation: two expressions set equal to each other (3x + 5 = 20)
• Solving: finding the value that makes an equation true
• Pattern: a rule that describes how a sequence or relationship works

3. The History of Algebra

The ideas in pre-algebra were developed over thousands of years across multiple civilizations.

• ~1700 BCE — Babylonian tablets show solved equations without using letters
• 820 CE — Al-Khwarizmi writes "Al-Kitab al-mukhtasar fi hisab al-jabr" — the word "algebra" comes from "al-jabr"
• 1557 — Robert Recorde invents the equals sign (=)
• 1600s — Descartes introduces the x, y, z notation we still use today
• Modern algebra is used in every quantitative field from physics to economics

4. The Order of Operations

Mathematics has rules for which operations to perform in what order — without them, expressions are ambiguous.

• P — Parentheses first: whatever is inside ( ) or [ ]
• E — Exponents next: powers and roots
• MD — Multiplication and Division, left to right
• AS — Addition and Subtraction, left to right
• Remember: PEMDAS (or "Please Excuse My Dear Aunt Sally")

5. Properties of Numbers

Several mathematical properties describe how numbers behave — knowing them makes algebra much easier.

• Commutative: a + b = b + a and a × b = b × a (order does not matter for + and ×)
• Associative: (a + b) + c = a + (b + c) (grouping does not matter for + and ×)
• Distributive: a(b + c) = ab + ac (multiply across addition)
• Identity: a + 0 = a and a × 1 = a
• Inverse: a + (−a) = 0 and a × (1/a) = 1

6. Key Pre-Algebra Concepts

The main topics of pre-algebra, each building on the last.

• Integers: positive and negative whole numbers on the number line
• Fractions and decimals: parts of wholes; converting between forms
• Ratios and proportions: comparing quantities; scaling up and down
• Percentages: fractions out of 100; used everywhere in real life
• Negative numbers: temperatures, sea level, debt, coordinates

7. Pre-Algebra in Real Life

Variables and proportional reasoning appear constantly in everyday decisions.

• Recipes: if a recipe serves 4 and you need to serve 10, how much of each ingredient?
• Sales tax: if something costs $24.99 and tax is 8%, what is the total?
• Discounts: a $80 jacket is 25% off — how much does it cost?
• Speed: you drive 65 mph for 3 hours — how far did you go?
• Savings: you have $40 and save $15 per week — after how many weeks will you have $100?

8. Arithmetic Thinking vs. Algebraic Thinking

Two fundamentally different ways of approaching math problems.

9. Equations vs. Expressions

Two things that look similar but are fundamentally different.

10. Common Pre-Algebra Mistakes

The errors that most reliably cause problems — and how to avoid them.

• Forgetting the order of operations: always PEMDAS
• Mixing up expressions and equations
• Losing the negative sign when distributing: −2(x − 3) = −2x + 6, not −2x − 6
• Dividing instead of multiplying when scaling recipes or maps
• Treating percent as a whole number: 25% = 0.25, not 25

11. Getting Good at Pre-Algebra

The habits that make the difference between struggling and succeeding.

• Practice a little every day: math is a skill, not a subject to memorize
• Write out every step — do not skip in your head
• Check every answer by substituting back into the original equation
• When stuck, simplify: make up numbers to understand the pattern
• Ask "does this answer make sense?" before moving on

12. Solve Your First Equation

Work through a complete equation step by step.

• Problem: You want to buy a $75 pair of shoes. You have $30 saved, and you earn $9 per hour babysitting. How many hours do you need to work?
• Set up the equation: 30 + 9h = 75
• Subtract 30 from both sides: 9h = 45
• Divide both sides by 9: h = 5
• Check: 30 + 9(5) = 30 + 45 = 75 ✓

13. What We Covered

Pre-algebra is the bridge from arithmetic to algebra — from calculating specific numbers to reasoning about unknowns.

• Variables represent unknown or changing quantities
• Expressions are math phrases; equations are math sentences with equals signs
• Order of operations (PEMDAS) prevents ambiguity
• Key topics: integers, fractions, ratios, percentages, negative numbers
• Solve equations by doing the same operation to both sides, then check your answer

Mastery quiz

1. What is a variable in pre-algebra?
   • A number that never changes
   • A letter that stands in for a number you don't know yet
   • A type of fraction
   • The answer to an equation
2. Which of these is an equation, not just an expression?
   • 3x + 5
   • 3x + 5 = 20
   • 2 + 3 × 4
   • one-half
3. In the order of operations (PEMDAS), which is done FIRST?
   • Addition
   • Multiplication
   • Parentheses
   • Subtraction
4. You have 30 dollars and earn 9 dollars per hour. To reach 75 dollars, the equation is 30 + 9h = 75. How many hours must you work?
   • 3 hours
   • 4 hours
   • 5 hours
   • 9 hours