Lesson 1.1
Variables & Simple Phrases
Before you can solve equations, you need to speak the language. In this lesson you'll learn how to translate everyday English words like "sum," "difference," and "product" into clean algebraic expressions.
Introduction
Algebra is a language. Just like every language has vocabulary, algebra has special words that tell you which operation to perform. A variable is a letter that represents an unknown number, and an algebraic expression is a phrase built from numbers, variables, and operations.
Past Knowledge
You know how to add, subtract, multiply, and divide whole numbers and simple fractions.
Today's Goal
Translate English phrases into algebraic expressions using the four key words: sum, difference, product, quotient.
Future Success
Every equation you'll ever solve starts as an expression. This skill is the foundation of all of algebra.
Key Concepts
1. What Is a Variable?
A variable is a letter (usually , , or ) that stands for a number we don't know yet. Think of it as a blank space waiting to be filled in.
2. The Four Key Words
Each arithmetic operation has a set of English words that map to it. Here's your translation dictionary:
| Operation | Key Word | English Phrase | Algebra |
|---|---|---|---|
| Addition | Sum | "The sum of a number and 5" | |
| Subtraction | Difference | "The difference of a number and 3" | |
| Multiplication | Product | "The product of 7 and a number" | |
| Division | Quotient | "The quotient of a number and 4" |
3. Other Common Words
Addition words: sum, plus, increased by, more than, total, added to
Subtraction words: difference, minus, decreased by, fewer than, less
Multiplication words: product, times, of, twice, double, triple
Division words: quotient, divided by, ratio, per, split into
Worked Examples
Example 1: Single-Operation Phrases
BasicTranslate each phrase into an algebraic expression.
"The sum of a number and 12"
"Sum" means addition. Let the unknown number be .
"The product of 5 and a number"
"Product" means multiplication. We write the number in front of the variable.
"The quotient of a number and 9"
"Quotient" means division. The first quantity goes on top.
Example 2: Two-Operation Phrases
IntermediateTranslate: "Three more than twice a number."
Identify the Pieces
"Twice a number" → (multiplication first). "Three more than" → add .
Build the Expression
Start with the multiplication, then add: "more than" means the is added on.
Example 3: Multi-Operation Phrase
AdvancedTranslate: "The quotient of the sum of a number and 7, and 4."
Work from the Inside Out
"The sum of a number and 7" translates to . This is a group inside the larger phrase.
Apply the Outer Operation
"The quotient of ... and 4" means divide the entire first group by .
Common Pitfalls
Writing "5x" as "x5"
In algebra, the number always goes in front of the variable. Write , never .
Forgetting Multiplication Is Invisible
"The product of 3 and a number" is , not . The multiplication sign is implied when a number and variable sit together.
Real-Life Applications
Every time you see a price tag that says "$5 per item," you're seeing an algebraic expression in disguise: the total cost is , where is the number of items. Translating words into math is the skill that lets scientists, engineers, and economists build the formulas and models that power modern technology.
Practice Quiz
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