Writing Expressions from Words
The world speaks in sentences. Algebra listens, then translates.
Why This Skill Exists
Real problems arrive in language, not in symbols. Before any equation can be solved, the situation must be translated from words into algebra. This lesson builds that translation muscle — the ability to hear a phrase and know exactly which algebraic structure it maps to.
twice a number”
5 more = + 5
The Vocabulary of Operations
Every operation has a family of English trigger words. Recognising them is the core skill of this lesson.
| Operation | Symbol | Trigger Words & Phrases |
|---|---|---|
| Addition | + | plus, sum, more than, increased by, added to, total, combined, exceeds, greater than, gain |
| Subtraction | − | minus, difference, less than, decreased by, subtracted from, reduced by, fewer, lost, shorter than, taken from |
| Multiplication | × | times, product, multiplied by, twice (×2), triple (×3), double, of (with fractions/%), per, every |
| Division | ÷ | divided by, quotient, ratio, per, split equally, half (÷2), a third (÷3), shared among |
| Power | xⁿ | squared (²), cubed (³), to the power of, raised to |
Some phrases reverse the order from how you read them. These require particular care:
Choosing a Variable
Before translating, you must define your variable. Write a clear statement of what the variable represents — including its units where relevant. This statement anchors the entire expression.
| Situation | Definition | Variable |
|---|---|---|
| Unknown number | Let n = the number | n |
| Price of an item | Let p = the price in rands | p |
| Time elapsed | Let t = time in hours | t |
| Person’s age | Let a = Maya’s age in years | a |
| Length of side | Let s = side length in cm | s |
Once the primary variable is defined, related quantities can often be expressed in terms of it — removing the need for additional variables:
| English | Expression (let n = the number) |
|---|---|
| Three times the number | 3n |
| Four less than the number | n − 4 |
| The next consecutive integer | n + 1 |
| The square of the number | n² |
| Half the number, increased by 3 | n/2 + 3 |
Single-Operation Translations
| Phrase | Translation | Reasoning |
|---|---|---|
| “the sum of x and 9” | x + 9 | “sum” → addition |
| “twelve decreased by y” | 12 − y | “decreased by” → subtract; 12 is the base |
| “the product of 7 and m” | 7m | “product” → multiply |
| “k divided by 5” | k/5 | “divided by” → k is the numerator |
| “n less than 20” | 20 − n | “less than” reverses order: 20 is base, n is subtracted |
| “triple the quantity” | 3q | “triple” → multiply by 3 |
Multi-Operation Translations
Most real-world situations involve more than one operation. Translate phrase by phrase, left to right, grouping with brackets where the structure demands it.
Translating Real-World Scenarios
Word problems often describe a situation with multiple quantities. The method is always: define the variable, identify related quantities, translate each part, then combine.
“Maya is 3 years older than twice her brother Luca’s age.”
“A shirt costs R80 more than half the price of a jacket.”
“A rectangle’s length is 5 cm more than three times its width. Write an expression for the perimeter.”
Translating Back — Reading an Expression
You should also be able to move in the other direction: given an algebraic expression, produce a natural English sentence that describes it. There are often several valid translations.
| Expression | One Valid Translation | Alternative |
|---|---|---|
| 2n + 7 | “seven more than twice a number” | “seven added to the product of two and n” |
| x/3 − 1 | “one less than a third of x” | “x divided by three, decreased by one” |
| (a + b)² | “the square of the sum of a and b” | “the quantity a plus b, squared” |
| 4(3x − 2) | “four times the difference of three x and two” | “the product of four and the quantity 3x minus 2” |
Practice Set
Lesson Checklist
- Identify the operation signalled by key trigger words and phrases
- Define variables clearly with units and context statements
- Translate single-operation and multi-operation phrases into expressions
- Handle order-sensitive phrases (“less than”, “subtracted from”, “ratio of”)
- Use brackets correctly when a sub-expression forms a unit
- Express related quantities in terms of a single defined variable
- Write English descriptions of given algebraic expressions
The final lesson of Stage I — consolidating substitution, BODMAS, and expression-writing into a unified problem-solving workflow that prepares you for equations.