Operations & Algebraic Thinking

Writing Numerical Expressions

Translating a verbal description into a numerical expression.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Order of Operations

Turning Words into an Expression

Suppose someone tells you: 'add 8 and 7, then multiply the result by 2.' Before reading further, try writing that as a numerical expression yourself. Did you write 8 + 7 × 2, or (8 + 7) × 2? Only one of them actually says what the sentence describes — which one, and why?

Writing 8 + 7 × 2 without parentheses would multiply only the 7 by 2 first, following the usual order of operations — that's not what the sentence said. The sentence describes finding a sum first, and then multiplying that whole sum, so it needs parentheses: (8 + 7) × 2. Certain words in a description are strong signals for exactly this kind of grouping.

Phrases like 'the sum of,' 'the difference of,' or 'the result' followed by another operation usually mean that a calculation must be treated as one finished quantity before anything else happens to it — which is exactly what parentheses are for.

Worked Example — Translating a Phrase with a Grouped Result

Write an expression for: 'subtract 5 from 12, then divide the result by the sum of 3 and 4.' Subtracting 5 from 12 is (12 − 5). The sum of 3 and 4 is (3 + 4). Dividing the first result by the second gives (12 − 5) ÷ (3 + 4).

Worked Example — Translating a Phrase Without Extra Grouping

Write an expression for: 'multiply 6 by 9, then add 4.' Multiplying 6 by 9 is 6 × 9. The word 'then add' attaches directly to that product without needing the product itself grouped further, so no extra parentheses are needed: 6 × 9 + 4.

Tip

Read a description one operation at a time, in the order it's described, and ask after each step: does this result get used as a single combined quantity in the next operation? If yes, wrap it in parentheses.

Common Mistakes

Writing every calculation in a word description with parentheses, even when the natural order of operations already produces the right result without them.
Add parentheses only where the description genuinely requires a step to happen before what the order of operations would normally do — extra parentheses aren't wrong, but check they're actually needed by comparing against what the sentence means.
Reversing the order of a subtraction or division phrase, such as writing '5 − 12' for 'subtract 5 from 12.'
'Subtract A from B' always means B − A, not A − B — the number being subtracted from comes second in the phrase but first in the expression.

Key Takeaways

Translating a verbal description into an expression means matching each phrase to an operation, in the order described.
Phrases like 'the sum of' or 'the result' often signal that parentheses are needed around that part of the calculation.
'Subtract A from B' translates to B − A, not A − B.

Summary

Writing numerical expressions turns a description into precise mathematical notation, using the order of operations deliberately. The next lesson uses two related rules to generate two number patterns at once, and compares what happens between them.

Track your progress

← Order of Operations Back to Operations & Algebraic Thinking Generating Numerical Patterns→