Words into Symbols
Translating a situation into algebra means defining the variable precisely, converting the words operation by operation, and reading the result back as the original sentence. · 9 min
Most algebra in the wild arrives as words: a price, a fee, a number nobody has told you yet. Turning those words into symbols is a skill with three steps, and the first is the one most often skipped. Before anything else, define the variable precisely — 'let n be the number of tickets', never 'let n be tickets'. A letter stands for a number, so its definition must name a number.
Guess before you learn
Translate the phrase '7 less than n' into symbols.
'7 less than n' starts at n and drops by 7: n − 7. If you wrote 7 − n, keep the pencil mark — the phrase lists the 7 first, and nearly everyone follows the word order the first time. 'Less than' is the classic order-reversing phrase, and it gets special attention below.
9–12
3–5
Words carry operation clues. 'Sum' and 'more than' mean add. 'Difference' and 'fewer than' mean subtract. 'Twice' and 'times' mean multiply. 'Shared equally' means divide. So 'twice the number, plus 3' becomes 2n + 3.
One clue needs care: '3 fewer than the number' means start with the number and take 3 away — n − 3, with the n written first even though the words said 3 first.
6–8
The discipline has three steps. One: define the variable precisely — 'let n be the number of tickets', a number, with its unit understood. Two: translate operation by operation, phrase by phrase. Three: read the finished symbols back as a sentence and check it says what the problem said.
Watch the order-reversers: '7 less than n' is n − 7, and '9 subtracted from n' is n − 9 — both name the 7 or 9 first but subtract it second. And the word 'is' writes the equals sign, turning an expression into an equation: 'twice n is 14' becomes 2n = 14.
9–12
Precision in step one prevents a famous error. 'There are six times as many students as professors.' Define s as the number of students and p as the number of professors; the equation is s = 6p. Writers who let the letters stand for the people themselves reliably produce 6s = p — the multiplier lands on the wrong side because the letters were never numbers.
A translation is also testable. Substitute an easy case: 2 professors should mean 12 students, and s = 6p passes while 6s = p fails. Keep the expression–equation distinction sharp too: 9n + 4 describes a cost; 9n + 4 = 40 claims that cost equals 40 and can be solved.
K–2
'Two more than the hidden number.' The hidden number is n. Two more means add 2. Write n + 2. The words told you what to write.
Now read it back: n + 2 says 'the hidden number, then two more.' The words and the symbols tell the same story.
Undergrad
Translation is modeling in miniature: choose the quantities, assign symbols, encode the stated relations. The students-and-professors error is a type error — treating a symbol as a label for objects rather than a measure of them. Units are the compile-time check: in s = 6p, both sides count people; an equation that adds dollars to tickets has already failed.
One situation supports several correct models: total cost as C = 9n + 4, or the implicit relation C − 9n = 4, or a table of pairs. They are linked by algebraic rewriting, and choosing the form that makes the question easy is half the craft of applied mathematics.
Postgrad
Formalization maps ambiguous, context-laden natural language into a compositional formal language: fix a signature — constants, function symbols, relations — then interpret the prose inside it. The hard cases are exactly the non-compositional ones: 'less than' as subtraction versus inequality, and quantifier scope ('every ticket has a fee') that prose leaves unresolved.
The read-back test of step three is a round trip: prose to formula to prose, checking the translation is faithful. Automated word-problem solvers stumble precisely where the mapping breaks compositionality — which is why the schoolroom habit of reading the symbols back aloud is not a formality but the whole verification.
defining the variable
The written sentence that pins the letter to a number: 'let n be the number of tickets.' Every translation starts here.
Whole sentences translate the same way, one phrase at a time — and the word 'is' marks the spot where the equals sign goes. When the symbols are assembled, run the read-back test: say the expression out loud in plain words and check it matches the original sentence. A translation you cannot read back is not finished. Try the full pipeline on a running example: tickets cost $9 each, plus a single $4 booking fee.
Translate: 'Three times a number, decreased by 5, is 16.' — the steps fade as you master them
Let n be the number.
3n
3n − 5
3n − 5 = 16
Why is this true?
Why must the variable be defined as a number rather than an object?
Because every operation in the expression — multiplying, adding, comparing — acts on numbers. A letter defined as 'tickets' has nothing for 9 × to act on; a letter defined as 'the number of tickets' does, and the units of every later line stay checkable.
Define the letter as a number, translate phrase by phrase with an eye on the order-reversers, and read the result back. Notice what the worked example produced: 3n − 5 = 16 — not just an expression but an equation, a sentence with a claim in it. Next folio takes up exactly that claim: what the equals sign promises, and how to use the promise to find the number.
Practice — new ink and old, interleaved
1.'Less than' and 'subtracted from' share a trap. Without looking back: what is it?
Both phrases reverse the written order — '7 less than n' and '7 subtracted from n' each mean n − 7, with the named number subtracted second.
How close were you? Grade yourself honestly — it sets your review date.
2.A pencil costs 2 dollars. In one sentence with an expression in it: what do p pencils cost, and why?
3.Which pair is a pair of like terms?
4.Translate: '12 less than twice a number.'
5.Without looking back: what is a variable, and what does 7m mean?
A variable is a letter standing for a number not yet fixed; 7m means 7 × m.
How close were you? Grade yourself honestly — it sets your review date.
6.Using the ticket translation 9n + 4, find the total cost of 7 tickets.
7.Put the steps for translating 'Twice a number, increased by 9, is 31' in order.
- Let n be the number
- Twice the number: 2n
- Increased by 9: 2n + 9
- 'Is 31' writes the equation: 2n + 9 = 31