Writing Algebraic Expressions

English is full of words that map directly onto mathematical operations, and learning to recognize them is like learning to read road signs — once you know them, navigation becomes automatic. For addition, the reliable signals are: 'sum,' 'increased by,' 'more than,' 'plus,' 'total,' 'added to.' For subtraction: 'difference,' 'decreased by,' 'minus,' 'fewer than,' 'less than.'

Most of these are straightforward — 'a number $n$ increased by $10$' becomes $n + 10$ in the same left-to-right order the words appear. But 'less than' is a trap, and it catches nearly every student at least once. Here's why it flips: the phrase '$7$ less than $x$' is telling you the result of starting with $x$ and removing $7$. The thing you're taking from ($x$) comes second in the English phrase but first in the math. So '$7$ less than $x$' → $x - 7$, not $7 - x$.

Compare these two concretely: if $x = 20$, then '$7$ less than $20$' is $13$. Writing $x - 7 = 20 - 7 = 13$ gives the right answer. Writing gives a negative number, which clearly doesn't match the meaning of 'less than' in this context. The wrong order isn't just a notation error — it produces a completely different (and wrong) value.

The same flip applies to 'subtracted from': '$5$ subtracted from $m$' means $m - 5$, not $5 - m$. Whenever you see 'less than' or 'subtracted from,' mentally reverse the order of the two quantities before you write the expression.

Mental model: 'less than' works like the phrase 'shorter than.' If you say 'Maria is $3$ inches shorter than Kai,' you start with Kai's height and subtract $3$ — Kai comes first in the math even though Maria came first in the sentence.

Multiplication signals include 'product,' 'times,' 'twice' (which specifically means $\times 2$), 'triple' ($\times 3$), and 'of' when it appears between a number and a variable (as in 'half of $x$,' which is $\frac{x}{2}$). Division signals include 'quotient,' 'divided by,' 'per,' and 'for every.'

Now, why don't we use the $\times$ symbol for multiplication in algebra? Because $x$ is the most common variable in algebra, and $\times$ looks almost identical to $x$ when handwritten. Writing $3 \times x$ creates genuine confusion about whether you're multiplying by a variable or just writing a symbol. The solution mathematicians settled on: put the coefficient (the number) directly before the variable with no symbol between them. So becomes . This isn't laziness — it's a deliberate notational choice that makes expressions faster to read and write as they grow in complexity.

For division, the fraction bar is preferred over the $\div$ symbol for the same readability reason: $\frac{z}{3}$ makes the structure — 'z divided by 3' — visually obvious in a way that $z \div 3$ doesn't when expressions get long. A fraction bar also interacts cleanly with the order of operations: the numerator and denominator are each implicitly grouped, so unambiguously means 'the whole quantity , divided by .'

A common mistake with coefficients: students sometimes write the variable before the number, like $x3$ instead of $3x$. Both are mathematically equivalent (multiplication is commutative), but convention places the number first because it mirrors how we speak ('three $x$') and because it makes expressions like $3x + 5$ scan left-to-right naturally.

Mental model: the coefficient is the multiplier and the variable is the thing being multiplied. Keeping the multiplier on the left is like keeping the price-per-unit on the left in a bill: '$3$ items at $x$ dollars each' → $3x$.

Many real-world phrases require more than one operation, and the order in which you apply those operations changes the result. This is where parentheses become essential — not as optional decoration, but as structural notation that changes meaning.

Consider 'twice the sum of a number $n$ and $6$.' The phrase 'the sum of $n$ and $6$' is a grouped unit — it has to be computed together before anything else happens to it. That grouping is written with parentheses: $(n + 6)$. Then 'twice' that result means multiply the entire grouped quantity by $2$: $2(n + 6)$.

Now compare this to the expression $2n + 6$, which reads as 'twice a number, plus six.' Let's see the difference at $n = 4$: $2(4 + 6) = 2(10) = 20$, versus . Same numbers, same operations, completely different answers — because the parentheses changed what the was multiplying. Without parentheses, the only multiplies . With parentheses, the multiplies the entire sum. Choosing wrong here isn't a minor slip; it produces a structurally different expression that models a different situation.

The rule for when to use parentheses: whenever a phrase contains a subgroup that must be computed before the outer operation, that subgroup needs parentheses. Phrases like 'the sum of,' 'the difference of,' or 'the quantity' in a word problem are explicit signals that a group is being formed. Treat them exactly the way you'd treat a clause in a sentence — as a unit that belongs together.

A useful test: after writing your expression, say it back in plain English. If $2(n + 6)$ reads as 'twice the sum of $n$ and $6$' and that matches the original phrase, your parentheses are right. If it reads differently, revise.

Mental model: parentheses are like the bowl in a recipe. Before you double the batter, you first mix all the ingredients inside the bowl. The bowl (parentheses) tells you what gets combined before the external operation (doubling) is applied.