Why does "less than" flip the order but "minus" does not?

Because of how English builds the comparison. "5 less than n" describes a number that sits 5 below n, so n is the starting point and you subtract from it: n - 5. "n minus 5" is read straight across left to right and gives the same answer, but the words come in the other order. The safest test is substitution: pick n = 20, decide in plain language what the answer should be, and see which expression produces it.

Is 3n the same as n3?

They mean the same product because multiplication is commutative, but mathematicians always write the number first: 3n. Writing n3 is unconventional and easy to misread as a different notation. Get in the habit of putting the coefficient in front of the variable now, since it makes combining like terms in later lessons much easier to see.

How do I know when an expression needs parentheses?

Parentheses are needed when a whole sum or difference is being multiplied or divided. Signal words are "the sum of," "the difference of," "the quantity," and "total." "Twice the sum of x and 6" becomes 2(x + 6) because the addition happens first. If the phrase instead says "twice a number, increased by 6," the doubling happens first and no parentheses are needed: 2x + 6.

What is the difference between a term and a factor?

Terms are added or subtracted; factors are multiplied. In 5x + 7, the terms are 5x and 7. Inside the term 5x, the factors are 5 and x. Keeping these straight matters later: you combine like terms by adding, but you factor an expression by pulling out a common multiplier.

M7MATH-5.1

Writing Algebraic Expressions

Learn to define a variable and turn word phrases and real situations into algebraic expressions, name terms, coefficients and constants, and handle tricky "less than" phrases.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Writing Algebraic Expressions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Writing Algebraic Expressions free

What this lesson covers

Algebra starts with a simple move: giving a name to a number you do not know yet. Once you call that unknown

n

h

c

, you can describe a whole situation in one short line instead of a paragraph. That line is an algebraic expression.

In this lesson you will define variables clearly, translate everyday phrases into symbols, and name the parts of an expression — its terms, coefficients, and constants. You will also learn to slow down on two order-sensitive phrases that trip up almost everyone the first time: "less than" and "divided into." Both of them flip the order of the numbers compared to how the words are read. Getting these right now makes the next lessons in this unit — evaluating expressions, combining like terms, and factoring — much smoother, because you will already trust what your expression means.

Defining a Variable: Say What It Stands For

A variable is a letter that represents a quantity you do not know or a quantity that can change. Writing

n

by itself is not enough — you have to say what

n

means, including units when there are any. Compare these two starts to the same problem:

"Let

n

= Maya's number" is weak if the problem is about money and hours. "Let

h

= the number of hours Maya worked" is strong, because anyone reading it knows exactly what to plug in later.

A few habits that make your work clear:

Choose a letter that hints at the quantity —

h

for hours,

c

for cost,

b

for books. Avoid the letter

l

(looks like 1) and

o

(looks like 0).

Define the variable as a number, not a thing. Write "let

b

= the number of books" rather than "let

b

= books." This matters because you multiply and add numbers, not objects.

Use only one variable per unknown quantity. If a problem has two unknowns that are related — say, one number is 5 more than another — pick a variable for one of them and describe the other in terms of it:

x

and

x + 5

.

Where students go wrong: they skip the definition entirely and jump to symbols. Then, three steps later, they cannot remember whether

t

meant total cost or total time. Writing the definition takes ten seconds and prevents the most common source of confusion in the whole unit.

Translating Words into Symbols

Most word phrases fall into predictable families. The table below groups the signal words by operation.

Operation	Words and phrases	Example phrase	Expression
Addition	sum, plus, more than, increased by, total, together	7 more than a number $n$	$n + 7$
Subtraction	difference, minus, less than, decreased by, fewer than, take away	a number $n$ decreased by 4	$n - 4$
Multiplication	product, times, of, twice, triple, each, per	triple a number $n$	$3n$
Division	quotient, per, split equally, divided by, divided into, half of	a number $n$ split among 5	$\frac{n}{5}$

Addition and multiplication are commutative, so order does not matter for them: "7 more than

n

" and "

n

plus 7" both give

n + 7

. Subtraction and division are not commutative —

9 - 4

is not

4 - 9

— so order is everything.

Watch for phrases that need grouping symbols. "Twice the sum of a number and 3" means you add first, then double:

2(n + 3)

. Compare that with "twice a number, plus 3," which is

2n + 3

. Whenever you see "the sum of..." or "the difference of..." appearing inside another operation, that inner phrase gets parentheses.

One more habit: read your finished expression back in words. If "

2n + 3

" reads back as "double the number, then add 3" and the problem said to add first, you have caught your own mistake before anyone else does.

The Order-Sensitive Phrases: "Less Than" and "Divided Into"

Two phrases reverse the order in which the numbers appear. They cause more errors than any other part of this lesson, so learn them as special cases.

"Less than" reverses. "5 less than a number

n

" means you start with

n

and take 5 away:

n - 5

. The very natural-looking answer

5 - n

is wrong. Test it with real numbers: if a number is 20, then 5 less than it is 15, and

20 - 5 = 15

works while

5 - 20 = -15

does not. "Fewer than" behaves the same way. So do "subtracted from" phrases: "8 subtracted from

x

" is

x - 8

.

Compare that with "less" or "minus" without the word "than," which do not reverse: "

n

less 5" and "

n

minus 5" both give

n - 5

— same answer, but reached by reading left to right.

"Divided into" reverses too. "A number

n

divided into 6 equal groups" is fine as

\frac{n}{6}

, but the phrase "6 divided into

n

" means

n \div 6

, or

\frac{n}{6}

— the 6 is doing the dividing. Think of long division: when you say "6 divided into 24," you are asking how many 6s fit in 24, which is

24 \div 6 = 4

. "Divided by," on the other hand, reads straight through: "

n

divided by 6" is

\frac{n}{6}

Phrase	Expression	Reversed?
5 less than $n$	$n - 5$	yes
$n$ less than 5	$5 - n$	yes
5 subtracted from $n$	$n - 5$	yes
$n$ minus 5	$n - 5$	no
$n$ divided by 5	$\frac{n}{5}$	no
5 divided into $n$	$\frac{n}{5}$	yes

The reliable check for all of these: substitute a friendly number and ask whether the result makes sense in the story.

Naming the Parts: Terms, Coefficients, and Constants

Once an expression is written, you should be able to describe its structure. Consider

4x + 7y - 9

.

A term is a part of the expression separated by plus or minus signs. Here there are three terms:

4x

7y

, and

-9

. Notice that the sign in front travels with the term, so the third term is negative nine, not nine.

A coefficient is the number multiplied by a variable in a term. The coefficient of

x

is 4; the coefficient of

y

is 7. When a variable appears alone, as in

x + 6

, its coefficient is 1, because

x

means

1 \cdot x

. In

-m

, the coefficient is

-1

.

A constant is a term with no variable — a fixed number. In

4x + 7y - 9

, the constant is

-9

.

A factor is different from a term: factors are multiplied. In the term

4x

, the factors are 4 and

x

. In

3(n + 2)

, the factors are 3 and the quantity

(n + 2)

Expression	Number of terms	Coefficients	Constant
$5n + 12$	2	5	12
$x - 8$	2	1	$-8$
$\frac{p}{3}$	1	$\frac{1}{3}$	none
$9 - 2w + w$	3	$-2$ and $1$	9

That third row surprises people:

\frac{p}{3}

is the same as

\frac{1}{3}p

, so its coefficient is

\frac{1}{3}

. Being able to see a fraction as multiplication by a unit fraction is exactly the skill you will need when you start combining like terms.

Expressions from Real Situations

Real-world problems usually contain a rate (something that repeats) and sometimes a one-time amount. The rate becomes a coefficient; the one-time amount becomes a constant.

A bike-share charges a 4 dollar unlock fee plus 2 dollars for every 10 minutes. Let

t

= the number of 10-minute blocks used. Cost in dollars:

2t + 4

. The 2 is the coefficient because it repeats with each block; the 4 is the constant because it happens once no matter what.

A cafeteria has 250 milk cartons and sells

c

cartons. Remaining:

250 - c

. Here the constant comes first because you start with 250 and subtract.

Three students share

p

pretzels equally, and each also brought 2 of their own. Each student's total:

\frac{p}{3} + 2

.

When you translate a situation, ask three questions in order. What is unknown, and what letter will name it? Does something repeat — that is a multiplication. Is there a fixed starting or ending amount — that is a constant.

Where students go wrong: attaching the constant to the repeating part, writing

2(t + 4)

for the bike-share example. Read it back: that would charge 4 extra dollars per block, not once. Substituting

t = 1

gives 12 dollars instead of the correct 6 dollars, which exposes the error immediately.

Key terms

Variable.: A letter that stands for an unknown number or a number that can change, such as $h$ for the number of hours worked.
Algebraic expression.: A combination of numbers, variables, and operations with no equal sign, such as $3n - 5$ .
Term.: A part of an expression separated by addition or subtraction signs; in $6x - 4$ the terms are $6x$ and $-4$ .
Coefficient.: The number multiplied by a variable in a term. In $7y$ the coefficient is 7; in $x$ it is 1; in $\frac{p}{3}$ it is $\frac{1}{3}$ .
Constant.: A term with no variable, so its value never changes, such as the $-9$ in $2x - 9$ .
Factor.: A quantity being multiplied. In $4x$ the factors are 4 and $x$ ; in $5(n+1)$ they are 5 and $(n+1)$ .
Commutative.: A property of addition and multiplication meaning order does not change the result; subtraction and division are not commutative.
Coefficient of 1.: The understood multiplier on a variable written alone, since $x$ means $1 \cdot x$ and $-x$ means $-1 \cdot x$ .

Worked example

A rock-climbing gym charges a 15 dollar day pass plus 6 dollars for each hour of wall time. Ravi buys a day pass, climbs for some hours, and then a 3 dollar equipment credit is subtracted from his bill. Define a variable, write an expression for his total cost in dollars, then name the terms, coefficients, and constants.

Step 1 — Define the variable. The unknown quantity is how long Ravi climbs, so let

h

= the number of hours of wall time. Notice this is defined as a number of hours, not as "time."

Step 2 — Find the repeating part. The 6 dollars happens once per hour, so it repeats: that is

6 \times h = 6h

. The 6 becomes a coefficient.

Step 3 — Find the one-time amounts. The day pass is 15 dollars no matter how long he climbs, so it is

+15

. The equipment credit is 3 dollars taken off once, so it is

-3

.

Step 4 — Assemble the expression. Total cost in dollars:

6h + 15 - 3

. You may leave it in this form to show the story, or simplify the two constants to get

6h + 12

.

Step 5 — Check by substituting. If Ravi climbs 2 hours, the expression gives

6(2) + 12 = 24

dollars. Checking against the story: 15 for the pass, 12 for two hours, minus 3 credit, which is

15 + 12 - 3 = 24

dollars. The expression matches.

Step 6 — Name the parts of

6h + 12

. There are 2 terms:

6h

and

12

. The coefficient of

h

is 6. The constant is 12. In the term

6h

, the factors are 6 and

h

.

A common wrong answer here is

6(h + 15) - 3

, which would charge the day pass for every single hour. Substituting

h = 2

gives 123 dollars — far off — which is exactly why the substitution check in Step 5 is worth doing every time.

Practice questions

Which expression represents "9 less than the product of 4 and a number

n

$9 - 4n$
$4n - 9$
$4(n - 9)$
$\frac{4n}{9}$

Answer: $4n - 9$

Work from the inside out. "The product of 4 and a number" is

4n

. Then "9 less than" that product reverses the reading order: you start with

4n

and subtract 9, giving

4n - 9

. The choice

9 - 4n

is the classic reversal error — check it with

n = 5

: 9 less than the product

20

should be 11, and

4(5) - 9 = 11

, while

9 - 20 = -11

. The choice

4(n-9)

subtracts first and then multiplies, which the wording does not say.

For the expression

8 - m + \frac{k}{2}

, state the number of terms, the coefficient of each variable, and the constant.

Answer: Three terms: $8$ , $-m$ , and $\frac{k}{2}$ . The coefficient of $m$ is $-1$ and the coefficient of $k$ is $\frac{1}{2}$ . The constant is 8.

Terms are separated by the plus and minus signs, and each sign belongs to the term that follows it, so the second term is

-m

, not

m

. A variable written alone has an understood multiplier of 1, so with the negative sign the coefficient of

m

-1

. Dividing by 2 is the same as multiplying by

\frac{1}{2}

, so

\frac{k}{2} = \frac{1}{2}k

and the coefficient of

k

\frac{1}{2}

. The only term without a variable is 8, so that is the constant.

A phone plan costs 25 dollars per month plus 10 cents for each text message sent. Define a variable and write an expression for the monthly cost in dollars. Then explain why the expression is not

25(t + 0.10)

Answer: Let $t$ = the number of text messages sent in the month. Monthly cost in dollars: $0.10t + 25$ . The form $25(t + 0.10)$ is wrong because it multiplies the monthly fee by the number of texts.

The 10 cents repeats once per text, so it becomes the coefficient:

0.10t

. The 25 dollars happens once per month regardless of texting, so it is the constant. Substituting

t = 100

in the correct expression gives

0.10(100) + 25 = 35

dollars, which matches 10 dollars of texts plus the 25 dollar fee. The same substitution in

25(t + 0.10)

gives

25(100.10) = 2502.50

dollars, which is nowhere near the real bill — a quick sign that the parentheses attach the fee to the wrong part of the situation.

FAQ

Why does "less than" flip the order but "minus" does not?: Because of how English builds the comparison. "5 less than $n$ " describes a number that sits 5 below $n$ , so $n$ is the starting point and you subtract from it: $n - 5$ . " $n$ minus 5" is read straight across left to right and gives the same answer, but the words come in the other order. The safest test is substitution: pick $n = 20$ , decide in plain language what the answer should be, and see which expression produces it.
Is $3n$ the same as $n3$ ?: They mean the same product because multiplication is commutative, but mathematicians always write the number first: $3n$ . Writing $n3$ is unconventional and easy to misread as a different notation. Get in the habit of putting the coefficient in front of the variable now, since it makes combining like terms in later lessons much easier to see.
How do I know when an expression needs parentheses?: Parentheses are needed when a whole sum or difference is being multiplied or divided. Signal words are "the sum of," "the difference of," "the quantity," and "total." "Twice the sum of $x$ and 6" becomes $2(x + 6)$ because the addition happens first. If the phrase instead says "twice a number, increased by 6," the doubling happens first and no parentheses are needed: $2x + 6$ .
What is the difference between a term and a factor?: Terms are added or subtracted; factors are multiplied. In $5x + 7$ , the terms are $5x$ and 7. Inside the term $5x$ , the factors are 5 and $x$ . Keeping these straight matters later: you combine like terms by adding, but you factor an expression by pulling out a common multiplier.

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