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Writing Expressions

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Writing Expressions - Complete Interactive Lesson

Part 1: From Words to Math

✍️ Writing Expressions

Part 1 of 5 — From Words to Math

Topics in This Part

Section
What an Expression Is
Why We Use Variables
Translating Simple Phrases

🔑 Key Concept: Writing an expression means turning a word phrase (like "five more than a number") into math symbols (like $n + 5$ ). It's translation — English on one side, math on the other.

What Is an Expression?

An algebraic expression is a mix of numbers, variables, and operations — and it has no equals sign.

This is an expression	This is not (it has $=$ )
$n + 5$	$n + 5 = 12$

The Four Operations and Their Keywords

Each operation has its own set of "signal words." Learning these is the heart of writing expressions.

Operation	Symbol	Keywords
Add	$+$	sum, plus, more than, increased by, total, added to
Subtract	$-$	difference, minus, less than, decreased by, fewer, take away
Multiply	$\times$ or $\cdot$

Match the Keyword to the Operation 🔽

Which operation does each word signal?

Translating a Simple Phrase

A simple phrase has one number, one variable, and one operation. Pick the variable, find the keyword, and write the symbols in order.

Example: "a number plus 7"

The unknown number → call it $n$
"plus" → $+$
the 7 → $7$

$n + 7$

Example: "the product of 4 and a number"

"product" →

Concept Check 🎯

Now You Write One

Reading translations is one thing — producing them is the real skill. Type your answers in the next drill exactly as math: a variable, a symbol, and a number.

💡 Reminder: numbers go in front of variables ( $7n$ , never $n7$ ), and there's no $\times$ sign in the final answer.

Type the Expression ⌨️

Write each simple phrase using $n$ as the variable. Use + for plus, and write a number next to the variable for multiply (like 5n).

1) "a number plus 10" $\Rightarrow$ ? 2) "the product of 7 and a number" $\Rightarrow$ ?

You've Got the Basics

You can now spot a keyword, pick a variable, and write a one-operation expression.

🔑 Carry this forward: every expression is built from these same four operations — the only thing that gets trickier is the order of the words. That's exactly what Part 2 tackles.

Part 2: Order Matters: "Less Than" & "Divided By"

✍️ Writing Expressions

Part 2 of 5 — Order Matters: "Less Than" & "Divided By"

⚠️ The #1 trap in this whole topic: with subtraction and division, the order of the words can flip the order of the symbols. "5 less than $n$ " is not $5 - n$ . This part fixes that forever.

When Order Doesn't Matter (and When It Does)

Addition and multiplication can be written in either order — the answer is the same:

$n + 5 = 5 + n \qquad 4 \cdot n = n \cdot 4$

Part 3: Two-Step Phrases & Grouping

✍️ Writing Expressions

Part 3 of 5 — Two-Step Phrases & Grouping

🔑 Leveling up: Real phrases often pack two operations into one sentence — like "3 more than twice a number." The challenge is doing them in the right order and knowing when you need parentheses.

Two Operations in One Phrase

Build the expression in the order the words describe.

Example: "3 more than twice a number"

"twice a number" → $2n$
"3 more than" that → add 3

$2n + 3$

Example: "5 less than the product of 4 and a number"

"the product of 4 and a number" →

Part 4: Real-World Expressions

✍️ Writing Expressions

Part 4 of 5 — Real-World Expressions

🔑 The payoff: Expressions describe real situations — money, time, distance, sharing. The trick is to define your variable first ("let $n$ = the number of...") and then write the rule.

Step 1: Define the Variable

In a word problem, you choose what the variable stands for. Write it down so you don't lose track.

Example: tickets cost $8 each

Let $t$ = the number of tickets bought. Then the total cost in dollars is:

$8t$

If $t = 3$ tickets, the cost is , i.e. $24. The expression works for number of tickets.

Part 5: Mixed Practice & Mastery Check

✍️ Writing Expressions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) match keywords to operations, (2) handle the "less than" / "divided by" order trap, (3) write two-step phrases with parentheses, and (4) build expressions from real situations. Let's pull it all together.

Quick Reference

Phrase	Expression
the sum of a number and 5	$n + 5$
5 less than a number	$n - 5$
the product of 3 and a number

A variable is a letter (like $n$ , $x$ , or $y$ ) that stands for a number we don't know yet — or a number that can change.

💡 Why bother? If your age is $a$ , then "your age in 3 years" is just $a + 3$ . One short expression describes the rule for every possible age at once.

🔑 Spot the signal: When you see "the sum of," reach for $+$ . When you see "the product of," reach for $\times$ . The keyword tells you the operation before you write a single symbol.

the two things multiplied →

4

and

n

$4 \cdot n \;=\; 4n$

💡 We almost never write the $\times$ sign in algebra — $4 \cdot n$ is written $4n$ . The number always goes in front of the letter.

But subtraction and division are picky. Order changes the meaning:

$8 - 3 = 5 \quad\text{but}\quad 3 - 8 = -5$

So when you translate a subtraction or division phrase, you must read it carefully and keep the order straight.

🔑 Standard form: We still prefer to write multiplication as $5n$ (number first) and addition with the variable first, like $n + 5$ — but you may write either order for $+$ and $\times$ .

The "Less Than" / "More Than" Flip

These two phrases describe what to do to a starting number, so the starting number comes second in the words but acts on the value:

"5 less than a number"

Start with the number $n$ , then take 5 away:

$n - 5 \qquad (\text{NOT } 5 - n)$

"a number less 5"

The word less (without "than") just means minus, in order:

$n - 5$

⚠️ Read it backwards. "7 less than $x$ " → start at $x$ , go down 7 → $x - 7$ . The number you hear first ( $7$ ) is what gets subtracted, so it lands after the minus sign. "More than" flips the same way: "7 more than $x$ " → .

Watch the Order 🎯

"Divided By" vs. "Divides"

Division has the same order trap. We usually write division as a fraction.

"a number divided by 3"

The number is on top (it's being split):

$\dfrac{n}{3}$

"the quotient of 12 and a number"

Whatever is named first goes on top:

$\dfrac{12}{n}$

"half of a number"

"Half of" means multiply by $\dfrac{1}{2}$ , which is the same as dividing by 2:

$\dfrac{1}{2}n \;=\; \dfrac{n}{2}$

💡 Tip: In "the quotient of A and B," the first thing (A) is always the numerator (top) and the second (B) is the denominator (bottom).

Order in Subtraction & Division 🔽

Pick the correct translation. Watch the order!

Before You Type: The Order Rule

When you write the next answers, slow down on the two tricky operations:

You hear...	You write...
" $A$ less than a number"	number $-\,A$
"a number divided by $A$ "	$\dfrac{\text{number}}{A}$

⚠️ The number you say first in "less than" lands after the minus sign. Picture starting at the variable and stepping down.

Type the Expression ⌨️

Write each phrase as an expression. Use $n$ as the variable. Type subtraction with a minus sign (like n-4) and division as a slash (like n/4).

1) "6 less than a number" $\Rightarrow$ ? 2) "a number divided by 8" $\Rightarrow$ ? 3) "11 more than a number" $\Rightarrow$ ?

"5 less than" that → subtract 5

💡 Build the inside first. Translate the small chunk (twice a number, the product of...) before you attach the "more than" or "less than" part on the outside.

Two-Step Translation 🎯

When You Need Parentheses

Parentheses are needed when an operation acts on a whole sum or difference, not just one number. Listen for a comma or the word "quantity."

Compare these two:

Phrase	Expression
"2 times a number, plus 5"	$2n + 5$
"2 times the quantity ( a number plus 5 )"	$2(n + 5)$

In the first, you only double the number. In the second, you double the entire $(n + 5)$ .

Example: "double the sum of a number and 4"

The sum of a number and 4 is $(n + 4)$ . "Double" multiplies the whole thing:

$2(n + 4)$

⚠️ The phrase "the sum of" or "the difference of" usually means a group. "Triple the sum of a number and 1" must be $3(n + 1)$ , not $3n + 1$ .

Parentheses or Not? 🔽

Choose the correct expression for each phrase.

Your Turn to Build

For the next drill, work inside-out: translate the small chunk first, then attach the outer operation.

🔑 If the phrase says "the sum of" or "the difference of," build that piece in parentheses first — then decide what multiplies it.

Build It ⌨️

Write each as an expression using $n$ . Use * for "times" only if you need it; numbers in front of a variable can be written together (like 4n). Use parentheses where needed, e.g. 3(n+2).

1) "4 more than twice a number" $\Rightarrow$ ? 2) "the product of 3 and the sum of a number and 5" $\Rightarrow$ ?

Example: a pizza split equally among friends

A pizza has 12 slices, shared by $f$ friends. Each person gets:

$\dfrac{12}{f}\ \text{slices}$

💡 Always say what the variable means. "Let $f$ = number of friends" turns a fuzzy story into an exact expression.

Real Situations 🎯

Common Real-World Patterns

Situation	Variable means	Expression
$8 per ticket	$t$ = tickets	$8t$
Started with $20$ , spent some	$x$ = amount spent	$20 - x$
$5$ more than her age	$a$ = her age	$a + 5$
$36$ cookies shared equally	$k$ = kids	$\dfrac{36}{k}$
Twice as far as before	$d$ = old distance	$2d$

⚠️ "Per," "each," and "every" almost always mean multiply. "$8 per ticket × $t$ tickets" → $8t$ . But "shared among / split equally" means divide.

Choose the Expression 🔽

Match each situation to its expression.

From Story to Symbols

For the last drill, read the situation and ask two questions:

What does the variable stand for? (It's given to you here.)
Which keyword is hiding? ("per" → multiply, "absent / left / lost" → subtract.)

💡 Then write the rule the same way you would any phrase — the story is just a phrase in disguise.

Translate the Story ⌨️

Write each expression. Use the given variable. Use +, -, /, and write a number next to a variable for multiply (like 6h).

1) A worker earns $6 per hour. Let $h$ = hours worked. Total pay = ? 2) A class has 28 students. Let $a$ = absent students. Students present = ?

⚠️ Three things to never forget:

"Less than" and "subtracted from" flip the order.

"The sum/difference of ..." usually needs parentheses when something multiplies it.

Write $3 \cdot n$ as $\mathbf{3n}$ — number in front, no times sign.

Mixed Practice 🎯

Last Set Before the Quiz

Time to produce three expressions from scratch — one division, one two-step, and one grouped phrase. Build each piece, then assemble.

💡 Watch for the word "sum" in the third one — it signals parentheses.

Final Build ⌨️

Write each expression using $n$ . Use +, -, /, parentheses, and write numbers next to variables for multiply.

1) "the quotient of a number and 7" $\Rightarrow$ ? 2) "9 more than five times a number" $\Rightarrow$ ? 3) "twice the sum of a number and 3" $\Rightarrow$ ?

One Last Check

You've practiced keywords, the order trap, grouping, and real-world rules. The Exit Quiz below pulls from all four — read each phrase slowly and watch the order.

🔑 Final reminders: "less than" flips, "the sum of" groups, and a number always sits in front of its variable.

Exit Quiz ✅

Answer all three to finish the lesson.