Writing Expressions

✍️ Writing Expressions

Part 1 of 5 — From Words to Math

Topics in This Part

🔑 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.

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.

The Four Operations and Their Keywords

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

🔑 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.

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" → $\times$
• 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.

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.

✍️ 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:

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

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 $x$" → $x + 7$.

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:

⚠️ 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" $\Rightarrow$ ?

✍️ 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" → $4n$
• "5 less than" that → subtract 5

$4n - 5$

💡 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:

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" $\Rightarrow$ ?

✍️ 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 $8 \cdot 3 = 24$, i.e. $24. The expression $8t$ works for any number of tickets.

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

⚠️ "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:

1. What does the variable stand for? (It's given to you here.)
2. 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 = ?

✍️ 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

⚠️ Three things to never forget:

1. "Less than" and "subtracted from" flip the order.
2. "The sum/difference of ..." usually needs parentheses when something multiplies it.
3. 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.