Lesson 2 ~30 min Unit 2 · Patterns & Algebra +90 XP

Writing Algebraic Expressions

Learn the secret code for turning English into algebra. "A number plus five" becomes $n + 5$. Master the keywords, watch out for the traps, and never guess again.

Today's hook: If apples cost 80c each and you buy $n$ apples, how much do you spend? The answer is an algebraic expression — and you already know how to write it.

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Think First

warm-up

A movie ticket costs $15. Write an expression for the total cost of $n$ tickets. Then calculate the cost for 4 tickets. Show your thinking.

Record your answer in your workbook.

The Big Idea

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Writing algebraic expressions means turning English phrases into maths symbols. It's like being a translator between two languages. Once you know the keyword dictionary, you can convert almost any phrase into algebra.

The secret is a keyword map. Each English word for an operation has a direct symbol equivalent. "Sum" becomes $+$, "product" becomes $\times$, and so on. The variable (usually $n$ or $x$) represents the unknown number.

$\text{English} \rightarrow \text{algebra}$

Pick your variable

When a question says "a number", choose a letter like $n$ or $x$ to represent it.

Find the operation

Look for keywords: sum, difference, product, quotient, more than, less than.

Watch the order

"5 less than $n$" is $n - 5$, NOT $5 - n$. The order matters!

What You'll Master

objectives

Know

Keywords for each operation (+, -, ×, ÷)
That "less than" reverses the order
Shorthand for multiplication (juxtaposition)

Understand

Why word order affects the algebraic expression
That the same phrase can be written in multiple valid ways
How to check your expression makes sense

Can Do

Translate English phrases into algebraic expressions
Handle phrases with multiple operations
Identify and avoid common translation traps

Words You Need

vocabulary

SumThe result of adding. "The sum of $x$ and 5" = $x + 5$.

DifferenceThe result of subtracting. "The difference between $x$ and 5" = $x - 5$.

ProductThe result of multiplying. "The product of 3 and $x$" = $3x$.

QuotientThe result of dividing. "The quotient of $x$ and 2" = $\frac{x}{2}$.

Increased byMeans add. "$x$ increased by 4" = $x + 4$.

Decreased byMeans subtract. "$x$ decreased by 4" = $x - 4$.

Spot the Trap

heads-up

Wrong: "5 less than $n$" is written as $5 - n$

Right: "5 less than $n$" means START with $n$, THEN subtract 5, so it's $n - 5$.

Wrong: "$x$ divided by 3" is written as $3 \div x$

Right: "$x$ divided by 3" means $x$ goes on top: $\frac{x}{3}$ or $x \div 3$.

The Keyword Dictionary

+5 XP

Here is your complete dictionary for translating English into algebra. Print this table into your book — you'll use it for every lesson in this unit.

Each operation has its own set of keywords. When you spot one, write the matching symbol. The variable is whatever letter you chose for "the number."

$\text{keyword} \rightarrow \text{symbol}$

Operation	Symbol	Keywords	Example
Addition	$+$	sum, plus, more than, increased by, added to, total of	$x + 5$
Subtraction	$-$	difference, minus, less than, decreased by, subtracted from	$x - 5$
Multiplication	$\times$	product, times, multiplied by, double, triple, of	$3x$
Division	$\div$	quotient, divided by, shared equally, per, half of	$\frac{x}{2}$

The Order Traps

+5 XP

Some phrases are tricksters. They flip the order you expect. The two biggest culprits are "less than" and "subtracted from".

When you read "5 less than $n$", your brain wants to write $5 - n$. But "less than" means the first number subtracts from the second. So it's $n - 5$.

$n - 5 \neq 5 - n$

"Less than" = reverse

"A less than B" always means $B - A$. The second number comes first.

"Subtracted from" = reverse

"A subtracted from B" = $B - A$. Read it backwards to get it right.

Test with numbers

If $n = 10$, "5 less than $n$" = 5. So $n - 5 = 5$ ✓. $5 - n = -5$ ✗.

Multiplication Shorthand

+5 XP

In algebra, we write multiplication without the times sign. $3 \times n$ becomes $3n$. This is called juxtaposition — putting things next to each other means multiply.

Double a number = $2n$. Triple a number = $3n$. Quadruple = $4n$. And "of" almost always means multiply: "half of $n$" = $\frac{1}{2}n$.

$\text{double } n = 2n$

Multi-Step Phrases

+5 XP

Some phrases combine multiple operations. Break them down piece by piece, working from the inside out. Use brackets when you need to group operations.

Take "5 more than double a number". First find "double a number" = $2n$. Then "5 more than" that = $2n + 5$. Build it step by step like LEGO.

$2n + 5$

Work inside out

Find the innermost phrase first, then wrap the outer phrases around it.

Use brackets

"3 times the sum of $n$ and 4" = $3(n + 4)$. The brackets group the sum.

Check with a number

If $n = 3$, "5 more than double 3" = 11. Does $2(3) + 5 = 11$? Yes!

Phrase	Step-by-step	Expression
"3 more than double $n$"	double $n$ = $2n$, then +3	$2n + 3$
"4 less than triple $x$"	triple $x$ = $3x$, then -4	$3x - 4$
"sum of $n$ and 5, divided by 2"	$n + 5$ first, then ÷2	$\frac{n+5}{2}$
"product of 3 and ($x$ minus 1)"	$x - 1$ first, then ×3	$3(x - 1)$

Watch Me Solve It · Worked example

Watch Me Solve It · Complex phrase

+15 XP per step

PROBLEM

Write an algebraic expression for: "The product of 4 and the difference between a number and 3, increased by 7."

1

Identify the innermost phrase

"the difference between a number and 3"

Look for the phrase that's most deeply nested. "Difference between..." means subtraction. Let $n$ = the number.
2

Translate the difference

$n - 3$

"Difference between $n$ and 3" = $n - 3$. The number comes first (it's not "3 less than $n$" here, so normal order applies).
3

Translate "product of 4 and..."

$4(n - 3)$

"Product of 4 and (the difference)" means $4 \times (n - 3)$. We MUST use brackets because the whole difference gets multiplied.
4

Add "increased by 7"

$4(n - 3) + 7$

"Increased by 7" means + 7 at the end. Check: if $n = 5$, difference = 2, product = 8, +7 = 15. Expression gives $4(2) + 7 = 15$ ✓

Answer $4(n - 3) + 7$

Common Pitfalls

heads-up

Writing $5 - n$ for "5 less than $n$"

"Less than" reverses the order. The correct expression is $n - 5$. Test it: if $n = 10$, "5 less than 10" = 5, and $10 - 5 = 5$ ✓.

Fix: read the phrase backwards. "5 less than $n$" → start with $n$, then subtract 5 → $n - 5$.

Forgetting brackets in "product of 3 and ($n$ minus 2)"

Writing $3n - 2$ instead of $3(n - 2)$. Without brackets, only $n$ gets multiplied by 3, not the whole $n - 2$.

Fix: when a multiplication acts on a group, always use brackets. "Product of 3 and (something)" = $3(\text{something})$.

Writing $\frac{2}{n}$ for "2 divided by $n$"

Actually, $\frac{2}{n}$ IS correct! But students often write $\frac{n}{2}$ by mistake, swapping the numerator and denominator.

Fix: the first number in "A divided by B" goes on top. $\frac{A}{B}$. Remember: TOP = what you start with.

Copy Into Your Books

Operation Keywords

+ : sum, plus, more than, increased by, total
- : difference, minus, less than, decreased by
× : product, times, double, triple, of
÷ : quotient, divided by, per, shared equally

Order Traps

"A less than B" = $B - A$ (reversed!)
"A subtracted from B" = $B - A$ (reversed!)
"A more than B" = $B + A$ (reversed!)
Test with numbers if unsure

Multiplication Rules

Never write $\times$ in algebra
$3 \times n$ becomes $3n$
Double = $\times 2$, Triple = $\times 3$
Use brackets for grouping: $3(n + 2)$

Checking Your Answer

Pick a value for the variable
Calculate using the original words
Calculate using your expression
If they match, your expression is correct!

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems mixing all the skills from this lesson. Work each one, then reveal the answer to check.

1 Write an expression for "8 less than a number $n$."

$n - 8$ — "Less than" reverses the order. Start with $n$, then subtract 8. Don't write $8 - n$!$n - 8$
2 Write an expression for "Triple a number, increased by 5."

Triple = $3n$. Increased by 5 = $+5$. So: $3n + 5$.$3n + 5$
3 Write an expression for "The sum of $x$ and 4, divided by 2."

Sum = $x + 4$. Divided by 2 = $\div 2$. So: $\frac{x+4}{2}$ or $(x + 4) \div 2$. Brackets are needed!$\frac{x+4}{2}$
4 Write an expression for "The product of 5 and the difference between $n$ and 3."

Difference = $n - 3$. Product of 5 and that = $5 \times (n - 3)$. So: $5(n - 3)$.$5(n - 3)$

Complete in your workbook.

Quick Check · 5 questions

Which expression means "5 more than a number $n$"?

+10 XP

Which expression means "3 less than a number $x$"?

+10 XP

"The product of 4 and ($n$ minus 2)" is written as:

+10 XP

"Double $x$, decreased by 7" is written as:

+10 XP

A student writes "$x - 5$" for "5 subtracted from $x$." Is this correct?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Easy 2 MARKS

Q6. Write an algebraic expression for: "The sum of $n$ and 8, decreased by 3."

Answer in your workbook.

Create Medium 3 MARKS

Q7. Write expressions for: (a) "7 subtracted from $m$" (b) "The product of 4 and ($p$ minus 1)" (c) "Half of ($x$ increased by 6)"

Answer in your workbook.

Apply Hard 4 MARKS

Q8. A plumber charges a call-out fee of $40 plus $35 per hour. Write an expression for the total cost of a job lasting $h$ hours. Then calculate the cost for a 3-hour job.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — "5 more than $n$" = $n + 5$.

2. C — "3 less than $x$" = $x - 3$.

3. A — "Product of 4 and ($n$ minus 2)" = $4(n - 2)$.

4. D — "Double $x$, decreased by 7" = $2x - 7$.

5. C — "5 subtracted from $x$" = $x - 5$. The student is correct!

Show Your Working Model Answers

Q6 (2 marks): Sum of $n$ and 8 = $n + 8$ [1 mark]. Decreased by 3 = $-3$. Final: $(n + 8) - 3$ or $n + 5$ [1 mark].

Q7 (3 marks): (a) $m - 7$ [1 mark]. (b) $4(p - 1)$ [1 mark]. (c) $\frac{x + 6}{2}$ or $\frac{1}{2}(x + 6)$ [1 mark].

Q8 (4 marks): Call-out fee = $40. Hourly rate = $35 per hour. Expression: $40 + 35h$ or $35h + 40$ [2 marks]. For 3 hours: $35(3) + 40 = 105 + 40 = $145 [2 marks].

Stretch Challenge · +25 XP, +10 coins

The Perimeter Puzzle

A rectangle has length $(n + 3)$ and width $(n - 1)$. Write an expression for its perimeter. Simplify your answer.

Reveal solution

Perimeter = $2 \times \text{length} + 2 \times \text{width} = 2(n + 3) + 2(n - 1) = 2n + 6 + 2n - 2 =$ $4n + 4$.

Quick Review

Keywords

sum +, difference -, product ×, quotient ÷

Less than

Reverses order: "5 less than n" = $n - 5$

Subtracted from

Reverses order: "5 from n" = $n - 5$

Multiplication

Never write ×; $3n$ not $3 \times n$

Multi-step

Work inside out; use brackets for grouping

Check

Test with numbers to verify your expression

Interactive: Algebra Machine

Substitute numbers into algebraic expressions and see them evaluate step by step.

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