Lesson 2 ~35 min Unit 2 · Algebra +85 XP

Expanding Brackets

One simple law turns $3(x + 2)$ into $3x + 6$. Master it and you unlock half of algebra, solving equations, factorising, and beyond.

Today's hook: A meal deal is 3 × (burger + chips + drink). You don't add each item three times, you treat the meal as a bundle and multiply. Algebra does the same trick.

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Think First

warm-up

Before you read on, quickly: expand $4(x + 5)$. What does the 4 multiply? Try it, then check your reasoning as you go.

Record your answer in your workbook.

The Big Idea

+5 XP to read

The whole lesson is one rule, called the distributive law. It says: if you're multiplying a bracket by something, that something multiplies every term inside the bracket. No exceptions, no shortcuts.

An expansion takes a bracketed expression and rewrites it without the brackets, by multiplying the multiplier through each inner term.

Think of it as an area model: a rectangle of width $a$ and length $(b + c)$ has area $a \cdot b + a \cdot c$.

$a(b + c) = ab + ac$

Multiply EVERY term

If three things sit inside the bracket, the multiplier hits all three.

Keep the sign

Whatever sign sits in front of each inner term stays with it after multiplying.

Brackets are gone

After expanding, no brackets should remain unless they were nested inside.

What You'll Master

objectives

Know

The distributive law $a(b+c) = ab+ac$
What "expanding" / "distributing" means
The area-model interpretation

Understand

Why the multiplier hits every term
Why a negative multiplier flips every sign
Why brackets must come off before like terms can be combined

Can Do

Expand $a(b+c)$ with positive or negative multipliers
Expand and then collect like terms
Use brackets to write area / perimeter expressions

Words You Need

vocabulary

BracketThe pair $( \,\)$ grouping terms that act as one unit.

ExpandRewrite an expression with the brackets removed, by multiplying through.

Distributive lawThe rule that powers expansion: $a(b+c) = ab+ac$.

MultiplierThe factor sitting outside the bracket, the number or letter that's about to be distributed.

Inner termEach term sitting inside the bracket. Every one of them gets multiplied.

Area modelPicturing $a(b+c)$ as a rectangle split into two regions, each with its own area.

Spot the Trap

heads-up

Wrong: "$3(x + 2) = 3x + 2$", the 3 only multiplied the first term.

Right: $3(x + 2) = 3x + 6$. The 3 multiplies BOTH the $x$ AND the $2$.

Wrong: "$-2(x - 3) = -2x - 6$", kept the minus on the 6 by accident.

Right: $-2(x - 3) = -2x + 6$. A negative times a negative is positive.

Parts of the Whole

+5 XP

Before you can expand, learn to read what's in front of you. Every bracketed expression has three parts you must spot at a glance.

The multiplier sits outside the bracket. The inner terms sit inside the bracket and each carries its own sign. Your job: multiply the multiplier by every inner term, keeping signs intact.

$3(x + 2) = 3x + 6$

Multiplier

The factor outside ( ). Could be a number, letter, or both.

Inner terms

Everything inside ( ). Each one is its own little expression.

Sign goes too

The sign in front of each inner term comes along when you multiply.

The Distribution

+5 XP

Here's the move in slow motion. Picture two arrows leaving the multiplier and landing on each inner term.

To expand $5(2x - 3)$: the 5 multiplies the $2x$ to give $10x$, and the 5 multiplies the $-3$ (keeping its minus sign) to give $-15$. Add them: $10x - 15$.

$5(2x - 3) = 10x - 15$

Arrow each term

Draw lines from the multiplier to each inner term if it helps you not miss one.

Multiply coefficients

$5 \times 2x = 10x$. The 5 multiplies just the coefficient, the $x$ stays.

Drop the brackets

Once everything's distributed, the brackets are no longer needed.

Watch the Negative

+5 XP

A negative multiplier flips the sign of every inner term. This is the #1 place students lose marks. Slow down, write the bracket out as a multiplication first if you need to.

For $-2(x - 3)$: the $-2$ multiplies the $x$ (giving $-2x$) and the $-2$ multiplies the $-3$ (giving $+6$, because negative × negative = positive).

So $-2(x - 3) = -2x + 6$, not $-2x - 6$.

$-2(x - 3) = -2x + 6$

Negatives flip signs

A negative multiplier reverses the sign of every inner term.

Neg × neg = pos

$-2 \times -3 = +6$. Two minuses cancel each other.

Don't drop the minus

Common error: only applying the negative to the first term. Hit every one.

Combine and Conquer

+5 XP

When two brackets sit in the same expression, expand each one first, then collect like terms. Don't try to combine while brackets are still up, that's how mistakes sneak in.

The recipe is always the same: (1) expand each bracket separately, (2) collect like terms. Don't skip step 1, like-term collection only works once everything is bracket-free.

$2(x+3) + 3(x-1) = 5x + 3$

Expand FIRST

Both brackets come off before anything else happens.

Then collect

Group $x$-terms together, constants together. Sort and Sum.

Check signs twice

If one bracket has a negative multiplier, double-check the inner sign-flips.

Watch Me Solve It · 3 examples

Watch Me Solve It · Basic expansion

+15 XP per step

PROBLEM

Expand $4(2x + 5)$.

1

Identify the multiplier

multiplier = 4, inner terms: $2x$ and $+5$

Spot the 4 outside the bracket and the two terms inside.
2

Multiply through each inner term

$4 \times 2x = 8x$ and $4 \times 5 = 20$
3

Drop the brackets and join

$= 8x + 20$

No like terms to collect, these are unlike (one $x$, one constant).

Answer$8x + 20$

Watch Me Solve It · Negative multiplier

+15 XP per step

PROBLEM

Expand $-2(3y - 4)$.

1

Note the negative multiplier

multiplier = $-2$

A negative multiplier will flip the sign of every inner term.
2

Multiply each inner term

$-2 \times 3y = -6y$ and $-2 \times (-4) = +8$

Negative × negative = positive.
3

Combine

$= -6y + 8$

Answer$-6y + 8$

Watch Me Solve It · Two brackets, simplify

+15 XP per step

PROBLEM

Expand and simplify $3(x + 2) - 2(x - 4)$.

1

Expand the first bracket

$3(x + 2) = 3x + 6$
2

Expand the second bracket

$-2(x - 4) = -2x + 8$

$-2 \times -4 = +8$. Flip both inner signs.
3

Bring together

$3x + 6 - 2x + 8$
4

Collect like terms

$(3x - 2x) + (6 + 8) = x + 14$

Answer$x + 14$

Common Pitfalls

heads-up

Multiplying only the first term

$3(x + 2) \rightarrow 3x + 2$ ✗, the 3 missed the 2. The multiplier must hit every inner term.

Fix: draw arrows from the multiplier to each inner term before you write anything down.

Forgetting to flip with a negative multiplier

$-2(x - 3) \rightarrow -2x - 6$ ✗. A negative multiplier flips the sign of EVERY inner term.

Fix: $-2 \times -3 = +6$. Two negatives multiply to a positive.

Collecting like terms before expanding

$3(x + 2) + 5x$, you can't combine $3$ and $5x$ until you've expanded the bracket. The 3 isn't sitting alone, it's a multiplier.

Fix: expand FIRST. Then look for like terms.

Copy Into Your Books

The Rule

$a(b + c) = ab + ac$
$a(b - c) = ab - ac$
Multiplier hits every inner term

With Negatives

$-a(b + c) = -ab - ac$
$-a(b - c) = -ab + ac$
Two negatives → positive

Two Brackets

Expand BOTH first
Then collect like terms
Never combine over a bracket

Area Model

$a(b+c)$ = rectangle $a \times (b+c)$
Splits into $a \cdot b$ and $a \cdot c$
Useful for sanity-checking

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems mixing positive and negative multipliers. Work each, then reveal the answer.

1 Expand $3(x + 7)$.

$3 \times x = 3x$ and $3 \times 7 = 21$$= 3x + 21$
2 Expand $-5(2a - 1)$.

$-5 \times 2a = -10a$ and $-5 \times -1 = +5$$= -10a + 5$
3 Expand and simplify $4(x + 2) + 3(x - 1)$.

$4(x+2) = 4x + 8$, $3(x-1) = 3x - 3$. Sum: $4x + 3x + 8 - 3$$= 7x + 5$
4 Expand and simplify $5(2y - 3) - 2(y + 4)$.

$5(2y-3) = 10y - 15$, $-2(y+4) = -2y - 8$. Sum: $10y - 2y - 15 - 8$$= 8y - 23$

Complete in your workbook.

Quick Check · 5 questions

Expand $3(x + 4)$.

+10 XP

Expand $-2(y - 5)$.

+10 XP

Expand and simplify $5(2x - 3) + x$.

+10 XP

Which expansion is WRONG?

+10 XP

Find $4(x + 1) - 3(x - 2)$.

+10 XP

Show Your Working · 3 questions

Show Your Working

7 marks total

Apply Easy 2 MARKS

Q6. Expand and simplify $2(x + 4) + 5(x - 1)$.

Answer in your workbook.

Apply Medium 2 MARKS

Q7. A rectangle has length $(2x + 3)$ cm and width $4$ cm. Write and simplify an expression for the area.

Answer in your workbook.

Analyse Hard 3 MARKS

Q8. Simplify $3(2a - 5) - 4(a + 1)$.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B$3(x+4) = 3x + 12$.

2. C$-2 \times -5 = +10$, so $-2(y-5) = -2y + 10$.

3. A$5(2x-3) = 10x - 15$, plus the lone $x$: $11x - 15$.

4. D$-3(x-2)$ should give $-3x + 6$, not $-3x - 6$ (two negatives → positive).

5. A$4(x+1) - 3(x-2) = 4x + 4 - 3x + 6 = x + 10$.

Show Your Working Model Answers

Q6 (2 marks): $2(x+4) + 5(x-1) = 2x + 8 + 5x - 5$ [1 expanding] $= 7x + 3$ [1 collecting].

Q7 (2 marks): Area $= 4(2x + 3)$ [1 setup] $= 8x + 12$ cm² [1 expansion].

Q8 (3 marks): $3(2a-5) = 6a - 15$ [1]. $-4(a+1) = -4a - 4$ [1]. Sum: $6a - 4a - 15 - 4 = 2a - 19$ [1].

Stretch Challenge · +25 XP, +10 coins

Two Brackets, One Trick

Expand $(x + 2)(x + 3)$ by treating it as $x(x + 3) + 2(x + 3)$, the distributive law applied twice. Show every step.

Reveal solution

$(x+2)(x+3) = x(x+3) + 2(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$.

Quick Review

The Law

$a(b + c) = ab + ac$

Every term

Multiplier hits all inner terms

Keep signs

Sign of each term comes along

Neg × neg

Result is positive

Two brackets

Expand both, then collect

Area model

$a(b+c)$ = split rectangle

Interactive: Expansion Visualiser

See the distributive law as an area model, adjust the multiplier and inner terms and watch the rectangle change.

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