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Lesson 8~30 minUnit 2 · Patterns & Algebra+90 XP

Expanding Single Brackets

$2(x+3)$ becomes $2x+6$. Learn the distributive law, the most important rule in algebra, and never mess up bracket expansion again.

Today's hook: A rectangle has width 2 and length $(x+3)$. Its area is $2(x+3)$. But you can also split it into two smaller rectangles. What are their areas?
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Think First
warm-up

Expand $3(x + 4)$. Draw a diagram showing two rectangles that represent this expression.

Record in workbook.
1
The Big Idea
+5 XP to read

Expanding means multiplying the term outside the bracket by every term inside. This is the distributive law: $a(b+c) = ab + ac$.

$2(x+3)$: multiply 2 by $x$ = $2x$, then 2 by $+3$ = $+6$. Result: $2x + 6$. The outside term visits every inside term.

Area model for 2(x + 3): a rectangle of height 2 split into a part of length x (area 2x) and a part of length 3 (area 6), so 2(x + 3) = 2x + 6.x322x62 × x = 2x2 × 3 = 62(x + 3) = 2x + 6Area = width × length
The 2 multiplies both parts of the length: 2 × x = 2x and 2 × 3 = 6.
Text description
  • A rectangle of height 2, labelled 2 on its left edge.
  • Its length is split into two parts, labelled x and 3 along the top edge.
  • The x part contains 2x, with 2 × x = 2x written underneath.
  • The 3 part contains 6, with 2 × 3 = 6 written underneath.
  • Below: 2(x + 3) = 2x + 6, and Area = width × length.
$a(b+c) = ab + ac$
Multiply EVERY term
$3(x+2) = 3x + 6$. The 3 multiplies both $x$ AND $2$.
Watch the signs
$-2(x-3) = -2x + 6$. Negative times negative = positive.
Area model helps
Think of a rectangle split into parts. Total area = sum of parts.
2
What You'll Master
objectives

Know

  • The distributive law
  • How to expand positive and negative terms
  • The area model

Understand

  • Why every inside term gets multiplied
  • How signs affect expansion
  • That expansion and factorising are opposites

Can Do

  • Expand any single bracket expression
  • Handle negative multipliers
  • Check answers by substitution
3
Words You Need
vocabulary
ExpandRemove brackets by multiplying the outside term by every term inside.
Distributive Law$a(b+c) = ab + ac$. The outside term is shared with every inside term.
FactoriseThe opposite of expanding. Putting a common factor outside brackets.
Area ModelA rectangle diagram showing expansion as the sum of smaller areas.
MultiplierThe term outside the bracket. e.g. In $3(x+2)$, the multiplier is 3.
Terms InsideThe expressions within the brackets. e.g. In $2(x+5)$, the terms are $x$ and $+5$.
4
Spot the Trap
heads-up

Wrong: $3(x+2) = 3x + 2$

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

Wrong: $-2(x-3) = -2x - 6$

Right: $-2(x-3) = -2x + 6$. Neg × neg = pos!

5
The Distributive Law
+5 XP

Multiply the outside term by every term inside. This is the most important rule in algebra.

2(x+5) = 2x + 10 2 × x = 2x, 2 × 5 = 10 3(2x−1) = 6x − 3 3 × 2x = 6x, 3 × (−1) = −3 −4(x+3) = −4x − 12 −4 × x = −4x, −4 × 3 = −12 −2(3x−5) = −6x + 10 −2 × 3x = −6x   −2 × (−5) = +10 Neg × neg = pos. This is the most common mistake!
6
The Arrow Method
+5 XP

Draw arrows from the outside term to each inside term. Follow each arrow and multiply. You can't miss a term!

Arrow method: 3(2x − 5) = 6x − 15, with one arrow from 3 to 2x and one from 3 to −5.3(2x− 5)= 6x − 153 × 2x = 6x3 × (−5) = −15
One arrow per term inside the bracket: 3 × 2x = 6x and 3 × (−5) = −15, so 3(2x − 5) = 6x − 15.
Text description
  • The expression 3(2x − 5) is written across the page, followed by = 6x − 15.
  • A curved arrow runs over the top from the 3 to the term 2x, labelled 3 × 2x = 6x.
  • A second curved arrow runs underneath from the 3 to the term −5, labelled 3 × (−5) = −15.
Try It: Expand $-2(4x + 3)$ using arrows.Ans: $-8x - 6$
7
Checking Your Answer
+5 XP

Pick a value for $x$ and check both the original and expanded forms give the same answer.

3(x+2) when x=4 3(4+2) = 3(6) = 18 Original 3x + 6 when x=4 3(4) + 6 = 12 + 6 = 18 Expanded Both give 18 ✓ Expansion is correct!
8
Quick Reference Table
+5 XP
ExpressionExpansion
$2(x+3)$$2x + 6$
$3(2x-1)$$6x - 3$
$-4(x+2)$$-4x - 8$
$-2(3x-5)$$-6x + 10$
$x(x+3)$$x^2 + 3x$

Watch Me Solve It · Worked example

Watch Me Solve It · Negative expansion
+15 XP per step
Q
PROBLEM
Expand $-3(2x - 4)$.
  1. 1
    Identify the multiplier and inside terms
    Multiplier: $-3$. Inside: $2x$ and $-4$
    The bracket has two terms: $2x$ (positive) and $-4$ (negative).
  2. 2
    Multiply $-3$ by $2x$
    $-3 \times 2x = -6x$
    Neg × pos = neg. $3 \times 2 = 6$. So $-6x$.
  3. 3
    Multiply $-3$ by $-4$
    $-3 \times (-4) = +12$
    Neg × neg = pos! $3 \times 4 = 12$. This is the step everyone gets wrong.
  4. 4
    Combine the results
    $-6x + 12$
    Check: let $x=1$. Original: $-3(2-4) = -3(-2) = 6$. Answer: $-6+12 = 6$ ✓
Answer$-6x + 12$
9
Common Pitfalls
heads-up
$3(x+2) = 3x + 2$
Only multiplied the first term! The 2 needs multiplying too.
Fix: draw arrows. $3 \times x = 3x$ AND $3 \times 2 = 6$.
$-2(x-3) = -2x - 6$
Forgot that neg × neg = pos. $-2 \times (-3) = +6$.
Fix: track signs carefully. Write $-2 \times (-3) = +6$ explicitly.
$x(x+4) = x + 4x = 5x$
Forgot that $x \times x = x^2$, not $x$.
Fix: $x \times x = x^2$. The answer is $x^2 + 4x$.

How are you completing this lesson?

Brain Trainer · 4 problems

D
Brain Trainer · Mixed
4 problems

  1. 1 Expand $4(x + 3)$.

    $4x + 12$$4x + 12$

  2. 2 Expand $2(3x - 5)$.

    $6x - 10$$6x - 10$

  3. 3 Expand $-3(2x + 1)$.

    $-6x - 3$. All signs flip!$-6x - 3$

  4. 4 Expand $-2(x - 4)$.

    $-2x + 8$. Neg × neg = pos!$-2x + 8$
Complete in workbook.
1
Multiple Choice
+10 XP
$3(x + 4) = $
2
Multiple Choice
+10 XP
$2(3x - 1) = $
3
Multiple Choice
+10 XP
$-4(x + 2) = $
4
Multiple Choice
+10 XP
$-2(3x - 5) = $
5
Multiple Choice
+10 XP
$x(x + 3) = $

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Medium 2 MARKS

Q6. Expand $5(2x + 3)$.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. Expand $-3(4x - 2)$ and check by substituting $x=1$.

Answer in your workbook.
Apply Medium 4 MARKS

Q8. A rectangle has width $x$ and length $(x+5)$. Write and simplify an expression for its area. Then find the area when $x=3$.

Answer in your workbook.
Apply Hard 3 MARKS

Stretch. Expand $3(2x+1) - 2(x-4)$.

Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins

Distribute to every term

Distribute to every term. A club buys three identical packs. Each pack costs $2n+5$ dollars, so the total cost is $3(2n+5)$ dollars. A student writes this as $6n+5$.

Expand the correct total-cost expression and explain the error. Compare the two expressions when $n=2$.

Show your working first, then open the worked solution to check your reasoning.

Check the worked solution

The factor 3 multiplies both terms: $3(2n+5)=6n+15$. The student multiplied $2n$ by 3 but did not multiply the constant 5. At $n=2$, the pack costs 9 dollars and three packs cost 27 dollars. The correct expression gives $6(2)+15=27$; the student’s expression gives 17.

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