How to use this lesson
1. Complete the warm-up questions from memory
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
🔥
Warm-Up: Retrieval Practice⏱ Before You Start
Answer from memory — no notes! Get all 3 correct before moving on.Q1 Simplify: 3x + 4x
Q2 Expand: 2(x + 5)
Q3 Factorise: 6x + 12
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
The Big Idea: The Reverse ProcessExpanding is like delivering a parcel to every room in a house.
3(x + 4)
→
3x + 12
Factorising is the reverse — find what’s common and pull it outside.
6x + 12
→
6(x + 2)
The 3 “visits” both x and +4 inside the bracket
📖
Core DefinitionsExpand
Multiply a bracket out. Each term inside is multiplied by the term outside.
Factorise
The reverse — take out a common factor and write what remains in brackets.
Common Factor
A number or variable that divides into all terms.
Greatest Common Factor
The largest factor common to all terms. Always use this when factorising.
✏️
Worked ExamplesWorked Example 01Expanding (Positive Multiplier)
Expand: 4(x + 3)
1
Multiply 4 by each term:
4 × x = 4x and 4 × 3 = 12✓ Final Answer
4x + 12
Worked Example 02Expanding (Negative Multiplier)KEY SKILL
Expand: −2(y − 3)
1
Multiply −2 by each term:
−2 × y = −2y2
−2 × (−3) = +6 (negative × negative = positive)✓ Final Answer
−2y + 6
⚠️ This is the #1 exam error! Negative × negative = positive.
Worked Example 03Expand and SimplifyNEW
Expand and simplify: 4(x + 2) + 3x
1
Expand the bracket:
4x + 82
Expression becomes:
4x + 8 + 3x3
Collect like terms:
4x + 3x = 7x✓ Final Answer
7x + 8
Worked Example 04Factorising (Numerical HCF)
Factorise: 8x + 12
1
Find the greatest common factor: both 8 and 12 divide by
42
Factor out 4:
8x ÷ 4 = 2x, 12 ÷ 4 = 3✓ Final Answer
4(2x + 3)
⚠️ Check by expanding back: 4 × 2x = 8x, 4 × 3 = 12 ✔
Worked Example 05Factorising (Variable HCF)NEW
Factorise: 3b + 6bc
1
Both terms share
3b2
Factor out 3b:
3b ÷ 3b = 1, 6bc ÷ 3b = 2c✓ Final Answer
3b(1 + 2c)
⚠️ Check: 3b × 1 = 3b, 3b × 2c = 6bc ✔
Factorising: find the HCF, divide each term, write what remains in brackets
✅
Method Checklists📋 Expanding
✓
Multiply every term inside the bracket✓
Watch for negative signs✓
Simplify after expanding📋 Factorising
✓
Identify the greatest common factor✓
GCF may include numbers AND variables✓
Divide each term by the GCF✓
Write remaining expression inside brackets✓
Check by expanding back🧮
Arithmetic Support Stripnegative × negative = positive
negative × positive = negative
Always multiply EVERY term inside the bracket
💪
Tiered Practice🔓 How It Works
Tap any card to reveal the full working and answer. Work through Green first, then Amber, then Red.0 / 12
Problems revealed
Tap problems to reveal working and answers
🟢 Green — Core Skills
🟢 Green · TYPE YOUR ANSWER
Expand: 3(x + 7)
✍️ Type Your Answer
Expand: 3(x + 7)
3 × x = 3x, 3 × 7 = 21
✓ 3x + 21
Green · Q1
Expand: 5(x + 4)
5 × x = 5x
5 × 4 = 20
✓ 5x + 20
Tap to reveal answer
Green · Q2
Expand: −3(a − 2)
−3 × a = −3a
−3 × (−2) = +6
✓ −3a + 6
Tap to reveal answer
Green · Q3
Factorise: 10y + 5
Common factor = 5
10y ÷ 5 = 2y
5 ÷ 5 = 1
✓ 5(2y + 1)
Tap to reveal answer
Green · Q4
Factorise: 12c − 18
Common factor = 6
12c ÷ 6 = 2c
18 ÷ 6 = 3
✓ 6(2c − 3)
Tap to reveal answer
🟡 Amber — Multi-Step
🟡 Amber · TYPE YOUR ANSWER
Expand: −4(y − 2)
Amber · Q5
Expand and simplify: 3(x + 5) + 2x
3x + 15 + 2x
Collect: 5x + 15
✓ 5x + 15
Tap to reveal answer
Amber · Q6
Expand and simplify: −2(4y − 1) + 3y
−8y + 2 + 3y
Collect: −5y + 2
✓ −5y + 2
Tap to reveal answer
Amber · Q7
Factorise: 9a + 6
Common factor = 3
9a ÷ 3 = 3a
6 ÷ 3 = 2
✓ 3(3a + 2)
Tap to reveal answer
Amber · Q8
Factorise: 5x + 10x²
Common factor = 5x
5x ÷ 5x = 1
10x² ÷ 5x = 2x
✓ 5x(1 + 2x)
Tap to reveal answer
🔴 Red — Exam Style
🔴 Red · TYPE YOUR ANSWER
Factorise completely: 15x + 20
Red · 2 marks · Q9
Expand fully: 7(2x − 3)
7 × 2x = 14x
7 × (−3) = −21
✓ 14x − 21
Tap to reveal answer
Red · 2 marks · Q10
Factorise completely: 12x + 18
Common factor = 6
12x ÷ 6 = 2x
18 ÷ 6 = 3
Check: 6 × 2x = 12x ✓
✓ 6(2x + 3)
Tap to reveal answer
Red · 2 marks · Q11
Expand and simplify: 4(f + 2) + 3f
4f + 8 + 3f
Collect: 7f + 8
✓ 7f + 8
Tap to reveal answer
🔍
Mistake Detective⚠️ These mistakes cost marks
Study each one carefully — then look for them in your own work.❌ Wrong
4(x + 3) = 4x + 3
Only the first term was multiplied! The 3 must also be multiplied by 4.
✅ Correct
4(x + 3)
= 4 × x + 4 × 3
= 4x + 12
❌ Wrong
−2(x − 3) = −2x − 6
neg × neg = positive! −2 × −3 = +6, not −6.
✅ Correct
−2(x − 3)
= −2x + 6
🎓
Examiner Lens🎓 To Gain Full Marks
- Show full expansion steps — don’t skip the multiplication
- Include simplification when combining like terms
- State the common factor clearly when factorising
- Check factorising by expanding back (write “Check: ✔”)
- Watch for negative multipliers — they change all signs
🔺
Extension — Double Brackets (Grade 5)🔺 Stretch Challenge
This extends your single-bracket expanding to two brackets multiplied together. Attempt this once you're confident with everything above. The method uses exactly the same distributive skill — just applied twice.
Extension WEDouble Brackets — Distributive MethodGRADE 5
Expand: (x + 3)(x + 5)
1
Treat
(x + 5) as a single parcel. Distribute each term in the first bracket across it:= x(x + 5) + 3(x + 5)2
Expand each bracket separately (Unit 2 skill!):
= x² + 5x + 3x + 153
Collect like terms (Unit 1 skill!):
= x² + 8x + 15✓ Final Answer
x² + 8x + 15
💡 This is the distributive method — it works for ANY polynomial multiplication, not just two brackets. You already know both steps: expanding single brackets and collecting like terms.
Each term in the first bracket distributes across the entire second bracket — then collect like terms.
Extension WE 2Double Brackets — Harder
Expand: (2x − 1)(x + 4)
1
Distribute:
= 2x(x + 4) + (−1)(x + 4)2
Expand each:
= 2x² + 8x − x − 43
Collect:
= 2x² + 7x − 4✓ Final Answer
2x² + 7x − 4
⚠️ Watch the negative: (−1)(x + 4) = −x − 4, not −x + 4.
🟡 Extension Practice
Ext · Q1
Expand: (x + 2)(x + 6)
x(x+6) + 2(x+6)
= x² + 6x + 2x + 12
= x² + 8x + 12
= x² + 6x + 2x + 12
= x² + 8x + 12
✓ x² + 8x + 12
Tap to reveal
Ext · Q2
Expand: (x − 3)(x + 7)
x(x+7) + (−3)(x+7)
= x² + 7x − 3x − 21
= x² + 4x − 21
= x² + 7x − 3x − 21
= x² + 4x − 21
✓ x² + 4x − 21
Tap to reveal
Ext · Q3
Expand: (3x + 1)(x − 2)
3x(x−2) + 1(x−2)
= 3x² − 6x + x − 2
= 3x² − 5x − 2
= 3x² − 6x + x − 2
= 3x² − 5x − 2
✓ 3x² − 5x − 2
Tap to reveal
🏆
Mastery Checklist🏆 Unit 2 Mastery
✓
I can expand brackets correctly
✓
I can handle negative multipliers
✓
I can expand and then simplify
✓
I can factorise using numerical common factors
✓
I can factorise using variable common factors
Nice work — you’ve completed Unit 2!
8 units remaining. Consistent practice is what turns Grade 3 into Grade 4/5. Keep going.
Ready for Unit 3?
If you can do these 3 things confidently, you're ready to move on:
👨👩👧 For Parents
Your child has completed Unit 2 of a structured GCSE Foundation Maths revision programme covering Algebra and Graphs — key topics on every Foundation paper.
✔ AQA, Edexcel & OCR aligned
✔ 170+ practice questions
✔ Step-by-step explanations
Up Next
Unit 3: Indices & Powers →
Your Guru Academy is an independent resource and is not affiliated with, endorsed by, or connected to AQA, Edexcel (Pearson), OCR, or any exam board. All exam board names are the property of their respective owners. Content is designed to support Foundation GCSE revision but does not guarantee any specific grade outcome.