Unit 1: Algebra & Sequences – GCSE Foundation Maths Rescue System

GCSE Foundation Maths Revision Pathway

This lesson is part of a structured 10-unit revision system designed to help Foundation students build confidence from Grade 3 towards Grade 4/5.

You are currently studying

✅ Unit 1 — Algebra & Sequences

Topics covered across the full pathway

2 Expanding & Factorising 3 Indices & Powers 4 Linear Equations 5 Inequalities 6 Straight-Line Graphs 7 Non-Linear Graphs 8 Real-Life Graphs 9 Graph Applications 10 Coordinate Geometry

Each unit follows the same structure: warm-up, tiered practice, worked examples, Mistake Detective & Examiner Lens.

For Teachers

Foundation GCSE Revision (Grade 3→5)
AQA 8300 · Edexcel 1MA1 · OCR J560
Scheme Weeks 8–12 · Homework · Cover work · Independent revision

⏱ 15–20 minutes 📱 No login required 🎯 Grade 3–5 students

Learning Objectives

By the end of this lesson, students will be able to:
✓ Simplify algebraic expressions by collecting like terms
✓ Substitute positive and negative values into expressions
✓ Identify and continue arithmetic sequences
✓ Find the nth term of increasing and decreasing sequences

Suggested Use

· 15-minute lesson starter
· Homework task
· Revision for Grade 5 boundary students
· Cover lesson activity
· Tutor session warm-up

Students can complete this lesson independently on phones, tablets or computers. No login or app required.

Teachers often share this link with students for homework or revision.
Paste into Google Classroom, WhatsApp, or your school portal.

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

🔥

Warm-Up: Retrieval Practice

⏱ Before You Start

Answer from memory — no notes! Get all 3 correct before moving on.

Q1 Simplify: 3x + 5x

Q2 Substitute x = 4 into: 2x − 3

Q3 Find the next term: 7, 10, 13, ___

Lesson Progress

✓ Warm-up — complete

→ Worked examples & key concepts

· Tiered practice (Green → Amber → Red)

· Mistake Detective & Examiner Lens

· Mastery checklist

💡

The Big Idea: The Tidy Room

Algebra is like tidying a messy room — group the same things together!

3 books + 5 books

8 books ✓

3x + 5x

8x ✓

3 books + 4 pens

≠

can’t combine ✗

3x + 4y

≠

can’t combine ✗

Like terms must have the same variable AND the same power.

Group like terms by colour: x-terms (violet), y-terms (amber), constants (green)

📖

Core Definitions

Variable

A letter representing a number, e.g. x, y, a

Expression

A mathematical phrase without an equals sign

Coefficient

The number multiplying a variable (in 5x, the coefficient is 5)

Like Terms

Terms with identical variables AND powers

Substitution

Replacing a variable with a given number value

Sequence

A list of numbers following a pattern or rule

Term-to-Term Rule

How you move from one term to the next (e.g. +3)

nth Term

A formula for finding any term in a sequence

✏️

Worked Examples

Worked Example 01Collecting Like Terms

Simplify: 4x + 3x − 2

Identify like terms: 4x and 3x are like terms. The −2 is a constant.

Add the coefficients: 4 + 3 = 7 so the x-terms become 7x

✓ Final Answer

7x − 2

Worked Example 02Two Variables

Simplify: 5a − 2b + 3a

Group like terms: 5a and 3a are both ‘a’ terms. −2b is a different category.

Combine: 5a + 3a = 8a

✓ Final Answer

8a − 2b

⚠️ 2b cannot combine with a-terms — different variables = different categories.

Worked Example 03Substitution (Positive)

Find the value of 3x + 4 when x = 2

Replace x with 2: 3(2) + 4

Calculate: 6 + 4

✓ Final Answer

Worked Example 04Substitution (Negative)KEY SKILL

Find the value of 4x − 3 when x = −5

Always use brackets: 4(−5) − 3

Calculate: −20 − 3

✓ Final Answer

−23

⚠️ Always use brackets when substituting negative numbers — it prevents sign errors.

Worked Example 04bSubstitution with x² (Negative)EXAM TRAP

Find the value of 2x² + 1 when x = −3

Replace x with (−3): 2(−3)² + 1

Square first: (−3)² = 9 → 2(9) + 1

Multiply: 18 + 1 = 19

✓ Final Answer

⚠️ Common trap: 2x² means 2 × (x²), NOT (2x)². So you square x first, then multiply by 2. If you did (2 × −3)² = (−6)² = 36 you'd get the wrong answer.

Worked Example 05Forming an ExpressionNEW

“Five more than twice a number x”

“Twice a number” → 2x

“Five more” → add 5 → 2x + 5

✓ Final Answer

2x + 5

📐 Part 2 — Sequences

We now use the same skills to find patterns in number sequences and build formulas.

Worked Example 06Term-to-Term Rule

Sequence: 4, 7, 10, 13, …

Find the difference between consecutive terms: 7 − 4 = 3

✓ Term-to-Term Rule

Add 3

Worked Example 07Finding the nth Term (Positive)

Sequence: 4, 7, 10, 13, … → Find the nth term

Common difference: +3

Formula starts with 3n

n = 1: 3(1) = 3 but first term is 4, so add 1

Check: n = 2 → 3(2) + 1 = 7 ✓

Common difference = +3 → nth term starts with 3n

✓ nth Term Formula

3n + 1

Worked Example 08nth Term (Negative Difference)NEW

Sequence: 10, 7, 4, 1, … → Find the nth term

Difference: 7 − 10 = −3 (decreasing)

Formula starts with −3n

n = 1: −3(1) = −3. First term is 10. Add 13.

Check: n = 2 → −6 + 13 = 7 ✓ | n = 3 → −9 + 13 = 4 ✓

✓ nth Term Formula

−3n + 13

Plug the position number into the formula to get each term

✅

Method Checklists

📋 Simplifying Expressions

✓

Identify like terms

✓

Add or subtract coefficients

✓

Keep variables unchanged

✓

Do not combine different variables

📋 Finding the nth Term

✓

Find the common difference

✓

Multiply difference by n

✓

Adjust to match the first term

✓

Check with n = 1 and n = 2

🧮

Arithmetic Support Strip

5x means 5 × x

x5 is incorrect notation

Use brackets for negative substitution

−(−2) = +2

Adding negatives → smaller numbers

nth term: dn ± adjustment

💪

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

Simplify: 5x + 3x

✍️ Type Your Answer

Simplify: 5x + 3x − 2

5x + 3x = 8x, constant stays → 8x − 2

✓ 8x − 2

Green · Q1

Simplify: 6x + 2x

6x + 2x = (6 + 2)x

✓ 8x

Tap to reveal answer

Green · Q2

Substitute x = 3 into: 4x − 1

4(3) − 1 = 12 − 1

✓ 11

Tap to reveal answer

Green · Q3

Find next term: 5, 9, 13, ___

Difference = +4 each time 13 + 4 = 17

✓ 17

Tap to reveal answer

Green · Q4

Write expression for: “three more than x”

“more than” means add x + 3

✓ x + 3

Tap to reveal answer

Green tier complete — keep building your exam skills!

This is Unit 1 of 10 in the Grade 3→5 Rescue System. Complete all 10 units to cover every Algebra & Graphs topic before Paper 1 on 14th May.

🟡 Amber — Multi-Step

🟡 Amber · TYPE YOUR ANSWER

Substitute x = −3 into: 4x + 7

Amber · Q5

Simplify: 3a + 4b + 2a − b

a terms: 3a + 2a = 5a b terms: 4b − b = 3b

✓ 5a + 3b

Tap to reveal answer

Amber · Q6

Substitute x = −2 into: 5x + 6

5(−2) + 6 = −10 + 6

✓ −4

Tap to reveal answer

Amber · Q7

Find nth term of: 2, 5, 8, 11

Difference = +3 → 3n n=1: 3(1)=3, first term=2 → −1

✓ 3n − 1

Tap to reveal answer

Amber · Q8

Find nth term of: 12, 9, 6, 3

Difference = −3 → −3n n=1: −3, first term=12 → +15 Check: n=2 → −6+15=9 ✓

✓ −3n + 15

Tap to reveal answer

🔴 Red — Exam Style (Full Working)

🔴 Red · TYPE YOUR ANSWER

Find the nth term of: 6, 10, 14, 18

Red · 2 marks · Q9

Simplify fully: 7x − 3 + 2x + 5

x terms: 7x + 2x = 9x constants: −3 + 5 = 2

✓ 9x + 2

Tap to reveal answer

Red · 2 marks · Q10

When x = −3, find: 4x + 1

4(−3) + 1 = −12 + 1

✓ −11

Tap to reveal answer

Red · 2 marks · Q11

Find nth term of: 15, 11, 7, 3

Difference = −4 → −4n n=1: −4(1)=−4, first term=15 Adjustment: 15−(−4) = 19 Check: n=2 → −8+19=11 ✓

✓ −4n + 19

Tap to reveal answer

🔍

Mistake Detective

⚠️ These mistakes cost marks

Study each one carefully — then look for them in your own work.

❌ Wrong

5x written as x5

Coefficient always comes before the variable

✅ Correct

❌ Wrong

3x + 4y = 7xy

Different variables cannot be combined

✅ Correct

3x + 4y stays as 3x + 4y

❌ Wrong

Sequence 10, 7, 4… nth term written as 3n − 7

Forgot the difference is negative

✅ Correct

Difference = −3 Formula: −3n + 13 Check: n=1 → −3+13 = 10 ✓

🎓

Examiner Lens

🎓 To Gain Full Marks

Show coefficient grouping clearly — don’t skip steps
Always use brackets when substituting negative numbers
Show “difference × n” explicitly when forming nth term
Verify at least one term (write “n = 2 → … ✓”)
Write the coefficient before the variable (5x not x5)

🏆

Mastery Checklist

🏆 Unit 1 Mastery

✓

I can simplify expressions with like terms

✓

I can substitute positive and negative numbers correctly

✓

I can form expressions from word descriptions

✓

I can find the nth term for a positive-difference sequence

✓

I can find the nth term for a negative-difference sequence

Ready for Unit 2?

If you can do these 3 things confidently, you're ready to move on:

Simplify expressions with like terms Substitute negative numbers into expressions Find the nth term of a sequence

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Nice work — you've completed Unit 1.

Students aiming for Grade 5 usually need practice across several algebra and graph topics before the exam. GCSE papers often include several of these topics.

🎯

Friend Challenge

Think you understood this unit? Challenge a friend to try it — then compare scores on the warm-up questions. The best way to revise is to explain it to someone else.

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Your Grade 3→5 Rescue System

10 units · Algebra, Graphs & Coordinate Geometry · Paper 1 is 14th May

10% complete — 9 units remaining

✅Unit 1: Algebra & SequencesCOMPLETE

🔒Unit 2: Expanding Brackets & Factorising 🔒Unit 3: Indices & Powers 🔒Unit 4: Linear Equations 🔒Unit 5: Linear Inequalities 🔒Unit 6: Straight-Line Graphs 🔒Unit 7: Non-Linear Graphs 🔒Unit 8: Real-Life Graphs 🔒Unit 9: Graph Applications 🔒Unit 10: Coordinate Geometry

👨‍👩‍👧 For Parents

Your child has completed Unit 1 of a structured GCSE Foundation Maths revision programme. Students aiming for Grade 5 often struggle with Algebra and Graphs — these topics can have a big impact on whether they reach a strong pass.

This unit is part of a 10-unit GCSE Foundation revision programme that gives structured practice across the main algebra and graph topics on typical Foundation papers, with instant feedback so students can learn from mistakes straight away.

✔ Aligned with AQA, Edexcel & OCR

✔ 170+ practice questions & 76 worked examples

✔ Step-by-step explanations

✔ One-time payment — no subscription

Paper 1 is on 14th May 2026. Starting now gives your child around 10 weeks of guided practice before the exam.

You've completed Unit 1 of 10

Continue the GCSE Maths Rescue System

The remaining 9 units complete the full Grade 3 → Grade 4/5 Foundation revision pathway — Expanding Brackets, Indices, Equations, Inequalities, Straight-Line Graphs, Non-Linear Graphs, Real-Life Graphs, Graph Applications & Coordinate Geometry.

Same structure as this unit — tiered practice, worked examples, instant feedback, Mistake Detective & Examiner Lens. Works offline on any device.

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💡 Students: show this page to your parent or tutor — one payment covers the full system

✓ AQA · Edexcel · OCR aligned

✓ 170+ practice questions & 76 worked examples

✓ No login required

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Unit 2: Expanding Brackets & Factorising →

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