Unit Overview

Algebra and Graphs

CIE IGCSE Mathematics· 5 min read 📊 25-30% of total assessment (tested in both core and extended papers)

1. Unit at a glance

We start with foundational algebraic manipulation skills, before moving to solving equations and inequalities, identifying patterns in sequences, and connecting algebraic relationships to their visual graph representations. The unit concludes with introductory calculus (differentiation) and formal function notation for extended tier candidates.

All subtopics build incrementally, with practical applications to real-world problems aligned with CIE 0580 assessment criteria for both core and extended tiers. Mastery of this unit is critical for scoring highly on both paper 2 and paper 4 assessments.

Work through the following subtopics in order to master all content in this unit:

Algebraic Manipulation & Indices

Master expanding, factorising, simplifying expressions, and the 7 laws of indices for positive, negative, and fractional exponents.

★★⏱ 8 min

Algebraic Fractions

Learn to add, subtract, multiply, divide, and simplify algebraic fractions by factorising numerators and denominators.

★★★⏱ 7 min

Equations, Inequalities & Variation

Solve linear, quadratic, simultaneous equations, linear inequalities, and apply direct and inverse variation formulas.

★★★⏱ 10 min

Sequences

Identify and generate nth terms for arithmetic, geometric, quadratic, and Fibonacci-style sequences.

★★⏱ 6 min

Graphs of Functions & Real-Life Graphs

Plot and interpret linear, quadratic, cubic, reciprocal graphs, plus real-world distance-time and speed-time graphs.

★★★⏱ 9 min

Differentiation

Learn to differentiate polynomial functions, find curve gradients, and identify maximum and minimum stationary points (extended only).

★★★★⏱ 7 min

Functions

Understand function notation, evaluate functions, find inverse functions, and calculate composite functions (extended only).

★★★★⏱ 8 min

2. Common Pitfalls

Wrong move:

Mixing up index laws when multiplying and dividing terms with the same base

Why:

Confusing adding vs subtracting exponents leads to incorrect simplified expressions, losing easy marks in short answer questions.

Correct move:

Remember $a^{m} \times a^{n} = a^{m + n}$ and $a^{m} \div a^{n} = a^{m - n}$ ; always confirm the base is identical before applying index laws.

Wrong move:

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number

Why:

This common error results in completely incorrect solutions to inequality problems, which are frequently tested across all papers.

Correct move:

Any operation involving a negative multiplier or divisor requires flipping the >, <, ≥, or ≤ sign to its opposite.

Wrong move:

Treating all sequences as arithmetic when calculating the nth term

Why:

Many assessment questions use quadratic or geometric sequences, so assuming a constant difference leads to wrong nth term formulas.

Correct move:

Always check for first differences (arithmetic), second differences (quadratic), or common ratio (geometric) before deriving the nth term.

3. Quick Reference Cheatsheet

Concept	Formula/Rule	Related Subtopic
Negative Index Law	$a^{- n} = \frac{1}{a ^{n}}$	Algebraic Manipulation & Indices
Algebraic Fraction Multiplication	$\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$ (cancel common factors first)	Algebraic Fractions
Quadratic Formula	$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ for $a x^{2} + b x + c = 0$	Equations, Inequalities & Variation
Arithmetic Sequence nth Term	$u_{n} = a + (n - 1) d$ where a = first term, d = common difference	Sequences
Linear Graph Gradient	$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$	Graphs of Functions & Real-Life Graphs
Power Rule for Differentiation	If $y = a x^{n}$ , then $\frac{d y}{d x} = an x^{n - 1}$	Differentiation
Composite Function Rule	$f g (x) = f (g (x))$ : apply function g first, then f to the result	Functions

What's Next

Start your learning with the first subtopic on algebraic manipulation, the foundational skill for all subsequent content in this unit. Once you have completed all 7 subtopics and practiced past paper questions for this unit, you will move on to Unit 3: Geometry, which builds on your algebraic skills to solve shape, space, and measure problems aligned with the CIE 0580 syllabus.