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Most Important O Level & IGCSE Maths Topics to Learn for an A*

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1. Algebra: The Backbone of the Entire Syllabus

  • Simplifying expressions

  • Expanding brackets

  • Factorising (single brackets, double brackets, trinomials, difference of squares)

  • Algebraic fractions

  • Solving linear equations

  • Solving quadratic equations by factorisation and formula

  • Forming equations from word problems

  • Inequalities

  • Compound inequalities

  • Rearranging formulas

Why this is essential

  • Used in Papers 1, 2, 4

  • Algebra unlocks mensuration, graphing, transformations, functions, vectors

  • Key to 40–50 percent of total exam marks

2. Number Skills & Arithmetic Precision

  • Fractions

  • Decimals

  • Percentages

  • Ratios & proportions

  • Standard form

  • Powers & roots

  • Estimation

  • Bounds and error intervals

  • Prime factorisation

  • HCF/LCM

  • Surds (4024)

Why important

  • Direct marks + accuracy marks

  • Non-calculator paper heavily depends on number fluency

  • Many long questions hide number traps

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

3. Graphs & Coordinate Geometry

  • Plotting linear graphs

  • Finding gradient and intercept

  • Equation of a line

  • Midpoints

  • Distance between two points

  • Parallel & perpendicular line identification

  • Quadratic graphs (shape, turning point, roots)

  • Inequalities on graphs

  • Graph interpretation questions

Why important

  • One of the most repeated sections

  • Easily earns predictable marks

  • Questions combine algebra and geometry

4. Simultaneous Equations

  • Linear–linear

  • Linear–quadratic

  • Substitution

  • Elimination

  • Graphical interpretation

Why essential

  • Appears almost every session

  • Has high method + accuracy marks

  • Required in real-life modelling questions

5. Vectors

  • Vector notation

  • Column vectors

  • Displacement vectors

  • Vector addition & subtraction

  • Scalar multiplication

  • Parallel & collinear vectors

  • Ratio division

  • Position vectors

Why essential

  • Vector geometry is a guaranteed appearance

  • Provides 6–8 marks with clean logic

  • Many students lose marks due to missing steps

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

6. Mensuration (Surface Area & Volume)

Key formulas to master

  • Cylinder: πr²h, 2πrh, 2πr²

  • Cone: ⅓πr²h, πrl

  • Sphere: 4πr², 4/3πr³

  • Pyramid: ⅓ × base area × height

  • Composite solids

  • Curved vs total surface area

Why crucial

  • High-weightage questions

  • 80 percent of students make errors due to incorrect formula substitution

  • Rarely tested lightly; usually multi-step

7. Geometry & Angle Properties

  • Angles on parallel lines

  • Polygon interior/exterior angles

  • Angle in a semicircle

  • Isosceles triangle rules

  • Similarity & congruence

  • Construction interpretations

  • Bearings

  • Symmetry (line and rotational)

Why essential

  • Pure marks if rules memorised

  • Fast-solving once angle relationships mastered

8. Trigonometry

  • SOHCAHTOA

  • Trig equations

  • Angle of elevation and depression

  • Applying Pythagoras with trig

  • Sine & cosine rule (4024)

  • Area of triangle ½ab sinC

Why essential

  • High-value questions

  • Calculator paper depends heavily on trig accuracy

  • One of the easiest A* scoring topics with practice

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

9. Probability

  • Basic probability

  • Combined events

  • Tree diagrams

  • Expected value (4024)

  • Relative frequency

  • Independent & dependent events

Why essential

  • Recurring problem-solving style

  • Very easy to lose marks through incorrect multiplication or addition

  • Probability = guaranteed 8–12 marks in any session

10. Statistics

  • Mean, median, mode

  • IQR (interquartile range)

  • Range

  • Cumulative frequency

  • Ogive interpretation

  • Box-and-whisker plots

  • Histograms (frequency density)

  • Comparing distributions

Why essential

  • High-structure questions

  • Examiners reward good reasoning

  • Students score well if they know formulas + interpretation rules

11. Transformations

  • Reflection

  • Rotation

  • Translation

  • Enlargement

  • Combination of transformations

  • Center of rotation

  • Scale factors (positive and negative)

Why essential

  • Fast marks

  • Highly predictable question styles

  • Requires diagram accuracy, not heavy theory

12. Sequences & Series

  • nth term for linear sequences

  • nth term for quadratic sequences

  • Differences and second differences

  • Term-to-term formula

  • Special patterns

Why important

  • Repeated in both short and long questions

  • One of the easiest A* scoring areas

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level & IGCSE Mathematics Free Material

13. Functions & Mapping

  • Function notation f(x)

  • Inverse functions f⁻¹(x)

  • Composite functions f(g(x))

  • Domain & range basics

  • Graph-based function interpretation

Why essential

  • Frequently asked in extended papers

  • Requires algebra fluency

  • Provides quick method marks when structure understood

14. Inequalities & Number Lines

  • Solving linear inequalities

  • Representing inequalities on number lines

  • Compound inequality solutions

  • Graphical inequalities (region shading)

Why essential

  • Appears across Paper 1 & Paper 2

  • Many lose marks from small sign errors

  • Easy A* topic once method memorised

15. Word Problems & Modelling Skills

  • Interpreting context-based questions

  • Setting up equations

  • Extracting data from diagrams, graphs, statements

  • Speed–distance–time

  • Ratio and proportion modelling

  • Geometry modelling (diagrams essential)

Why essential

  • Long questions often based on modelling

  • Represents exam reasoning skills

  • A* students excel in interpreting and structuring solutions

16. Financial Mathematics (Compound Interest, Depreciation)

  • Simple interest

  • Compound interest

  • Percentage change

  • Depreciation

  • Successive multipliers

  • Reverse percentages

Why essential

  • Commonly tested in calculator papers

  • Straightforward marks once formula memorised

17. Bounds, Limits & Error Analysis

  • Upper bound, lower bound

  • Using inequalities to express rounded data

  • Combining bounds operations

  • Maximum/minimum possible values

  • Estimation checks

Why essential

  • Students often misinterpret rounding levels

  • Examiners test this with short but tricky questions

18. Matrices (For 4024)

  • 2×2 matrices

  • Addition, subtraction

  • Matrix multiplication

  • Transformations via matrices

  • Determinants and inverse matrices

Why essential

  • High-value structured question

  • Mechanical once rules memorised

19. Set Theory (4024)

  • Venn diagrams

  • Intersection & union

  • Complement sets

  • Number of elements in sets

  • Shading regions

Why essential

  • Guaranteed appearance

  • Very high accuracy marks

  • Easy scoring once diagram logic understood

20. Using Calculator Efficiently for Problem Solving

  • Fraction button

  • Ans button

  • Standard form (EXP key)

  • Avoiding early rounding

  • Memory storage

  • Bracket accuracy

  • Type full expression instead of step-by-step

Why essential

  • Saves time

  • Prevents accuracy loss

  • Affects multi-step problems dramatically

How to Prioritise These Topics for an A*

  • Focus order

    • algebra

    • graphs

    • mensuration

    • trigonometry

    • vectors

    • statistics

    • transformations

    • probability

    • sequences

    • functions

  • Build mastery from high-frequency → medium → low-frequency topics

  • Do 5–10 questions per topic daily

  • Use past papers to reinforce problem-solving structure

Topic Mastery Checklist for A* Students

  • Can solve algebraic fractions confidently

  • Can handle transformations flawlessly

  • Can compute trig angles accurately

  • Can interpret graphs without confusion

  • Can work mensuration with correct formula choice

  • Can solve probability questions quickly

  • Can analyse cumulative frequency & histogram questions

  • Can cancel vectors and divide in ratios correctly

  • Can express answers in correct form (SF, units, inequality)

  • Can complete past papers above 85% consistently

Final Topic Importance Weight Summary

  • Algebra: extremely high

  • Graphs: high

  • Mensuration: high

  • Trigonometry: high

  • Vectors: high

  • Probability & statistics: high

  • Number skills: foundational

  • Functions, sequences, inequalities: medium-high

  • Transformations: medium but easy marks

  • Word problems: essential for long questions

  • Matrices & sets (4024): medium-high depending on paper

Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

Educate A Change focuses on truly bringing change in the educational sector worldwide. All our courses are designed to ensure success, and provide complete preparation without requiring any other external material. Therefore, it takes away the expense of academies, books, past paper compilations and other miscellaneous items. Instead, by just the courses we offer, the student can prepare fully for the best grades in their CAIEs, Edexcel and other high-school exams. The earnings from our course-sales are invested in social projects and growing the free educational activities of Educate A Change.