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Study Notes & Lessons

Original, plain-language mini-lessons written by a practising CAPS maths teacher — each one explains a topic, works through examples step by step, and flags the mistakes that cost marks.

Grade 8

⚖️
Whole Numbers · 8 min read

Properties of Whole Numbers

The commutative, associative and distributive properties, the identity roles of 0 and 1, and why dividing by 0 is undefined — the rules that let you rearrange any calculation, with a diagram, worked examples and an exam-style worksheet with memo.

Read the lesson
🌳
Whole Numbers · 8 min read

Factors, Multiples & Prime Factorisation

What factors and multiples really are (and how never to mix them up), what makes a number prime, and how to break any number into its prime factors with a factor tree — worked examples, a clear diagram and an exam-style worksheet with memo.

Read the lesson
🔗
Whole Numbers · 9 min read

HCF and LCM (Highest Common Factor & Lowest Common Multiple)

Highest Common Factor and Lowest Common Multiple made simple by reading them straight off the prime factors — with a Venn-diagram picture, exam-style worked examples and word problems, plus a worksheet with memo.

Read the lesson
➕
Integers · 9 min read

Adding and Subtracting Integers

Ordering integers on a number line, then the two adding rules and the 'add the opposite' trick for subtracting — with a number-line diagram, exam-style worked examples, a worksheet with memo and a quick quiz.

Read the lesson
✖️
Integers · 7 min read

Multiplying and Dividing Integers

The one sign rule that covers both × and ÷ — same signs give a positive, different signs give a negative — shown as a grid, with exam-style worked examples, a worksheet with memo and a quick quiz.

Read the lesson
🧮
Integers · 8 min read

Integers: Squares, Cubes, Roots & Mixed Operations

Squares, cubes, square roots and cube roots of integers, the crucial difference between (−3)2 and −32, and how to run BODMAS through a calculation containing negatives — worked examples, worksheet with memo and a quiz.

Read the lesson
🔄
Integers · 7 min read

Properties of Integers & Inverses

The commutative, associative and distributive properties applied to integers, plus additive and multiplicative inverses — what they are, how exams ask for them, with worked examples, a worksheet with memo and a quiz.

Read the lesson
💵
Financial Maths · 10 min read

Simple Interest & Budgets (Grade 8)

Working out simple interest on savings and loans, finding the total amount owed or saved, and drawing up a monthly budget that balances.

Read the lesson
🔢
Number & Integers · 5 min read

Order of Operations: Getting BODMAS Right

Do calculations in the wrong order and you get the wrong answer, even with perfect arithmetic. BODMAS is the rule that fixes that.

Read the lesson
🍕
Common Fractions · 7 min read

Equivalent Fractions & Simplifying

What the top and bottom of a fraction actually mean, how to build equivalent fractions, and how to simplify to lowest terms using the HCF — with a bar diagram, worked examples, a worksheet with memo and a quiz.

Read the lesson
➕
Common Fractions · 9 min read

Adding & Subtracting Fractions

Adding and subtracting fractions with the same denominator, with different denominators using the LCD, and with mixed numbers — shown with a bar diagram, exam-style worked examples, a worksheet with memo and a quiz.

Read the lesson
✖️
Common Fractions · 9 min read

Multiplying & Dividing Fractions

Multiplying straight across, the reciprocal, and why dividing by a fraction means multiplying by its flip — including mixed numbers and dividing whole numbers by fractions, with worked examples, a worksheet with memo and a quiz.

Read the lesson
√
Common Fractions · 6 min read

Squares, Cubes & Roots of Fractions

Squaring and cubing fractions, and finding square roots and cube roots of fractions — the rule is simply 'do it to the top and to the bottom', with worked examples, a worksheet with memo and a quiz.

Read the lesson
🍰
Common Fractions · 8 min read

Fractions of Quantities, Grouping & Sharing

Finding a fraction of an amount, working backwards from a fraction to the whole, and the grouping and sharing word problems that carry the marks in exams — worked examples, a worksheet with memo and a quiz.

Read the lesson
🔟
Decimal Fractions · 8 min read

Decimals: Place Value, Ordering & Rounding

What each digit after the decimal point is worth, how to compare and order decimals correctly, and how to round off to a given number of decimal places — with a place-value chart, worked examples, a worksheet with memo and a quiz.

Read the lesson
➕
Decimal Fractions · 6 min read

Adding & Subtracting Decimals

The one rule that makes decimal addition and subtraction reliable — line up the decimal points — plus filling gaps with zeros and money problems, with worked examples, a worksheet with memo and a quiz.

Read the lesson
✖️
Decimal Fractions · 9 min read

Multiplying & Dividing Decimals

Counting decimal places when multiplying, the shift trick for dividing by a decimal, multiplying and dividing by 10, 100 and 1000, and estimating the answer first — worked examples, a worksheet with memo and a quiz.

Read the lesson
√
Decimal Fractions · 6 min read

Squares, Cubes & Roots of Decimals

Squaring and cubing decimals, and finding square and cube roots of decimals, using the decimal-place count or by converting to fractions — with worked examples, a worksheet with memo and a quiz.

Read the lesson
🔷
Patterns · 8 min read

Extending Numeric & Geometric Patterns

Spotting what a pattern is doing — a constant difference, a constant ratio, or something else entirely — and extending it, including patterns given as diagrams or tables, with worked examples, a worksheet with memo and a quiz.

Read the lesson
🔑
Patterns · 9 min read

Pattern Rules: In Words and in Algebra

Turning a pattern into a general rule — first described in words, then written algebraically as Tₙ = dn + c — and using that rule to find far-off terms or to work backwards, with worked examples, a worksheet with memo and a quiz.

Read the lesson
⁵
Exponents · 9 min read

Exponents: Notation, Squares, Cubes & Roots

What exponential notation actually means, how to read and write it, squares, cubes, square roots and cube roots, and scientific notation — with a labelled diagram, worked examples and a quiz.

Read the lesson
📐
Exponents · 10 min read

The Laws of Exponents (Grade 8)

The five exponent laws in the Grade 8 CAPS syllabus — multiplying, dividing, a power of a power, a power of a product, and the zero exponent — each explained from first principles with worked examples, a worksheet and a quiz.

Read the lesson
🔤
Algebraic Expressions · 8 min read

Algebraic Language: Terms, Coefficients & Like Terms

The vocabulary of algebra — variables, constants, terms, coefficients and exponents — how to spot like terms, and how to add and subtract them correctly, with worked examples and a quiz.

Read the lesson
✳️
Algebraic Expressions · 9 min read

Expanding & Simplifying Algebraic Expressions

Multiplying a monomial into brackets, dividing expressions by a monomial, and substituting values to find the numerical value of an expression — the core Grade 8 algebra skills, with worked examples and a quiz.

Read the lesson
⚖️
Algebraic Equations · 10 min read

Solving Algebraic Equations (Grade 8)

Solving equations by inspection and by using inverse operations — one-step, two-step, brackets and equations with exponents — plus setting up equations from word problems, with worked examples and a quiz.

Read the lesson
🔀
Functions & Relationships · 8 min read

Functions & Relationships: Flow Diagrams, Tables and Formulae

Input values, output values and rules — how the same relationship can be shown as a flow diagram, a table or a formula, and how to find a missing input, output or rule, with a clear diagram and a quiz.

Read the lesson
📈
Graphs · 9 min read

Interpreting & Drawing Graphs

Reading global graphs — increasing, decreasing and constant sections, linear versus non-linear, maximum and minimum points, and discrete versus continuous data — plus plotting points on the Cartesian plane, with diagrams and a quiz.

Read the lesson
📋
Data Handling · 9 min read

Collecting & Organising Data

How a statistics investigation actually works — asking a good question, samples versus populations, questionnaires, and organising results with tally tables, stem-and-leaf displays and grouped intervals, with a worked diagram and a quiz.

Read the lesson
📏
Geometry of Straight Lines · 8 min read

Angle Relationships: Straight, Intersecting & Perpendicular Lines

Angles on a straight line, angles round a point, vertically opposite angles and perpendicular lines — the four relationships every geometry rider is built from, with a diagram, worked examples and a quiz.

Read the lesson
🔺
Geometry of 2D Shapes · 8 min read

Congruent & Similar 2D Shapes

The difference between congruent shapes (identical) and similar shapes (same shape, different size), how to spot each, and how to use proportion to find missing sides — with a clear diagram, worked examples and a quiz.

Read the lesson
📐
Pythagoras · 9 min read

The Theorem of Pythagoras (Grade 8)

The relationship between the three sides of a right-angled triangle — finding the hypotenuse, finding a shorter side, leaving answers in surd form, and testing whether a triangle is right-angled — with a labelled diagram, worked examples and a quiz.

Read the lesson
📏
Constructions · 9 min read

Construction of Geometric Figures

Using only a compass and ruler to bisect a line, bisect an angle, construct 60° and 90°, and build a triangle from given sides — the constructions that appear as their own exam question, with step-by-step diagrams.

Read the lesson
⚖️
Ratio & Rate · 14 min read

Ratio and Rate: Every Exam Question Type

All the ratio and rate skills a Grade 8–9 exam tests — simplifying, sharing in a ratio, finding a missing part, equivalent ratios, increasing/decreasing in a ratio, and unit rates — each worked in full, with a worksheet.

Read the lesson
％
Financial Maths · 15 min read

Percentages: The Complete Money-Maths Guide

Every percentage question in the Grade 8–9 syllabus — finding a percentage, increase and decrease, one number as a percentage of another, percentage change, VAT, profit and loss, and reverse percentages — each worked in full, with a worksheet.

Read the lesson
📐
Geometry · 13 min read

Angles on Parallel Lines: The Complete Exam Guide

Every angle relationship a parallel-lines question uses — corresponding, alternate, co-interior, plus straight-line and vertically-opposite angles — with worked examples for finding angles, solving for x, and multi-step reasoning, and a full worksheet.

Read the lesson
🔺
Geometry · 13 min read

Angles in Triangles and Quadrilaterals: Full Guide

Find any missing angle in triangles and quadrilaterals — including isosceles, right-angled, the exterior-angle rule, solving for x, and the special quadrilaterals — each worked in full, with a worksheet.

Read the lesson
📊
Data Handling · 13 min read

Mean, Median, Mode and Range: Complete Guide

The four everyday statistics, and every twist an exam adds — odd and even data sets, the median between two middle values, finding a missing value from the mean, and working back from a total — each worked in full, with a worksheet.

Read the lesson
🎲
Probability · 13 min read

Probability: The Complete Grade 8–9 Guide

Every probability question in the Grade 8–9 syllabus — single events, the complement (not happening), relative frequency, and simple two-event experiments — each worked in full, with a worksheet.

Read the lesson
⬛
Measurement · 14 min read

Area and Perimeter: Every Shape (Complete Guide)

Perimeter and area for every shape in the Grade 8–9 syllabus — rectangles, squares, triangles, circles (and half-circles), plus composite shapes and working backwards from the area — each worked in full, with a worksheet.

Read the lesson
🧊
Measurement · 14 min read

Surface Area and Volume: Complete Guide

Volume and surface area for every 3D object in the syllabus — cubes, rectangular prisms and triangular prisms — plus capacity, unit conversions and working backwards, each worked in full, with a worksheet.

Read the lesson
🔄
Geometry · 13 min read

Transformation Geometry: All Four Transformations

Every transformation an exam sets — reflection in the axes and y = x, translation, rotation about the origin, and enlargement — with the coordinate rule for each, worked examples and a worksheet.

Read the lesson

Grade 9

⚖️
Whole Numbers · 8 min read

Properties of Whole Numbers

The commutative, associative and distributive properties, the identity roles of 0 and 1, and why dividing by 0 is undefined — the rules that let you rearrange any calculation, with a diagram, worked examples and an exam-style worksheet with memo.

Read the lesson
🌳
Whole Numbers · 8 min read

Factors, Multiples & Prime Factorisation

What factors and multiples really are (and how never to mix them up), what makes a number prime, and how to break any number into its prime factors with a factor tree — worked examples, a clear diagram and an exam-style worksheet with memo.

Read the lesson
🔗
Whole Numbers · 9 min read

HCF and LCM (Highest Common Factor & Lowest Common Multiple)

Highest Common Factor and Lowest Common Multiple made simple by reading them straight off the prime factors — with a Venn-diagram picture, exam-style worked examples and word problems, plus a worksheet with memo.

Read the lesson
➕
Integers · 9 min read

Adding and Subtracting Integers

Ordering integers on a number line, then the two adding rules and the 'add the opposite' trick for subtracting — with a number-line diagram, exam-style worked examples, a worksheet with memo and a quick quiz.

Read the lesson
✖️
Integers · 7 min read

Multiplying and Dividing Integers

The one sign rule that covers both × and ÷ — same signs give a positive, different signs give a negative — shown as a grid, with exam-style worked examples, a worksheet with memo and a quick quiz.

Read the lesson
🧮
Integers · 8 min read

Integers: Squares, Cubes, Roots & Mixed Operations

Squares, cubes, square roots and cube roots of integers, the crucial difference between (−3)2 and −32, and how to run BODMAS through a calculation containing negatives — worked examples, worksheet with memo and a quiz.

Read the lesson
🔄
Integers · 7 min read

Properties of Integers & Inverses

The commutative, associative and distributive properties applied to integers, plus additive and multiplicative inverses — what they are, how exams ask for them, with worked examples, a worksheet with memo and a quiz.

Read the lesson
⚡
Exponents · 15 min read

The Laws of Exponents: Complete Exam Guide

Every exponent law with its own worked example — multiplying, dividing, powers of powers and products, zero and negative exponents — plus combined exam-style questions and a full worksheet with answers.

Read the lesson
🧮
Algebra · 16 min read

Factorising: The Complete Exam Guide (Every Type)

A full walkthrough of every kind of factorising you can be asked in an exam — common factors, difference of squares, all four trinomial sign patterns, leading coefficients and grouping — with a worked example for each and a worksheet to practise.

Read the lesson
⚖️
Equations · 15 min read

Solving Linear Equations: Every Type in the Exam

One balancing method solves every linear equation. Here it is applied to all the exam types — one-step, two-step, variables on both sides, brackets, negatives and fractions — plus word problems and a full worksheet.

Read the lesson
➗
Algebraic Expressions · 10 min read

Algebraic Fractions (Grade 9)

Simplifying algebraic fractions by factorising and cancelling, and adding, subtracting, multiplying and dividing them — including why you may only cancel factors.

Read the lesson
📉
Graphs · 6 min read

Straight-Line Graphs: Reading y = mx + c

Every straight line is described by y = mx + c. Once you know what m and c do, you can sketch any line and read any graph.

Read the lesson
📐
Geometry · 6 min read

Pythagoras' Theorem: Finding the Missing Side

In any right-angled triangle, a2 + b2 = c2. Here's how to use it to find a missing side — and the one thing you must check first.

Read the lesson
🔷
Patterns · 8 min read

Extending Numeric & Geometric Patterns

Spotting what a pattern is doing — a constant difference, a constant ratio, or something else entirely — and extending it, including patterns given as diagrams or tables, with worked examples, a worksheet with memo and a quiz.

Read the lesson
🔑
Patterns · 9 min read

Pattern Rules: In Words and in Algebra

Turning a pattern into a general rule — first described in words, then written algebraically as Tₙ = dn + c — and using that rule to find far-off terms or to work backwards, with worked examples, a worksheet with memo and a quiz.

Read the lesson
⁵
Exponents · 9 min read

Exponents: Notation, Squares, Cubes & Roots

What exponential notation actually means, how to read and write it, squares, cubes, square roots and cube roots, and scientific notation — with a labelled diagram, worked examples and a quiz.

Read the lesson
🔤
Algebraic Expressions · 8 min read

Algebraic Language: Terms, Coefficients & Like Terms

The vocabulary of algebra — variables, constants, terms, coefficients and exponents — how to spot like terms, and how to add and subtract them correctly, with worked examples and a quiz.

Read the lesson
✳️
Algebraic Expressions · 9 min read

Expanding & Simplifying Algebraic Expressions

Multiplying a monomial into brackets, dividing expressions by a monomial, and substituting values to find the numerical value of an expression — the core Grade 8 algebra skills, with worked examples and a quiz.

Read the lesson
⚖️
Algebraic Equations · 10 min read

Solving Algebraic Equations (Grade 8)

Solving equations by inspection and by using inverse operations — one-step, two-step, brackets and equations with exponents — plus setting up equations from word problems, with worked examples and a quiz.

Read the lesson
🔀
Functions & Relationships · 8 min read

Functions & Relationships: Flow Diagrams, Tables and Formulae

Input values, output values and rules — how the same relationship can be shown as a flow diagram, a table or a formula, and how to find a missing input, output or rule, with a clear diagram and a quiz.

Read the lesson
📈
Graphs · 9 min read

Interpreting & Drawing Graphs

Reading global graphs — increasing, decreasing and constant sections, linear versus non-linear, maximum and minimum points, and discrete versus continuous data — plus plotting points on the Cartesian plane, with diagrams and a quiz.

Read the lesson
📋
Data Handling · 9 min read

Collecting & Organising Data

How a statistics investigation actually works — asking a good question, samples versus populations, questionnaires, and organising results with tally tables, stem-and-leaf displays and grouped intervals, with a worked diagram and a quiz.

Read the lesson
📏
Geometry of Straight Lines · 8 min read

Angle Relationships: Straight, Intersecting & Perpendicular Lines

Angles on a straight line, angles round a point, vertically opposite angles and perpendicular lines — the four relationships every geometry rider is built from, with a diagram, worked examples and a quiz.

Read the lesson
🔺
Geometry of 2D Shapes · 8 min read

Congruent & Similar 2D Shapes

The difference between congruent shapes (identical) and similar shapes (same shape, different size), how to spot each, and how to use proportion to find missing sides — with a clear diagram, worked examples and a quiz.

Read the lesson
📐
Pythagoras · 9 min read

The Theorem of Pythagoras (Grade 8)

The relationship between the three sides of a right-angled triangle — finding the hypotenuse, finding a shorter side, leaving answers in surd form, and testing whether a triangle is right-angled — with a labelled diagram, worked examples and a quiz.

Read the lesson
📐
Geometry of 2D Shapes · 10 min read

Congruent & Similar Triangles: The Four Conditions

The four conditions that prove two triangles congruent — SSS, SAS, AAS and RHS — how to write the correspondence correctly, and how similarity differs. With drawn figures for every condition, worked proofs and a quiz.

Read the lesson
🧊
Geometry of 3-D Objects · 9 min read

Geometry of 3-D Objects: Faces, Vertices, Edges & Nets

Classifying 3-D objects by their faces, vertices and edges, the five Platonic solids, Euler's formula, the properties of spheres and cylinders, and how nets fold into solids — with drawn figures, worked examples and a quiz.

Read the lesson
📦
Surface Area & Volume · 10 min read

Surface Area & Volume of 3-D Objects (Grade 9)

Volume and surface area of cubes, rectangular prisms, triangular prisms and cylinders, SI unit conversions, and what happens to volume when you double the dimensions — with drawn solids, worked examples and a quiz.

Read the lesson
🔗
Functions & Relationships · 8 min read

Functions & Relationships (Grade 9)

Input values, output values and rules shown as flow diagrams, tables, formulae and equations — how to move between these equivalent forms and how to find a missing input, output or rule.

Read the lesson
📐
Area & Perimeter · 10 min read

Area & Perimeter of 2-D Shapes (Grade 9)

Area and perimeter formulae for squares, rectangles, triangles, parallelograms, trapeziums and circles, plus composite shapes, unit conversions and what happens to area when the dimensions double.

Read the lesson
🔄
Transformation Geometry · 10 min read

Transformation Geometry (Grade 9)

Reflecting, translating, rotating, enlarging and reducing whole figures on the Cartesian plane, the coordinate rules for each, working backwards from an image, and the effect of enlargement on perimeter and area.

Read the lesson
📊
Data Handling · 9 min read

Collect, Organise & Summarise Data (Grade 9)

Running a data investigation properly — samples and bias, grouping data into class intervals, drawing and reading histograms, and choosing between the mean, median and mode when the data has outliers.

Read the lesson
🔢
Whole Numbers · 8 min read

The Real Number System: Rational & Irrational Numbers

Naming and sorting numbers — natural, whole, integers, rational and irrational — how each set sits inside the next, and how to tell a rational number from an irrational one.

Read the lesson
⚖️
Ratio & Proportion · 8 min read

Direct & Indirect Proportion

Telling direct from indirect (inverse) proportion, setting up each correctly, and solving the classic exam problems — unit costs, workers and time, speed and journey time.

Read the lesson
🟰
Algebraic Equations · 9 min read

Solving Equations by Factorisation (Product = 0)

The zero-product rule, solving equations of the form (x + a)(x + b) = 0, factorising first when the equation is not yet in that form, and dividing polynomials by a monomial.

Read the lesson
✖️
Algebraic Expressions · 9 min read

Multiplying Binomials: (x + 2)(x + 3) and Beyond

Expanding the product of two binomials with the area model and the FOIL order, plus squaring a binomial and the difference of two squares — the Grade 9 algebra skill that factorising later reverses. With a diagram, worked examples and a quiz.

Read the lesson
🔬
Exponents · 8 min read

Negative Exponents & Scientific Notation (Grade 9)

What a negative exponent really means, how to rewrite it as a fraction, and how to write very large and very small numbers in scientific notation — the Grade 9 extension of the exponent laws, with worked examples and a quiz.

Read the lesson
💰
Financial Maths · 10 min read

Financial Maths: Interest, Hire Purchase & Exchange Rates

Simple and compound interest, hire purchase, rentals, exchange rates and commission — the real-world money maths of Grade 9, taught with the formulas, fully worked examples and a quiz.

Read the lesson
📏
Constructions · 9 min read

Construction of Geometric Figures

Using only a compass and ruler to bisect a line, bisect an angle, construct 60° and 90°, and build a triangle from given sides — the constructions that appear as their own exam question, with step-by-step diagrams.

Read the lesson
⚖️
Ratio & Rate · 14 min read

Ratio and Rate: Every Exam Question Type

All the ratio and rate skills a Grade 8–9 exam tests — simplifying, sharing in a ratio, finding a missing part, equivalent ratios, increasing/decreasing in a ratio, and unit rates — each worked in full, with a worksheet.

Read the lesson
％
Financial Maths · 15 min read

Percentages: The Complete Money-Maths Guide

Every percentage question in the Grade 8–9 syllabus — finding a percentage, increase and decrease, one number as a percentage of another, percentage change, VAT, profit and loss, and reverse percentages — each worked in full, with a worksheet.

Read the lesson
🔢
Patterns · 13 min read

Number Patterns: Finding and Using the Rule

Every linear-pattern question type — finding the rule Tₙ = dn + c, predicting a far term, working backwards to find which term equals a value, decreasing patterns, and patterns from diagrams — each worked in full, with a worksheet.

Read the lesson
📐
Geometry · 13 min read

Angles on Parallel Lines: The Complete Exam Guide

Every angle relationship a parallel-lines question uses — corresponding, alternate, co-interior, plus straight-line and vertically-opposite angles — with worked examples for finding angles, solving for x, and multi-step reasoning, and a full worksheet.

Read the lesson
🔺
Geometry · 13 min read

Angles in Triangles and Quadrilaterals: Full Guide

Find any missing angle in triangles and quadrilaterals — including isosceles, right-angled, the exterior-angle rule, solving for x, and the special quadrilaterals — each worked in full, with a worksheet.

Read the lesson
📊
Data Handling · 13 min read

Mean, Median, Mode and Range: Complete Guide

The four everyday statistics, and every twist an exam adds — odd and even data sets, the median between two middle values, finding a missing value from the mean, and working back from a total — each worked in full, with a worksheet.

Read the lesson
🎲
Probability · 13 min read

Probability: The Complete Grade 8–9 Guide

Every probability question in the Grade 8–9 syllabus — single events, the complement (not happening), relative frequency, and simple two-event experiments — each worked in full, with a worksheet.

Read the lesson
⬛
Measurement · 14 min read

Area and Perimeter: Every Shape (Complete Guide)

Perimeter and area for every shape in the Grade 8–9 syllabus — rectangles, squares, triangles, circles (and half-circles), plus composite shapes and working backwards from the area — each worked in full, with a worksheet.

Read the lesson
🧊
Measurement · 14 min read

Surface Area and Volume: Complete Guide

Volume and surface area for every 3D object in the syllabus — cubes, rectangular prisms and triangular prisms — plus capacity, unit conversions and working backwards, each worked in full, with a worksheet.

Read the lesson
🔄
Geometry · 13 min read

Transformation Geometry: All Four Transformations

Every transformation an exam sets — reflection in the axes and y = x, translation, rotation about the origin, and enlargement — with the coordinate rule for each, worked examples and a worksheet.

Read the lesson

Grade 10

🔢
Algebraic Expressions · 8 min read

Real Numbers, Surds & Rounding (Grade 10)

Rational versus irrational numbers, working out which two integers a surd lies between, and rounding to a given number of decimal places or significant figures — the opening skills of the Grade 10 syllabus.

Read the lesson
⚖️
Equations & Inequalities · 9 min read

Linear Inequalities & Interval Notation (Grade 10)

Solving linear inequalities, the one rule that catches everybody (dividing by a negative), showing the answer on a number line, and writing it in interval notation.

Read the lesson
🔗
Equations & Inequalities · 10 min read

Simultaneous Linear Equations (Grade 10)

Solving two equations in two unknowns by substitution and by elimination, what the solution means graphically, and the word problems that lead to a pair of equations.

Read the lesson
✖️
Algebraic Expressions · 8 min read

Multiplying a Binomial by a Trinomial (Grade 10)

Expanding products like (2x + 3)(x² − x + 4) reliably using a grid, then collecting like terms — plus the cube identities that come from it.

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🧩
Algebraic Expressions · 10 min read

Factorising: Grouping in Pairs & the Cubes (Grade 10)

A decision guide based on counting the terms, then the two new Grade 10 methods: grouping four terms in pairs, and the sum and difference of two cubes.

Read the lesson
🔧
Equations & Inequalities · 8 min read

Literal Equations: Changing the Subject of a Formula

Rearranging a formula to make a different letter the subject — the same inverse operations as solving an equation, but the answer is an expression instead of a number.

Read the lesson
ⁿ
Exponents · 11 min read

Laws of Exponents & Rational Exponents (Grade 10)

The exponent laws extended to negative and fractional powers, converting between roots and rational exponents, simplifying harder expressions, and solving exponential equations by matching bases.

Read the lesson
➗
Algebraic Expressions · 10 min read

Algebraic Fractions: Simplify, Add & Subtract (Grade 10)

Simplifying algebraic fractions by factorising, multiplying and dividing them, and adding or subtracting using a lowest common denominator — including denominators that factorise as cubes.

Read the lesson
▱
Euclidean Geometry · 11 min read

Properties of Quadrilaterals & Their Proofs (Grade 10)

Definitions and full property lists for the parallelogram, rectangle, rhombus, square, trapezium and kite — sides, angles and diagonals — plus how to prove a shape is a parallelogram.

Read the lesson
📐
Euclidean Geometry · 8 min read

The Midpoint Theorem (Grade 10)

The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length — the theorem, its converse, and how to use each in a proof.

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📈
Functions & Graphs · 10 min read

The Hyperbola & Exponential Graph (Grade 10)

Sketching y = a/x and y = abᵙ, finding asymptotes and intercepts, and reading the effect of a, b and a vertical shift q — the two curved graphs of Grade 10.

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📊
Statistics · 10 min read

Statistics: Five-Number Summary & Box Plots (Grade 10)

Mean, median and mode, the five-number summary, quartiles and the interquartile range, drawing and reading a box-and-whisker plot, and describing the spread and skewness of data.

Read the lesson
🎲
Probability · 9 min read

Probability: Venn Diagrams & the Addition Rule (Grade 10)

Sample spaces, complementary and mutually exclusive events, the addition rule, and reading a two-set Venn diagram to answer probability questions.

Read the lesson
📦
Measurement · 9 min read

Measurement: Surface Area, Volume & Scaling (Grade 10)

Surface area and volume of prisms, cylinders, cones, spheres and pyramids, and what happens to area and volume when every dimension is multiplied by a factor k.

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📐
Trigonometry · 9 min read

Trigonometry: Solving Triangles & Angles of Elevation

Using sin, cos and tan to find a missing side or angle in a right-angled triangle, and applying them to angles of elevation and depression in real problems.

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📈
Functions · 15 min read

Sketching the Parabola: The Complete Exam Guide

The full method for sketching any quadratic — plus every exam variation: upward and downward parabolas, no x-intercepts, and finding the equation from a graph — each worked in full, with a diagram and a worksheet.

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📐
Trigonometry · 15 min read

Trigonometry: SOH-CAH-TOA and Every Question Type

Every right-angled trig question in Grade 10 — naming sides, finding a side with sin/cos/tan, finding the hypotenuse, finding an angle with the inverse, special angles, and an angle-of-elevation application — each worked in full, with a diagram and worksheet.

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📐
Trigonometry · 11 min read

Special Angles, Reciprocal Ratios & Trig Equations (Grade 10)

The exact values for 0°, 30°, 45°, 60° and 90° without a calculator, the reciprocal ratios cosec, sec and cot, and solving simple trig equations for angles between 0° and 90°.

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📏
Euclidean Geometry · 11 min read

Congruent & Similar Triangles (Grade 10)

The four conditions for congruency (SSS, SAS, AAS and RHS), what similarity means, and how to set out a proof so that every statement earns its reason.

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〰️
Functions & Graphs · 12 min read

Trigonometric Graphs (Grade 10)

The shapes of y = sin x, y = cos x and y = tan x on [0° ; 360°], and what a and q do in y = a sin x + q — amplitude, vertical shift, range and asymptotes.

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🔢
Patterns · 10 min read

Number Patterns: Linear Sequences (Grade 10)

Finding the constant difference, writing the general term Tₙ = dn + c, calculating any term, and working backwards to find which term has a given value.

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📊
Statistics · 11 min read

Grouped Data, Percentiles & Modal Interval (Grade 10)

Estimating the mean of grouped data using midpoints, identifying the modal interval and the interval containing the median, and finding percentiles and the semi-interquartile range.

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🛒
Financial Maths · 11 min read

Hire Purchase, Exchange Rates & Population Growth (Grade 10)

Working out a hire-purchase agreement with a deposit and monthly instalments, converting between currencies, and using the compound formula for population growth.

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📉
Graphs · 6 min read

Straight-Line Graphs: Reading y = mx + c

Every straight line is described by y = mx + c. Once you know what m and c do, you can sketch any line and read any graph.

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💰
Financial Maths · 14 min read

Simple & Compound Interest: Every Exam Type

Both interest formulas and every question type an exam sets — finding the final amount, the interest earned, working back to the principal or rate, depreciation, and inflation — each worked in full, with a worksheet.

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🎯
Analytical Geometry · 15 min read

Analytical Geometry: All the Exam Question Types

The three core formulas — distance, midpoint, gradient — and every way an exam uses them: parallel and perpendicular lines, collinear points, and finding an unknown coordinate, each worked in full with a worksheet.

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🟰
Algebraic Equations · 9 min read

Solving Equations by Factorisation (Product = 0)

The zero-product rule, solving equations of the form (x + a)(x + b) = 0, factorising first when the equation is not yet in that form, and dividing polynomials by a monomial.

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Grade 11

√
Exponents & Surds · 10 min read

Surds: Simplifying & Rationalising (Grade 11)

Simplifying surds using perfect-square factors, adding and subtracting like surds, multiplying them out, and rationalising a denominator — including the conjugate method.

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📉
Equations & Inequalities · 9 min read

Quadratic Inequalities (Grade 11)

Solving inequalities such as x² − x − 6 > 0 by factorising, finding the critical values, and using a sign table or parabola sketch — with the answer in interval notation.

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⚡
Exponents & Surds · 11 min read

Exponent Laws & Exponential Equations (Grade 11)

The exponent laws including rational exponents, simplifying expressions with different bases, and solving exponential equations by making the bases the same or by factorising.

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🌀
Trigonometry · 12 min read

Trig Equations & General Solutions (Grade 11)

Finding the reference angle, writing the general solution with + k·360° or + k·180°, and then listing the specific solutions that fall inside a given interval.

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📐
Analytical Geometry · 11 min read

Analytical Geometry: Lines & Inclination (Grade 11)

Equation of a line through two points, lines parallel and perpendicular to a given line, and the angle of inclination θ where m = tanθ.

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🔺
Trigonometry · 12 min read

Sine, Cosine & Area Rules (Grade 11)

Choosing between the sine rule, the cosine rule and the area rule for non-right-angled triangles, and applying them to two-dimensional problems.

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💰
Financial Maths · 12 min read

Finance: Growth & Decay (Grade 11)

Compound growth and decay, straight-line versus reducing-balance depreciation, different compounding periods, and the difference between a nominal and an effective interest rate.

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🎲
Probability · 11 min read

Probability: Venn Diagrams & Rules (Grade 11)

The addition rule, mutually exclusive and complementary events, reading and filling in Venn diagrams, and testing whether two events are independent.

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📈
Functions · 12 min read

The Parabola: y = a(x + p)² + q (Grade 11)

The turning-point form of the parabola, what a, p and q each do to the graph, finding intercepts and the axis of symmetry, and writing down domain and range.

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📉
Functions · 12 min read

Hyperbola & Exponential Graphs (Grade 11)

y = a/(x + p) + q and y = a·bx+p + q — locating the asymptotes, finding intercepts, sketching, and stating domain and range.

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〰️
Trigonometry · 13 min read

Trigonometric Graphs (Grade 11)

y = a sin(kx + p), y = a cos(kx + p) and y = a tan(kx + p) — amplitude, period, horizontal shift, asymptotes, and reading the graphs to answer questions.

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📊
Statistics · 13 min read

Statistics: Spread, Ogives & Box Plots (Grade 11)

The five-number summary and box-and-whisker diagram, variance and standard deviation, drawing and reading an ogive, recognising skewness, and identifying outliers.

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⚖️
Equations & Inequalities · 10 min read

Simultaneous Equations: One Linear, One Quadratic (Grade 11)

Solving a linear equation together with a quadratic one by substitution, why there are usually two solution pairs, and what those pairs mean on a graph.

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📝
Equations & Inequalities · 10 min read

Word Problems with Quadratics (Grade 11)

Turning a sentence into an equation, solving it, and — the step that costs the most marks — rejecting the answer that does not make sense in the context.

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🌳
Probability · 11 min read

Tree Diagrams & Two-Way Tables (Grade 11)

Drawing a two-stage tree diagram, the difference between drawing with and without replacement, reading a two-way contingency table, and testing independence from one.

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🔍
Equations · 14 min read

The Discriminant & Nature of Roots: Full Guide

b2 − 4ac reveals everything about a quadratic's roots before you solve. Every exam type — classifying roots, rational vs irrational, and finding k for equal / real / non-real roots — worked in full, with a worksheet.

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🔢
Patterns & Sequences · 11 min read

Quadratic Number Patterns (Grade 11 & 12)

Recognising a quadratic pattern from its constant second difference, finding Tₙ = an² + bn + c, and answering the standard NSC follow-ups about which term equals a given value.

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⭕
Euclidean Geometry · 15 min read

Circle Geometry: Every Theorem, With Examples

The theorems that decide Paper 2 marks — centre vs circumference, semicircle, same segment, cyclic quadrilateral, tangent–radius and tangent–chord — each with the reason to quote, a worked example, a diagram and a worksheet.

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✖️
Equations · 15 min read

Solving Quadratic Equations: Every Method & Type

All the ways to solve a quadratic — factorising, the formula, rearranging first, a leading coefficient, and quadratics hidden in fractions — plus a worked example for each and a full worksheet.

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🔺
Trigonometry · 15 min read

Trig Identities & Reduction Formulae: Full Guide

The identities and reduction rules that crack Paper 2 trig — simplifying, proving identities, using the CAST diagram, reducing angles and finding exact values — each worked in full, with a worksheet.

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Grade 12

🔢
Patterns & Sequences · 11 min read

Quadratic Number Patterns (Grade 11 & 12)

Recognising a quadratic pattern from its constant second difference, finding Tₙ = an² + bn + c, and answering the standard NSC follow-ups about which term equals a given value.

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∑
Patterns & Sequences · 12 min read

Series, Sigma Notation & Convergence (Grade 12)

Sum formulae for arithmetic and geometric series, reading and writing sigma notation, and the sum to infinity — including the values of the ratio for which a series converges.

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🔁
Functions · 11 min read

Inverse Functions (Grade 12)

Finding the inverse of a line, a parabola and an exponential function, why y = x² needs a restricted domain, and how to state whether an inverse is a function.

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🧮
Algebra · 10 min read

Polynomials: Remainder & Factor Theorem (Grade 12)

Using the remainder theorem to find a remainder without dividing, the factor theorem to find the first factor of a cubic, and then factorising and solving the cubic completely.

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📈
Calculus · 12 min read

Sketching Cubic Graphs (Grade 12)

Using the derivative to find stationary points, deciding which is a maximum and which a minimum, locating the point of inflection, and putting it all together in a sketch.

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🎯
Calculus · 12 min read

Optimisation & Rates of Change (Grade 12)

Building a formula from a described situation, differentiating it to find the maximum or minimum, and interpreting a derivative as a rate of change — including velocity and acceleration.

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⭕
Analytical Geometry · 11 min read

The Circle & Its Tangent (Grade 12)

The equation (x − a)² + (y − b)² = r², completing the square to find the centre and radius from the general form, and finding the tangent at a point on the circle.

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📐
Euclidean Geometry · 11 min read

Proportionality & Similarity Theorems (Grade 12)

The proportion theorem for a line parallel to a side of a triangle, similar triangles and the AAA condition, and how to lay out a proportion so the ratios cannot be flipped.

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🔺
Trigonometry · 12 min read

Compound & Double Angle Identities (Grade 12)

The formulae for sin(A ± B) and cos(A ± B), the three forms of cos2A, using them to find exact values without a calculator, and proving identities.

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🏦
Financial Maths · 12 min read

Annuities & Loan Repayments (Grade 12)

Future value for savings and present value for loans, working out a monthly instalment, and finding the outstanding balance part-way through a loan.

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📈
Statistics · 11 min read

Regression & Correlation (Grade 12)

Drawing a scatter plot, finding the least squares regression line y = A + Bx on the calculator, using it to predict, and interpreting the correlation coefficient r.

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🔢
Probability · 12 min read

Counting Principles & Probability (Grade 12)

The fundamental counting principle, arrangements with n!, handling items that must stay together or that repeat, and using counting to work out a probability.

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📡
Trigonometry · 12 min read

Three-Dimensional Trigonometry (Grade 12)

Applying the sine, cosine and area rules to problems set in three dimensions — identifying the right triangle to work in, finding the shared side first, and then answering the question.

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📐
Functions · 9 min read

Average Gradient & Average Rate of Change (Grade 12)

The gradient of the straight line joining two points on a curve, why it is an average rather than the gradient at a point, and how it becomes the derivative as the two points close together.

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🚀
Calculus · 15 min read

Differentiation: The Rules and How Exams Use Them

The power rule plus every way an exam applies it — differentiating polynomials, rewriting surds and fractions first, first principles, gradients and tangents, and turning points — each worked in full, with a worksheet.

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🔢
Patterns & Sequences · 15 min read

Arithmetic & Geometric Sequences: Full Guide

Both sequence types and every exam question — finding a term, finding which term equals a value, the sum of a series, and the sum to infinity — each worked in full, with a worksheet.

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🔁
Functions · 14 min read

Logarithms: The Complete Grade 12 Guide

Logs as the reverse of exponents, plus every exam type — evaluating, converting between log and exponent form, the log laws, change of base, and solving exponential equations with logs — each worked in full, with a worksheet.

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