Grade 9 Mathematics

🔢 Properties of numbers Grade 9 uses the real number system to organise natural numbers, whole numbers, integers, rational numbers and irrational numbers. The sets are connected: each smaller set sits inside a larger one. Start lesson → 🔢 Calculations with whole numbers Estimation, rounding and compensation help you judge an answer and calculate efficiently. Start lesson → 🔢 Multiples and factors Multiples come from repeated multiplication, while factors divide a number exactly. Start lesson → 🔢 Ratio, rate and proportion Ratio compares quantities, rate compares different units, and proportion describes consistent change. Start lesson → 🔢 Problems in financial contexts Percentages connect directly to discount, profit, loss, interest and commission. Start lesson →

➕ Numbers smaller than 0 Negative numbers extend the number line below zero and describe values relative to a reference point. Start lesson → ➕ Adding and subtracting integers Adding and subtracting integers can be understood as signed moves on a number line. Start lesson → ➕ Multiplying and dividing integers For integer multiplication and division, decide the sign before calculating the magnitude. Start lesson → ➕ Powers, roots and word problems Powers repeat multiplication and roots reverse powers; brackets determine whether a negative sign belongs to the base. Start lesson →

🍫 Equivalent fractions Equivalent fractions look different but represent exactly the same value. Start lesson → 🍫 Adding and subtracting fractions Fractions can be added or subtracted once they are written with equal-sized parts. Start lesson → 🍫 Multiplying and dividing fractions Multiplication finds a fraction of a fraction; division asks how many of one fractional amount fit in another. Start lesson → 🍫 Equivalent forms Fractions, decimals and percentages are different notations for the same rational value. Start lesson →

🔟 Equivalent forms Decimal notation expresses fractions using place values based on powers of ten. Start lesson → 🔟 Calculations with decimals Decimal calculations work like whole-number calculations when place value is protected. Start lesson → 🔟 Solving problems Decimal problem solving starts by identifying the quantity, operation and unit before calculating. Grade 9 learners must keep place value accurate and interpret the result in context. Start lesson → 🔟 More problems More decimal problems combine several operations, conversions or real-life decisions. The focus is choosing a sensible sequence of steps and keeping precision until the final answer. Start lesson → 🔟 Decimals in algebraic expressions Decimal coefficients behave like other numbers in algebraic expressions. Start lesson →

⚡ Revision Exponents are shorthand for repeated multiplication and follow laws based on that meaning. Start lesson → ⚡ Integer exponents A negative exponent means reciprocal, not a negative answer. Start lesson → ⚡ Solving simple exponential equations A simple exponential equation has the unknown in an exponent. When both sides can be written using the same positive base, the exponents can be compared. Start lesson → ⚡ Scientific notation Scientific notation writes a non-zero number as a coefficient whose absolute value is at least 1 but less than 10, multiplied by an integer power of 10. In Grade 9 CAPS, negative exponents extend the notation to very small numbers. Start lesson →

🧩 Geometric patterns A geometric pattern grows through pictures or arrangements. Grade 9 learners describe what changes from figure to figure and connect the picture to numbers. Start lesson → 🧩 More patterns Not every pattern grows by the same amount. This lesson compares different growth structures and asks learners to justify a rule rather than guessing from one step. Start lesson → 🧩 Different kinds of patterns in sequences Sequences can have a constant difference, a constant ratio, or another structure. Grade 9 learners classify the pattern only after checking how consecutive terms are related. Start lesson → 🧩 Formulae for sequences A sequence rule can describe how to get the next term from the previous term, or it can connect a term directly to its position number. Grade 9 patterns are not limited to arithmetic sequences. Start lesson →

🔁 Find output numbers for given input numbers A function maps each allowed input to exactly one output using a rule. Start lesson → 🔁 Different ways to represent the same relationship A relationship between input and output can be described verbally, with a flow diagram, a table, a formula or an equation. The goal is to recognise that these forms can encode the same rule. Start lesson → 🔁 Different representations of the same relationship This lesson moves between tables, formulae and graphs. A correct representation must preserve the same ordered pairs, not merely look similar. Start lesson →

🧮 Algebraic language Algebra translates operation instructions into symbols, using variables for values that can change. Start lesson → 🧮 Properties of operations Operation properties explain why expressions can be rearranged or expanded without changing their value. Start lesson → 🧮 Combining like terms in algebraic expressions Like terms have the same variable part, so their coefficients can be combined. Start lesson → 🧮 Multiplication of algebraic expressions Multiplying algebraic expressions requires every factor to multiply every term it acts on. Start lesson → 🧮 Dividing polynomials by integers and monomials Dividing a polynomial by a monomial means dividing every term by the common divisor. Start lesson → 🧮 Products and squares of binomials A binomial product is expanded by multiplying every term in one bracket by every term in the other. Start lesson → 🧮 Substitution into algebraic expressions Substitution replaces a variable with a given value and then evaluates the resulting expression. Start lesson → 🧮 Introduction Factorisation rewrites an algebraic expression as a product. It is the reverse of expanding brackets. Start lesson → 🧮 Factors of expressions of the form ab + ac When every term contains a common numerical or algebraic factor, that factor can be taken outside brackets. Start lesson → 🧮 Factors of expressions of the form x² + (b + c)x + bc A trinomial x²+(b+c)x+bc can be factorised by finding two numbers whose product is the constant term and whose sum is the x-coefficient. Start lesson → 🧮 Factors of expressions of the form a² − b² A difference of two perfect squares factorises into two conjugate brackets. Start lesson → 🧮 Simplification of algebraic fractions Algebraic fractions simplify by factorising first and cancelling common factors, not by cancelling terms joined by addition or subtraction. Start lesson →

⚖️ Solving equations by inspection Solving by inspection means testing or recognising the value that makes the left-hand side equal to the right-hand side. The emphasis is on the meaning of a solution before formal inverse steps. Start lesson → ⚖️ Solving equations using additive and multiplicative inverses Inverse operations solve equations systematically by undoing operations while keeping the equation balanced. The same valid operation is applied to both sides. Start lesson → ⚖️ Setting up equations Setting up an equation means translating a verbal relationship into symbols. Define the unknown first, then express two equal quantities on opposite sides of the equal sign. Start lesson → ⚖️ Equations and situations Once an equation models a situation, its solution must be interpreted in the original context. A mathematically correct value can still be unsuitable if it violates the real situation. Start lesson → ⚖️ Solving equations by using the laws of exponents Some Grade 9 equations first need exponent laws before the bases can be compared. Simplify products, quotients or powers of powers, then solve the exponent equation. Start lesson → ⚖️ Introduction An equation is true for particular values of the variable. Solving means finding those values. Start lesson → ⚖️ Solving by factorisation Part 1 Part 1 starts with equations that are already written as a product equal to zero. The key idea is the zero-product rule: if A×B=0, at least one factor must be zero. Start lesson → ⚖️ Solving by factorisation Part 2 Part 2 begins with a quadratic expression that must first be factorised. The zero-product rule can only be used after the equation is written as factors multiplied together and equal to zero. Start lesson → ⚖️ Solving by factorisation Part 3 Part 3 combines factorisation choices with equation solving. Learners must recognise the useful factorisation form, factor completely, and then interpret every valid solution. Start lesson → ⚖️ Set up equations to solve problems A real situation can be solved by defining an unknown, translating the relationships into an equation, solving, and interpreting the result. Start lesson → ⚖️ Equations and ordered pairs An equation in two variables describes many ordered pairs. A pair is a solution when its x- and y-values make the equation true. Start lesson →

📐 Constructing perpendicular lines A perpendicular construction creates a line meeting another at exactly 90° using equal-radius arcs. Start lesson → 📐 Bisecting angles To bisect an angle is to construct two equal angles. Compass arcs create points that are equally placed from the arms, and a ruler joins the vertex to the arc intersection. Start lesson → 📐 Constructing special angles without a protractor Special angles can be constructed from reliable 60°, 90° and bisection constructions without measuring with a protractor. Start lesson → 📐 Angle bisectors in triangles The three internal angle bisectors of a triangle meet at one point, called the incenter. That point is the same perpendicular distance from all three sides. Start lesson → 📐 Interior and exterior angles in triangles An exterior angle of a triangle is formed when one side is extended. It is supplementary to its adjacent interior angle and equals the sum of the two opposite interior angles. Start lesson → 📐 Constructing congruent triangles A construction is uniquely determined only when the given measurements fix one triangle. Grade 9 explores which combinations such as SSS, SAS, AAS and RHS produce congruent triangles. Start lesson → 📐 Diagonals of quadrilaterals The diagonals of a quadrilateral reveal different properties depending on the type of quadrilateral. Grade 9 uses constructions to investigate which diagonal facts are guaranteed. Start lesson → 📐 Angles in polygons Polygon interior-angle sums come from splitting the polygon into triangles. Start lesson →

🔷 Revision: Classification of triangles Triangles can be classified by side equality and by angle size. Start lesson → 🔷 Finding unknown angles in triangles Unknown triangle angles are found by combining facts such as the 180° angle sum, equal base angles in an isosceles triangle, straight-line angles and exterior-angle relationships. Start lesson → 🔷 Quadrilaterals Quadrilaterals are classified by side, angle and diagonal properties. Start lesson → 🔷 Congruent triangles Congruent triangles have exactly the same size and shape. A proof must match corresponding vertices and justify congruence using a valid condition such as SSS, SAS, AAS or RHS. Start lesson → 🔷 Similar triangles Similar triangles have the same shape; corresponding angles are equal and corresponding sides share a scale factor. Start lesson → 🔷 Extension questions Extension geometry problems combine more than one theorem. The challenge is deciding which fact unlocks the next fact and writing a complete reason chain. Start lesson →

📏 Angle relationships This lesson focuses on recognising the standard angle relationships created by straight, intersecting, perpendicular and parallel lines. Start lesson → 📏 Identify and name angles Angle naming uses three letters with the vertex in the middle. This lesson links correct notation to the angle relationships in line diagrams. Start lesson → 📏 Solving problems Straight-line geometry problems combine angle facts from intersecting lines, perpendicular lines and parallel lines cut by a transversal. Each calculated angle must be supported by a named relationship. Start lesson →

📐 Investigating the sides of a right-angled triangle Pythagoras' Theorem can be discovered by comparing the areas of squares built on the three sides of a right-angled triangle. Start lesson → 📐 Checking for right-angled triangles The converse of Pythagoras tests whether three side lengths form a right-angled triangle. Start lesson → 📐 Finding missing sides When a triangle is known to be right-angled, Pythagoras' Theorem can find a missing side. The algebra changes depending on whether the missing side is the hypotenuse or a shorter side. Start lesson → 📐 More practice using Pythagoras’ Theorem Pythagoras can be embedded inside rectangles, coordinate grids and composite figures. The main skill is spotting the right triangle before calculating. Start lesson →

📦 Area and perimeter of squares and rectangles Perimeter measures distance around a shape; area measures surface inside it. Start lesson → 📦 Area and perimeter of composite figures A composite shape can be split into familiar shapes whose areas can be combined. Start lesson → 📦 Area and perimeter of circles Circle measurements depend on radius or diameter and π. Start lesson → 📦 Converting between units Unit conversion changes the unit label and numerical value while preserving the physical quantity. Start lesson → 📦 Area of other quadrilaterals Areas of parallelograms, kites and trapeziums can be understood by rearranging or combining simpler shapes. Start lesson → 📦 Doubling dimensions of a 2D shape Scaling all lengths by k scales perimeter by k and area by k². Start lesson →

🔁 From formulas to words, tables and graphs A relationship between two variables can be expressed in words, a formula, a flow diagram, a table or a graph without changing the relationship itself. Start lesson → 🔁 Tables and graphs Tables and graphs show pairs of values that belong to the same relationship. Start lesson →

📈 Global graphs A graph gives a global picture of how two quantities vary together, including trends, intercepts and regions of increase or decrease. Start lesson → 📈 Changes at different rates Graphs reveal whether quantities change at a constant or changing rate. Start lesson → 📈 Draw graphs from tables of ordered pairs A graph can be constructed from a table by plotting each ordered pair accurately on labelled axes. Start lesson → 📈 Gradient Gradient measures the steepness and direction of a straight line by comparing vertical change to horizontal change. Start lesson → 📈 Finding the formula for a graph For a straight-line relationship, the formula can be recovered from its gradient and where it crosses the y-axis. Start lesson → 📈 x- and y-intercepts Intercepts are the points where a graph crosses an axis. Start lesson → 📈 Graphs of non-linear functions Non-linear functions do not form one straight line. Their rate of change varies across the graph. Start lesson →

📦 Surface area Surface area is the total area of all outside faces or surfaces of a 3D object. Start lesson → 📦 Volume Volume measures the three-dimensional space occupied by an object. Start lesson → 📦 Capacity Capacity describes how much a container can hold and is closely related to volume. Start lesson → 📦 Doubling dimensions and the effect on volume When every linear dimension is scaled by k, volume is scaled by k³. Start lesson →

🪞 Points on a coordinate system A coordinate system locates points using horizontal x-values and vertical y-values. Start lesson → 🪞 Reflection (flip) A reflection flips a figure across a mirror line while preserving lengths and angles. Start lesson → 🪞 Translation (slide) A translation slides every point by the same horizontal and vertical displacement. Start lesson → 🪞 Enlargement (expansion) and reduction (shrinking) An enlargement or reduction changes size from a centre using a scale factor while preserving shape. Start lesson →

🧊 Classifying 3D objects 3D objects are classified using faces or surfaces, edges and vertices. Start lesson → 🧊 Nets and models of prisms and pyramids A net is a flat pattern that folds to form a 3D object. Start lesson → 🧊 Platonic solids Platonic solids are convex polyhedra made from one type of congruent regular polygon with the same number of faces meeting at every vertex. Start lesson → 🧊 Euler’s formula Euler’s formula links the numbers of faces, vertices and edges of many polyhedra. Start lesson → 🧊 Cylinders A cylinder has two congruent parallel circular bases joined by one curved surface. Grade 9 also uses cylinder measurements in the CAPS surface-area, volume and capacity work. Start lesson → 🧊 Spheres A sphere is a 3D object with one curved surface whose points are all the same distance from its centre. Start lesson →

📋 Collecting data Good data collection starts with a clear question and a method that represents the population fairly. Start lesson → 📋 Organising data Organising data makes patterns visible using ordered lists, frequency tables, grouped intervals or stem-and-leaf displays. Start lesson → 📋 Summarising data Summary statistics describe the centre and spread of a data set. Start lesson →

📊 Bar graphs and double bar graphs Bar graphs compare frequencies across categories, while double bar graphs compare two related groups in the same categories. Start lesson → 📊 Histograms A histogram displays grouped continuous numerical data using touching bars. Start lesson → 📊 Pie charts A pie chart represents each category as a sector of a 360° circle. Start lesson → 📊 Broken-line graphs A broken-line graph connects ordered observations to show how a numerical quantity changes, often over time. Start lesson → 📊 Scatter plots A scatter plot displays paired numerical data to investigate whether two variables are associated. Start lesson →

🔎 Which graph is best? Different graphs suit different data and questions, so the representation should be chosen deliberately. Start lesson → 🔎 The effects of summary statistics on how data is reported Mean, median, mode and range can tell different stories about the same data, especially when extreme values are present. Start lesson → 🔎 Misleading graphs Graphs can mislead through distorted scales, truncated axes, unequal intervals, oversized pictures or missing labels. Start lesson → 🔎 Analysing extreme values and outliers An outlier is an unusually distant data value that can strongly affect some summaries and interpretations. Start lesson →

🎲 Simple events Probability measures how likely an event is on a scale from 0 to 1. Start lesson → 🎲 Compound events A compound event combines two or more stages or events and can be organised using tables or tree diagrams. Start lesson →