Mathematics

Divisibility rules are quick tests that tell you whether one number divides evenly into another without having to do the full division.

Long division is a step-by-step written method for dividing large numbers, repeating the cycle of divide, multiply, subtract, and bring down.

Long multiplication breaks a multi-digit multiplication into smaller steps using partial products, then adds them together.

BODMAS gives the correct sequence for evaluating expressions with multiple operations: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

Column addition lines up numbers by place value and adds from right to left, carrying any overflow into the next column.

Column subtraction lines up numbers by place value and subtracts from right to left, borrowing from the next column when needed.

An inequality compares two expressions using <, >, ≤ or ≥. Solving an inequality is like solving an equation, except that multiplying or dividing by a negative number reverses the inequality sign.

A linear equation is an algebraic equation where each term is either a constant or a single variable raised to the first power.

Substitution means replacing a letter (variable) in an expression with a given number and then calculating the result. It is used to evaluate formulae and check solutions to equations.

A system of equations is a set of two or more equations with the same variables, solved simultaneously.

A two-step equation needs exactly two operations to isolate the variable. Undo addition or subtraction first, then undo multiplication or division.

How to translate word problems into algebraic equations and solve them step by step.

To add or subtract fractions, first rewrite them with a common denominator (LCD), then add or subtract the numerators.

To divide by a fraction, multiply by its reciprocal — flip the second fraction and multiply.

Two fractions are equivalent when they represent the same value. Multiply or divide both the numerator and denominator by the same non-zero number.

Multiply fractions by multiplying numerator by numerator and denominator by denominator. No common denominator needed.

How to subtract fractions with the same and different denominators, including mixed numbers.

Foundation of mathematics covering numbers, expressions, equations, and problem-solving skills that build all higher mathematics.

Use the symbols <, >, and = to compare natural numbers, and arrange them in ascending or descending order using place value and the number line.

Every digit in a number has a place value — its position tells you how much it is worth. Learn to read, write, and expand large numbers up to millions.

Rounding replaces a number with a nearby "round" number that is easier to work with. Learn to round to the nearest 10, 100, 1 000, and 10 000.

Percent change measures how much a value has increased or decreased relative to its original value, expressed as a percentage.

A percentage is a fraction out of 100. To find a percentage of a number, multiply the number by the percentage divided by 100.

Simple interest is calculated on the original principal only. The formula I = PRT links the interest, principal, rate, and time. Compound interest is briefly introduced as interest on interest.

VAT (Value Added Tax) is a percentage added to a price. Discounts are percentages subtracted from a price. Knowing how to calculate both is an essential everyday maths skill.

Completing the square rewrites a quadratic expression ax² + bx + c in the form a(x + p)² + q. This technique is used to solve quadratics, find the vertex of a parabola, and derive the quadratic formula.

The discriminant of a quadratic equation determines how many and what type of roots the equation has.

The quadratic formula solves any quadratic equation ax² + bx + c = 0 by finding the roots in terms of a, b, and c.

Two quantities are in direct proportion when they increase and decrease at the same rate. If one doubles, the other doubles too. The relationship is written y = kx, where k is the constant of proportionality.

Two quantities are in inverse proportion when one increases as the other decreases by the same factor. The product of the two quantities is always constant, and the formula is y = k/x.

To divide a quantity in a given ratio, find the value of one part by dividing the total by the sum of the ratio parts, then multiply each ratio number by one part.

A ratio compares two quantities. A proportion states that two ratios are equal. Proportions are solved using cross-multiplication.

Angles are classified by their measure in degrees — acute, right, obtuse, straight, and reflex. Understanding angle types is the foundation of geometry.

Angles on a straight line sum to 180°. Angles around a full point sum to 360°. Vertically opposite angles are always equal.

When a transversal crosses two parallel lines, it creates pairs of equal angles (corresponding and alternate angles) and supplementary angle pairs (co-interior angles that add up to 180°).

The sum of the interior angles of any polygon depends only on the number of sides. For an n-sided polygon the sum is (n − 2) × 180°. Regular polygons have all angles equal, so each interior angle = sum ÷ n.

The angle sum property of triangles and exterior angle theorem with examples.

A compound shape is made up of two or more simple shapes joined together. To find the total area, split the compound shape into rectangles, triangles, or other familiar shapes, calculate each area separately, and add them up.

Explore the units used to measure length, mass, time, and volume, and practise converting between them using the correct conversion factors.

Formulas for the area and perimeter of a parallelogram with worked examples.

Formulas for the area and perimeter of a rectangle, including worked examples.

Formula cards for the area and perimeter of all common 2D shapes.

Formulas for the area, perimeter, and diagonal of a square, with worked examples.

Formulas for the area and perimeter of a trapezoid with worked examples.

How to calculate the area and perimeter of triangles using base-height and Heron's formula.

An arc is part of the circumference; a sector is a pie-slice of a circle. Both are calculated as a fraction of the full circle, where the fraction = angle ÷ 360.

Formulas for the area and circumference of a circle, and how to apply them.

The circumference is the distance around the outside of a circle. It equals π multiplied by the diameter, or 2π multiplied by the radius.

An angle is formed by two rays sharing a vertex. Learn to identify angle types, use a protractor to measure angles, and draw angles of any given size.

A triangle is a three-sided polygon. Learn the triangle inequality, how to construct triangles from given measurements using compass and ruler, and the main types of triangles.

A line segment is a straight path between two fixed points. Learn how to draw segments of a given length, measure them, and find the midpoint using a compass.

The coordinate plane uses two perpendicular number lines (axes) to locate any point using an ordered pair (x, y). It is divided into four quadrants.

The gradient (slope) of a line measures how steep it is — how much the line rises or falls for every unit it moves to the right. Gradient = change in y ÷ change in x (rise ÷ run).

The midpoint of a line segment is the point exactly halfway between its endpoints. The distance formula uses Pythagoras' theorem to find the straight-line distance between any two points on a coordinate grid.

Every straight-line graph can be written as y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). Knowing these two values is enough to draw or describe any straight line.

Study of shapes, space, and measurement covering angles, areas, volumes, coordinates, and spatial reasoning.

Formulas for the volume and surface area of a cone with worked examples.

How to calculate the surface area and volume of a cube and rectangular box.

Formulas for the volume and surface area of a cuboid (rectangular box) with worked examples.

Formulas for computing the surface area and volume of a cylinder.

Formula cards for the volume and surface area of all common 3D solids.

Formulas for the volume and surface area of a sphere with worked examples.

A shape has line symmetry when it can be folded along a line so that both halves match exactly. Learn to identify axes of symmetry in common shapes, regular polygons, and everyday letters.

A shape has rotational symmetry when it looks the same after being rotated less than a full turn about a fixed centre point. Explore the order of rotational symmetry and point (central) symmetry.

Study of triangle relationships and periodic functions using angles, trigonometric ratios, and their applications.

The Pythagorean theorem relates the three sides of a right triangle: a² + b² = c².

Angles of elevation (looking up) and depression (looking down) are measured from the horizontal. Combined with trigonometry, they allow us to find heights, distances, and positions in the real world.

The three primary trigonometric ratios: sine, cosine, and tangent in a right triangle.

SOHCAHTOA is a mnemonic for the three trigonometric ratios in a right-angled triangle. It is used to find missing side lengths when an angle and one side are known.

When you know two sides of a right-angled triangle, you can use the inverse trigonometric functions (arcsin, arccos, arctan) to find a missing angle.