Complete GCSE and IGCSE Maths Masterclass: Higher Level

Explore the basics of algebra, learning how to add, subtract, and multiply with sign rules, and how variables, terms, and powers form algebraic expressions.

Explore powers and roots through seven rules for exponents, including multiplying and dividing like bases, handling negative powers, and extracting roots, with practical algebraic examples.

Explore multiplying out brackets in algebra, starting with single brackets and progressing to double and triple brackets, using sign rules and practice examples to reinforce expansion.

Learn factorizing by taking common from two-term expressions and rewriting as brackets, and apply the dots (difference of squares) concept to factorize more complex expressions.

Master surds by applying six rules for multiplying, dividing, and simplifying; include sums, products, and rationalizing denominators, plus bracket expansion and expressing results as a sqrt b.

Learners discover how to solve linear and simple quadratic equations for x by shifting constants to the right, removing denominators with multiplication, and taking square roots when needed.

Master rearranging formulas and making the subject of the formula by isolating the variable through multiplication, division, addition, subtraction, and square roots in algebra.

Learn the factorization method to solve quadratic equations by converting them into two linear equations, using product and sum to find x within the standard form ax^2+bx+c=0.

Master solving quadratic equations with the quadratic formula and compare it to factorization. Use the standard form ax^2+bx+c=0 and x=(-b±√(b^2-4ac))/(2a) for exact solutions.

Learn to solve quadratic equations in standard form using the completing the square method, shifting constants, forming a perfect square, and deriving the two roots.

Explore algebraic fractions through simplification and solving, including factoring, completing the square, cross-multiplication, and canceling common factors to simplify complex expressions and divisions.

Learn to identify linear and quadratic sequences, compute their nth terms using the common difference and second difference, and apply examples from the first and second terms.

Explore solving inequalities graphically by converting each inequality to a quadratic or linear equation, drawing corresponding lines (solid for ≤ and dotted for <), and shading the solution region.

Learn to solve linear equations, categorize linear, quadratic, and cubic equations, and apply methods for quadratic equations such as factorization, quadratic formula, and completing the square, isolating the unknown.

Learn to solve simultaneous equations with two variables using elimination and substitution, forming ax+by=c and handling both linear and mixed linear-quadratic cases.

Apply the natural numbers sum formula, n(n+1)/2, to quickly find sums like 1 to 10 and 1 to 100, yielding 55 and 5050.

Discover the quick method to add natural numbers using the sum formula n(n+1)/2, demonstrated with the first 20 numbers, yielding 210.

Learn to find the area of a complex shape by dividing it into three rectangles, calculating each area (4×3, 14×5, 4×8), and summing to 114 square centimeters.

Learn to solve algebra questions for GCSE and IGCSE through a step-by-step linear equation, shifting terms to isolate x and find x = 3/10.

Explore linear equations and their straight-line graphs, and identify lines through axes and the origin. Learn to compute gradient using two-point formulas and rise over run.

Determine a line’s equation by using the slope and y-intercept from y = mx + c, with two points or a graph, and convert to standard form.

Draw graphs from equations using three methods: table of values, y = mx + c form, and intercepts, with examples and interpreting slope and intercept.

parallel lines share the same gradient and follow y = mx + c; perpendicular lines have negative reciprocal slopes, used to find equations through a point.

Explore how quadratic graphs form u-shaped or inverted u-shaped forms based on the x^2 coefficient, identify the line of symmetry, and plot by completing a value table and finding intercepts.

Explore harder graphs, including cubic and circle graphs, plus exponential graphs with base k. Learn to classify equations by max power and sketch x^3 and tangents to circles.

Explore reciprocal graphs and equations, including symmetry about y=x and y=-x, quadrant placement, and undefined points at x=0; compare sin x, cos x, and tan x graphs.

Learn to solve simultaneous equations using graphs by plotting a circle and a line, finding intersection points to obtain x and y, and apply to sine and quadratic equations.

Explore how graph transformation translates a function along the y and x axes using f(x) notation, including shifts by adding or subtracting constants and simple quadratic examples.

Explore distance-time and velocity-time graphs, learn how gradient reveals speed or acceleration, identify motion, rest, and deceleration, and estimate distance via area under the curve.

Explore gradients on real-life graphs by reading axis units like meters per second, liters per second, and people per minute, and learn about average gradients and tangent lines.

Explore ratio, proportion, and rate of change, including converting between ratio and fraction, simplifying, scaling, and applying direct proportion and percentage concepts in everyday problems.

Master direct and inverse proportion by turning proportional statements into equations with a constant k, and applying these concepts to problems such as the cost of cheese pasties.

Explore the fundamentals of percentage, learn to find percent of a number, handle percent increases and decreases, and convert numbers to percentages with practical GCSE-style examples.

Explore percentage changes in value, learning the change over original formula and applying it to gains and losses in prices, weights, and stocks through worked examples.

Explore percentage concepts across problems, from simple interest at 3% per year to gym demographics and cube volume. Apply 10% side-length increases and 33.1% volume rise to practice.

Explore compound growth and decay with the compound interest concept, compare it to simple interest, and apply the formula N = N0 × multiplier^n using 1.08 and 0.91.

Master metric and imperial unit conversions, covering length, mass, volume, area, and speed, with practical worked examples including pounds to kilograms.

Master speed, density, and pressure using their core formulas—speed equals distance over time, density equals mass over volume, and pressure equals force over area—through worked examples.

Convert 23% to 0.23, use 0.77, and multiply by 20,000 to obtain the 2026 price of $15,400, a method applicable to stocks or cryptocurrency.

learn to solve percentage increase problems by converting 45% to decimal, adding 1 to get 1.45, and multiplying a $400 base to get $580 for bitcoin in 2016.

Master geometry with rules: triangle angles sum to 180, straight line angles sum to 180, quadrilaterals sum to 360, angles around a point sum to 360, isosceles triangle properties.

Examine parallel lines and angle relationships, including alternate, allied, and corresponding angles, plus vertically opposite angles and straight-line sums, with a triangle sum proof.

Analyze geometry problems by identifying angles in triangles, using the triangle angle sum of 180 degrees, and applying isosceles, right-angle, and parallel-line rules to find x, y, and other angles.

Explore polygons and their interior and exterior angles. Apply regular polygon formulas to find interior angles, exterior angles, and the number of sides.

Explore triangles and quadrilaterals, including equilateral, right angle, isosceles, and scalene triangles, and learn quadrilateral properties, 360-degree angle sum, and shapes like square, rectangle, rhombus, parallelogram, trapezium, and kite.

Master congruent shapes and similar shapes using SSS, two angles and one side, two sides and one angle, and RHS; then apply scale factors to find missing lengths.

Explore the four transformations—translation, rotation, reflection, and enlargement—on the coordinate plane, using vectors, axes, angles, lines, and scale factors to describe and locate changes.

Explore area and circumference of circles, using radius and diameter formulas with pi 3.14. Learn sector, arc, and segment calculations, including area of sector and area of segment.

Explore three-dimensional shapes and surface area and volume, including cubes, cylinders, cones, spheres and hemispheres, with key formulas and terminology like vertices, faces and edges.

Draw a line to separate the shape into a square and a rectangle, compute each area using given sides, and sum to obtain the total area 2x^2+24x+46.

Apply the straight-line angle rule to a geometry problem: set x plus 121 equal to 180 degrees, then solve to obtain x = 59 degrees.

solve a geometry problem using pythagoras and tangent to find theta in a right triangle formed by a 5 m height and a 15 m by 12 m rectangle.

Master trigonometric ratios in right-angled triangles—sine, cosine, tangent—with opposite, adjacent, and hypotenuse; learn the soh cah toa mnemonic and common values for 30, 45, 60 degrees.

Explore the sine and cosine rules to solve any triangle by using labeled sides and angles, and apply the area formula with two sides and the included angle.

Explore three-dimensional Pythagoras theorem and its use on cuboids and pyramids to find diagonals. Apply d^2 = a^2 + b^2 + c^2 and basic trigonometry in three dimensions.

Explore vectors as quantities with size and direction, distinguished from scalars, and learn notation, arrow representation, and operations like scalar multiplication, addition, and subtraction on triangles and parallelograms.

Explore probability basics: outcomes between 0 and 1, apply the formula of favorable over total outcomes, and see how probabilities add to one using coins and balls.

Learn to count outcomes with sample space diagrams and the product rule, applying them to coin flips, spinners, and dice to find probabilities.

Explore how to distinguish fair and biased dice, understand probability, and apply relative and expected frequencies, frequency trees, and repeated experiments to assess fairness.

Analyze probability experiments with frequency trees, compute relative and expected frequencies, and apply dice and GCSE student data to practice interpreting outcomes.

Learn the and/or rules of probability, distinguishing independent and dependent events, and applying the formulas for and/or scenarios using bags, dice, and card examples.

Learn to build a tree diagram for three coin flips, map the eight-outcome sample space, and calculate probabilities with independent events and complements.

Explore conditional probability by distinguishing dependent and independent events, and solving problems with and without replacement while applying the conditional formula p(a and b) = p(a) p(b|a).

Learn the basics of sets and Venn diagrams, including elements, union, intersection, and complement, and apply these concepts to probability within the probability and statistics chapter.