Complete Edexcel IGCSE Higher Tier Maths Review – 60 Topics Explained

The video presents a comprehensive review of the Edexcel IGCSE Higher Tier Maths course, broken down into 60 topics. Each topic is illustrated with examples drawn from past‑paper questions, and the presenter recommends a three‑step learning routine: watch the example, attempt the practice question via the provided link, then review the solution.

Number

Prime factorisation is tackled with tree diagrams, while the highest common factor (HCF) and lowest common multiple (LCM) are most efficiently found using Venn diagrams. Fraction operations follow the standard rules: addition and subtraction require a common denominator, multiplication uses “top × top, bottom × bottom”, and division follows the “keep, flip, change” method. Percentages are handled by converting the percentage to a decimal multiplier; increases and decreases use (1\pm\frac{\%}{100}), and reverse percentages are found by dividing by the same multiplier. Compound interest uses the formula
[ \text{Amount}=P\left(1+\frac{r}{100}\right)^{n} ]
with depreciation represented by a minus sign inside the brackets. Ratios are shared by adding the parts, dividing the total by this sum, and then multiplying by each part; differences in ratio are treated by equating the numerical difference to the given value. Standard form expresses a number as a coefficient between 1 and 10 multiplied by a power of 10, with negative exponents indicating values less than 1. Bounds are calculated by determining the lower and upper limits of a measurement, and recurring decimals are converted to fractions.

Algebra

Sequences are described by the nth‑term formulas (a_n = a + (n-1)d) for linear sequences and a quadratic form involving (n^2) for quadratic sequences; the sum of the first n terms is (S_n = \frac{n}{2}(2a+(n-1)d)). Bracket expansion progresses from single brackets (simple distribution) to double brackets using the “Lobster Claw” (FOIL) method, and finally to triple brackets by expanding two at a time. Factorising begins with common numerical and algebraic factors, proceeds to quadratic factorisation by finding two numbers that multiply to (ac) and add to (b), and includes the difference of squares identity (a^2-b^2=(a-b)(a+b)). Linear equations are solved by isolating the variable with inverse operations; when variables appear on both sides, the smallest term is moved first. Inequalities are solved similarly, with the sign reversing when multiplying or dividing by a negative number, and are represented on a number line with open or closed circles. Regions are defined by inequalities such as (x=-2) or (y=0.5x+1), indicating the side of the line that satisfies the condition. Simultaneous equations are tackled by elimination (matching coefficients) or substitution (making one variable the subject). Indices follow the standard laws, and fractional powers express roots. Rearranging formulae makes a chosen variable the subject, while direct and inverse proportion use (y=kx) and (y=k/x) respectively. Algebraic fractions are simplified, combined, and used in equations. Completing the square rewrites quadratics into ((x+\frac{b}{2})^2) form, facilitating solution of equations and graphing. Functions are evaluated by substitution, composed as (f(g(x))), and inverted by swapping (x) and (y). Differentiation applies the power rule (d(ax^n)/dx = a n x^{n-1}); constants differentiate to zero, and stationary points occur where (dy/dx=0). Kinematics links displacement, velocity and acceleration through successive differentiation.

Geometry & Mensuration

Construction of a perpendicular bisector uses arcs drawn from two points on a line; an angle bisector uses arcs from the vertex. Transformations include reflections in the lines (x=a) (vertical) and (y=b) (horizontal), translations defined by a vector ((\Delta x,\Delta y)), anticlockwise rotations about a centre, and enlargements specified by a centre and scale factor (k). Angle relationships are summarised: opposite angles are equal, alternate (Z‑shaped) and corresponding (F‑shaped) angles are equal, co‑interior angles sum to 180°, and angles on a straight line also sum to 180°. Polygon angle formulas give exterior angle (=360°/n) and interior angle calculations, allowing the number of sides to be deduced. Bearings are measured clockwise from North using three‑figure notation. The Pythagoras theorem finds hypotenuse or legs of right‑angled triangles, while trigonometry (SOH CAH TOA) computes missing sides or angles. Area and perimeter formulas cover triangles, parallelograms, trapezia, circles and composite shapes; volume and surface‑area formulas address triangular prisms, cylinders, cones, spheres and hemispheres, with slant height (l=\sqrt{r^2+h^2}). Similar shapes use scale factor (k) for lengths, (k^2) for areas and (k^3) for volumes. Circle theorems include angles at the centre, angles in the same segment, the semicircle theorem, cyclic quadrilaterals, tangents, and the alternate segment theorem. The intersecting chords/secants theorem (power of a point) extends these ideas to chords and secants. 3‑D Pythagoras and trigonometry handle diagonals in cuboids.

Statistics & Probability

Probability fundamentals state that the sum of all probabilities in an experiment equals 1. Probability tables help locate specific outcomes, and frequencies are estimated by multiplying the probability by the number of trials. Tree diagrams illustrate combined events (AND, OR) and conditional probability. Averages—mean, median, mode, range and inter‑quartile range—are calculated from raw data or frequency tables. Cumulative frequency graphs are plotted, interpreted, and used to find the median and IQR. Set theory notation (∪, ∩, ′) is applied to numbers, and Venn diagrams are filled in to compute probabilities of overlapping events. Histograms display frequency density, are drawn with appropriate class widths, and are interpreted for distribution shape.

Proofs and Advanced Topics

Algebraic proofs verify statements such as “the sum of two even numbers is even” or properties of consecutive integers. Recurring decimals are converted to fractions by setting (x) equal to the decimal and solving for (x). Surds are simplified by extracting the largest square factor, expanded within brackets, and rationalised using conjugates. Graph transformations describe shifts, reflections, stretches and compressions of functions. Vector operations include finding a vector between two points, reversing direction (multiplying by –1), and halving a vector by dividing each component by 2. The sine and cosine rules solve oblique triangles, and the triangle area formula (\frac{1}{2}ab\sin C) links side lengths and included angle.