GCSE Maths Study Guides

Algebra uses letters and symbols to represent variables, formulate equations, and solve mathematical problems systematically. For GCSE Maths, fluency with expanding brackets, factorising expressions, rearranging formulae, and solving linear, quadratic, and simultaneous equations underpins both Foundation and Higher papers.

Trigonometry links the interior angles of a triangle to the lengths of its sides. In GCSE Maths, right-angled triangles are solved using Pythagoras' theorem and the sine, cosine, and tangent ratios (SOH CAH TOA), while non-right-angled triangles on Higher tier require the Sine Rule, the Cosine Rule, and Area = ½ab sin C.

Probability and statistics provide methods for collecting, summarising, and interpreting data alongside calculating the likelihood of events. Key GCSE topics include averages from frequency tables, cumulative frequency and box plots, histograms, Venn diagrams, and probability tree diagrams for independent and conditional events.

A quadratic expression takes the form ax² + bx + c (where a ≠ 0) and plots as a symmetrical parabola. In GCSE Maths, you solve quadratic equations by factorising into two brackets, using the quadratic formula, or completing the square to find the roots, y-intercept, and turning point.

Geometry studies the properties of angles, lines, polygons, and circles. Securing marks in GCSE geometry relies on stating exact geometric reasons alongside your calculations, including parallel-line angle facts, interior and exterior polygon formulas, circle theorems, and triangle congruence criteria.

Laws of indices provide the algebraic rules for multiplying, dividing, and raising powers with the same base, extending on the Higher Tier to negative indices (which take the reciprocal, x⁻ⁿ = 1/xⁿ) and fractional indices (where the denominator is the root and the numerator is the power, x^(m/n) = (ⁿ√x)ᵐ). Surds are irrational roots of integers (such as √2 or √5) that cannot be written as exact fractions and must be manipulated in exact form on Non-Calculator Paper 1. Examiners regularly test evaluating combined negative fractional powers of fractions, simplifying surd expressions, and rationalising two-term denominators using the difference of two squares conjugate.

Standard form expresses very large or very small numbers concisely in the format A x 10ⁿ, where 1 ≤ A < 10 and n is an integer. Whenever continuous quantities (such as length, mass, or time) are rounded to a given degree of accuracy, their true value lies within an error interval bounded by a Lower Bound (LB) and an Upper Bound (UB), found by adding and subtracting half of the rounding unit. On the Higher Tier, examiners regularly test propagating upper and lower bounds through division and subtraction formulas (such as speed = distance / time) and finding the level of accuracy to which a value is known.

Fast, accurate percentage calculations in GCSE Maths rely on converting percentage increases and decreases into single decimal multipliers: adding r% multiplies by (1 + r/100), and subtracting r% multiplies by (1 - r/100). Repeated percentage changes, such as bank compound interest, population growth, or car value depreciation, are modelled using the exponential formula Total = Initial x (Multiplier)ⁿ. Examiners regularly test reverse percentages (finding the original value before a percentage change was applied by dividing by the multiplier) and multi-year compound growth problems.

Ratio and proportion problems bridge arithmetic and algebra throughout GCSE Maths: ratio questions frequently require combining two linked ratios (such as A:B and B:C) via a lowest common multiple or forming an algebraic equation when items are added to or removed from a ratio. On the Higher Tier, proportional relationships are expressed as algebraic equations using a constant of proportionality k: direct proportion takes the form y = kx (or y = kx², y = k√x), whereas inverse proportion takes the form y = k / x (or y = k / x²). Examiners consistently test finding the constant k from a pair of values and using the resulting formula to calculate unknown values or percentage changes.

Simultaneous equations are pairs of equations containing two unknown variables (usually x and y) that must be satisfied at the same time. Two linear simultaneous equations are solved by scaling the equations so one variable has matching coefficients and using elimination (remembering 'SSS: Same Signs Subtract, Different Signs Add'), whereas a linear equation paired with a non-linear quadratic or circle equation (such as x² + y² = r²) on the Higher Tier must be solved by substitution. Geometrically, the solutions to any pair of simultaneous equations represent the exact (x, y) coordinates where their graphs intersect.

Inequalities compare the relative size of algebraic expressions using the symbols <, >, ≤, and ≥. Linear inequalities are solved using the same inverse operations as linear equations (with the key rule that multiplying or dividing both sides by a negative number flips the inequality sign), and can be represented on a number line or as a shaded region in the (x, y) coordinate plane. On the Higher Tier, examiners always test quadratic inequalities (ax² + bx + c > 0 or < 0), which require finding the critical values and sketching a parabola to determine whether the solution is a single connected interval or two separate tails.

A sequence is an ordered list of numbers following a mathematical rule that connects each term to its position n (the position-to-term or nth term rule) or to the previous term (term-to-term / iteration rule). Linear (arithmetic) sequences have a constant first difference d between consecutive terms, giving the nth term formula dn + c, whereas quadratic sequences on the Higher Tier have a constant second difference 2a, giving the formula an² + bn + c. Examiners frequently test finding linear and quadratic nth terms, proving algebraically whether a given number is a term in a sequence, and working with geometric progressions (with surd or fractional common ratios) and Fibonacci-style sequences.

Every non-vertical straight line on a coordinate grid can be written in the form y = mx + c, where m is the gradient (rate of change of y with respect to x) and c is the y-intercept (where the line crosses the y-axis at (0, c)). Parallel lines always have the exact same gradient (m₁ = m₂), whereas on the Higher Tier, two perpendicular lines intersecting at 90° have gradients that are negative reciprocals of each other (m₁ x m₂ = -1, or m₂ = -1 / m₁). Examiners frequently combine perpendicular gradients with midpoints to find the equation of a perpendicular bisector, or with circle geometry (x² + y² = r²) to find the equation of a tangent to a circle at a given point.

On the GCSE Maths Higher Tier, a function f(x) is an algebraic rule that maps an input x to a single output; combining two functions in sequence creates a composite function fg(x), and reversing the mapping creates the inverse function f⁻¹(x). In particular, in the composite function fg(x), you always apply the inner function g(x) first (working from right to left) and substitute it into the outer function f! This topic also covers graph transformations, where changes outside the bracket affect the y-coordinates vertically as expected, while changes inside the bracket affect the x-coordinates horizontally in the opposite direction.

On the GCSE Maths Higher Tier, Circle Theorems are tested in multi-step geometric problems where half of the marks are awarded for calculating the unknown angles and the other half for writing the exact, word-perfect geometrical reason for each step. There are eight core circle theorems governing angles subtended by arcs, cyclic quadrilaterals, chords, and tangents (culminating in the Alternate Segment Theorem). Examiners strictly penalise vague language such as 'bowtie rule', 'arrow theorem', or 'opposite angles in a cyclic quadrilateral are equal' (they add up to 180°, they are not equal).

While Pythagoras' Theorem (a² + b² = c²) and SOHCAHTOA apply strictly to right-angled triangles (including right-angled triangles extracted from inside 3D cuboids and pyramids), the GCSE Maths Higher Tier extends trigonometry to any non-right-angled triangle using the Sine Rule, the Cosine Rule, and the area formula Area = ½ab sin C. Choosing the right formula comes down to checking whether you have a matching opposite 'Side-Angle pair': if you have a matching pair, use the Sine Rule; if you do not have a matching pair (SAS or SSS), use the Cosine Rule. Examiners frequently combine these rules in multi-step bearings, area of segments, and 3D space diagonal problems.

A vector is a quantity that has both magnitude (length/size) and direction, represented either as a column vector (x above y) or in bold/arrow notation such as **a**, **b**, or $ec{AB}$. Reversing the direction of a vector negates it ($ec{BA} = -ec{AB}$), and multiplying a vector by a scalar k scales its length by k while keeping it parallel. At the back of the GCSE Maths Higher Tier paper, 4-to-5-mark vector geometry questions test finding paths through shapes divided in ratios (e.g., AP : PB = 2 : 3 means $ec{AP} = rac{2}{5}ec{AB}$), proving two vectors are parallel, and proving three points lie on the same straight line (collinear).

Calculating the volume and total surface area of 3D solids requires distinguishing between prisms and cylinders (where Volume = Cross-Sectional Area x Length) and pointed or curved 3D solids (pyramids, cones, spheres, hemispheres, and frustums). When finding the total surface area of a solid cone, hemisphere, or half-cylinder, you must always remember to add the flat circular base(s) to the curved surface area, and use Pythagoras' Theorem ($l^2 = r^2 + h^2$) to convert between a cone's vertical height $h$ and its slant height $l$. On the Higher Tier, examiners frequently test finding the volume of a frustum (a cone with its top sliced off) and equating the volumes of two melted solids to solve for an unknown radius or height in terms of $\pi$.

Two shapes are congruent if they are the exact same size and shape (proven for triangles using the four conditions SSS, SAS, ASA, or RHS), whereas two shapes are mathematically similar if one is an enlargement of the other (corresponding angles are equal and corresponding lengths are in the same ratio). On the Higher Tier, if the linear length scale factor (or ratio of corresponding lengths) between two similar 3D solids is $k$ (or $a : b$), then the area scale factor is $k^2$ ($a^2 : b^2$) and the volume/mass scale factor is $k^3$ ($a^3 : b^3$). Examiners regularly test converting between area and volume ratios by taking the square root or cube root first to find the linear ratio $a : b$ before cubing or squaring.

Probability measures the likelihood of an event occurring on a scale from 0 (impossible) to 1 (certain), where the probabilities of all mutually exclusive exhaustive outcomes sum to 1. Multi-stage events are modelled using probability tree diagrams by multiplying along branches for 'and' ($P(A \text{ and } B) = P(A) \times P(B)$) and adding between mutually exclusive branch outcomes for 'OR', while overlapping categories are organised using Venn diagrams and set notation ($A \cap B$, $A \cup B$, $A'$). On the Higher Tier, examiners heavily test conditional probability ('without replacement' where the denominator decreases by 1 on the second pick) and Grade 9 algebraic tree diagrams that form quadratic equations.

On the GCSE Maths Higher Tier, grouped continuous data with unequal class intervals is displayed using histograms, where the vertical y-axis is Frequency Density ($ ext{FD} = ext{Frequency} div ext{Class Width}$) and the area of each bar (not its height) represents the frequency. To estimate medians, quartiles, and the interquartile range (IQR), grouped frequencies are added as a running total and plotted at the upper endpoint of each class interval to form a cumulative frequency curve. Examiners frequently ask students to construct a box plot directly from a cumulative frequency curve and compare two distributions in context using both a measure of average (the median) and a measure of spread (the IQR).

At the top end of the GCSE Maths Higher Tier (Grades 7 to 9), algebraic fractions and formal algebraic proof test complete fluency in factorising quadratics, expanding brackets, and manipulating identities. Simplifying an algebraic fraction always starts by factorising the numerator and denominator completely (often using the difference of two squares $a^2 - b^2 = (a - b)(a + b)$) so whole common brackets can be cancelled. In formal algebraic proof, testing a few numerical examples earns zero marks; instead, you must represent integers algebraically ($2n$ for an even number, $2n + 1$ for an odd number, or $n, n + 1, n + 2$ for consecutive integers), expand and simplify, and factorise out the required integer multiple.