This is where maths becomes a language: equations that describe lines and curves, triangles that measure heights you cannot reach, and patterns that continue for ever. Each topic starts from its key idea and works through the standard method step by step, with graded practice, so you can explain a method as well as use it.
Prime factorisation, HCF and LCM, and decimal expansions.
Every whole number is a product of primes in exactly one way. HCF and LCM come straight from the prime factors.
Zeroes of quadratics, their sum and product, and reading them off a graph.
The zeroes of a polynomial are where its graph crosses the x-axis. For ax² + bx + c, the zeroes add to −b/a and multiply to c/a.
Solve two equations together, and see when lines meet, never meet or overlap.
Two linear equations are two straight lines. Where they cross is the one pair (x, y) that works for both.
Factorising, the discriminant, the nature of roots and word problems.
A quadratic equation ax² + bx + c = 0 has at most two roots. The discriminant b² − 4ac tells you how many real roots before you solve.
Number patterns that grow by the same amount: nth term and sum.
In an arithmetic progression, each term is the one before plus a fixed difference d.
Similar triangles, the basic proportionality theorem and Pythagoras.
Similar triangles have the same shape: equal angles, and sides in the same ratio.
Distance between points, midpoints and the section formula.
The distance formula is just Pythagoras on the grid.
sin, cos and tan, the special angles, and identities.
In a right-angled triangle, the ratios of the sides depend only on the angle. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
Find heights of towers and trees using angles of elevation.
Draw the right triangle: the height is opposite the angle and the ground distance is adjacent. Then tan θ = height ÷ distance.
Tangents meet the radius at 90°, and two tangents from a point are equal.
A tangent is at 90° to the radius at the point of contact. That makes a right-angled triangle you can solve with Pythagoras.
Sectors, arcs and rings.
A sector is a slice of the circle. Its area and arc are the same fraction, θ/360, of the full circle.
Mean, modal class and median class from grouped frequency tables.
For grouped data, use each class mark (the middle of the class) as the value.
Cards, coins and two dice.
P(E) = favourable outcomes ÷ total outcomes, when all outcomes are equally likely. P(not E) = 1 − P(E).
