SQA National 5 Mathematics

National 5 Mathematics formula reference

Quick reference cards for common National 5 Mathematics rules, formulae and method choices.

Indices

a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m−n, (a^m)ⁿ = a^mn, a⁰ = 1 and a⁻ⁿ = 1aⁿ.

Simplify using square factors: √72 = √(36 × 2) = 6√2. Combine like surds and rationalise simple denominators.

Factorise first, then cancel common factors only. For example x² − 9x + 3 = x − 3, with x ≠ −3.

y = mx + c. Gradient m = (y₂ − y₁) ÷ (x₂ − x₁). c is the y-intercept where x = 0

Quadratic formula

x =-b ± √(b² − 4ac)2a

Factorise where possible.

For ax² + bx + c = 0, b² − 4ac tells you the root type: positive gives two roots, zero gives one repeated root, negative gives no real roots.

For a right-angled triangle, a² + b² = c², where c is the hypotenuse.

SOH CAH TOA: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.

a / sin A = b / sin B = c / sin C

use when you have a matching angle-side pair.

a² = b² + c² − 2bc cos A

use for SAS or SSS in non-right-angled triangles.

Area = 12ab sin C, where C is the included angle between sides a and b

Arc length = θ360 × πd

sector area = θ360 × πr²

If OA = a and OB = b, then AB = b − a. For v = (x, y), magnitude |v| = √(x² + y²).

Length scale factor k gives area scale factor k² and volume scale factor k³. Match corresponding sides before calculating k.

A smaller standard deviation means values are generally closer to the mean.

P(not A) = 1 − P(A)

for tree diagrams, multiply along branches and add separate routes.

For reverse percentages, divide by the multiplier. After a 20% increase, multiplier = 1.2, so original = final ÷ 1.2.