What does Area 1 Expressions and Formulae cover in SQA National 5 Mathematics?

Area 1 is the algebra, number and measure toolkit of the course. It covers simplifying and working with surds and the laws of indices, writing numbers in scientific notation, expanding brackets and factorising (common factor, difference of two squares and trinomials), completing the square, simplifying and combining algebraic fractions, calculating the gradient of a line, and finding the length of an arc, the area of a sector and the volume of standard solids. These skills underpin the Relationships and Applications work that follows.

What are the three factorising methods in National 5 Maths?

The three factorising methods examined are taking out a common factor (the reverse of expanding a single bracket), the difference of two squares where a squared term minus a squared term factorises as a sum times a difference, and factorising trinomials such as x squared plus bx plus c into two brackets, including non-unitary trinomials where the coefficient of x squared is not one. You should always check for a common factor first.

What volume formulae do I need for National 5 Maths?

You need the volume of a prism as cross-section area times length, a cylinder as pi r squared h, a cone as one third pi r squared h, a pyramid as one third of the base area times height, and a sphere as four thirds pi r cubed. The cone and pyramid each hold a third of the matching cylinder or prism. The sphere and cone formulae are usually given on the formula sheet, but you should know them anyway.

How do you complete the square in National 5 Maths?

To complete the square on x squared plus bx plus c, halve the coefficient of x to get p, write the bracket x plus p all squared, then subtract p squared to cancel the extra constant it introduces and add the original constant. The result is the form x plus p all squared plus q, which shows the turning point of the parabola at minus p, q, and q is the minimum value of the expression.

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SQA National 5 Mathematics Area 1 Expressions and Formulae: surds, indices, algebra, completing the square, fractions, gradient and measure

A deep-dive SQA National 5 Mathematics guide to Area 1 Expressions and Formulae. Covers surds and indices, scientific notation, expanding and factorising, completing the square, algebraic fractions, the gradient of a line, and the arc, sector and volume of solids.

Generated by Claude Opus 4.815 min readNational 5Updated 2026-06-14

Reviewed by: AI editorial process; not yet individually human-reviewed

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What Area 1 actually demands
Surds and indices
Scientific notation
Expanding and factorising
Completing the square
Algebraic fractions
Gradient
Arcs, sectors and volume
How Area 1 is examined
Check your knowledge

What Area 1 actually demands

Expressions and Formulae is the algebra, number and measure toolkit of National 5 Mathematics. The examiners test fluent manipulation, exact non-calculator work, and accurate substitution into measure formulae. This guide walks through all seven topics of the area, then sets out the patterns the SQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Surds and indices

The area opens with surds and indices. Simplify a surd by pulling out the largest square factor, so $\sqrt{48} = 4\sqrt{3}$ ; add and subtract only like surds; rationalise a denominator by multiplying top and bottom by the surd. The laws of indices, $a^m \times a^n = a^{m+n}$ , $a^{-n} = \tfrac{1}{a^n}$ and $a^{m/n} = \sqrt[n]{a^m}$ , let you simplify powers and switch between root and index form. This is core Paper 1 territory.

Scientific notation

Scientific notation writes any number as $a \times 10^n$ with $1 \le a < 10$ . A positive power is a large number, a negative power is a small one. To multiply, multiply the front numbers and add the powers; to divide, divide and subtract; then adjust so the front number is between $1$ and $10$ .

Expanding and factorising

Algebraic expressions covers expanding single and double brackets and the three factorising methods: common factor, difference of two squares $a^2 - b^2 = (a + b)(a - b)$ , and trinomials including non-unitary ones. Always check for a common factor first; the word "fully" demands it.

Completing the square

Completing the square rewrites $x^2 + bx + c$ as $(x + p)^2 + q$ by halving the coefficient of $x$ and correcting the constant. The form reveals the turning point at $(-p, q)$ and the minimum value $q$ , all without calculus.

Algebraic fractions

Algebraic fractions mirror ordinary fractions, with one extra step: factorise top and bottom before cancelling. Multiply tops and bottoms, divide by flipping the second fraction, and add or subtract over a common denominator.

Gradient

Gradient is the steepness of a line, $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ (rise over run). Positive rises, negative falls, zero is horizontal, and a vertical line has an undefined gradient. Gradient leads into the equation of a straight line in the Relationships area.

Arcs, sectors and volume

Arcs, sectors and volume treats an arc and a sector as the fraction $\dfrac{\theta}{360}$ of a circle, and applies the volume formulae for the prism, cylinder, cone, pyramid and sphere. Keep answers as multiples of $\pi$ until the final rounding.

How Area 1 is examined

A typical SQA profile for Expressions and Formulae:

Exact non-calculator work. Paper 1 rewards surd and fraction answers and index laws done by hand.
Fluent algebra. Expanding, factorising and completing the square appear in almost every paper.
Accurate measure. Arc, sector and volume questions reward correct substitution and rounding on Paper 2.

Check your knowledge

A mix of recall and method questions covering Area 1. Attempt them, then check against the solutions.

Simplify $\sqrt{75}$ . (1 mark)
Write $0.0036$ in scientific notation. (1 mark)
Factorise fully $3x^2 - 27$ . (2 marks)
Express $x^2 + 6x + 2$ in the form $(x + p)^2 + q$ . (2 marks)
Find the volume of a cylinder of radius $3$ cm and height $10$ cm, in terms of $\pi$ . (2 marks)

Solutions to check-your-knowledge questions

Step 1: Simplify a surd by extracting the largest square factor

Find the largest perfect square that divides the number under the root, separate the surd into two factors, and simplify the square root of the perfect square:

\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}.

Step 2: Write a small decimal in scientific notation

Scientific notation requires the front number to satisfy $1 \le a < 10$ . Count how many places the decimal point moves to the right to reach $3.6$ ; each place is a power of $10^{-1}$ , giving a negative exponent:

0.0036 = 3.6 \times 10^{-3}.

Step 3: Factorise fully by taking a common factor then applying the difference of two squares

Always check for a common factor first. Here $3$ divides both terms, leaving a difference of two squares that factorises into a sum and a difference:

3x^2 - 27 = 3(x^2 - 9) = 3(x + 3)(x - 3).

Step 4: Complete the square on a quadratic expression

Halve the coefficient of $x$ to find the number inside the bracket, write the squared bracket, then subtract the square of that number and add the original constant to correct the value:

x^2 + 6x + 2: \quad \text{half of } 6 = 3, \quad (x + 3)^2 - 9 + 2 = (x + 3)^2 - 7.

Step 5: Find the volume of a cylinder in terms of pi

Substitute the radius and height directly into the cylinder volume formula, evaluating $r^2$ before multiplying, and leave the answer as a multiple of $\pi$ :

V = \pi r^2 h = \pi \times 9 \times 10 = 90\pi \text{ cm}^3.

Sources & how we know this

SQA National 5 Mathematics Course Specification — SQA (2018)