ESAT Maths 1 topics: the full list

What topics are in ESAT Maths 1?

ESAT Mathematics 1 covers seven areas: units, number, ratio and proportion, algebra, geometry, statistics and probability. The list below follows the order of UAT-UK's ESAT content specification, put into plain English.

Mathematics 1 is the ESAT module every candidate takes. It has 27 multiple-choice questions in 40 minutes, with no calculator. Every other ESAT module assumes you know its content, so a gap here can slow you down in Physics or Chemistry as well.

The specification says some questions use this knowledge directly and others apply it in less familiar settings, so you need to recognise a topic when the question doesn't name it.

Units

It's the shortest section, but units come up inside questions across the test.

• Standard units of mass, length, time and money
• Compound units such as speed, density, pressure, rates of pay and unit prices
• Converting between related units, including area and volume units
• SI prefixes from nano- to giga-, and units with negative powers (from the specification's general notes)

Number

This is the arithmetic every other module leans on, all of it done without a calculator.

• Ordering integers, decimals and fractions, including negatives, and the inequality symbols
• The four operations with integers, decimals, fractions and mixed numbers, and place value
• Primes, factors, multiples, highest common factor, lowest common multiple and prime factorisation
• Inverse operations, cancelling, and the order of operations
• Systematic listing, including multiplying the number of choices at each stage to count outcomes
• Squares, square roots (positive and negative), cubes and cube roots
• Index laws, including fractional and negative powers, and standard form
• Converting between fractions, decimals and percentages, including recurring decimals
• Exact work with fractions, surds and multiples of pi, simplifying surds and rationalising denominators
• Upper and lower bounds, rounding and error intervals
• Estimating answers, including ones involving pi or surds

Ratio, proportion and percentages

Mixing and concentration problems sit in this section, so it matters for Chemistry as well.

• Scale factors, scale drawings and maps
• One quantity as a fraction or a percentage of another
• Ratio notation, sharing in a ratio, and ratio in real settings such as mixing, concentration and conversion
• Percentage change, reverse percentages, percentages over 100% and simple interest
• Direct and inverse proportion, including their graphs and equations with powers
• Comparing lengths, areas and volumes of similar shapes
• Growth and decay, compound interest and repeated (iterative) processes

Algebra, graphs and sequences

This section and geometry are the two longest on the list.

• Algebraic notation, substituting into formulae, and the difference between an equation and an identity
• Index laws in algebra
• Expanding brackets and factorising quadratics, including the difference of two squares
• Simplifying and combining algebraic fractions
• Rearranging formulae and showing two expressions are equivalent
• Straight lines: gradient, intercept, parallel and perpendicular lines, and the equation through two points
• Quadratic graphs: roots, intercepts, and turning points by completing the square
• Recognising and sketching linear, quadratic, simple cubic, reciprocal and exponential graphs, and sin, cos and tan graphs in degrees for any angle
• Gradients and areas under graphs in real settings, such as speed-time graphs
• Simultaneous equations, including one linear and one quadratic, and forming equations from a situation
• Solving quadratics by factorising, completing the square and the quadratic formula
• Linear inequalities in one or two variables, shown on a number line or a graph
• Sequences: term-to-term and position-to-term rules, and the nth term of linear and quadratic sequences

Geometry, trigonometry and vectors

Some formulae here you need to know and some are given. The section after probability says which.

• Angle facts, including parallel lines and the interior and exterior angles of polygons
• Properties of special quadrilaterals and triangles
• Congruent triangles (SSS, SAS, ASA, RHS) and similar shapes
• Rotations, reflections, translations and enlargements, including fractional and negative scale factors
• Pythagoras' theorem in two and three dimensions
• Circle vocabulary and the standard circle theorems
• Geometry problems on coordinate axes
• 3D shapes, plans and elevations, scale drawings and three-figure bearings
• Perimeter, area and volume, including composite shapes, arc lengths and sectors
• Right-angled trigonometry in 2D and 3D, and exact values of sin, cos and tan for common angles
• Vectors: adding, subtracting, multiplying by a number, and using them in geometric arguments

Statistics

You may be asked to read a chart or diagram as well as calculate from one.

• Tables and charts for categorical, discrete and time series data
• Histograms with equal and unequal class widths, frequency density and cumulative frequency graphs
• Averages and range, estimates from grouped data, quartiles and interquartile range, and comparing data sets
• Scatter graphs, correlation and causation, lines of best fit, and the risks of reading beyond the data

Probability

The specification expects you to work with tables, trees and Venn diagrams here, and you don't need formal set notation.

• Frequency trees and tables for experiments, and expected outcomes
• Relative frequency, theoretical probability and the 0 to 1 scale
• Probabilities of an exhaustive set of outcomes adding to one
• Listing outcomes with tables, grids, Venn diagrams and tree diagrams
• When to add and when to multiply probabilities, and conditional probability, including events that do and don't depend on earlier outcomes

What isn't on the list, and which formulae you're given

You won't need the sine or cosine rules. The specification says you aren't expected to recall or use them. Differentiation and integration aren't on the Mathematics 1 list either.

Formulae for spheres, pyramids and cones are given if a question needs them. The specification lists the circumference and area of a circle, the volume of a cylinder, Pythagoras' theorem, the trigonometric ratios and the quadratic formula as ones you should know.

How should you use the list?

Mark each line secure, rusty or not yet taught. A secure topic only needs to turn up in mixed practice now and then. A rusty one needs an evening with your notes and a handful of questions. A topic you haven't been taught needs a proper lesson before you practise it.

Check each mark with a few questions without a calculator, because a heading can look familiar long after the method has gone. Then sit a timed Mathematics 1 module. Some of your mistakes may sit in topics you marked secure, such as reversing a percentage change or a sign in a rearranged formula, so move those lines back to rusty.

The free ESAT paper on this site is a full Mathematics 1 module you can use for this. Its score is Noetra's own estimate, not an official result.

Questions students ask

Is ESAT Maths 1 GCSE level?

The ESAT specification doesn't give it a level. The same seven headings appear in Part 2 of the TMUA specification, which UAT-UK describes as almost all covered within a Higher Level GCSE maths course.

Do I need the sine and cosine rules for ESAT Maths 1?

No. The specification says candidates aren't expected to recall or use the sine or cosine rules.

Which formulae are given in ESAT Maths 1?

Formulae for spheres, pyramids and cones are given if needed. You should know the others yourself, including the circle formulae, the volume of a cylinder and the quadratic formula.

Do the other ESAT modules use Maths 1 content?

Yes. The specification says every module assumes knowledge of the Mathematics 1 content.

Official sources

Current details were checked against these primary sources on the date shown above.

• UAT-UK: ESAT format and official guidance
• UAT-UK: ESAT specification, guide and archives