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# GCSE mathematics

**AQA GCSE Mathematics Exam**

**Introduction**

GCSE Mathematics exam is divided into 3 papers.

Paper 1 is a non-calculator paper with Paper 2 and 3 being calculator. Students can choose between doing the Foundation papers (grades 1-5) or the higher tier (grades 4 – 9). All questions papers must be taken from the same tier. Each paper will test the student on the topics within Number, Algebra, Ratio, proportion and rates of change, Geometry and measures, Probability and Statistics.

**Assessment details**

** Paper 1**

Non-Calculator

Written exam: 1 hour 30 minutes

80 marks

Questions will be a mix of short, single mark questions to multi-step problems. The mathematical demand increases as the student progresses through the paper.

**Paper 2**

Calculators allowed

Written exam: 1 hour 30 minutes

80 marks

Questions will be a mix of short, single mark questions to multi-step problems. The mathematical demand increases as the student progresses through the paper.

**Paper 3**

Calculators allowed

Written exam: 1 hour 30 minutes

80 marks

__Number__

Number makes up 25% of the Foundation Tier and 15% of the Higher Tier. Higher content only will be denoted with a *.

- Order positive and negative integers, decimals and fractions.
- Use the symbols =, ≠, , ≤, ≥
- Apply the four operations, including formal written methods, to integers, decimals, fractions and mixed numbers (positive and negative).
- Understand and use place value
- Knowledge and application of order of operations (BIDMAS)
- Prime numbers, factors, multiples, prime factorisation, HCF, LCM.
- Apply systematic listing strategies including use of product rule for counting.
- Use positive integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5.
- Estimating roots of any given positive number. *
- Calculate exactly with fractions..
- Calculating with surds. *
- Standard form.
- Work interchangeably with terminating decimals and their corresponding fractions.
- Change recurring decimals into their corresponding fractions and vice versa.*
- Identify and work with fractions in ratio problem.
- Interpret fractions and percentages as operators.
- Use standard units of mass, length, time, money and other measures (including standard compound measures) using decimal quantities where appropriate.
- Estimate answers check calculations using approximation and estimation, including answers obtained using technology.
- Round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures).
- Upper and lower bounds. *

__Algebra__

Algebra makes up 20% of the Foundation Tier and 30% of the Higher Tier. Topics in the Higher content only will be denoted with a *.

- Use and interpret algebraic notation.
- Substitute numerical values into formulae and expressions, including scientific formulae.
- Understand and use the concepts and vocabulary of expressions, equations, formulae, inequalities, terms and factors.
- Simplify and manipulate algebraic expressions.
- Collecting like terms, multiplying a single term over a bracket, taking out common factors, simplifying expressions.
- Expand products of two binomials.
- Factorising quadratic expressions including difference of two squares.
- Understand and use standard mathematical formulae.
- Rearrange formulae to change the subject.
- Know the difference between an equation and an identity.
- Argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments.
- Where appropriate, interpret simple expressions as functions with inputs and outputs.
- Work with coordinates in all four quadrants.
- Plot graphs of equations that correspond to straight-line graphs in the coordinate plane.
- Identify and interpret gradients and intercepts of linear functions graphically and algebraically.
- Identify and interpret roots, intercepts and turning points of quadratic functions graphically.
- Deduce roots algebraically.
- Recognise, sketch and interpret graphs of linear functions and quadratic functions.
- Plot and interpret graphs, and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration.
- Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other nonlinear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts.*
- Recognise and use the equation of a circle with centre at the origin find the equation of a tangent to a circle at a given point.*
- Solve linear equations in one unknown algebraically.
- Find approximate solutions using a graph.
- Solve quadratic equations algebraically by factorising.

– Including those that require rearrangement.*

– Including completing the square and by using the quadratic formula.* - Find the approximate solutions using a graph.
- Solve two simultaneous equations in two variables (linear/linear) algebraically.

– Including linear/quadratic.* - Find approximate solutions using a graph.
- Find approximate solutions to equations numerically using iteration.*
- Translate simple situations or procedures into algebraic expressions or formulae.
- Derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution.
- Solve linear inequalities in one variable.

– Solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable. - Represent the solution set on a number line.

– Represent the solution set on a number line, using set notation and on a graph. - Generate terms of a sequence from either a term-to-term or a position-to-term rule.
- Recognise and use sequences of triangular, square and cube numbers and simple arithmetic progressions, Fibonacci-type, quadratic and simple geometric progressions.
- Deduce expressions to calculate the nth term of a linear sequence.
- Nth term for quadratic.

__Ratio__

Ratio makes up 25% of the Foundation Tier and 20% of the Higher Tier. Topics in the Higher Tier content only will be denoted with a *.

- Change freely between related standard units (eg time, length, area, volume/capacity, mass) and compound units (eg speed, rates of pay, prices) in numerical contexts.
- Use scale factors, scale diagrams and maps.
- express one quantity as a fraction of another, where the fraction is less than 1 or greater than 1.
- Use ratio notation, including reduction to simplest form.
- Divide a given quantity into two parts in a given part : part or part : whole ratio.
- Express the division of a quantity into two parts as a ratio.
- Apply ratio to real contexts and problems (such as those involving conversion, comparison, scaling, mixing, concentrations).
- Express a multiplicative relationship between two quantities as a ratio or a fraction
- Understand and use proportion as equality of ratios.
- Relate ratios to fractions and to linear functions.
- Define percentage as ‘number of parts per hundred’.
- Interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively.
- Express one quantity as a percentage of another.
- Compare two quantities using percentages.
- Work with percentages greater than 100%.
- Solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics.
- Solve problems involving direct and inverse proportion, including graphical and algebraic representations.
- Use compound units such as speed, rates of pay, unit pricing.
- Compare lengths, areas and volumes using ratio notation.
- Understand that X is inversely proportional to Y is equivalent to X is proportional to 1 Y.
- Interpret equations that describe direct and inverse proportion.

– Construct and interpret equations that describe direct and inverse proportion.* - Interpret the gradient of a straight-line graph as a rate of change recognise and interpret graphs that illustrate direct and inverse proportion.
- Interpret the gradient at a point on a curve as the instantaneous rate of change.*
- Apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts.*
- Set up, solve and interpret the answers in growth and decay problems, including compound interest.

– And work with general iterative processes.*

__Geometry__

Geometry makes up 15% of the Foundation Tier and 20% of the Higher Tier. Topics in the Higher Tier content only will be denoted with a *.

- Use conventional terms and notations: points, lines, vertices, edges, planes, parallel lines, perpendicular lines, right angles, polygons, regular polygons and polygons with reflection and/or rotation symmetries.
- Use the standard conventions for labelling and referring to the sides and angles of triangles.
- Draw diagrams from written description.
- Use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle).
- Use these to construct given figures and solve loci problems.
- Know that the perpendicular distance from a point to a line is the shortest distance to the line.
- Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles.
- Understand and use alternate and corresponding angles on parallel lines.
- Derive and use the sum of angles in a triangle (eg to deduce and use the angle sum in any polygon, and to derive properties of regular polygons).
- Derive and apply the properties and definitions of: special types of quadrilaterals, including square, rectangle, parallelogram, trapezium, kite and rhombus.
- And triangles and other plane figures using appropriate language.
- Use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS).
- Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results about angles and sides, including Pythagoras’ theorem and the fact that the base angles of an isosceles triangle are equal, and use known results to obtain simple proofs.
- Identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement.

– Including fractional scale factors.

– Including negative scale factors.* - Describe the changes and invariance achieved by combinations of rotations, reflections and translations.
- Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference.

– Including: tangent, arc, sector and segment. - Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results.
- Solve geometrical problems on coordinate axes.
- Identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres.
- Interpret plans and elevations of 3D shapes.

– Construct and interpret plans and elevations of 3D shapes. - Use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money etc.).
- Measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings.
- Know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders).
- Know the formulae for circumference of a circle, area of a circle and calculate area and perimeter of composite shapes.
- Calculate arc lengths, angles and areas of sectors of circles.
- Apply the concepts of congruence and similarity, including the relationships between lengths in similar figures.

– Including the relationships between lengths, areas and volumes in similar figures. - Know the formulae for Pythagoras’ theorem and trigonometric ratios.
- Apply them to find angels and lengths in right-angled triangles in two dimensional figures.

– Apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures. - Exact trig values.
- Know and apply the sine rule and cosine rule.*
- Know and apply the area of a triangle using sine.
- Describe translation as 2D vectors.
- Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors.

– Use vectors to construct geometric arguments and proofs.

__Probability and Statistics__

Probability and statistics makes up 15% of the Foundation Tier and 15% of the Higher Tier. Topics in the Higher Tier content only will be denoted with a *.

- Record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees.
- Apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments.
- Relate relative expected frequencies to theoretical probability, using appropriate language and the 0 to 1 probability scale.
- Apply the property that the probabilities of an exhaustive set of outcomes sum to 1.
- Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to 1.
- Understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size.
- Enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams.

– Including using tree diagrams. - Construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities.
- Content Higher content only calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions.
- Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams.*
- Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling.
- Interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data, and know their appropriate use.

– Including tables and line graphs for time series data. - Construct and interpret diagrams for grouped discrete data and continuous data, ie histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use.*
- Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:

– Appropriate graphical representation involving discrete, continuous and grouped data.

– Including box plots.* - Appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers).

– Including quartiles and inter-quartile range.* - Apply statistics to describe a population.
- Use and interpret scatter graphs of bivariate data.
- Recognise correlation.

– Know that it does not indicate causation.

– Draw estimated lines of best fit.

– Make predictions.

– Interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

## Assessment details

Assessment objectives

The exams will assess the following AOs in the context of the content set out in the Subject content section.

**AO1:** Use and apply standard techniques.

Students should be able to:

- accurately recall facts, terminology and definitions.
- use and interpret notation correctly.
- accurately carry out routine procedures or set tasks requiring multi-step solutions.

**AO2:** Reason, interpret and communicate mathematically.

Students should be able to:

- make deductions, inferences and draw conclusions from mathematical information.
- construct chains of reasoning to achieve a given result.
- interpret and communicate information accurately.
- present arguments and proofs.
- assess the validity of an argument and critically evaluate a given way of presenting information.

**AO3:** Solve problems within mathematics and in other contexts.

Students should be able to:

- translate problems in mathematical or non-mathematical contexts into a process or a series of mathematical processes.
- make and use connections between different parts of mathematics.
- interpret results in the context of the given problem.
- evaluate methods used and results obtained.
- evaluate solutions to identify how they may have been affected by assumptions made.