GCSEmathsworksheets.com is the premier website for randomly generated printable GCSE maths worksheets.
Parts of this website use Javascript and may not function correctly if you have it disabled.
To print a worksheet select the “Print” option from your web browser menu, usually located in the top right corner of the screen. (The Print option may be hidden under the “Share…” submenu for mobile phone users).
Each worksheet has a link to view answers at the bottom of the page.
To generate a new worksheet, refresh the page using your web browser.
NEW (see changelog for past updates)
3rd December 2019: R11 – Speed. distance & time and Density worksheets
Select a topic from the syllabus below, using the quick links provided:
1. Number
– Fractions, decimals and percentages
– Measures and accuracy
2. Algebra
– Graphs
– Solving equations and inequalities
– Sequences
3. Ratio, proportion and rates of change
4. Geometry and measures (under construction)
– Mensuration and calculation (under construction)
– Vectors (under construction)
5. Probability (under construction)
6. Statistics (under construction)
1. Number
What students need to learn:
N1 order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥
N7 calculate with roots, and with integer and fractional indices
N8 calculate exactly with fractions, surds and multiples of π; simplify surd expressions involving squares (e.g. √12 = √(4 × 3) = √4 × √3 = 2√3) and rationalise denominators
N9 calculate with and interpret standard form A × 10n, where 1 ≤ A < 10 and n is an integer
Fractions, decimals and percentages
N11 identify and work with fractions in ratio problems
N12 interpret fractions and percentages as operators
Measures and accuracy
N15 round numbers and measures to an appropriate degree of accuracy (e.g. to a specified number of decimal places or significant figures); use inequality notation to specify simple error intervals due to truncation or rounding
N16 apply and interpret limits of accuracy, including upper and lower bounds
2. Algebra
What students need to learn:
A1 use and interpret algebraic manipulation, including:
• ab in place of a × b
• 3y in place of y + y + y and 3 × y
• a2 in place of a × a, a3 in place of a × a × a, a2b in place of a × a × b
• a/b in place of a ÷ b
• coefficients written as fractions rather than as decimals
• brackets
A2 substitute numerical values into formulae and expressions, including scientific formulae
A4 simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:
● collecting like terms
● multiplying a single term over a bracket
● taking out common factors
● expanding products of two or more binomials
● factorising quadratic expressions of the form x2 + bx + c, including the difference of two squares; factorising quadratic expressions of the form ax2 + bx + c
● simplifying expressions involving sums, products and powers, including the laws of indices
A5 understand and use standard mathematical formulae; rearrange formulae to change the subject
A6 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 and proofs
A7 where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the “inverse function”; interpret the succession of two functions as a ‘composite function’ (the use of formal function notation is expected)
Graphs
A8 work with coordinates in all four quadrants
A13 sketch translations and reflections of a given function
A15 calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts (this does not include calculus)
A16 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
Solving equations and inequalities
A20 find approximate solutions to equations numerically using iteration
A21 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
A22 solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph
Sequences
A23 generate terms of a sequence from either a term-to-term or a position-to-term rule
A24 recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a rational number > 0 or a surd) and other sequences
A25 deduce expressions to calculate the nth term of linear and quadratic sequences
3. Ratio, proportion and rates of change
What students need to learn:
R2 use scale factors, scale diagrams and maps
R4 use ratio notation, including reduction to simplest form
R6 express a multiplicative relationship between two quantities as a ratio or a fraction
R7 understand and use proportion as equality of ratios
R8 relate ratios to fractions and to linear functions
R11 use compound units such as speed, rates of pay, unit pricing, density and pressure
R12 compare lengths, areas and volumes using ratio notation; make links to similarity (including trigonometric ratios) and scale factors
R13 understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; construct and interpret equations that describe direct and inverse proportion
R14 interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion
R15 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 (this does not include calculus)
R16 set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes
4. Geometry and measures
What students need to learn:
G1 use conventional terms and notation: 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
G2 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
G3 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 (e.g. to deduce and use the angle sum in any polygon, and to derive properties of regular polygons)
G4 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
G5 use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)
G6 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
G7 identify, describe and construct congruent and similar shapes, including on coordinate axes, by considering rotation, reflection, translation and enlargement (including fractional and negative scale factors)
G8 describe the changes and invariance achieved by combinations of rotations, reflections and translations
G9 identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment
G10 apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
G11 solve geometrical problems on coordinate axes
G12 identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres
G13 construct and interpret plans and elevations of 3D shapes
Mensuration and calculation
G14 use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money, etc.)
G15 measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings
G16 know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)
G17 know the formulae: circumference of a circle = 2πr = πd, area of a circle = πr2; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids
G18 calculate arc lengths, angles and areas of sectors of circles
G19 apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures
G20 know the formulae for: Pythagoras’ theorem a^2+b^2=c^2, and the
trigonometric ratios, sinθ=opposite/hypotenuse, cosθ=adjacent/hypotenuse, and tanθ=opposite/adjacent; apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures
G21 know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°;
know the exact value of tan θ for θ = 0°, 30°, 45° and 60°
G22
know and apply the sine rule a/(sin A)=b/(sin B)=c/(sin C) , and cosine rule a^2=b^2+c^2-2bc cos A , to find unknown lengths and angles
G23 know and apply Area = 1/2ab sin C to calculate the area, sides or angles of any triangle
Vectors
G24 describe translations as 2D vectors
G25 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
5. Probability
What students need to learn:
P1 record, describe and analyse the frequency of outcomes of probability experiments using tables and frequency trees
P2 apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
P3 relate relative expected frequencies to theoretical probability, using appropriate language and the 0-1 probability scale
P4 apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one
P5 understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
P6 enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams
P7 construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
P8 calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
P9 calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams
6. Statistics
What students need to learn:
S1 infer properties of populations or distributions from a sample, while knowing the limitations of sampling
S2 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, tables and line graphs for time series data and know their appropriate use
S3 construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use
S4 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 quartiles and inter-quartile range)
S5 apply statistics to describe a population
S6 use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends while knowing the dangers of so doing