The problems below have been selected because they are closely linked to the curriculum, while providing opportunities for all students to work and think like mathematicians. They have detailed Teachers' Resources offering guidance on how they can be used in the classroom.
Please email any comments to secondary.nrich@maths.org
Looking for primary problems? See the NRICH Primary Curriculum Map.
Key
Articles are indicated by an 'A'.
Tasks badged
are suitable for the whole class;
Tasks badged
are suitable for the majority;
Tasks badged
are for those who like a serious challenge.
Highlight problems
Number
| Pre-Secondary | Age 11 - 12 | Age 12 - 13 | Age 13 - 14 | Age 14 - 15 | Age 15 - 16 | Extension |
|---|---|---|---|---|---|---|
Place Value, Integers, Ordering & Rounding | ||||||
| Understand and use place value for decimals, measures and integers of any size | ||||||
| Order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols $=, ≠, <, >, ≤, ≥$ | ||||||
| Round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures] | ||||||
| Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation $a<x≤b$ | Apply and interpret limits of accuracy when rounding or truncating (including upper and lower bounds) | |||||
| Short problems: Place Value, Integers, Ordering & Rounding | ||||||
Factors, Multiples & Primes | ||||||
| Use concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property | ||||||
| Appreciate the infinite nature of the sets of integers, real and rational numbers | ||||||
| Short problems: Factors, Multiples & Primes | ||||||
Powers and Roots | ||||||
| Use integer powers and associated real roots (square, cube and higher), recognise powers of $2, 3, 4, 5$ and distinguish between exact representations of roots and their decimal approximations | Estimate powers and roots of any given positive number | |||||
| Calculate with roots, and with integer (and fractional) indices | ||||||
| Interpret and compare numbers in standard form $A \times 10^{n} \hspace{2mm}1≤A<10$, where $n$ is a positive or negative integer or zero | ||||||
| Short problems: Powers and Roots | ||||||
Fractions, Decimals & Percentages | ||||||
| Work interchangeably with terminating decimals and their corresponding fractions (such as $3.5$ and $\frac{7}{2}$ or $0.375$ and $\frac{3}{8}$) | Change recurring decimals into their corresponding fractions and vice versa | |||||
| Interpret fractions and percentages as operators | ||||||
| Define percentage as 'number of parts per hundred', interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express one quantity as a percentage of another, compare two quantities using percentages, and work with percentages greater than $100%$ | ||||||
| Short problems: Fractions, Decimals & Percentages | ||||||
Number Operations and Calculation Methods | ||||||
| Use the four operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative | |||||
| Use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals | ||||||
| Recognise and use relationships between operations including inverse operations | ||||||
| Calculate exactly with fractions, surds, and multiples of $π$; simplify surd expressions involving squares and rationalise denominators | ||||||
| Use a calculator and other technologies to calculate results accurately and then interpret them appropriately | ||||||
| Short problems: Number Operations and Calculation Methods | ||||||
Ratio, Proportion & Rates of Change | ||||||
| Change freely between related standard units [for example time, length, area, volume/capacity, mass] | ||||||
| Use scale factors, scale diagrams and maps | ||||||
| Express one quantity as a fraction of another, where the fraction is less than $1$ and greater than $1$ | ||||||
| Use ratio notation, including reduction to simplest form | Compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios) | |||||
| 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 | ||||||
| Understand that a multiplicative relationship between two quantities can be expressed as a ratio or a fraction | ||||||
| Relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions | ||||||
| Solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics | ||||||
| Solve problems involving direct and inverse proportion, including graphical and algebraic representations | Understand that $X$ is inversely proportional to $Y$ is equivalent to $X$ is proportional to $\frac{1}{Y}$; 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 instantaneous and average rate of change (gradients of tangents and chords) 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 | ||||||
| Use compound units such as speed, unit pricing and density to solve problems | Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts | |||||
| Short problems: Ratio, Proportion & Rates of Change | ||||||
Algebra
| Pre-Secondary | Age 11 - 12 | Age 12 - 13 | Age 13 - 14 | Age 14 - 15 | Age 15 - 16 | Extension |
|---|---|---|---|---|---|---|
Patterns and Sequences | ||||||
| Generate terms of a sequence from either a term-to-term or a position-to-term rule | Deduce expressions to calculate the $n$th term of linear and quadratic sequences | ||||
| Recognise arithmetic sequences and find the $n$th term | ||||||
| Recognise geometric sequences and appreciate other sequences that arise | Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ($r^n$ where $n$ is an integer, and $r$ is a positive rational number or a surd) and other sequences | |||||
| Short problems: Patterns and Sequences | ||||||
Creating & Manipulating Linear & Quadratic Expressions | ||||||
Use and interpret algebraic notation, including:
| ||||||
Simplify and manipulate algebraic expressions to maintain equivalence by:
| Simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:
| |||||
| Short problems: Creating & Manipulating Linear & Quadratic Expressions | ||||||
Equations & Formulae | ||||||
| Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors | 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 | ||||
| Use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement) | ||||||
| Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs | Find approximate solutions to equations numerically using iteration | |||||
| Solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph | ||||||
| Solve linear inequalities in one or two variables, and quadratic inequalities in one variable; represent the solution set on a number line, using set notation, and on a graph | ||||||
| Solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square, and by using the quadratic formula; find approximate solutions using a graph | ||||||
| Substitute numerical values into formulae and expressions, including scientific formulae | 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 | |||||
| Understand and use standard mathematical formulae; rearrange formulae to change the subject | ||||||
| Short problems: Equations & Formulae | ||||||
Functions and Graphs | ||||||
| Where appropriate, interpret simple expressions as functions with inputs and outputs; interpret the reverse process as the 'inverse functions'; interpret the succession of two functions as a 'composite function' | ||||||
| Work with coordinates in all four quadrants | 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 | |||||
| Recognise, sketch and produce graphs of linear and quadratic functions of one variable with appropriate scaling, using equations in $x$ and $y$ and in the Cartesian plane | Plot and interpret graphs (including reciprocal graphs and exponential 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 | |||||
| Interpret mathematical relationships both algebraically and graphically | 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 | |||||
| Reduce a given linear equation in two variables to the standard form $y = mx + c$; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically | Use the form $y = mx + c$ to identify parallel and perpendicular lines; find the equation of the line through two given points, or through one point with a given gradient | |||||
| Use linear and quadratic graphs to estimate values of $y$ for given values of $x$ and vice versa and to find approximate solutions of simultaneous equations | Recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function $y = \frac{1}{x}$ with $x ≠ 0$, the exponential function $y = k^x$ for positive values of $k$, and the trigonometric functions (with arguments in degrees) $y = \sin x$, $y = \cos x$ and $y = \tan x$ for angles of any size | |||||
| Find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs | Sketch translations and reflections of the graph of a given function | |||||
| Identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically, and turning points by completing the square | ||||||
| Short problems: Functions and Graphs | ||||||
Geometry
| Pre-Secondary | Age 11 - 12 | Age 12 - 13 | Age 13 - 14 | Age 14 - 15 | Age 15 - 16 | Extension |
|---|---|---|---|---|---|---|
Angles, Polygons and Geometrical Proof | ||||||
| Derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures (for example, equal lengths and angles) using appropriate language and technologies | Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment | ||||
| Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric | ||||||
| Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles | ||||||
| Draw and measure line segments and angles in geometric figures, including interpreting scale drawings | ||||||
| Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons | ||||||
| Understand and use the relationship between parallel lines and alternate and corresponding angles | ||||||
| Use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles | ||||||
| Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids | Apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures | |||||
| Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras' Theorem, and use known results to obtain simple proofs | Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results | |||||
| Interpret mathematical relationships both algebraically and geometrically | ||||||
| Short problems: Angles, Polygons and Geometrical Proof | ||||||
Construction | ||||||
| Derive and use 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); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line | ||||||
| Short problems: Construction | ||||||
Perimeter, Area and Volume | ||||||
| Derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders) | Calculate surface areas and volumes of spheres, pyramids, cones and composite solids | ||||
| Calculate and solve problems involving: perimeters of 2D shapes (including circles), areas of circles and composite shapes | Calculate arc lengths, angles and area of sectors of circles | |||||
| Short problems: Perimeter, Area and Volume | ||||||
3D Shapes | ||||||
| Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D | Construct and interpret plans and elevations of 3D shapes | |||||
| Short problems: 3D Shapes | ||||||
Transformations | ||||||
| Identify properties of, and describe the results of, translations, rotations and reflections applied to given figures | Describe the changes and invariance achieved by combinations of rotations, reflections and translations | |||||
| Interpret and use fractional and negative scale factors for enlargements | ||||||
| Short problems: Transformations | ||||||
Vectors | ||||||
| Describe translations 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 | ||||||
| Short problems: Vectors | ||||||
Pythagoras' Theorem & Trigonometry | ||||||
| Use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles | Apply Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles (and, where possible, general triangles) in two and three dimensional figures; interpret and use bearings | |||||
| Know the exact values of $\sin \theta$ and $\cos \theta$ for $\theta = 0^\circ, 30^\circ, 45^\circ, 60^\circ$ and $90^{\circ}$; know the exact value of $\tan \theta$ for $\theta = 0^\circ, 30^\circ, 45^\circ$ and $60^\circ$ | ||||||
| Know and apply the sine rule and cosine rule to find unknown lengths and angles | ||||||
| Know and apply $Area = \frac{1}{2} ab \sin C$ to calculate the area, sides or angles of any triangle | ||||||
| Short problems: Pythagoras' Theorem & Trigonometry | ||||||
Statistics and probability
| Pre-Secondary | Age 11 - 12 | Age 12 - 13 | Age 13 - 14 | Age 14 - 15 | Age 15 - 16 | Extension |
|---|---|---|---|---|---|---|
Statistics | ||||||
| Describe, interpret and compare observed distributions of a single variable through: appropriate graphical representation involving discrete, continuous and grouped data; and appropriate measures of central tendancy (mean, mode, median) and spread (range, consideration of outliers) | ||||||
| Construct and interpret appropriate tables, charts, and diagrams, including frequency tables, bar charts, pie charts, and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data | Interpret and construct tables and line graphs for time series data | |||||
| 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 | ||||||
| Describe simple mathematical relationships between two variables (bivariate data) in observational and experimental contexts and illustrate using scatter graphs | 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 predications; interpolate and extrapolate apparent trends whilst knowing the dangers of doing so | |||||
| Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling | ||||||
Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:
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| Apply statistics to describe a population | ||||||
| Short problems: Statistics | ||||||
Probability | ||||||
| Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the $0$ - $1$ probability scale | Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size | |||||
| Understand that the probabilities of all possible outcomes sum to $1$ | Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to $1$ | |||||
| Enumerate sets and unions/ intersections of sets systematically, using tables, grids and Venn diagrams | ||||||
| Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities | 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 | ||||||
| Short problems: Probability | ||||||