How to learn Key stage 3: Mathematics

The national curriculum in England: Key stages 1 and 2 framework document, Key stage 3

0 of 70 concepts ready to learn · Markdown, for your AI

1. Algebra

Coordinates (age 11+): Plot and read coordinates in all four quadrants of the Cartesian plane, using positive and negative x- and y-values to describe positions precisely
Algebraic Notation: Use and interpret algebraic notation including: ab for a × b, 3y for y + y + y, a² for a × a, a/b for a ÷ b, coefficients as fractions, and brackets for grouping; read and write algebraic expressions fluently
Substituting into Formulae: Substitute numerical values into formulae and expressions including scientific formulae; evaluate expressions by replacing variables with given values and computing the result using correct order of operations
Solving Linear Equations: Use algebraic methods to solve linear equations in one variable, including equations that require rearrangement, expanding brackets, and collecting terms on both sides; solve equations with rational number coefficients
Algebraic Transformations: Model situations or procedures by translating them into algebraic expressions or formulae and by using graphs; move between word problems, algebraic representations, tables, and graphical representations
Linear Function Graphs: Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-intercept in context
Plotting Linear Graphs: Plot linear graphs by generating a table of values, reduce a two-variable linear equation to the form y = mx + c, and calculate gradients from two points on a line
Simultaneous Equations: Understand that two linear equations can be solved simultaneously by finding the point where their graphs intersect, and interpret this graphically and algebraically as the pair of values satisfying both equations
Estimating answers (age 13+): Use graphs of linear and quadratic functions to estimate output values for given inputs, find approximate solutions to equations, and interpret graphical information in real-world contexts
Quadratic Graphs: Recognise that quadratic functions produce curved (parabolic) graphs, distinguish them from linear graphs, and use plotted quadratic graphs to estimate values and find approximate solutions
Generating Sequences: Generate terms of a sequence from a term-to-term rule (e.g., 'add 3 each time') or a position-to-term rule (e.g., '2n + 1'), and identify whether a sequence is arithmetic, geometric, or neither
Nth-Term Rules: Find the nth-term expression for an arithmetic sequence by identifying the common difference and the zero-term, and use it to determine any term in the sequence or test whether a given number belongs to the sequence
Expressions & Equations Vocabulary: Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors; distinguish between an expression (no equals sign), an equation (equals sign), and an inequality (inequality sign)
Collecting Like Terms: Simplify algebraic expressions by collecting like terms — combine terms with the same variable and power (e.g., 3a + 2b + 5a = 8a + 2b) while maintaining equivalence
Expanding Single Brackets: Expand (multiply out) a single term over a bracket using the distributive property, e.g., 3(2x + 5) = 6x + 15; expand expressions involving negative multipliers
Factorising Expressions: Factorise algebraic expressions by taking out common factors — identify the highest common factor of all terms and write the expression as a product, e.g., 6x + 9 = 3(2x + 3)
Expanding Double Brackets: Expand products of two or more binomials, e.g., (x + 3)(x - 2) = x² + x - 6, using the grid method or FOIL; simplify the result by collecting like terms
Simple formulae: Understand and use standard mathematical formulae; rearrange formulae to change the subject, performing inverse operations to isolate a different variable
Numbers on a number line: Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations

2. Geometry and measures

Angles in triangles (age 11+): Derive and apply formulae for the area of triangles, parallelograms, and trapezia, and for the volume of cuboids and other prisms (including cylinders), connecting each formula to its geometric reasoning
Understanding angles (age 11+): Apply the properties of angles at a point (360°), on a straight line (180°), and vertically opposite angles to find unknown angles in multi-step problems
Angles with parallel lines: Understand and use the relationship between parallel lines cut by a transversal: corresponding angles, alternate interior angles, and co-interior (same-side interior) angles; use these to find unknown angles
Angle sums in triangles and polygons: Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons
Types of angles (age 13+): Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts
Trigonometry basics: Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem
3-D shapes (age 11+): Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections
Circles: Circumference & Area: Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts
Measuring angles (age 11+): Draw and measure line segments and angles accurately using ruler and protractor, and interpret scale drawings to extract real measurements
Understanding angles (age 12+): Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle
Types of angles (age 11+): Use conventional geometric terms and notation to describe, sketch, and draw points, lines, parallel and perpendicular lines, right angles, regular polygons, and reflectively/rotationally symmetric polygons
Angles in triangles (age 12+): Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent
Properties of triangles and quadrilaterals: Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths
Coordinate Transformations: Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original
Coordinates (age 12+): Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids

3. Number

Fractions on a number line: Understand and use place value for decimals, measures, and integers of any size; extend the number system to include all positive and negative integers, decimals, and fractions on a single number line
Percentages as Fractions: Define percentage as 'number of parts per hundred'; interpret percentages and percentage changes as a fraction or a decimal; express one quantity as a percentage of another; compare quantities using percentages; work with percentages greater than 100%
Multiplying fractions (age 11+): Interpret fractions and percentages as operators — find a fraction or percentage of an amount by multiplying, understanding that 'of' means multiply (e.g., 3/4 of 200 = 3/4 × 200 = 150)
Decimal place value (age 11+): Round numbers and measures to an appropriate degree of accuracy including to a specified number of decimal places or significant figures
Estimating by rounding: Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b; understand upper and lower bounds of rounded values
Ratio (age 11+): Use conventional notation for the priority of operations including brackets, powers, roots, and reciprocals; apply BIDMAS/BODMAS consistently to evaluate complex numerical expressions
Square and cube numbers: Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations
Decimals and fractions (age 11+): Work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and 7/2 or 0.375 and 3/8); convert fluently between the two forms
Number Sets & Infinity: Appreciate the infinite nature of the sets of integers, real numbers, and rational numbers; position integers on a number line and distinguish between rational and irrational numbers
Positive and Negative Numbers: Understand positive and negative numbers as describing quantities with opposite directions or values; use them in context such as temperature, floors in a building, and bank balances
Fractions on a number line (age 11+): Order positive and negative integers, decimals, and fractions on a number line; use the symbols =, ≠, <, >, ≤, ≥ to compare values including negative numbers and mixed representations
Factors, multiples, and primes (age 11+): Use the concepts and vocabulary of prime numbers, factors, multiples, common factors, common multiples, highest common factor (HCF), lowest common multiple (LCM), and prime factorisation including product notation and the unique factorisation property
Dividing fractions: Divide a fraction by a fraction using the 'keep-change-flip' method and visual models; interpret and solve word problems involving division of fractions by fractions
Mixed & Improper Fractions: Use the four operations with formal written methods applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative
Sign Rules for Multiplication: Multiply and divide with positive and negative integers and rational numbers, understanding the rules for the sign of the product or quotient
Using inverse operations: Recognise and use relationships between operations including inverse operations; use these relationships to check answers and simplify calculations
Powers of Ten Notation: Interpret and compare numbers in standard form A × 10ⁿ where 1 ≤ A < 10 and n is an integer; convert between ordinary numbers and standard form

4. Probability

The Probability Scale: Understand probability as a measure on a scale from 0 (impossible) to 1 (certain); use the language of probability including likely, unlikely, certain, and impossible
Experimental probability: Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability
Complementary events: Understand and apply the rule that probabilities of all mutually exclusive outcomes sum to one; use this to find the probability of a complementary event (P(not A) = 1 − P(A))
Sets & Venn Diagrams: Enumerate sets and their unions and intersections systematically using tables, grids, and Venn diagrams to organise and count outcomes
Venn Diagrams and Counting Outcomes: Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events
Tree diagrams: Generate theoretical sample spaces for single and combined events using listing, tables, and tree diagrams, and calculate theoretical probabilities as the number of favourable outcomes divided by the total number of equally likely outcomes

5. Ratio, proportion and rates of change

Unit Conversions: Convert freely between related standard units of measurement (time, length, area, volume/capacity, mass) using decimal notation to up to three decimal places where appropriate
Ratio Notation: Use ratio notation to describe the relationship between two or more quantities, simplify ratios to their simplest form, and convert between ratio and fraction representations
Dividing Quantities by Ratio: Divide a given quantity into two parts in a given part:part or part:whole ratio, and express the division as a fraction of the whole
One Quantity as a Fraction: Express one quantity as a fraction of another where the result may be less than 1 or greater than 1, and interpret the meaning in context
Ratio Notation and Relationships: Understand that a multiplicative relationship between two quantities can be expressed as a ratio; use ratio notation; simplify ratios
Proportion: Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations
Compound Units: Use compound units such as speed (distance ÷ time), unit pricing (cost ÷ quantity), and density (mass ÷ volume); solve problems involving compound units
Scale and similar shapes (age 11+): Use scale factors to interpret and create scale diagrams and maps, calculating real-life distances from map measurements and vice versa
Percentages (age 12+): Solve problems involving percentage increase, percentage decrease, finding the original value after a percentage change, and calculating simple interest

6. Statistics

Pictograms and tally charts (age 11+): Construct and interpret frequency tables, bar charts, pie charts, pictograms, and vertical line charts for both categorical and grouped numerical data, choosing appropriate representations for the data type
Comparing measurements: Describe, interpret, and compare distributions of a single variable using appropriate measures of central tendency (mean, median, mode) and spread (range), including the effect of outliers
Scatter Graphs & Correlation: Describe simple mathematical relationships between two variables using scatter graphs, identify positive, negative, or no correlation, and use a line of best fit to make predictions
Scatter Graphs: Plot bivariate data on a scatter graph with correctly labelled axes and appropriate scales; describe the correlation (positive, negative, none) and draw an estimated line of best fit where appropriate