Maths

• Counting and numbers to 20: Counting objects, reading and writing numbers to 20, one more and one less.
• Patterns and sorting: Sorting by colour, shape and size, and continuing repeating patterns.
• Number bonds to 10 and 20: Pairs that make 10 and 20, and the facts that come from them.
• Odd and even numbers: Spotting odd and even numbers and counting in twos.
• Place value: What each digit is worth, from ones up to millions and down to thousandths.
• Adding and subtracting: Counting on and back, number lines and mental strategies.
• Column addition and subtraction: Written methods with carrying and exchanging.
• Mental maths strategies: Partitioning, compensating, doubling and halving to calculate in your head.
• Times tables: Multiplication facts up to 12 × 12.
• Multiplying and dividing: Short and long multiplication, and division with remainders.
• Long division: Dividing by two-digit numbers, and remainders as fractions or decimals.
• Roman numerals: Reading and writing I, V, X, L, C, D and M.
• Factors, multiples and primes: Factors, multiples, primes, HCF and LCM.
• Negative numbers: Ordering and calculating with numbers below zero.
• Order of operations: Brackets, powers, then × and ÷, then + and −.
• Rounding and estimating: Rounding to the nearest 10, decimal places and significant figures, and estimating.
• Upper and lower bounds: The biggest and smallest a rounded value could be, and bounds in calculations.
• Powers and roots: Squares, cubes, square roots and cube roots.
• Laws of indices: Multiplying and dividing powers, zero, negative and fractional indices.
• Standard form: Very large and very small numbers like 3 × 10⁸ (scientific notation).
• Surds: Simplifying, multiplying and rationalising roots that don’t simplify.
• Recurring decimals: Turning recurring decimals into fractions.
• Fractions: What a fraction means, and finding fractions of shapes and numbers.
• Equivalent fractions: Simplifying and finding fractions that are the same size.
• Mixed numbers and improper fractions: Switching between mixed numbers and improper fractions.
• Adding and subtracting fractions: Using a common denominator.
• Multiplying and dividing fractions: Including mixed numbers.
• Decimals: Reading, ordering and calculating with decimals.
• Percentages: Finding a percentage of an amount.
• Fractions, decimals and percentages: Switching between all three and putting them in order.
• Percentage change: Increases, decreases, multipliers and reverse percentages.
• Ratio: Simplifying ratios and sharing in a ratio.
• Proportion: Direct and inverse proportion, recipes and best buys.
• Speed and other rates: Speed = distance ÷ time, and other rates.
• Proportion equations: Writing and using y = kx, y = k/x and other proportion formulas.
• Simple and compound interest: Interest, depreciation, hire purchase and inflation.
• Growth and decay: Repeated percentage change, half-lives and exponential models.
• Annuities, loans and investments: Future and present value, monthly payments and sinking funds.
• Expressions: Collecting like terms and using letters for numbers.
• Substitution and formulae: Putting numbers into expressions and formulas.
• Expanding and factorising: Brackets in and out, including two brackets.
• Linear equations: Finding the unknown, like 3x + 5 = 20.
• Changing the subject: Rearranging formulas to make another letter the subject.
• Inequalities: Solving inequalities and showing them on number lines and graphs.
• Sequences: Term-to-term rules, nth terms and quadratic sequences.
• Straight-line graphs: y = mx + c, gradients and intercepts.
• Simultaneous equations: Two equations, two unknowns, including one linear and one quadratic.
• Quadratic equations: Factorising, the quadratic formula and the discriminant.
• Completing the square: Writing quadratics as (x + a)² + b and finding turning points.
• Quadratic graphs: Roots, intercepts, turning points and sketching parabolas.
• Other graphs: Cubic, reciprocal (hyperbola) and exponential graphs.
• Algebraic fractions: Simplifying, adding and solving with algebraic fractions.
• Functions: Function notation, domain and range, composite and inverse functions.
• Transforming graphs: Translations, stretches and reflections of y = f(x).
• Algebraic proof: Showing that something is always true using algebra.
• Polynomials and the factor theorem: Dividing polynomials, the factor and remainder theorems and cubic equations.
• Exponentials and logarithms: Laws of logs, e and solving exponential equations.
• Arithmetic and geometric series: Sums of sequences, sigma notation and sums to infinity.
• The binomial expansion: Expanding (a + b)ⁿ with Pascal’s triangle and nCr.
• Partial fractions: Splitting a fraction into simpler fractions.
• Iteration and numerical methods: Change of sign, iteration and the Newton–Raphson method.
• Limits and continuity: What happens as x gets closer and closer to a value.
• Differentiation: Gradients of curves, first principles and the power rule.
• Chain, product and quotient rules: Differentiating composite functions, products and quotients, trig, eˣ and ln x.
• Stationary points and optimisation: Tangents, turning points, curve sketching and maximising or minimising.
• Integration: Integration as the reverse of differentiation, and areas under curves.
• Integration by substitution and by parts: Substitution, integration by parts and partial fractions.
• Areas and volumes by integration: Areas between curves and volumes of revolution.
• Differential equations: Separating variables, growth and decay models, first and second order.
• Taylor and Maclaurin series: Writing functions as power series and estimating with them.
• Functions of several variables: Partial derivatives, gradients and double integrals.
• Shapes and solids: Names and properties of 2D shapes and 3D solids.
• Position and direction: Left, right, turns, compass directions and simple grids.
• Symmetry: Lines of symmetry and rotational symmetry.
• Angles: Right, acute, obtuse and reflex angles, and using a protractor.
• Properties of quadrilaterals: Squares, rectangles, rhombuses, parallelograms, kites and trapeziums.
• Angles on a line and round a point: They add to 180° and 360°, and vertically opposite angles.
• Angles in triangles: The three angles always add up to 180°.
• Angles in parallel lines: Alternate, corresponding and co-interior angles.
• Angles in polygons: Interior and exterior angles of any polygon.
• Perimeter and area: Rectangles, triangles, parallelograms, trapeziums and compound shapes.
• Circles: Circumference and area using π.
• Arcs and sectors: Arc lengths and sector areas.
• Volume and surface area: Cuboids, prisms and cylinders.
• Cones, spheres and pyramids: Volume and surface area of pyramids, cones, spheres and frustums.
• Coordinates: Plotting points in all four quadrants and finding midpoints.
• Transformations: Reflection, rotation, translation and enlargement.
• Scale drawings and maps: Using scales on maps and plans.
• Bearings: Three-figure bearings and finding directions.
• Constructions and loci: Bisectors, accurate drawings and loci.
• Congruence and similarity: Congruent triangles, similar shapes and scale factors for length, area and volume.
• Pythagoras’ theorem: a² + b² = c² in right-angled triangles, in 2D and 3D.
• Circle theorems: Angles in circles, tangents and cyclic quadrilaterals.
• Geometric proof: Proving facts about shapes step by step with reasons.
• Coordinate geometry: Distance, midpoint, parallel and perpendicular lines, and the equation of a circle.
• Vectors: Adding, scaling and using vectors in proofs.
• Right-angled trigonometry: SOH CAH TOA: sides and angles in right-angled triangles.
• Exact trig values: The values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°.
• Sine and cosine rules: Any triangle, and its area ½ab sin C.
• Trig graphs: Graphs of sine, cosine and tangent, and how they change.
• Trig identities and equations: Using identities to simplify and solve trig equations.
• Radians: Measuring angles in radians, arcs, sectors and small-angle approximations.
• Comparing and measuring: Longer and shorter, heavier and lighter, and measuring with rulers, scales and jugs.
• Telling the time: Clocks, 24-hour time, durations and timetables.
• Money: Coins, notes, change and budgets.
• Units and conversions: Metric units, imperial units and converting area and volume units.
• Density and compound measures: Density, pressure, rates and converting compound units.
• Tally charts and pictograms: Collecting data with tallies and showing it with pictures.
• Charts and graphs: Bar charts, pie charts, line graphs and frequency tables.
• Averages and range: Mean, median, mode and range, including from tables.
• Collecting data and sampling: Questionnaires, samples, bias and stratified sampling.
• Quartiles and box plots: Quartiles, interquartile range, outliers and box plots.
• Cumulative frequency and histograms: Grouped data, cumulative frequency curves and histograms with unequal widths.
• Scatter graphs: Correlation and lines of best fit.
• Variance and standard deviation: How spread out data is around the mean.
• Correlation and regression: The least squares line and the correlation coefficient.
• The normal distribution: Bell curves, z-scores and probabilities from the normal distribution.
• Hypothesis testing: Testing a claim with data: null hypothesis, significance and p-values.
• Probability: Chance on a 0–1 scale, expected outcomes and relative frequency.
• Venn diagrams and sets: Sets, set notation and probability from Venn diagrams.
• Tree diagrams and combined events: Independent and dependent events, AND and OR.
• Conditional probability: Probability given that something has already happened.
• Counting: permutations and combinations: The counting principle, factorials, arrangements and choices.
• The binomial distribution: The probability of r successes in n trials.
• Discrete random variables: Probability distributions, expectation and variance.
• Matrices: Adding and multiplying matrices, and matrices as transformations.
• Determinants and inverses: The determinant, the inverse and what they mean.
• Systems of linear equations: Solving three or more equations with matrices and row reduction.
• Eigenvalues and eigenvectors: Vectors that a matrix only stretches, and what that tells you.
• Complex numbers: i² = −1, Argand diagrams, modulus and argument.
• Vectors in 3D, lines and planes: Vector equations of lines and planes, dot and cross products.
• Proof by induction and contradiction: Proof by induction, contradiction and counterexample.
• Binary and number bases: Counting in base 2, 8 and 16 and converting between bases.
• Sets and logic: Statements, truth tables, sets and quantifiers.
• Number theory: Divisibility, the Euclidean algorithm and modular arithmetic.
• Graphs and networks: Vertices and edges, shortest paths, trees and route problems.
• Recurrence relations: Sequences defined by earlier terms, and solving them.
• Constant acceleration (SUVAT): The five equations of motion and motion graphs.
• Forces and Newton’s laws: Resolving forces, F = ma, connected particles and friction.
• Projectiles: Things thrown through the air, split into horizontal and vertical motion.
• Moments: Turning effects, equilibrium of beams and centres of mass.