Mathematics

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every section of 1MA1

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1. NNumber

   Learning requirements for number.

   • N1Compare and order signed integers, decimals and fractions using equality and inequality symbols.
   • N2Calculate with signed integers, decimals, fractions and mixed numbers using place value and written methods.
   • N3Use inverse operations and the correct order of operations with brackets, powers, roots and reciprocals.
   • N4Find primes, factors, multiples, HCF and LCM; express an integer as a unique product of prime factors.
   • N5List possibilities systematically without omissions or duplicates.
   • N6Calculate positive whole-number powers and their real roots, and recognise common powers.
   • N7Calculate expressions involving roots and integer powers.
   • N8Keep calculations exact when working with fractions and multiples of pi.
   • N9Read, write and calculate with numbers in standard form.
   • N10Convert between terminating decimals and equivalent fractions.
   • N11Use fractions to interpret and solve ratio problems.
   • N12Find a fraction or percentage of a quantity by treating it as a multiplier.
   • N13Use appropriate standard and compound units, including decimal measurements.
   • N14Estimate results and check the plausibility of calculated answers, including calculator output.
   • N15Round to decimal places or significant figures and write error intervals for rounded or truncated values.
   • N16Interpret the precision of measurements and the limits caused by rounding.

2. RRatio, proportion and rates of change

   Ratio, proportion and rates of change

   • R1Convert between related standard and compound units in numerical and algebraic problems.
   • R5Divide quantities in a ratio and explain the resulting proportions.
   • R2Interpret and calculate with scale factors, scale drawings and maps.
   • R6Express multiplicative relationships between quantities using ratios.
   • R3Express one amount as a fraction of another, including improper fractions.
   • R7Connect ratios with fractions and linear relationships.
   • R4Write ratios and simplify them using equivalent forms.
   • R8Connect ratios and fractions to linear functions.
   • R9Use percentage multipliers, comparisons, changes, original values and simple interest, including percentages above 100.
   • R10Solve direct and inverse proportion problems using graphs and algebra.
   • R11Calculate with compound measures including speed, pay rates, unit prices, density and pressure.
   • R12Compare lengths, areas and volumes as ratios and connect these to scale factors and similarity.
   • R13Interpret direct and inverse proportion equations, including the reciprocal relationship in inverse proportion.
   • R14Interpret straight-line gradients as rates and recognise direct and inverse proportion graphs.
   • R15Use chords and tangents to estimate average and instantaneous rates of change without calculus.
   • R16Model growth and decay with repeated multipliers, including compound interest, and interpret the result.

3. AAlgebra

   Algebra

   • A1Read and write conventional algebraic notation for products, powers, fractions, coefficients and brackets.
   • A4Expand, factorise and rearrange algebraic expressions, including quadratics.
   • A18Solve quadratic equations by factorising, completing the square and the quadratic formula; estimate solutions graphically.
   • A2Substitute numerical values into expressions and formulae, including scientific formulae.
   • A19Solve simultaneous equations algebraically and estimate their solutions on graphs.
   • A3Distinguish expressions, equations, identities, formulae and inequalities, and identify terms and factors.
   • A21Represent contextual problems as equations, solve them and interpret the solutions.
   • A5Use mathematical formulae and rearrange them to make a different variable the subject.
   • A6Distinguish equations from identities and justify algebraic equivalence with mathematical arguments.
   • A7Interpret an algebraic expression as a rule linking an input to an output.
   • A8Plot and interpret coordinates in every quadrant.
   • A9Plot straight lines, identify parallel gradients and find a line equation from points or a point and gradient.
   • A10Find and interpret a straight line’s gradient and intercepts from its equation or graph.
   • A11Identify quadratic roots, intercepts and turning points on graphs; find roots algebraically.
   • A12Recognise and sketch linear, quadratic, simple cubic and reciprocal graphs.
   • A13Sketch how translating or reflecting a function changes its graph.
   • A14Use contextual graphs, including reciprocal and unfamiliar functions, to estimate solutions to motion and other problems.
   • A15Estimate gradients and areas beneath non-linear graphs and interpret their meaning in practical contexts.
   • A16Use the equation of a circle centred at the origin and find the tangent at a given point.
   • A17Solve linear equations including unknowns on both sides, and estimate solutions from graphs.
   • A20Use iteration to obtain approximate numerical solutions of equations.
   • A22Solve a linear inequality in one variable and show its solutions on a number line.
   • A23Generate a sequence from a term-to-term rule or a rule using its position.
   • A24Recognise figurate, arithmetic, Fibonacci-type, quadratic and geometric sequences with positive rational ratios.
   • A25Find and use an expression for the nth term of a linear sequence.

4. GGeometry and measures

   Geometry and measures

   • G1Use geometric vocabulary and labelling conventions, recognise symmetry and draw a figure from a description.
   • G12Identify the faces, surfaces, edges and vertices of three-dimensional shapes.
   • G16Calculate areas of triangles and quadrilaterals, and volumes of cuboids and prisms.
   • G2Construct bisectors and perpendiculars with ruler and compass; solve loci problems using shortest-distance properties.
   • G17Calculate circle measurements and areas and volumes involving curved solids.
   • G3Use angle facts for points, lines, parallel lines, triangles and polygons to derive unknown angles.
   • G18Calculate arc lengths, sector angles and sector areas.
   • G4Explain and apply defining properties of triangles, quadrilaterals and other plane shapes.
   • G5Use SSS, SAS, ASA and RHS to establish triangle congruence.
   • G6Build simple geometric proofs using angle facts, congruence, similarity and established shape properties.
   • G7Construct and describe rotations, reflections, translations and enlargements, including fractional scale factors.
   • G8Describe combined rotations, reflections and translations, including what stays invariant.
   • G9Identify and use circle parts and their properties, including chords, tangents, arcs, sectors and segments.
   • G10Apply and prove circle theorems about angles, radii, tangents and chords, and derive related results.
   • G11Solve geometrical problems using coordinate axes.
   • G13Draw and interpret plans and elevations of three-dimensional objects.
   • G14Select and use standard units for geometric and everyday measurements.
   • G15Measure lengths and angles accurately and interpret bearings, maps and scale drawings.
   • G19Use congruence and similarity to relate corresponding lengths of shapes.
   • G20Use Pythagoras and sine, cosine and tangent ratios to find lengths and angles in two-dimensional right triangles.
   • G21Recall exact sine and cosine values at 0, 30, 45, 60 and 90 degrees, and tangent values at 0, 30, 45 and 60 degrees.
   • G22Use the sine and cosine rules to calculate unknown sides and angles in triangles.
   • G23Use half the product of two sides and the sine of their included angle to solve triangle area problems.
   • G24Represent a translation with a two-dimensional vector.
   • G25Add and subtract vectors and multiply by scalars using diagrams and column notation.

5. PProbability

   Learning requirements for probability.

   • P1Record and analyse outcomes of probability experiments in tables and frequency trees.
   • P2Use randomness, fairness and equally likely outcomes to predict expected frequencies.
   • P3Compare relative frequencies with theoretical probabilities on the zero-to-one scale.
   • P4Use total probability of one and add probabilities of mutually exclusive outcomes in an exhaustive set.
   • P5Explain why larger unbiased samples tend to give relative frequencies closer to theoretical probabilities.
   • P6List sets and combinations systematically with tables, grids, Venn diagrams and trees.
   • P7Construct sample spaces for single and combined equally likely experiments and calculate probabilities.
   • P8Calculate probabilities of independent and dependent combined events using trees and stated assumptions.
   • P9Calculate conditional probabilities using two-way tables, trees, Venn diagrams and expected frequencies.

6. SStatistics

   Learning requirements for statistics.

   • S1Infer population features from a sample while explaining the limitations of sampling.
   • S2Choose, construct and interpret frequency tables, bar and pie charts, pictograms, vertical-line charts and time-series graphs.
   • S3Construct and interpret histograms with unequal class widths and cumulative frequency graphs for grouped data.
   • S4Compare distributions using appropriate graphs, averages, modal classes, ranges and possible outliers.
   • S5Use statistical summaries to describe a population.
   • S6Interpret scatter graphs, correlation and lines of best fit; distinguish correlation from causation and assess extrapolation risks.

practice preview — fresh every time

1. 2 marks · foundation tier

   Mia buys 2 sandwiches at £0.60 each and pays with a £5 note. How much change should Mia get? Give your answer in pounds (£).

   show the answer

   £3.80

   2 × £0.60 = £1.20. Change: £5 − £1.20 = £3.80.

2. 2 marks · foundation tier

   A train travels 200 km in 5 hours. Work out the average speed in km/h.

   show the answer

   40

   Speed = distance ÷ time = 200 ÷ 5 = 40 km/h.