Year 10 (GCSE) Maths Master Checklist — Comprehensive Topics (Foundation & Higher)

Below is a comprehensive, organised checklist of the subject matter a 15‑year‑old doing Year 10 (GCSE) maths should cover to “master” the course. I’ve grouped topics into strands and indicated items that are typically Higher‑tier (H) — most other items are required for both Foundation and Higher. If you want a version strictly for Foundation or a syllabus-mapped list for a specific exam board (AQA, Edexcel, OCR), tell me and I’ll adapt it.

Number

• Integers, primes, factors, multiples, HCF and LCM
• Place value, ordering and rounding
• Fractions: simplifying, equivalence, mixed numbers, arithmetic with fractions
• Decimals: arithmetic, converting to/from fractions
• Percentages: percentage of amounts, percentage change, reverse percentages, percentage points
• Ratio and proportion: sharing in a ratio, dividing quantities, unitary method
• Direct and inverse proportion (including scale factors)
• Standard form (scientific notation) and orders of magnitude
• Index laws (positive, zero, negative indices)
• Surds and exact form (simplifying, operations with surds) (H)
• Recurring and terminating decimals; converting recurring to fraction
• Estimation and bounds; upper and lower bounds

Algebra

• Algebraic notation and manipulation; rules of indices
• Simplifying expressions; collecting like terms
• Expanding brackets, including double brackets
• Factorising: common factor, quadratics, difference of squares
• Substitution into expressions and formulae
• Rearranging formulae (changing the subject)
• Solving linear equations (one-step, two-step, multi-step)
• Solving quadratic equations by factorising, completing the square, quadratic formula (H)
• Algebraic fractions: simplification and basic operations (H)
• Simultaneous equations (linear–linear by elimination/substitution; linear–quadratic) (H for linear–quadratic)
• Inequalities: solving and representing on number lines, linear inequalities, quadratic inequalities (H)
• Sequences: nth term of arithmetic sequences, arithmetic series (sum) (H for formula use)
• Geometric sequences and growth/decay (H)
• Iteration and simple functional iteration (H)
• Algebraic proof and reasoning (basic)

Graphs and Functions

• Coordinates in 2D; plotting points
• Straight-line graphs: gradient, intercept, equation of a line, parallel/perpendicular (H)
• Quadratic, cubic and reciprocal graphs (shape and sketching)
• Transformations of graphs: translations, reflections, stretches (H)
• Graphs of simple functions and using graphs to solve equations/inequalities
• Distance–time and velocity–time graphs (interpreting gradients and areas)
• Function notation and simple function manipulation (H)

Geometry and Measures

• Properties of angles: parallel lines, interior/exterior angles, corresponding angles
• Angles in polygons; interior and exterior angle sums
• Triangle properties: types, isosceles, equilateral; angle sum
• Pythagoras’ theorem (including 3D applications)
• Basic trigonometry: sin, cos, tan for right-angled triangles (including using inverse functions)
• Trigonometric applications in non-right triangles (sine and cosine rules) (H)
• Circle geometry: radius, diameter, circumference, arc length, sector area
• Circle theorems (angles in same segment, cyclic quadrilaterals, tangent–radius, etc.) (H)
• Perimeter, area of 2D shapes (triangles, rectangles, parallelograms, trapezia, circles)
• Surface area and volume of 3D solids: prisms, cylinders, cones, spheres, pyramids (H for some)
• Units and unit conversion (including compound units)
• Bearings, loci and constructions (compass and straightedge)
• Similarity and congruence; scale factors; enlargement and similarity proofs

Trigonometry, Vectors and Transformations

• Vector basics: notation, addition, subtraction, scalar multiplication, position vectors (H)
• Transformations: translations, rotations, reflections, enlargements and combinations
• Transformations in coordinate geometry (H when combined with algebra)
• Geometric proof involving transformations (H)

Probability and Statistics

• Probability basics: sample space, outcomes, theoretical probability
• Combined events: independent events, mutually exclusive events, and use of Venn diagrams and tree diagrams
• Conditional probability (H)
• Expected value (simple cases)
• Collecting and representing data: frequency tables, bar charts, pie charts, pictograms
• Averages: mean, median, mode; range; when to use which measure
• Grouped data: calculating estimated mean from frequency tables
• Cumulative frequency, percentiles, quartiles, box plots
• Histograms (including with unequal class widths)
• Scatter plots, correlation and lines of best fit (interpretation and drawing)
• Interpreting statistical measures and graphical data; misrepresentation of data

Applied and Problem-solving Skills

• Translating word problems into algebraic expressions and equations
• Using mathematical reasoning to solve multi-step problems
• Working with contextualised problems: finance (interest, tax, budgeting), scale maps, bearings, rates (density, speed)
• Accuracy, significant figures and decimal places; propagation of error (bounds)
• Exam techniques: checking answers, units, interpreting results in context

Calculator and Technology Skills

• Correct and efficient use of scientific/graphical calculator for standard GCSE tasks
• Using calculator to solve equations, plot graphs, statistical functions (depends on allowed calculator)
• Using simple software/tools (graph plotters) where permitted in learning

Typical Higher‑only topics (summary)

• Surds, rationalising denominators
• Advanced indices (fractional indices)
• Algebraic fractions and more complex factorisation
• Quadratic formula and completing the square (full use)
• Simultaneous linear–quadratic equations
• Trigonometry beyond right triangles: sine and cosine rules
• Circle theorems
• Vectors (more advanced)
• Conditional probability and more complex probability problems
• More advanced graphs and transformations

How I can help next

• Produce a term-by-term Year 10 syllabus or lesson plan
• Produce a topic-by-topic checklist mapped to AQA/Edexcel/OCR GCSE
• Create practice question sets and worked solutions for any topic
• Make a revision timetable prioritising weaker areas

Which would you like me to do next — a Foundation vs Higher split, a week-by-week plan for Year 10, or mapping to a particular exam board?