Skills

TMUA Skills are focused mathematical ideas and techniques you can work on individually—from subtle details of familiar topics to useful ways of approaching problems. Each skill offers a way to build your understanding and make unfamiliar situations more manageable.

Each skill will have at least one teaching video, hosted on YouTube and free for everyone! The rest, I’m afraid, is part of PLUS: tracking your progress in each skill, guidance on what to work on next, and dedicated practice questions to help you work on that skill in isolation before returning to full papers. Videos and practice questions are being added gradually as we develop each skill.

We’re adding resources skill by skill! So far, 13 of 60 skills have teaching videos, and 2 of 60 have practice questions.

Algebra · 17

Flexible variablesAnswering suitable multiple-choice questions by testing convenient admissible values of variables, choosing values that distinguish the options without a full general calculation.In 54 past-paper questionsPractice being written
Recognising and constructing squaresSpotting squares in algebraic and surd expressions, such as $3+2\sqrt{2}$ , and creating useful forms by completing the square.In 57 past-paper questionsPractice being written
Substitutions in identitiesChoosing values that isolate constants, eliminate terms or reveal relationships in identities, particularly polynomial identities.In 4 past-paper questionsPractice being written
Factors and remainders of polynomialsUsing the factor and remainder theorems to identify factors, find remainders and determine unknown coefficients, including factorisation of expressions in several variables.In 42 past-paper questionsPractice being written
Binomial expansionUsing the binomial theorem to expand expressions, identify particular terms or coefficients, and determine unknown quantities from an expansion.In 35 past-paper questionsPractice being written
Coefficients by countingFinding coefficients by counting ways to select terms from polynomial factors, including expansions such as $(1+2x+x^2)^6$ .In 12 past-paper questionsPractice being written
Sign analysis for inequalitiesSolving inequalities by locating zeros and undefined points, analysing signs between them, and handling repeated factors and endpoints.In 27 past-paper questionsPractice being written
Fractional powersUnderstanding real-valued powers such as $x^{a/b}$ , particularly negative inputs, domain restrictions and when familiar laws of indices are valid.In 15 past-paper questionsPractice being written
Logarithms and exponentialsUsing logarithm and exponential laws to simplify expressions and solve equations, including simultaneous equations, and understanding the graphs and behaviour of logarithmic and exponential functions.In 76 past-paper questionsPractice being written
Logarithmic domainsDetermining valid arguments and bases for logarithmic expressions, and checking that equation transformations and candidate solutions respect the original domain.In 9 past-paper questionsPractice being written
$\sqrt{x^2}=|x|$ Understanding why $\sqrt{x^2}=|x|$ , rather than $x$ in general, and applying this when simplifying expressions whose sign must be considered.In 8 past-paper questionsPractice being written
Quadratic discriminantUsing the quadratic discriminant to determine the number of real roots and establish conditions, together with the relationships $\alpha+\beta=-b/a$ and $\alpha\beta=c/a$ between the roots and coefficients of $ax^2+bx+c=0$ , where $a\ne0$ .In 59 past-paper questionsPractice being written
Simultaneous equationsSolving simultaneous equations using substitution, elimination and useful rearrangements, including linear and non-linear systems and determining how many solutions are possible. Also include algebraic problems that can be framed as simultaneous equations.In 68 past-paper questionsPractice being written
Transforming equations and inequalitiesUnderstanding when operations on equations and inequalities preserve their meaning, including squaring, multiplying or dividing by expressions, and applying functions with appropriate sign and domain conditions.In 42 past-paper questionsPractice being written
Comparing numerical expressionsComparing or ordering numerical expressions involving roots, powers, logarithms and familiar constants, using useful rewritings, bounds, reference values or sensible numerical approximations.In 8 past-paper questionsPractice being written
Simultaneous inequalitiesSolving inequalities that must hold simultaneously, combining their solution sets and handling endpoints, excluded values and cases where no common solution exists.In 16 past-paper questionsPractice being written
Deductions from inequalitiesDrawing conclusions from given inequalities by combining them, establishing bounds and checking the sign conditions needed for valid deductions.In 34 past-paper questionsPractice being written

Sequences and number · 9

Arithmetic sequencesUsing sequence structure beyond standard formulas, including symmetric naming such as $m-d,m,m+d$ , mean and median relationships, and interpreting terms as points on a straight line.In 32 past-paper questionsPractice being written
Geometric sequencesUsing sequence structure beyond standard formulas, including representations such as $a/r,a,ar$ , relationships between terms, and useful approaches involving ratios and products.In 35 past-paper questionsPractice being written
Number theory: primes and factorisationUsing prime factorisation to reason about divisibility, factors and multiples, including the restrictions on prime exponents in perfect powers.In 24 past-paper questionsPractice being written
Number theory: remaindersUsing remainders to simplify calculations, identify cycles in powers and determine which numerical values are possible.In 17 past-paper questionsPractice being written
Ratio and proportionUsing ratios and proportional relationships to compare quantities, combine information and determine unknown values, including direct and inverse proportion and changes in scale.In 24 past-paper questionsPractice being written
CombinatoricsCounting arrangements and selections with the multiplication principle, permutations, combinations and division for repeated objects, splitting into cases so that every outcome is counted exactly once.In 10 past-paper questionsPractice being written
Probability and Venn diagramsCalculating probabilities and using Venn diagrams to organise information about events and sets, including overlaps, complements, conditional probability and independence.In 14 past-paper questionsPractice being written
Non-standard sequences and sumsWorking with sequences and sums beyond standard arithmetic and geometric forms, including unfamiliar recurrences, integral definitions and patterns that emerge through rearranging or combining terms.In 75 past-paper questionsPractice being written
Systematic enumerationOrganising possibilities into an exhaustive set of cases, so that a counting or reasoning problem can be solved with confidence that nothing has been missed.In 18 past-paper questionsPractice being written

Functions and graphs · 11

Even and odd functionsUsing $f(-x)=f(x)$ and $f(-x)=-f(x)$ in graphical symmetry, combinations of functions and definite integration.In 34 past-paper questionsPractice ready
Standard graph transformationsUsing horizontal and vertical translations, stretches and axis reflections, with particular attention to their order and combined effects. Also their effects on area integration.In 45 past-paper questionsPractice being written
90° graph rotationsFinding and interpreting equations after clockwise or anticlockwise rotations through 90° about the origin, including quadratic and cubic graphs.In 5 past-paper questionsPractice being written
Modulus graphsConstructing and interpreting $|f(x)|$ , $f(|x|)$ and $|y|=f(x)$ using reflections, retained sections and restrictions.In 34 past-paper questionsPractice being written
Functions of periodic functionsSketching functions such as $f(\sin x)$ and $f(\cos x)$ using the outer function and the range and repetition of the periodic input, with extensions to other periodic inner functions.In 15 past-paper questionsPractice being written
Graph intersections and inequalitiesUsing intersections and relative positions of graphs to solve equations and inequalities or determine the number of solutions, including choosing rearrangements that make the graphs easier to interpret.In 107 past-paper questionsPractice being written
Abstract functionsReasoning about functions known only through stated properties or unusual rules, including functional equations and recursive definitions, to work out values, properties and consequences.In 43 past-paper questionsPractice being written
Graphs of rational functionsSketching and interpreting rational functions using domain restrictions, asymptotes, behaviour near denominator zeros and at large inputs, and the distinction between holes and vertical asymptotes.In 6 past-paper questionsPractice being written
Recognising structure in composite functionsRecognising inner expressions and relationships in composite functions, and treating a shared expression as a variable to simplify comparisons, equations and other reasoning.In 29 past-paper questionsPractice being written
Monotonicity and transformationsUnderstanding how transformations and compositions change increasing or decreasing behaviour, and using this to compare values and find extrema on restricted domains.In 23 past-paper questionsPractice being written
Regions defined by inequalitiesInterpreting and constructing regions in a coordinate plane from inequalities, combining conditions and identifying which boundaries and points are included.In 14 past-paper questionsPractice being written

Calculus · 7

DifferentiationUsing standard differentiation techniques to find derivatives, equations of tangents and normals, and stationary points, including finding and classifying maxima and minima.In 97 past-paper questionsPractice being written
IntegrationUsing standard integration techniques to find indefinite and definite integrals, determine constants of integration, and calculate areas under curves or between curves.In 103 past-paper questionsPractice being written
Fundamental Theorem of CalculusConnecting differentiation and definite integration, including differentiating functions defined by integrals and handling variable limits.In 11 past-paper questionsPractice being written
Integrating floor and fractional-part functionsEvaluating definite integrals involving floor, ceiling and fractional-part functions by understanding their piecewise behaviour and splitting the interval appropriately.In 8 past-paper questionsPractice being written
Graph reflections with integrationReflecting graphs in lines such as $x=a$ and $y=b$ , finding the resulting equations, and using reflections to simplify areas and definite integrals.In 21 past-paper questionsPractice being written
Stationary points and multiplicitiesUnderstanding polynomial turning points and points of inflection, especially stationary points of inflection, and using root multiplicities and derivative behaviour to interpret graph shapes and numbers of solutions.In 48 past-paper questionsPractice being written
Trapezium ruleUsing and interpreting the trapezium rule to approximate definite integrals and areas, including reasoning about the estimate from the graph and the chosen strips.In 13 past-paper questionsPractice being written

Geometry and trigonometry · 10

Similar trianglesRecognising similar triangles, matching corresponding sides and using proportional lengths to solve geometric problems.In 24 past-paper questionsPractice being written
SSA ambiguous trianglesDetermining when two sides and a non-included angle give zero, one or two valid triangles.In 14 past-paper questionsPractice ready
Circle geometryCombining coordinate equations, algebra and geometry to solve circle problems, with standard circle theorems assumed and an emphasis on choosing an effective approach.In 42 past-paper questionsPractice being written
Equations of straight linesFinding and interpreting equations of straight lines from points, gradients and intercepts, and using parallel or perpendicular lines to solve coordinate geometry problems.In 39 past-paper questionsPractice being written
Position vectors and linesUsing position vectors, parallelism and expressions for points on lines to solve geometric problems, including moving points and intersections.In 3 past-paper questionsPractice being written
Shortest paths on 3-D surfacesUnfolding three-dimensional surfaces into plane figures to find shortest paths, checking that the resulting route is valid on the original shape.In 6 past-paper questionsPractice being written
Standard trigonometric equationsSolving sine, cosine and tangent equations with linear arguments, including forms such as $\sin A=c$ and $\sin A=\sin B$ , and finding all solutions in a specified interval.In 48 past-paper questionsPractice being written
Non-standard trigonometric equationsSolving trigonometric equations that require algebraic or trigonometric manipulation and factorisation before reducing to standard trigonometric equations.In 29 past-paper questionsPractice being written
Geometric conditions and guaranteesDeciding which properties guarantee a particular type of two-dimensional shape or a relationship such as triangle congruence, including recognising when the information is insufficient.In 13 past-paper questionsPractice being written
GeometryWorking out lengths, angles and areas in harder geometry problems by finding a helpful construction, splitting a figure into simpler pieces or reasoning about where an extreme position must occur.In 23 past-paper questionsPractice being written

Logic · 6

Logic: CounterexamplesIdentifying whether a proposed example is a valid counterexample to a mathematical statement, checking that it satisfies the assumptions but contradicts the conclusion.In 38 past-paper questionsPractice being written
Logic: EquivalenceRecognising conditions that imply one another, including equivalent reformulations and contrapositives, and distinguishing equivalence from one-way implication.In 36 past-paper questionsPractice being written
Logic: Necessity and sufficiencyDistinguishing necessary and sufficient conditions, interpreting the direction of implication, and identifying conditions that must follow from—or guarantee—a given statement.In 76 past-paper questionsPractice being written
Logic: What could be trueIdentifying statements that could be true under given conditions, by finding a valid example or checking that the statement is compatible with all the information provided.In 20 past-paper questionsPractice being written
Logic: NegationNegating statements involving inequalities, “and”, “or”, and quantifiers such as “for every” and “there exists”, including nested quantifiers.In 14 past-paper questionsPractice being written
Deduction from constraintsCombining rules and restrictions to determine what is possible or what must be true, particularly in problems involving arrangements, codes, objects or spatial information.In 49 past-paper questionsPractice being written