Worksheets & practice

1.1

Working across scales

• Calculate with very large or very small quantities and express the result in standard form.
• Keep the leading number from 1 up to, but not including, 10; translate a calculator's E display into a power of 10.

1.2

Patterns with a constant step

• Recover the first term and common difference from information about an arithmetic sequence.
• Find a particular term or a total of consecutive terms, including sums written with Σ.
• Build an arithmetic model for a context, make predictions and judge whether a constant step is a reasonable approximation.

1.3

Patterns with a constant multiplier

• Identify the first term and common ratio of a geometric sequence, including from incomplete information.
• Calculate terms and finite totals, and move between an expanded sum and Σ notation.
• Use a geometric model to describe repeated percentage change and interpret predictions in context.

1.4

Money over time

• Model compound growth and annual depreciation; distinguish a percentage change from the multiplier used in the model.
• Match the interest rate and number of periods to annual, half-yearly, quarterly or monthly compounding; use financial technology when appropriate.
• Adjust an investment's future value for inflation to describe its purchasing power.

1.5

Powers and logarithmic notation

• Simplify products, quotients and powers involving integer exponents, including zero and negative exponents.
• Translate an exponential statement into a logarithmic statement and explain which exponent a logarithm represents.
• Evaluate common and natural logarithms with technology, remembering that a real logarithm needs a positive argument.

1.6

Writing a convincing algebraic argument

• Establish a numerical or algebraic claim through a logical chain of justified steps.
• Prove an identity by transforming one side into the other; distinguish an identity from an equation true only for particular values.
• Check a proposed result and explain what the check does, and does not, establish.

1.7

Rewriting powers and solving exponential equations

• Connect rational powers with roots and simplify expressions, taking the principal non-negative root when an even root is intended.
• Expand or combine logarithms using their product, quotient and power rules; check positive arguments and valid bases.
• Change the logarithm base when useful, and solve exponential equations with common bases or logarithms.

1.8

A finite total from infinitely many terms

• Decide whether a geometric series converges by checking that the absolute value of its ratio is less than 1.
• Use the sum to infinity to find a total or an unknown parameter, and reject values that fail the convergence condition.

1.9

Expanding a positive integer power

• Use binomial coefficients to expand a power of a two-term expression or extract a required term or coefficient.
• Calculate combination coefficients from factorials, Pascal's triangle or technology, keeping powers and signs consistent.

1.10

Counting choices and extending binomial expansions

Additional HL

• Choose between multiplication, permutations and combinations by deciding whether order matters and whether choices are restricted.
• Generate terms of a binomial expansion with a negative or fractional power, first factoring out a suitable constant.
• State the interval where the infinite expansion is valid and use a suitable truncation for an approximation.
• Keep counting practice within the assessed scope: arrangements of identical objects and circular arrangements are outside this section.

1.11

Splitting an algebraic fraction

Additional HL

• Rewrite a proper rational expression as partial fractions and find the constants by substitution or matching coefficients.
• Handle denominators containing up to two distinct linear factors, and verify the result by recombining the fractions.

1.12

Locating complex numbers

Additional HL

• Work with i² = −1 and identify the real part, imaginary part and conjugate of a + bi.
• Plot a complex number on an Argand diagram and find its modulus and argument with the correct quadrant.

1.13

Choosing a useful complex-number form

Additional HL

• Convert between Cartesian form, r(cos θ + i sin θ), r cis θ and r e^(iθ).
• Choose an efficient form for addition, multiplication or division, then interpret the calculation geometrically.
• Connect multiplication and division with changes in modulus and argument, including rotations and stretches.

1.14

Complex powers, roots and polynomial equations

Additional HL

• Use conjugate pairs to find missing non-real roots of a polynomial with real coefficients.
• Apply de Moivre's theorem to complex powers and prove its positive-integer case by induction.
• Find all distinct nth roots of a complex number and represent their equal angular spacing on an Argand diagram.

1.15

Choosing a proof strategy

Additional HL

• Write a complete induction proof with a base case, a clear assumption and an inductive step that uses that assumption.
• Build a proof by contradiction, identifying exactly which assumption leads to the contradiction.
• Disprove a universal claim with a counterexample and explain why that example violates the claim.

1.16

Solving and classifying linear systems

Additional HL

• Solve up to three simultaneous linear equations in three unknowns using algebra and technology.
• Distinguish a unique solution, an inconsistent system and a system with infinitely many solutions.
• Express an infinite family of solutions using a free parameter and interpret parameter-dependent cases.

2.1

Describing straight lines

• Find a line's gradient and intercepts from points, an equation or a diagram.
• Build and rearrange line equations in gradient-intercept, point-gradient and general form.
• Use gradient relationships to construct parallel or perpendicular lines and interpret a gradient in its context.

2.2

Inputs, outputs and reversing a function

• Interpret function notation and determine sensible domains and ranges from a formula, graph or model.
• Explain how an inverse reverses a one-to-one function; interchange its domain and range.
• Relate a function and its inverse through reflection in y = x, without confusing an inverse with a reciprocal.

2.3

Turning function information into a graph

• Produce a labelled sketch from a formula, description or calculator display, showing the important shape and features.
• Use graphing technology with a suitable viewing window, including for sums or differences of functions.
• Distinguish a carefully scaled drawing from a sketch that communicates the essential behaviour.

2.4

Reading a graph with technology

• Locate roots, intercepts, turning points, extreme values and intersections accurately with graphing tools.
• Recognise symmetry and describe horizontal or vertical asymptotes from the graph.
• Check that the viewing window includes the features relevant to the question.

2.5

Combining and undoing functions

• Evaluate and form composite functions in the stated order, checking that intermediate outputs are allowed inputs.
• Find an inverse algebraically when the function is one-to-one and verify it by composition.
• Recognise the identity function and use the domain-range relationship when describing inverse functions.

2.6

Three useful views of a quadratic

• Move between expanded, factorised and completed-square forms of a quadratic.
• Read the intercepts, vertex and axis of symmetry from a suitable form and sketch the parabola.
• Explain how the leading coefficient affects the direction and vertical stretch of the graph.

2.7

Quadratic roots and solution intervals

• Solve a quadratic using factorisation, completing the square or the quadratic formula, choosing an efficient method.
• Solve quadratic inequalities and show the correct intervals, including or excluding endpoints as required.
• Use the discriminant to decide how many real roots occur and to find parameter conditions.

2.8

Reciprocals and linear-over-linear functions

• Explain the shape and self-inverse property of y = 1/x.
• Sketch a rational function with a linear numerator and denominator, identifying intercepts and excluded input values.
• Find and label its vertical and horizontal asymptotes, checking any factor cancellation before claiming an asymptote.

2.9

Exponential and logarithmic graphs

• Sketch exponential and logarithmic functions, including base e, with their domains, ranges, intercepts and asymptotes.
• Explain the inverse relationship between these families and compare growth or decay for different valid bases.
• Rewrite a^x as e^(x ln a) and use inverse identities to simplify expressions.

2.10

Finding solutions in mathematical models

• Choose an algebraic, graphical or numerical method to solve an equation, including equations requiring technology.
• Use a substitution where it turns an exponential equation into a familiar algebraic form.
• Interpret solutions in a real situation, respecting the model's domain and the required accuracy.

2.11

Moving and stretching graphs

• Describe translations, reflections in either axis, and vertical or horizontal stretches using function notation.
• Apply a sequence of transformations in the correct order and track key coordinates, domains, ranges and asymptotes.
• Recognise that f(qx) gives a horizontal stretch with factor 1/q for q > 0; the general combined input ax + b belongs to the HL extension.

2.12

Reading and factoring polynomials

Additional HL

• Connect a polynomial's factors and roots with its graph, including repeated roots.
• Use the factor and remainder theorems to find unknown coefficients, test factors or solve polynomial equations.
• Use the sum and product of all roots, including complex roots, to relate solutions to the coefficients.

2.13

Rational graphs with quadratic terms

Additional HL

• Analyse rational functions with a linear numerator over a quadratic denominator, or a quadratic numerator over a linear denominator.
• Find intercepts, domain restrictions and horizontal, vertical or oblique asymptotes, using algebra to support the sketch.
• Use technology to investigate behaviour and check that a sketch shows each branch correctly.

2.14

Symmetry, repetition and invertibility

Additional HL

• Test whether a function is even or odd algebraically and relate the result to graph symmetry.
• Identify periodic behaviour and explain what a period means.
• Restrict a domain when necessary to obtain an inverse; find that inverse and recognise self-inverse functions.

2.15

Comparing two functions

Additional HL

• Find where one function lies above or below another by studying their difference or their graphs.
• Solve polynomial inequalities up to degree 3 analytically and express the answer with correct interval endpoints.
• Use technology for more complicated comparisons and respect discontinuities and domain restrictions.

2.16

Building new graphs from an existing graph

Additional HL

• Predict the graphs of |f(x)|, f(|x|), 1/f(x) and [f(x)]² from the behaviour of f.
• Handle combined input transformations f(ax + b), tracking fixed points, intercepts and asymptotes where relevant.
• Solve modulus equations and inequalities through appropriate cases, graphs or technology, checking the original conditions.

3.1

Measuring three-dimensional objects

• Calculate the distance and midpoint between points given by three coordinates.
• Find volumes and surface areas of solids, including right pyramids, right cones, spheres, hemispheres and combined shapes.
• Identify useful right-angled triangles to find lengths and angles between intersecting lines or between a line and a plane.

3.2

Choosing a triangle method

• Choose sine, cosine or tangent for a right-angled triangle and use an inverse ratio when finding an angle.
• Choose the sine rule or cosine rule for an appropriate non-right-angled triangle.
• Calculate a triangle's area from two sides and their included angle; label a supporting diagram.

3.3

Turning a situation into a triangle

• Translate a written description into a clear diagram with the relevant lengths and angles marked.
• Solve problems involving bearings, elevation, depression and distance using Pythagoras or trigonometry.
• Interpret the result with suitable units and check that it matches the geometry.

3.4

Angles measured by arc length

• Convert between degrees and radians, using exact multiples of π when appropriate.
• Use radian measure to calculate arc lengths, sector areas and unknown circle dimensions.

3.5

Extending trigonometry around the circle

• Use the unit circle to determine signs, related angles and exact values based on 0, π/6, π/4, π/3 and π/2.
• Connect tangent with sine divided by cosine and with the gradient of a line through the origin.
• Check for two possible triangles when applying the sine rule in an ambiguous case.

3.6

Using identities instead of finding an angle

• Use sin² θ + cos² θ = 1 to connect ratios and select signs from the angle's quadrant.
• Apply the sine and cosine double-angle identities to simplify expressions or determine exact values.
• Find a related trigonometric ratio without first calculating the angle.

3.7

Modelling a repeating pattern

• Sketch sine, cosine and tangent graphs and identify their periods, zeros and relevant asymptotes.
• Interpret amplitude, horizontal shift, vertical shift and period in a transformed trigonometric model.
• Build or use a sinusoidal model for a repeating situation, matching units and interpreting parameters in context.

3.8

Finding every trigonometric solution in an interval

• Solve trigonometric equations by algebra or graphing and list all solutions in the specified finite interval.
• Use identities or substitution to reduce an equation to a quadratic in a trigonometric ratio.
• Check endpoints, angle units and rejected values; an unrestricted general solution is outside this section's requirements.

3.9

Reciprocal and inverse trigonometric functions

Additional HL

• Use sec, cosec and cot as reciprocal ratios, and work with their associated Pythagorean identities.
• Distinguish inverse trigonometric functions from reciprocals and use their principal ranges correctly.
• Sketch arcsin, arccos and arctan with accurate domains, ranges and key features.

3.10

Combining angles

Additional HL

• Use compound-angle identities to simplify expressions, establish results and find exact values.
• Derive double-angle identities from the compound-angle formulae and include the double-angle identity for tangent.

3.11

Explaining trigonometric symmetry

Additional HL

• Relate values at transformed angles using unit-circle reasoning, identities and graph symmetry.
• Use these relationships to simplify expressions and justify equivalent trigonometric forms.

3.12

Representing position and displacement

Additional HL

• Write vectors using components or i, j and k; distinguish a point's position vector from a displacement between points.
• Add, subtract and scale vectors algebraically and geometrically, including zero, opposite and parallel vectors.
• Find magnitudes and unit vectors and use vector relationships to prove geometrical results.

3.13

Angles from a scalar product

Additional HL

• Calculate a dot product using components or magnitudes and an included angle.
• Use scalar products to find angles and test perpendicularity or parallelism for non-zero vectors.
• Apply the algebraic properties of the dot product to simplify vector expressions and justify results.

3.14

Describing a line in space

Additional HL

• Write and convert a line's vector, parametric and Cartesian equations, identifying a point and a direction vector.
• Use direction vectors to calculate the angle between lines.
• Interpret a time parameter as motion at constant velocity and connect the velocity vector's magnitude with speed.

3.15

Deciding how two lines meet

Additional HL

• Classify a pair of lines as coincident, parallel, intersecting or skew.
• Find an intersection using independent parameters and verify the point in both line equations.
• Explain why non-parallel lines in three dimensions need not intersect.

3.16

Perpendicular directions and area

Additional HL

• Calculate a cross product, determine its direction and use its magnitude to connect with the included angle.
• Apply cross-product properties, including the sign change when reversing the order.
• Use the cross product to find a normal vector, test parallelism and calculate triangle or parallelogram area.

3.17

Describing a plane

Additional HL

• Construct a plane from a point and two independent directions, or from a point and a normal vector.
• Move between parametric vector, scalar-product and Cartesian forms of the plane equation.
• Check whether a point or a direction is compatible with the plane.

3.18

Intersections and angles involving planes

Additional HL

• Find and interpret intersections of a line with a plane, two planes, or three planes by solving suitable equations.
• Recognise when an intersection is empty, a point, a line or an entire plane.
• Use normal and direction vectors to find plane-plane and line-plane angles, taking the correct complementary angle when needed.

4.1

Asking whether data can be trusted

• Distinguish populations from samples and discrete from continuous data; assess the credibility of the collection process.
• Compare simple random, systematic, stratified, quota and convenience sampling, identifying likely sources of bias.
• Investigate missing values and recording mistakes; identify outliers using the 1.5 IQR fences without automatically discarding valid observations.

4.2

Displaying and comparing distributions

• Organise discrete or continuous data in frequency tables and equal-width frequency histograms.
• Construct and read cumulative-frequency graphs to estimate medians, quartiles, percentiles and spread.
• Draw box plots with marked outliers and compare centre, spread and symmetry; consider whether a roughly normal shape is plausible.
• Use equal class widths for histogram work here; frequency-density histograms are outside the assessed section.

4.3

Choosing numerical summaries

• Calculate and interpret mean, median and mode; estimate a grouped mean using class midpoints and identify the modal class.
• Use technology for quartiles, variance and standard deviation, and interpret these alongside the interquartile range.
• Predict how adding a constant or multiplying every observation changes measures of centre and spread.
• Recognise that different quartile conventions can produce slightly different answers for discrete data.

4.4

Predicting y from x

• Describe a scatter plot's linear association and use technology to obtain and interpret Pearson's correlation coefficient.
• Draw a sensible best-fit line through the mean point and obtain the y-on-x regression equation with technology.
• Interpret the slope and intercept and make appropriate predictions of y, treating extrapolation and reverse prediction cautiously.
• Explain why strong correlation does not establish causation and why Pearson's r can mislead for a non-linear pattern.

4.5

Building a probability model

• Describe the sample space and events for a trial, checking whether the outcomes really are equally likely.
• Calculate theoretical probabilities and complements, and compare them with experimental relative frequencies.
• Use probability to predict an expected count over repeated trials, without interpreting that count as a guarantee.

4.6

Combining and conditioning events

• Choose a tree, Venn diagram, outcome table or sample-space diagram to organise a multi-stage probability problem.
• Calculate probabilities involving inclusive 'or', 'and', complements and conditional information.
• Distinguish independence from mutual exclusivity and update probabilities correctly when sampling without replacement.

4.7

A random quantity and its long-run average

• Build or interpret a discrete probability distribution from a table or rule and ensure its probabilities total 1.
• Calculate an expected value by weighting each possible value with its probability.
• Interpret expected net gain, including entry costs, to judge whether a game is fair.

4.8

Counting successes with a binomial model

• Check that a fixed number of independent trials each has two outcomes and the same success probability.
• Use binomial technology for individual, cumulative and interval probabilities, handling 'at least' and 'more than' carefully.
• Find and interpret the model's mean and variance, np and np(1 − p).

4.9

Areas under a normal curve

• Sketch a normal model with its mean and standard deviation and use symmetry and the approximate 68%, 95%, 99.7% pattern.
• Translate a probability statement into a shaded region and calculate its area with technology.
• Use inverse-normal technology to find a boundary or percentile when the mean and standard deviation are known.

4.10

Predicting x from y

• Find the x-on-y regression equation with technology and use it to predict x from a specified y.
• Choose the regression direction to match the variable being predicted; do not treat the two regression lines as interchangeable.

4.11

Checking independence with probability formulae

• Apply the conditional-probability formula to connect a restricted sample space with an intersection probability.
• Test independence using P(A ∩ B) = P(A)P(B) or an equivalent conditional-probability comparison, where defined.
• Explain the conclusion in the context of the two events.

4.12

Standard scores and unknown normal parameters

• Convert between a value and its z-score and interpret the sign and size of that score.
• Use inverse-normal results to form equations for an unknown mean, standard deviation or both.
• Solve the parameter equations and check that the standard deviation is positive and the probabilities are consistent.

4.13

Reversing a conditional probability

Additional HL

• Use Bayes' theorem to revise the probability of a source or cause after observing an outcome.
• Build the denominator from all relevant cases in a partition of up to three events, using a tree when helpful.

4.14

Comparing discrete and continuous random variables

Additional HL

• Calculate E(X), E(X²), variance and standard deviation from a discrete distribution or a continuous density, including piecewise densities.
• Check that a density is non-negative with total area 1, then use integrals for interval probabilities; a density value itself may exceed 1.
• Find a continuous distribution's median from cumulative area and its mode from the maximum density.
• Interpret fair net gains and use E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X) for transformed variables.

5.1

From a changing secant to an instantaneous rate

• Estimate a limit from a table or graph and describe what value the function approaches.
• Explain a derivative as a tangent gradient or an instantaneous rate, using notation suited to the variables.
• Connect the tangent gradient with a limiting sequence of secant gradients; formal limit techniques are reserved for HL.

5.2

Where a function rises or falls

• Use the sign of the first derivative to identify intervals of increasing or decreasing behaviour.
• Relate positive, negative and zero gradients to the graphs of a function and its derivative.

5.3

Differentiating integer powers

• Apply the power rule to constant multiples of integer powers, including negative powers after suitable rewriting.
• Differentiate sums of these terms and simplify the resulting derivative.

5.4

Equations touching and crossing a curve

• Find the gradient at a specified point using a derivative or technology, then write the tangent equation.
• Find the normal equation using perpendicularity, recognising horizontal and vertical special cases.

5.5

Recovering a function and measuring area

• Reverse the power rule for sums of integer powers other than x^−1, including the arbitrary constant.
• Use a given function value to determine that constant.
• Write a definite integral for the area beneath a positive curve and evaluate it with technology, connecting it with antiderivatives.

5.6

Choosing a differentiation rule

• Differentiate rational powers, sine, cosine, e^x and ln x, including their sums and constant multiples.
• Identify inner and outer functions and apply the chain rule to a composite expression.
• Use product and quotient rules when required, combining rules and simplifying carefully.

5.7

Reading a function through its derivatives

• Calculate second derivatives and use either f″(x) or d²y/dx² notation correctly.
• Connect the shapes of f, f′ and f″, including how changes in gradient show on the derivative graphs.

5.8

Classifying key points and finding an optimum

• Find stationary points and classify local maxima or minima using a sign change in f′ or a valid second-derivative test.
• Distinguish concave-up and concave-down intervals and confirm a change in concavity at an inflection point.
• Set up and solve an optimisation problem within a meaningful domain, comparing relevant boundary values.
• Recognise that f″ = 0 alone does not prove an inflection point, and that an inflection point need not be stationary.

5.9

Connecting position, velocity and acceleration

• Move between displacement, velocity and acceleration using differentiation or integration and any initial conditions.
• Distinguish speed from velocity and identify changes of direction.
• Calculate displacement from the integral of velocity and total distance from the integral of speed, splitting at sign changes when useful.

5.10

Recognising antiderivative patterns

• Integrate rational powers, sine, cosine, exponentials and 1/x, using ln|x| for the last case.
• Integrate these functions with a linear inner expression, including the required constant factor.
• Recognise a multiple of an inner derivative and use the reverse chain rule or an equivalent substitution.

5.11

Definite integrals and enclosed regions

• Evaluate a definite integral from an antiderivative, or use technology when an analytic antiderivative is unsuitable.
• Separate signed accumulation from geometrical area and split an interval where the curve changes sign.
• Find areas between curves by locating intersections and integrating the appropriate difference, writing the setup before calculating.

5.12

Limits, first principles and higher derivatives

Additional HL

• Describe convergence, divergence, continuity and differentiability informally and explain how these ideas connect.
• Use the limiting difference quotient to derive the derivative of a polynomial.
• Calculate and interpret higher derivatives using nth-derivative notation; formal continuity or differentiability tests are outside the stated examination requirements.

5.13

Resolving indeterminate limits

Additional HL

• Recognise a 0/0 or ∞/∞ quotient before applying l'Hôpital's rule, repeating it where valid and necessary.
• Use suitable Maclaurin expansions as an alternative way to evaluate limits at finite points or infinity.
• Check the transformed expression and the existence of the resulting limit instead of treating the rule as automatic.

5.14

Implicit relationships and linked rates

Additional HL

• Differentiate a relation that defines y implicitly, applying the chain rule to every term involving y.
• Connect rates of change through an equation between changing quantities and interpret signs and units.
• Solve optimisation problems requiring implicit differentiation or the chain rule and check possible endpoint optima.

5.15

Extending the derivative and integral toolkit

Additional HL

• Differentiate tan, sec, cosec, cot, a^x, logₐ x, arcsin, arccos and arctan.
• Recognise and integrate their derivative patterns, including compositions with a linear expression.
• Rearrange suitable integrands using partial fractions and interpret an indefinite integral as a family of curves.

5.16

Substitution and integration by parts

Additional HL

• Carry out a substitution consistently, changing the differential and either the limits or the final variable.
• Choose and apply integration by parts, including repeated applications and cases where the original integral reappears.
• Recognise the examination scope: a substitution is supplied when the integrand is not a direct reverse-chain-rule pattern.

5.17

Integrating across and rotating a region

Additional HL

• Set up an integral with respect to y for a region bounded by a curve and the y-axis.
• Find volumes formed by rotating a region about the x-axis or y-axis, using appropriate outer and inner radii.
• Choose bounds and the variable of integration to match the geometry, then interpret the volume in cubic units.

5.18

Solving first-order differential equations

Additional HL

• Generate an approximate solution to dy/dx = f(x, y) with Euler's method and a stated constant step size.
• Separate variables and integrate, using an initial condition when supplied and retaining relevant constant solutions.
• Solve a homogeneous equation dy/dx = f(y/x) using y = vx and the product rule.
• Use an integrating factor for a linear equation y′ + P(x)y = Q(x), and check the resulting solution.

5.19

Approximating functions with power series

Additional HL

• Obtain Maclaurin expansions for e^x, sin x, cos x, arctan x, ln(1 + x) and (1 + x)^p with rational p.
• Generate related expansions through substitution, multiplication, differentiation or integration, keeping the required order of terms.
• Use a differential equation to determine series coefficients and apply an expansion only within an appropriate validity range.