14 lessons covering the pure mathematics, mechanics and statistics areas of a UK A-level Mathematics course, with exam skills and interactive explorers.
A study guide to A-level Mathematics, not a full course or textbook. The paper structure and content areas follow the AQA A-level Mathematics 7357 pages; other boards (Edexcel, OCR, WJEC) cover the same content in different paper layouts. Method and formulae are standard A-level mathematics. Lab numbers are practice values, not exam data. Check your own board's specification and formula sheet.
['GCSE Mathematics at higher tier or equivalent']
Course outline
Proof and algebra
Use proof methods and handle indices and surds exactly.
Quadratics and the discriminant
Solve quadratics three ways and use the discriminant to count roots.
Functions and graph transformations
Use function notation, composites, inverses and the four basic transformations.
Coordinate geometry: lines and circles
Work with straight lines, midpoints, distances and circle equations.
Sequences, series and the binomial expansion
Sum arithmetic and geometric series and expand brackets with the binomial theorem.
Trigonometry: radians, identities and equations
Work in radians and solve trigonometric equations.
Exponentials and logarithms
Model growth and decay and use logarithms to solve exponential equations.
Differentiation
Differentiate powers, find tangents and normals and locate stationary points.
Integration and numerical methods
Integrate for areas, and use the trapezium rule and Newton-Raphson.
Vectors and kinematics
Use vectors and the constant acceleration equations.
Forces, Newton's laws and moments
Resolve forces, apply F = ma and take moments.
Statistics: sampling, data and probability
Choose samples, summarize data and use probability rules.
Distributions and hypothesis testing
Use the binomial and normal distributions and run a hypothesis test.
Exam skills and using the specification
Plan across three papers and show full method.
Sources and curriculum note
Reviewed October 7, 2026. Confirm the specification for your board and exam year.
Read every lesson below. The interactive reader above contains the same explanations, with visual tools and quizzes.
1. Proof and algebra
Learning goal: Use proof methods and handle indices and surds exactly.
Proof means showing a statement is always true, or false, with logical steps. Proof by deduction starts from known facts and algebra, for example showing that the sum of two consecutive integers is odd: n + (n + 1) = 2n + 1. Proof by exhaustion checks every case. Proof by contradiction assumes the opposite and reaches an impossibility, as in the classic proof that the square root of 2 is irrational.
A single counterexample disproves a general claim. To disprove 'n squared + n + 41 is always prime', test values until one fails: n = 40 gives 40 x 40 + 40 + 41 = 1681, which equals 41 x 41. State the value and show the check.
Indices follow the laws a^m x a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn), a^0 = 1 and a^(-n) = 1 / a^n. A fractional power means a root: a^(1/2) is the square root of a. Surds are exact roots such as the square root of 12, which simplifies to 2 times the square root of 3. Rationalize a denominator by multiplying top and bottom by the conjugate.
On the exam, write the logical connectors ('therefore', 'since') and finish by stating what you proved. Marks go to method, not only the final line.
Worked example
Disprove a claim.
Pick a value
Calculate
Compare with the claim
State the counterexample
Practice problem and solution
Simplify: what is 2 to the power 3 times 2 to the power 4, written as a single number? Enter a number.
2^3 x 2^4 = 2^7 = 128.
Mental model: Proof needs logic and a clear conclusion; one counterexample disproves; keep surds exact.
Common trap: Testing a few cases and calling it proof.
2. Quadratics and the discriminant
Learning goal: Solve quadratics three ways and use the discriminant to count roots.
A quadratic has the form ax^2 + bx + c = 0 with a not zero. Solve by factorizing when possible, by completing the square, or with the formula x = (-b plus or minus the square root of (b^2 - 4ac)) / (2a). Completing the square gives a(x + b/(2a))^2 + c - b^2/(4a), which shows the vertex of the graph.
The discriminant is b^2 - 4ac. If it is positive there are two distinct real roots, if zero one repeated root, and if negative no real roots (the graph does not meet the x-axis). The discriminant also decides whether a line meets a curve: substitute, form a quadratic and test it.
Sketch a parabola using the roots, the y-intercept (x = 0) and the vertex. If a is positive it opens up and has a minimum; if negative it opens down and has a maximum. Quadratic inequalities are solved by sketching, then reading where the graph is above or below the axis.
Check answers by substituting back, and give exact forms when the question asks for surds.
Worked example
Solve and check.
Pick method
Find roots
Substitute back
State both
Practice problem and solution
Find the discriminant of x^2 - 6x + 5. Enter a number.
b^2 - 4ac = 36 - 20 = 16.
Mental model: The discriminant counts real roots; complete the square for the vertex.
Common trap: Forgetting to check the sign of a.
3. Functions and graph transformations
Learning goal: Use function notation, composites, inverses and the four basic transformations.
A function takes each input from its domain to exactly one output in its range. A composite function fg(x) means f(g(x)): apply g first. The inverse function reverses the mapping and exists when the function is one-to-one; its graph is a reflection in y = x. To find an inverse, swap x and y and rearrange.
Transformations change a graph y = f(x). y = f(x) + a moves it up by a. y = f(x + a) moves it left by a. y = a f(x) stretches it vertically by factor a. y = f(ax) stretches it horizontally by factor 1/a. Replacing y by -f(x) reflects in the x-axis and f(-x) reflects in the y-axis.
Modulus functions |f(x)| reflect any part below the x-axis upward. Solve modulus equations by splitting into cases, and check each answer. Piecewise functions use different rules on different intervals.
Describe transformations in words and with a vector or factor, and say which direction is which. Sign mistakes inside brackets are the most common error.
Worked example
Describe a move.
Inside or outside brackets
Direction
Factor
State in words
Practice problem and solution
If f(x) = 2x + 1 and g(x) = x^2, what is f(g(3))? Enter a number.
g(3) = 9, then f(9) = 2 x 9 + 1 = 19.
Mental model: Inside the bracket moves horizontally and opposite; outside changes vertically.
Common trap: Putting the shift in the wrong direction.
4. Coordinate geometry: lines and circles
Learning goal: Work with straight lines, midpoints, distances and circle equations.
The gradient between (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). The line through a point with gradient m is y - y1 = m(x - x1). The midpoint is the average of the coordinates and the distance is the square root of the sum of squared differences. Parallel lines have equal gradients. Perpendicular lines have gradients with product -1.
A circle with centre (a, b) and radius r has equation (x - a)^2 + (y - b)^2 = r^2. Complete the square on x and y to find the centre and radius from an expanded equation. A tangent is perpendicular to the radius at the point of contact.
To find where a line meets a circle, substitute the line into the circle equation and solve the quadratic. The discriminant tells you whether the line crosses, touches or misses the circle. The angle in a semicircle is a right angle, and the perpendicular from the centre to a chord bisects the chord.
Sketch every problem. A picture shows whether a distance is a radius, whether a point is inside the circle and which gradient is needed.
Worked example
Find a circle.
Centre
Radius
Equation
Check a point
Practice problem and solution
What is the distance between (0, 0) and (3, 4)? Enter a number.
sqrt(9 + 16) = 5.
Mental model: Gradients and distances first; circle equations come from the centre and radius.
Common trap: Forgetting the negative reciprocal.
5. Sequences, series and the binomial expansion
Learning goal: Sum arithmetic and geometric series and expand brackets with the binomial theorem.
An arithmetic sequence adds a common difference d: the nth term is a + (n - 1)d and the sum of n terms is n/2 times (2a + (n - 1)d). A geometric sequence multiplies by a common ratio r: the nth term is a r^(n-1) and the sum is a(1 - r^n)/(1 - r) for r not equal to 1.
A geometric series converges when |r| < 1, and then the sum to infinity is a/(1 - r). For example, 1 + 1/2 + 1/4 + ... = 2. Sigma notation writes sums compactly. Recurrence relations define each term from the one before, for example u(n+1) = 2u(n) + 1.
The binomial expansion of (a + b)^n has terms C(n, k) a^(n-k) b^k, where C(n, k) = n! / (k!(n-k)!). For a positive integer n there are n + 1 terms. For (1 + x)^n with other values of n the expansion is infinite and valid for |x| < 1.
Check a series with a small case: n = 1 or n = 2. Many errors come from using n instead of n - 1.
Worked example
Test a formula.
Use n = 1
Use n = 2
Compare
Trust it
Practice problem and solution
What is the sum to infinity of 6 + 3 + 1.5 + ...? Enter a number.
a = 6, r = 1/2, so 6 / (1 - 1/2) = 12.
Mental model: Know both sum formulas, the convergence test, and C(n, k) for binomial terms.
Common trap: Using n instead of n - 1 in the nth term.
6. Trigonometry: radians, identities and equations
Learning goal: Work in radians and solve trigonometric equations.
A radian is the angle at the centre of a circle where the arc length equals the radius. A full turn is 2 pi radians, which equals 360 degrees, so 180 degrees is pi radians. Arc length is r times theta and sector area is half r^2 times theta, with theta in radians.
Use exact values for 30, 45 and 60 degrees: sin 30 = 1/2, sin 45 = root 2 over 2 and sin 60 = root 3 over 2. Key identities are tan x = sin x / cos x and sin^2 x + cos^2 x = 1. The sine and cosine rules solve non-right triangles, and the area of a triangle is half ab sin C.
Solve equations such as sin x = 0.5 over a given interval by finding the principal value and then the other values from symmetry: sin x = sin (180 - x). For equations in the form of a quadratic in sin x or cos x, substitute and factorize. For small angles in radians, sin x is about x.
Check the calculator mode, degrees or radians, before every question. State all solutions in the interval and no others.
Worked example
Solve in an interval.
Find principal value
Find second value
Add periods
Stay in range
Practice problem and solution
A sector has radius 4 and angle 2 radians. What is its arc length? Enter a number.
r x theta = 4 x 2 = 8.
Mental model: Radians make arc and sector formulas simple; solve with symmetry and check the interval.
Common trap: Calculator in the wrong mode.
7. Exponentials and logarithms
Learning goal: Model growth and decay and use logarithms to solve exponential equations.
An exponential function has the variable in the power, such as y = a b^x. When b > 1 it grows and when 0 < b < 1 it decays. The number e, about 2.718, is the base for which the gradient of e^x equals e^x, so e^x is its own derivative.
A logarithm reverses an exponential: log_b y = x means b^x = y. The laws are log a + log b = log (ab), log a - log b = log (a / b) and n log a = log (a^n). Natural logarithm ln uses base e. To solve 2^x = 7, take logs: x = ln 7 / ln 2, about 2.807.
Many real relationships fit y = a b^x or y = a x^n. Taking logs turns them into straight lines: log y = log a + x log b, so plotting log y against x gives gradient log b. Plotting log y against log x suits a power law.
In modeling questions, interpret a and b: a is the starting value and b is the growth factor per unit of x. Always comment on the limits of the model.
Worked example
Linearize data.
Choose the model
Take logs
Plot
Read gradient
Practice problem and solution
A population of 200 grows by a factor 1.5 each year. What is the population after 2 years? Enter a number.
200 x 1.5 x 1.5 = 450.
Mental model: Logs reverse exponentials; linearize models by taking logs.
Common trap: Applying the log to a sum as if it were a product.
8. Differentiation
Learning goal: Differentiate powers, find tangents and normals and locate stationary points.
The derivative dy/dx gives the gradient of a curve at a point. For y = x^n the derivative is n x^(n-1). Differentiate each term of a polynomial separately and constants disappear. For the derivative from first principles, take the limit of (f(x + h) - f(x)) / h as h tends to zero.
The tangent at a point has the gradient of the curve there. The normal is perpendicular, with gradient -1 over that value. Stationary points have dy/dx = 0. To classify them, use the second derivative: d2y/dx2 > 0 is a minimum, < 0 is a maximum, and = 0 needs another test. Increasing means dy/dx > 0.
The chain rule, product rule and quotient rule handle composite functions and products. Useful derivatives: e^(kx) gives k e^(kx), ln x gives 1/x, sin x gives cos x (radians) and cos x gives -sin x. Optimization problems write a quantity as a function of one variable, differentiate and find the best value.
Always check whether the question wants the maximum value or the x-value where it occurs. Substitute back to find the value.
Worked example
Classify a point.
Find dy/dx
Solve = 0
Find d2y/dx2
Conclude
Practice problem and solution
A curve is y = x^3 - 3x. What is the gradient at x = 2? Enter a number.
dy/dx = 3x^2 - 3 = 12 - 3 = 9.
Mental model: Differentiate with the power rule, set to zero for stationary points and classify with the second derivative.
Common trap: Forgetting to substitute back for the value.
9. Integration and numerical methods
Learning goal: Integrate for areas, and use the trapezium rule and Newton-Raphson.
Integration reverses differentiation: the integral of x^n is x^(n+1)/(n+1) plus a constant, for n not equal to -1. A definite integral evaluates the antiderivative at the limits and subtracts. It gives the signed area between the curve and the x-axis, so regions below the axis count as negative.
The area between two curves is the integral of the upper minus the lower. Useful integrals: e^(kx) gives e^(kx)/k, 1/x gives ln |x|, cos x gives sin x and sin x gives -cos x. Integration by substitution and by parts handle more complex cases.
When no exact integral is available, use the trapezium rule: area is about h/2 times (first + last + 2 times the sum of the middle ordinates), with strip width h. The estimate is too high for a convex curve and too low for a concave curve.
Newton-Raphson finds roots of f(x) = 0: x(n+1) = x(n) - f(x(n)) / f'(x(n)). It converges quickly near a root but can fail when the gradient is near zero. Change of sign shows a root lies in an interval, provided the function is continuous.
Worked example
Estimate an area.
Choose strips
Find ordinates
Apply the rule
Compare
Practice problem and solution
What is the integral of 3x^2 from 0 to 2? Enter a number.
[x^3] from 0 to 2 = 8 - 0 = 8.
Mental model: Integrate by raising the power; use trapezium and Newton-Raphson when exact methods fail.
Common trap: Forgetting the constant of integration.
10. Vectors and kinematics
Learning goal: Use vectors and the constant acceleration equations.
A vector has magnitude and direction. In two dimensions write it as a column or as xi + yj. Its magnitude is the square root of x^2 + y^2. Add vectors component by component, and a position vector gives the location of a point from the origin. The vector from A to B is b minus a.
Kinematics describes motion. For constant acceleration use v = u + at, s = ut + 1/2 at^2, v^2 = u^2 + 2as and s = (u + v)t / 2. Here u is the initial velocity, v the final velocity, a the acceleration, t the time and s the displacement. Choose a positive direction and keep signs consistent.
On a velocity-time graph the gradient is acceleration and the area under the graph is displacement. On a displacement-time graph the gradient is velocity. With variable acceleration use calculus: v = ds/dt and a = dv/dt. Falling bodies use a = g, taken as 9.8 meters per second squared unless the question states another value.
Write down the five suvat quantities, mark the three you know, and choose the equation without the unknown one.
Worked example
Solve a suvat problem.
Choose direction
List u, v, a, s, t
Select equation
Check units
Practice problem and solution
A car starts at 0 m/s and accelerates at 2 m/s^2 for 5 s. How far does it travel? Enter a number.
s = ut + 1/2 at^2 = 0 + 0.5 x 2 x 25 = 25.
Mental model: Use suvat for constant acceleration and calculus for variable acceleration.
Common trap: Mixing up the sign of g.
11. Forces, Newton's laws and moments
Learning goal: Resolve forces, apply F = ma and take moments.
Newton's first law says an object keeps its velocity unless a resultant force acts. The second law says resultant force F = ma, with force in newtons, mass in kilograms and acceleration in meters per second squared. The third law says forces between two bodies are equal and opposite. Weight is mg, with g about 9.8 meters per second squared.
Resolve a force F at angle theta to the horizontal into components F cos theta horizontally and F sin theta vertically. In equilibrium, the resultant force is zero in every direction. On a slope, resolve parallel and perpendicular to the surface. Friction is at most mu times the normal reaction R, and equals mu R when limiting.
For connected particles, such as two masses over a pulley, write F = ma for each mass and add the equations. Tension is the same on both sides of a light string over a smooth pulley. Check the direction of motion before choosing signs.
The moment of a force about a point is force times the perpendicular distance. A body is in equilibrium when the sum of moments is zero and the resultant force is zero. For a uniform beam, treat the weight as acting at the midpoint.
Worked example
Take moments.
Choose a point
Find distances
Sum clockwise and anticlockwise
Equate
Practice problem and solution
A resultant force of 12 N acts on a 3 kg mass. What is the acceleration? Enter a number.
a = F / m = 12 / 3 = 4.
Mental model: F = ma, resolve forces, equilibrium needs zero force and zero moments.
Common trap: Using the wrong perpendicular distance for a moment.
12. Statistics: sampling, data and probability
Learning goal: Choose samples, summarize data and use probability rules.
A population is the whole group and a sample is the part you measure. A random sample gives every member an equal chance. Opportunity, quota and self-selected samples can be biased. AQA A-level Mathematics uses a prescribed large data set, so check the specification and practice with it.
Summarize data with mean, median and mode for location, and with range, interquartile range and standard deviation for spread. Standard deviation is the square root of the variance, which is the mean of squares minus the square of the mean. Outliers are often defined as more than 1.5 times the IQR beyond a quartile. Histograms use frequency density, which is frequency divided by class width.
Probability uses P(A or B) = P(A) + P(B) - P(A and B), and for independent events P(A and B) = P(A) P(B). Conditional probability P(A | B) = P(A and B) / P(B). Venn diagrams and tree diagrams organize the cases.
Correlation does not prove causation. State the context, and say what you would check before concluding anything from a sample.
Worked example
Compute a probability.
Define events
Choose a rule
Calculate
Check range
Practice problem and solution
P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)? Enter a number.
0.5 + 0.4 - 0.2 = 0.7.
Mental model: Choose unbiased samples, summarize with the right measures, and use probability rules carefully.
Common trap: Adding probabilities without subtracting the overlap.
13. Distributions and hypothesis testing
Learning goal: Use the binomial and normal distributions and run a hypothesis test.
The binomial distribution models the number of successes X in n independent trials, each with probability p. P(X = r) = C(n, r) p^r (1 - p)^(n - r). The mean is np and the variance is np(1 - p). Use it when there are two outcomes, a fixed number of trials, constant p and independence.
The normal distribution is symmetric and bell-shaped, set by its mean and standard deviation. About 68 percent of values are within one standard deviation of the mean, 95 percent within two and 99.7 percent within three. Convert to z = (x - mean) / standard deviation to use tables or a calculator.
A hypothesis test starts with a null hypothesis H0, such as p = 0.5, and an alternative H1, such as p > 0.5. Calculate the probability of a result at least as extreme as observed, assuming H0 is true. If it is below the significance level, such as 5 percent, reject H0. The conclusion is a statement about the evidence, not proof.
Write the conclusion in context and with the level: 'There is sufficient evidence at the 5 percent level to suggest that ...'. A one-tailed test checks one direction and a two-tailed test checks both, splitting the significance level.
Worked example
Run a test.
Hypotheses
Distribution
Probability
Conclusion
Practice problem and solution
X follows a binomial distribution with n = 20 and p = 0.3. What is the mean? Enter a number.
np = 20 x 0.3 = 6.
Mental model: Binomial needs fixed n, constant p and independence; test by comparing the probability with the level.
Common trap: Claiming a test proves the null hypothesis.
14. Exam skills and using the specification
Learning goal: Plan across three papers and show full method.
AQA A-level Mathematics (7357) has three written papers, each 2 hours and 100 marks, and each worth one third of the A-level. Paper 1 is pure mathematics: proof, algebra, coordinate geometry, sequences, trigonometry, exponentials and logarithms, differentiation, integration and numerical methods. Paper 2 adds vectors and mechanics, and Paper 3 adds statistics. Other exam boards cover similar content in different paper structures, so check your board.
Marks are split between assessment objectives: about 50 percent for using and applying standard techniques, 25 percent for reasoning and proof, and 25 percent for problem solving and modeling. Show every step, because method marks are awarded even when the final answer is wrong. Underline the command word: show that, find, hence, prove and explain each need different answers.
In 'show that' questions the answer is given, so each step must be explicit. 'Hence' means use the earlier result. Round only at the end and give exact answers if asked. Check the units and the reasonableness of a result.
Allow about one minute per mark. Leave a hard question and return, but write down any partial method first. Practice past papers under timed conditions and mark them with the official scheme.
Worked example
Plan a paper.
Scan
Easy first
Hard later
Check
Practice problem and solution
A 2-hour paper has 100 marks. About how many minutes per mark? Enter a number.
120 / 100 = 1.2.
Mental model: Three papers of 100 marks; show method; plan about a minute per mark.
Common trap: Giving only an answer with no working.