SumClash Lab: watch maths solved, one step at a time

SumClash Lab is a free maths lab: type any question and watch it solved one step at a time, practise with problems the Lab marks for you, and explore live graphs, matrices, vectors and distributions.

Watch it solved

Order of operations: Watch BODMAS pick which operation to do next, one step at a time. (KS3 / GCSE)
Powers and indices: See the base and the power in two colours, and watch a power become repeated multiplication. Includes the index laws. (KS3 / GCSE)
Number bases: Convert between decimal and binary, octal, hexadecimal or any base: repeated division for the whole part, repeated multiplication for the fraction. (KS3 / A-level)
Fractions: Add, subtract, multiply, divide and simplify fractions, with bars that show why. (KS3 / GCSE)
Column addition: Add in columns with carrying, one place value at a time. (KS2 / KS3)
Column subtraction: Subtract in columns, borrowing across zeros when needed. (KS2 / KS3)
Long multiplication: Multiply digit by digit, carry, and add the partial products. (KS2 / KS3)
Long division: The bus-stop method: divide, multiply, subtract, bring down. (KS2 / KS3)
Linear equations: Keep the balance: do the same to both sides until x stands alone, then check. (KS3 / GCSE)
Quadratic equations: Factorise, complete the square or use the formula, then check each root. (GCSE / A-level)
Simultaneous equations: Eliminate one letter, find the other, and see the two lines cross. (GCSE)
Expanding brackets: Multiply out brackets with an area model, so no term is missed. (GCSE)
Differentiation: Every rule applied in order — product, chain, quotient — then verified numerically. (A-level)
Integration: Find the antiderivative, then prove it by differentiating back. (A-level)
Percentages: A percentage of an amount, increase and decrease, percentage change and finding the whole, each with a check. (KS3 / GCSE)
Ratio and proportion: Share an amount in a ratio, simplify a ratio, solve a : b = c : ? and use the unitary method. (KS3 / GCSE)
Inequalities: Solve a linear inequality, flip the sign when dividing by a negative, and test the answer on a number line. (GCSE)
Sequences and series: Spot whether a sequence is arithmetic, geometric or quadratic, then find its n-th term and the sum of the first n terms. (GCSE / A-level)
Mean, median, mode and spread: Mean, median, mode, range, quartiles, variance and standard deviation of a data set, with the deviations table. (KS3 / A-level)
Counting: permutations and combinations: Factorials, permutations (order matters), combinations (order does not) and arrangements of a word with repeated letters. (A-level)
Bayes' theorem: How likely is a positive test to be real? A table of 100,000 people, then the formula, then what it means. (A-level)
Speed, distance, time and work rate: Find speed, distance or time with units that match, and how long workers take on a job together. (KS3 / GCSE)
Interest, loans and savings: Simple and compound interest, depreciation, a loan's monthly payment and a savings annuity, each checked a second way. (GCSE / University)
Coordinate geometry: Distance, midpoint, gradient and the equation of the line through two points, with the graph. (GCSE)
Area, perimeter and volume: Circles, cylinders, cones, spheres, cuboids and more: exact answers in π and as decimals, with the right units. (KS3 / GCSE)
Triangles: sine and cosine rules: Solve any triangle with the sine and cosine rules, or a right-angled one with SOH CAH TOA and Pythagoras, with a scale drawing. (GCSE / A-level)
Limits: Where does a function head as x approaches a number or infinity? Direct substitution, a two-sided table and the graph. (A-level)
Differential equations: Euler's method: Follow the slopes of dy/dx = f(x, y) step by step from a starting point, and see how close the estimate is. (University)
Numerical methods: Newton's method: Find a root with tangent lines: each guess is the last tangent's crossing of the axis. A convergence table and graph. (University)
Complex numbers: Add, subtract, multiply and divide with i² = −1, then modulus, argument and polar form, on an Argand diagram. (A-level)
Matrices: Determinant, inverse, transpose, products and eigenvalues of 2×2 and 3×3 matrices, with a check. (A-level / University)
Vectors: dot and cross product: Dot product, cross product, lengths, the angle between two vectors, projection and parallelogram area. (A-level)
Logic: truth tables: A truth table for any formula in p, q, r with ¬ ∧ ∨ → ↔, and whether it is a tautology, a contradiction or an equivalence. (University)
Sets and Venn diagrams: Union, intersection, difference, symmetric difference and complements of two sets, with a Venn diagram and a counting check. (KS4 / University)
Graph theory: Degrees, the handshaking lemma, connectedness, trees and Euler paths and circuits, for a list of edges or a complete graph K_n. (University)

Practice

Practice and daily challenge: new problems the Lab marks for you, getting harder as you improve.

Explore with live graphs

Calculus sandbox: Type any function, drag a point along it and watch the slope, tangent, secant and area under the curve update live, with the derivative worked out step by step.
Matrices and transformations: Change a 2×2 matrix and see exactly how it stretches, rotates, reflects or shears the plane, with determinant, inverse and eigenvectors.
Complex plane: Plot complex numbers on an Argand diagram, add, multiply and divide them, and see roots of unity spread round a circle.
Distributions: Normal, binomial and Poisson distributions as areas under a curve, with simulated random draws settling onto the exact probability.
Projectile motion: Launch a projectile, change speed, angle and gravity, and read off range, maximum height and time of flight with the SUVAT equations.
Vectors in 3D: Orbit a 3D scene and explore the dot product, cross product, angle between vectors and the parallelogram they span.
Parametric curves: Draw a curve from x(t) and y(t), then trace a moving point and read its velocity, speed, tangent and arc length.
Implicit curves: Plot F(x, y) = c, slide a point along the curve and see its tangent and gradient without solving for y.

Guided lessons

Adding and simplifying fractions: Make the pieces the same size, add, then simplify.
Area, perimeter and volume: Circles, cylinders, cones and spheres: exact answers in terms of π, then decimals with the right units.
Averages and spread: Mean, median, mode, range, variance and standard deviation, worked on real data.
Bayes' theorem: when a positive test is not enough: Why a good test can still give mostly false alarms, shown with a table of 100,000 people.
Complex numbers by hand: Add, multiply and divide with i² = −1, then find modulus and argument.
Complex numbers on the plane: See a + bi as a point, add and multiply them, and watch roots spread evenly round a circle.
Coordinate geometry: Distance, midpoint, gradient and the equation of a line through two points.
Counting: arrangements and selections: Factorials, permutations, combinations and arranging words with repeated letters.
Derivative from first principles: Slide a second point towards P and watch the secant slope settle on the tangent slope.
Euler's method: following the slope: Trace the solution of a differential equation in small steps and measure the error.
Expanding brackets: Multiply everything in one bracket by everything in the other, then tidy up.
Solving quadratics by factorising: Turn x² - 5x + 6 = 0 into two brackets and read off both answers.
Graph theory basics: Degrees, the handshaking lemma, connectedness, trees and Euler paths.
Curves defined by an equation: See an equation in x and y as a curve, and find tangents without solving for y.
Powers and index laws: What 2^5 really means, and the three laws that make big powers easy.
Solving inequalities: Solve a linear inequality, and learn the one rule that is different from an equation.
Interest, loans and savings: Simple and compound interest, monthly loan payments and regular savings, with every answer checked a second way.
Limits: what a function heads towards: Approach a value from both sides, and see when a limit exists, and when it does not.
Logic and truth tables: Build truth tables, and spot tautologies, contradictions and equivalent statements.
Matrices by hand: Determinant, inverse, products and eigenvalues, with every entry worked out.
What a matrix does to the plane: See a 2×2 matrix as a recipe for moving i-hat and j-hat, and watch determinants and eigenvectors appear.
Newton's method: finding roots with tangents: Home in on a root by following tangent lines, and watch the number of correct digits double.
The normal distribution: Probability as area: the 68-95-99.7 rule, what σ does, and how random draws settle onto the exact answer.
Counting in other bases: Binary, base 4 and beyond: how the same number looks different in another base.
Order of operations (BODMAS): Why 2 + 3 × 4 is 14 and not 20, and how to always work in the right order.
Curves from a moving point: Draw a curve by letting a point move with time, and read its velocity, speed and tangent.
Percentages without panic: Percentage of an amount, increases and decreases, percentage change and working back to the whole.
Projectile motion: Split a throw into horizontal and vertical motion, then change gravity and see what changes.
Ratio and proportion: Share an amount in a ratio, simplify a ratio and use the unitary method.
Sequences and series: Spot the rule of an arithmetic, geometric or quadratic sequence, then find the n-th term and the sum.
Sets and Venn diagrams: Union, intersection, difference and complements, with a Venn diagram and a counting check.
Simultaneous equations: Find the pair of numbers that makes two equations true at the same time.
Solving linear equations: Keep an equation balanced and watch x get all on its own, step by step.
Speed, distance, time and work: Link speed, distance and time with one formula, then see how workers share a job.
Triangles: sine rule, cosine rule, SOH CAH TOA: Solve right-angled triangles, then any triangle, and draw it to scale.
Dot and cross products in 3D: Use the dot product to find angles and perpendicular vectors, and the cross product to find areas and normals.
Vectors: dot and cross products: Lengths, the dot product, the angle between vectors and the cross product, by hand.