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Mathematics

From numbers to functions — learn to think logically

Learn the core topics of school mathematics step by step, from natural numbers to quadratic equations, trigonometry and probability. Every lesson has worked examples, interactive models and a quiz, so that you understand the formulas and can apply them instead of just memorising them.

82 lessons11 modules≈ 33.9 hGrades 5–11BeginnerIntermediateAdvancedUniversity
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Natural numbers and the order of operations

Course content

All tests · 17Formulas & shortcuts
1

Numbers and operations

Beginner

Calculate confidently with natural numbers, divisibility, fractions, percentages and negative numbers.

11 lessons
  1. 1Natural numbers and the order of operationsBrackets, powers, multiplication and division, addition and subtraction: learn to evaluate any expression in the right order.
  2. 2Divisibility rules, prime and composite numbersDivisors and multiples, divisibility of a sum and a product, the rules for 2, 3, 4, 5, 9, 10 and 25 and why they work, prime and composite numbers, the sieve of Eratosthenes, and prime factorisation with a factor tree or a division column.
  3. 3Greatest common divisor and least common multipleFinding the GCD by listing divisors, by prime factorisation and with Euclid’s algorithm, coprime numbers, the LCM, the rule GCD · LCM = a · b, three numbers, and word problems about sharing, cutting, meeting again and common denominators.
  4. 4Remainders, digits and last-digit problemsDivision with remainder a = bq + r and finding the dividend, remainders of sums, products and powers, the cycles of last digits of powers, writing a number by its digits and digit problems, divisibility tasks with unknown digits, coprime numbers and rounding — the DİM topic “Natural numbers”.
  5. 5Common fractions: meaning, simplifying and comparingA fraction as part of a whole and as a division, proper and improper fractions, mixed numbers, the basic property of a fraction, simplifying, a common denominator via the LCM, comparing fractions and placing them on the number line.
  6. 6Operations with fractionsAdding and subtracting fractions with like and unlike denominators and mixed numbers, cancelling before multiplying, reciprocals and division, finding a fraction of a number and a number from its fraction, multi-step expressions and word problems.
  7. 7Decimal fractionsPlace value after the decimal point; reading, comparing and rounding decimals; the four operations; converting between common fractions and decimals; and a first look at repeating decimals.
  8. 8PercentagesPercent as a hundredth; converting between fractions, decimals and percentages; the three basic problems; increases and decreases with a multiplier and successive changes; simple and compound interest; mixtures and percentage points.
  9. 9Positive and negative numbers, the number line and absolute valueThe number line, integers and rational numbers, opposite numbers, absolute value as the distance to zero, comparing and ordering signed numbers, and real-life uses: temperature, depth and bank balance.
  10. 10Operations with positive and negative numbersAdding numbers with the same and with different signs (with the number-line model), subtraction as adding the opposite, the sign rules for multiplication and division, powers of negative numbers — (−a)ⁿ versus −aⁿ, the laws and order of operations, and multi-step expressions and word problems with negative fractions and decimals.
  11. 11Sets and operations on setsSets and elements, finite, infinite and empty sets, subsets and their number 2ⁿ, union, intersection and difference on Euler–Venn diagrams, the number of elements in a union of two and three sets, number sets and intervals — with DİM-style closed and coded tasks.
2

Algebra basics

Intermediate

Work with letters, solve equations and systems of equations, and draw graphs of linear functions.

8 lessons
  1. 12Algebraic expressions, monomials and polynomialsVariables and the value of an expression, terms and coefficients, like terms and expanding brackets; monomials in standard form and their degree; adding, subtracting and multiplying polynomials — with step-by-step worked examples.
  2. 13Special products (short multiplication formulas)The square of a sum and of a difference, the difference of two squares, the cube of a sum and of a difference, the sum and difference of cubes — with area and volume pictures; using the formulas in both directions, mental arithmetic, a first look at completing the square, and typical mistakes.
  3. 14Factoring polynomialsFour ways to write a polynomial as a product — taking out the common factor, grouping, the special product formulas and factoring a quadratic trinomial — and what they are good for: simplifying, solving equations of the form ab = 0, proving divisibility and mental arithmetic.
  4. 15Algebraic fractions (rational expressions)Allowed values of the variable, the basic property of a fraction and cancelling by factoring, the common denominator, the four operations with algebraic fractions, long simplification chains, and fractional equations with the check for excluded values.
  5. 16Linear equationsThe root of an equation, the key properties of equations, solving ax + b = 0 step by step and solving word problems with equations.
  6. 17Systems of linear equationsLearn to solve a system of two linear equations in two unknowns by substitution, by elimination and graphically.
  7. 18Word problems with equations and systemsFive steps to build the model of a word problem; motion (towards each other, catching up, on a river, a train through a tunnel), joint work and pipes, mixtures and alloys, percents and bonuses, problems given by a table and digit problems; rejecting roots that do not fit and writing the full solution of a DİM written task.
  8. 19Linear functions and their graphsThe idea of a function, the coordinate plane, the function y = kx + b and how k and b change its graph.
3

Geometry

Intermediate

Angles and triangles, the area and perimeter of shapes, and the Pythagorean theorem.

8 lessons
  1. 20Angles and parallel linesMeasuring and classifying angles, the angle bisector, adjacent and vertical angles, perpendicular lines, the angles formed when a transversal crosses two parallel lines (corresponding, alternate and co-interior), the tests for parallel lines, and clock-hand problems.
  2. 21TrianglesClassifying triangles by angles and by sides, a proof that the angles add up to 180°, the exterior angle theorem, the angles of isosceles and equilateral triangles, the triangle inequality, the larger side opposite the larger angle, and the median, angle bisector, altitude and midline — with many angle-chasing problems.
  3. 22Congruent trianglesThe three criteria that tell when two triangles are congruent (SAS, ASA and SSS), the criteria for right triangles, proofs with congruent triangles (the isosceles triangle, the perpendicular bisector) and the basic compass-and-ruler constructions — with step-by-step solved problems.
  4. 23Quadrilaterals: parallelogram, rectangle, rhombus, square and trapezoidThe angle sum of a polygon (n − 2) · 180°, the properties and tests of the parallelogram, rectangle, rhombus and square, Thales’s theorem, the midline of a triangle and of a trapezoid, and the isosceles trapezoid — with proofs through congruent triangles and many solved problems.
  5. 24Perimeter and area of polygonsThe difference between perimeter and area, units of area and converting them, the area formulas of the rectangle, square, parallelogram, triangle, rhombus and trapezoid — each with a short derivation — composite shapes, and room, floor and tiling problems.
  6. 25Circumference and area of a circleCircle and disc, radius, diameter and chord, the history and approximations of π, the formulas C = 2πr and A = πr² with their derivations, arc length, the area of a sector and of a ring, what happens when the radius is doubled, and wheel, running-track and pizza problems.
  7. 26The Pythagorean theoremThe relationship between the legs and the hypotenuse of a right triangle, square roots and everyday uses of the theorem.
  8. 27Similar trianglesProportional segments and the generalised Thales theorem, the three similarity tests, the ratios of perimeters (k) and of areas (k²), proportional segments in a right triangle (h² = pq, a² = cp), the angle bisector property and measuring heights with shadows — with step-by-step solved problems.
4

Higher-grade topics

Advanced

Quadratic equations, arithmetic and geometric progressions, the basics of trigonometry, probability and statistics.

7 lessons
  1. 28Quadratic equationsQuadratic equations and their coefficients, incomplete equations, completing the square, the discriminant and the quadratic formula, the D/4 shortcut for an even second coefficient, Vieta’s theorem and its converse, factoring a quadratic trinomial, biquadratic equations and word problems — with step-by-step worked examples.
  2. 29The quadratic function and its graphThe function y = ax², shifting the parabola y = a(x − m)² + n, the vertex x₀ = −b/(2a), the axis of symmetry, the intercepts, a step-by-step graphing routine, how a, b and c shape the graph, greatest and least value problems, and finding the formula from points.
  3. 30Sequences and arithmetic progressionsGeneral-term and recursive formulas of a sequence; the common difference, nth term and characteristic property of an arithmetic progression, finding it from two terms, and the sum of the first n terms (Gauss’s trick).
  4. 31Geometric progressionsThe common ratio, nth term, characteristic property and sum of the first n terms of a geometric progression; the sum of an infinite geometric series, turning repeating decimals into fractions, and growth and decay problems.
  5. 32Trigonometry basicsSine, cosine and tangent of an acute angle, the values for special angles, the unit circle and the basic trigonometric identity.
  6. 33Basic statistics: data, charts and averagesCollecting data and frequency tables; bar, pie and line charts and histograms; the mean and the weighted mean, median, mode, range, variance and standard deviation; outliers and misleading graphs.
  7. 34Basics of probabilityRandom, certain and impossible events, classical probability, relative frequency and the law of large numbers, complementary events, the addition and multiplication rules, tree diagrams, counting with combinations and geometric probability.
5

High-school maths in depth

Advanced

Ratio, powers and roots, inequalities, logarithms, trigonometric identities, vectors and combinatorics.

18 lessons
  1. 35Ratio and proportionRatios, the main property of a proportion, direct and inverse proportion, dividing a quantity in a given ratio, scale and percentage problems.
  2. 36Powers with integer exponentsNatural, zero and negative exponents, all the laws of exponents, the sign of a power of a negative number, comparing powers, rewriting with a common base, and standard form.
  3. 37Square roots, nth roots and rational exponentsThe square root, irrational and real numbers, √(a²) = |a|, the rules for roots, moving factors out of and into a root, comparing radicals, nth roots, rational exponents, rationalising denominators and the graph of y = √x.
  4. 38Linear inequalities and systemsProperties of inequalities, intervals and their notation, solving linear inequalities, systems and double inequalities, counting integer solutions, simple inequalities with absolute value, and word problems about tariffs and budgets.
  5. 39Equations and inequalities with absolute valueAbsolute value as a distance, the equations |f(x)| = a, |f(x)| = g(x) (with the condition g ≥ 0) and |f| = |g|, equations with several absolute values by the interval method, the inequalities |f| < g, |f| > g and with two absolute values, and counting the roots of an equation with a parameter from the graphs y = |f(x)| and y = f(|x|) — with DİM’s typical answers: the sum of the roots, the number of integer solutions.
  6. 40Quadratic inequalities and the interval methodSolving quadratic inequalities with a parabola sketch (D > 0, D = 0, D < 0), the interval method for products and quotients, even-power factors, rational inequalities, systems of linear and quadratic inequalities, and the domain of expressions with roots and fractions.
  7. 41Parameter problems: investigating the quadraticWhat a parameter is and the special case a = 0, the number of roots, a quadratic that keeps its sign for all x, signs of the roots with Vieta’s theorem, reciprocal roots, the minimum of x₁² + x₂², both roots greater than a number, the position of the parabola’s vertex and linear systems with a parameter — answers written as intervals, with DİM-style closed and coded tasks.
  8. 42Radical (irrational) equations and inequalitiesEquations with the variable under a root sign: the domain condition, squaring both sides and checking for extraneous roots, solving √f = g with an equivalent system, substitution, equations with two radicals and the simplest radical inequalities.
  9. 43Functions and their propertiesThe definition of a function and ways to give one, domain and range, zeros and sign intervals, increasing and decreasing, even and odd functions, periodicity, graph transformations (shifts, stretches, reflections, |f(x)|), the graphs of y = k/x, √x, |x| and x³, and the inverse function.
  10. 44Exponential functions, equations and inequalitiesThe exponential function y = aˣ, its graph and properties, the number e, growth and decay models (compound interest, doubling, half-life), four methods for exponential equations, and exponential inequalities.
  11. 45Logarithms, their properties and the logarithmic functionThe definition of the logarithm and the basic identity, common (log) and natural (ln) logarithms, the product, quotient and power rules, change of base, the function y = logₐx, comparing logarithms, and applications: pH, decibels, the earthquake scale and doubling time.
  12. 46Logarithmic equations and inequalitiesFinding the domain, the simplest equations, equating arguments, using the laws, the substitution t = logₐx, taking logarithms (equations like x^(log x)), extraneous and lost roots, solving logarithmic inequalities as a system with the domain, and mixed exam-style problems.
  13. 47Radian measure, basic identities and reduction formulasRadian measure, sine and cosine of any angle on the unit circle, signs by quadrant, the basic identities, finding all ratios from one, reduction formulas with the “horse rule”, and simplifying expressions and proving identities.
  14. 48Addition, double-angle and half-angle formulasSine, cosine and tangent of a sum and a difference (with a derivation), double-angle, power-reduction and half-angle formulas, sum-to-product and product-to-sum formulas, a·sin x + b·cos x = R·sin(x + φ), exact values for 15°, 75° and 22.5°, and exam-style simplifications.
  15. 49Trigonometric functions and their graphsThe functions y = sin x, cos x, tan x and cot x: domain, range, period, parity, intervals of increase and decrease, and zeros; sketching their graphs, the transformations y = A·sin(kx + b) + c (amplitude, period, phase shift), and harmonic oscillations.
  16. 50Trigonometric equations and inequalitiesarcsin, arccos and arctan; the solution formulas for sin x = a, cos x = a and tan x = a and their special cases; equations that reduce to quadratics, factoring, homogeneous equations, using identities, selecting roots in an interval, and the simplest trigonometric inequalities on the unit circle.
  17. 51VectorsCoordinates and length of a vector, adding, subtracting and scaling vectors, collinearity, the dot product and the angle between vectors.
  18. 52Combinatorics: counting methodsThe rules of sum and product, factorials, permutations, arrangements and combinations, the binomial theorem and Pascal's triangle.
6

Geometry: going deeper

Advanced

The circle and its elements, the laws of sines and cosines, the coordinate method and geometric transformations.

3 lessons
  1. 53Circle theorems: chords, tangents and inscribed anglesProperties of chords and diameters, why a tangent is perpendicular to the radius, equal tangent segments, central and inscribed angles, products of chord and secant segments, the incircle and circumcircle of a triangle, and cyclic and tangential quadrilaterals — with solved problems.
  2. 54Law of sines and law of cosines: solving trianglesSine and cosine of angles from 0° to 180°, the triangle area S = ½ab · sin C, the law of sines (with 2R), the law of cosines, Heron’s formula and solving a triangle in every case — with surveying and navigation problems.
  3. 55Coordinate geometry: equations of a line and a circleDistance between two points and the midpoint of a segment, the general and slope forms of a line, the line through two points, parallel and perpendicular lines, the distance from a point to a line, the equation of a circle and where a line meets a circle — with worked examples.
7

Solid geometry: figures in space

Advanced

Lines and planes in space, prisms and pyramids, cylinders, cones and spheres: angles, distances, surface areas and volumes — with DİM-style tasks.

3 lessons
  1. 56Lines and planes in spaceThe axioms of solid geometry, how lines and planes can lie, skew lines, the perpendicular, the oblique and its projection, the three-perpendiculars theorem, the angle between a line and a plane, the dihedral angle and distances in space — with DİM-style tasks.
  2. 57Polyhedra: prisms and pyramidsThe elements of a polyhedron and Euler’s theorem, the rectangular box and the cube (grade 9 exam), the lateral and total surface area and the volume of prisms and pyramids, where the foot of the height lies, sections, the frustum and similar solids — with DİM-style tasks.
  3. 58Solids of revolution: cylinder, cone and sphereCylinder, cone, frustum of a cone and sphere: their elements, axial and plane sections, lateral and total surface area and volume; the spherical segment and sector, inscribed and circumscribed solids, liquid-level problems — with DİM-style tasks.
8

The derivative and its applications

Advanced

The definition and meaning of the derivative, tangent lines, the table of derivatives, differentiation rules, the chain rule, investigating functions and optimisation — one topic per lesson.

8 lessons
  1. 59Limits of sequences and functionsThe limit of a sequence, monotone bounded sequences and the number e, limits of functions at a point and at infinity, resolving 0/0 and ∞/∞, the standard trigonometric limits and continuity problems with a parameter — in the style of the DİM entrance exam.
  2. 60Increments and the definition of the derivativeIncrements of the argument and of the function, the average rate of change and the secant, the idea of a limit as Δx → 0, the definition and notations of the derivative, a 3-step method for finding derivatives from the definition, and how differentiability is related to continuity.
  3. 61The geometric and physical meaning of the derivative. The tangent lineThe derivative as the slope of the tangent and as a rate of change: the tangent-line equation, horizontal and parallel tangents, tangents perpendicular to a line or at a given angle, the normal line, velocity, acceleration, electric current and marginal cost — with worked examples.
  4. 62The table of derivativesAll 16 lines of the table of derivatives — powers and roots, eˣ and aˣ, ln x and logₐ x, sin, cos, tan, cot: where each formula comes from, when it holds, and many worked values.
  5. 63Derivatives of sums, products and quotientsThe derivatives of sums, differences, constant multiples, products and quotients: where the rules come from, how to apply them and when it pays to simplify first — with many worked examples.
  6. 64The derivative of a composite function (the chain rule)How to recognise the inner and outer functions, the chain rule (f(g(x)))′ = f′(g(x)) · g′(x), the special case f(kx + b), composite versions of the basic formulas, functions with several layers, and the chain rule together with the product and quotient rules — with step-by-step worked examples.
  7. 65Increasing and decreasing functions. ExtremaUse the sign of the derivative to find where a function increases and decreases, its critical points and its maximum and minimum points: Fermat’s theorem, the sign chart, the second-derivative test and sketching a graph with the derivative — with many worked examples.
  8. 66Greatest and least values on a closed interval. Optimisation problemsHow an extremum differs from the greatest or least value, the Weierstrass theorem, the “critical points inside + endpoints” algorithm and optimisation problems on area, box volume, cans, distance and profit, solved step by step.
9

Antiderivatives and the integral

Advanced

Antiderivatives and the indefinite integral, the Newton–Leibniz formula, the area of a curvilinear trapezoid and of figures — with DİM entrance tasks.

2 lessons
  1. 67The antiderivativeThe antiderivative and the indefinite integral, the general form F(x) + C, the antiderivative whose graph passes through a given point, the table of antiderivatives and the three rules of integration, and finding the law of motion from the velocity — with DİM-style tasks.
  2. 68The integral and the area of a curvilinear trapezoidThe curvilinear trapezoid and its area, the definite integral and the Newton–Leibniz formula, properties of the integral and problems with a parameter, areas of figures bounded by curves, distance from velocity and the work of a variable force — with DİM-style closed, coded and written tasks.
10

Calculus

University

Limits, derivatives and their applications, integrals, series and differential equations.

7 lessons
  1. 69Limits and continuityThe intuitive and ε–δ definitions of a limit, one-sided limits, limit laws, the remarkable limits, indeterminate forms, continuity, types of discontinuity and the intermediate value theorem.
  2. 70Derivatives and the rules of differentiationThe derivative as a limit, its geometric and physical meaning, the table of derivatives, the sum, product, quotient and chain rules, implicit differentiation and higher derivatives.
  3. 71Applications of derivativesThe tangent line, linear and Taylor approximation, monotonicity, extrema, convexity and inflection points, optimisation problems, L'Hôpital's rule and related rates.
  4. 72Integrals: antiderivatives and the definite integralAntiderivatives, the table of indefinite integrals, Riemann sums, the definite integral as an area, the Fundamental Theorem of Calculus, properties of integrals, the area between curves and the average value of a function.
  5. 73Integration techniquesSubstitution, integration by parts and the LIATE rule, partial fractions, trigonometric integrals, improper integrals, and applications: volumes of revolution and work.
  6. 74Sequences and seriesLimits of sequences, series and partial sums, the geometric and harmonic series, convergence tests (divergence, comparison, ratio, integral), power series and the Taylor and Maclaurin series of eˣ, sin x, cos x and ln(1 + x).
  7. 75Differential equationsOrdinary differential equations and their order, separable equations, first-order linear equations and the integrating factor, exponential growth, decay and cooling, second-order equations with constant coefficients and the harmonic oscillator.
11

Linear algebra, probability and discrete maths

University

Matrices, determinants, eigenvalues, complex numbers, random variables, statistical inference and logic.

7 lessons
  1. 76Matrices and matrix operationsMatrix notation, addition, multiplication by a number and by another matrix, the identity and the transpose; matrices as transformations of the plane; systems written as Ax = b, Gaussian elimination and the rank of a matrix.
  2. 77Determinants and inverse matrices2 × 2 and 3 × 3 determinants by Sarrus' rule and cofactor expansion, their geometric meaning and properties; the inverse matrix via the adjugate and via Gauss–Jordan, Cramer's rule and when A⁻¹ exists.
  3. 78Eigenvalues and eigenvectorsThe definition Av = λv, the characteristic equation det(A − λI) = 0, worked 2 × 2 examples, diagonalisation A = PDP⁻¹ and the ideas behind Markov chains, principal component analysis and Google's PageRank.
  4. 79Complex numbersThe imaginary unit i² = −1, the algebraic form and operations, the conjugate, modulus and argument, trigonometric and exponential forms, Euler's formula, De Moivre's formula, roots of unity and quadratics with a negative discriminant.
  5. 80Random variables and distributionsDiscrete and continuous random variables, expectation and variance, the binomial, Poisson, uniform and normal distributions, the 68–95–99.7 rule and z-scores, the law of large numbers and the central limit theorem.
  6. 81Statistical inference: estimation and hypothesis testingFrom a sample to the population: estimators, confidence intervals for the mean, hypothesis testing with p-values, significance levels and type I/II errors, the t-test, correlation and least-squares linear regression.
  7. 82Logic, sets and graphsPropositional logic and truth tables, quantifiers, proof methods (direct, contradiction, induction), set operations and Venn diagrams, relations and functions (injective, surjective, bijective) and graph basics: degrees, paths, trees, Euler and Hamilton paths.

School subjects

Primary MathsGrades 1–4: counting, adding and subtracting, times tables, shapesLife KnowledgeGrades 1–4: nature, health, safety, family and societyPhysicsMotion, energy, light and the laws of the UniverseChemistryThe world of atoms, substances and reactions