1 min czytania
Fractions: simplifying, adding, multiplying and dividing
What a fraction means, how to simplify one, and the four operations — with a common denominator only when you need it.
Centrum nauki: proste lekcje matematyki
Proste wyjaśnienia tematów, o które uczniowie pytają najczęściej. Każda lekcja zaczyna od idei, rozwiązuje przykład krok po kroku i sprawdza wynik silnikiem matematycznym Sumzo — a potem prowadzi do kalkulatora i ćwiczeń.
Arytmetyka i liczby
1 min czytania
Prime factors, GCD and LCM
Break numbers into primes, then use the prime factorisation to find the greatest common divisor and least common multiple.
2 min czytania
Square roots explained: perfect squares, surds and estimating
What a square root is, the perfect squares to know, how to simplify surds such as √180, and how to estimate roots like √65 without a calculator.
Algebra
5 min czytania
Equations: what they are and how to solve them
What an equation is, what its solution means, and how to solve equations step by step — from one-step equations to fractions and two variables.
1 min czytania
Solving linear equations
Use inverse operations to get the unknown on its own, keep the equation balanced, and always check your answer.
2 min czytania
Quadratic equations: how to solve them
Factorising, completing the square, the quadratic formula and the discriminant — and how to tell which method to use.
1 min czytania
Slope and the equation of a line
What slope measures, how to find it from two points, and how to write a line as y = mx + b.
4 min czytania
How to simplify radicals (simplest radical form)
How to simplify radicals by taking out perfect-square factors, what simplest radical form means, and how to simplify radical expressions — with checked examples.
2 min czytania
Rational equations: how to solve them
What a rational equation is, how to solve rational equations by multiplying by the lowest common denominator, and why every answer must be checked.
3 min czytania
Factoring polynomials: a step-by-step guide
The steps for factoring polynomials — common factor first, then two, three or four terms — with worked examples, each checked by multiplying back out.
3 min czytania
Solving inequalities: how to solve and graph them
Solving inequalities like equations — with one extra rule — and showing the answer on a number line, in interval notation and as a shaded region.
2 min czytania
What is a solution in math? Examples with one, none or many
A solution is a value that makes an equation true. How to check one, and examples with one solution, no solution, all real numbers and two solutions.
2 min czytania
How to solve a word problem: 5 steps that work
A five-step method for solving word problems, the key words that turn into maths, and worked examples — ages, consecutive numbers, discounts and tickets.
2 min czytania
How to graph an equation: lines, parabolas and more
Three ways to graph an equation — a table of values, slope and intercept, and key points of a parabola — with worked examples and how to match a graph to its equation.
3 min czytania
Functions: how to graph a function, domain, range and f(g(x))
Function notation, how to graph a function from its formula, how to find its domain and range, and how composite functions such as f(g(x)) work.
2 min czytania
Polynomials: how to simplify, expand and solve them
What a polynomial is, how to simplify and expand polynomials, and how to solve polynomial equations by factoring — with checked examples.
2 min czytania
Rational expressions: simplify, multiply, divide, add and subtract
What rational expressions are, their excluded values, and how to simplify, multiply, divide, add and subtract them — then solve rational equations.
2 min czytania
How to solve literal equations (rearranging formulas)
A literal equation has several letters. Solving it means isolating one of them — the same moves as an ordinary equation, with letters instead of numbers.
1 min czytania
How to simplify expressions and complex fractions
How to simplify the following kinds of expression: like terms, brackets, exponents and complex fractions (a fraction inside a fraction), each with checked examples.
1 min czytania
Find x: how to solve for x in equations, angles and triangles
The questions that say “find x” — equations, angles on a line, triangle angles, Pythagoras and similar shapes — and the one idea behind all of them.
2 min czytania
Systems of equations and graphing systems of linear inequalities
How to find the solution to a system of equations three ways, then how to graph a system of inequalities: boundary lines, shading and the overlap.
Geometria
1 min czytania
The Pythagorean theorem
How a² + b² = c² finds a missing side of any right triangle, plus the triples worth remembering.
1 min czytania
Area of common shapes and circles
Area formulas for rectangles, triangles, trapeziums and circles — and why the triangle formula has a half in it.
2 min czytania
Area of a trapezium (trapezoid): formula and examples
The formula for the area of a trapezium (a trapezoid in the US), why it works, and worked examples — including finding a missing height or side.
2 min czytania
Types of triangles: scalene, isosceles and equilateral
Scalene, isosceles and equilateral triangles — how to tell them apart, their angles, and how to find their area, including the area of an equilateral triangle.
1 min czytania
Polygons and quadrilaterals: is a square a quadrilateral?
Is a square a quadrilateral? Yes — and a rectangle, a rhombus and a parallelogram. How the family fits together, plus polygon angle sums and areas.
1 min czytania
Triangular prism volume, cylinders, cones and other 3D shapes
How to find the volume and surface area of a triangular prism, and the volume of cylinders and cones — one idea (area of the cross-section × length) behind most of them.
Trygonometria
1 min czytania
Trigonometry basics: SOH CAH TOA and exact values
The three trig ratios, the two special triangles that give exact values, and how degrees relate to radians.
Statystyka
1 min czytania
Averages and spread: mean, median, mode, range and standard deviation
Three ways to describe a typical value, two ways to describe how spread out data is, and when to use each.
2 min czytania
Statistics basics: z-scores, percentiles, quartiles and combinations
Three statistics skills that come up again and again: turning a z-score into a percentile, finding quartiles in a data set, and counting with the combination formula.
Analiza matematyczna
1 min czytania
Derivatives and the power rule
A derivative is the slope of a curve at a point. The power rule lets you differentiate any polynomial in one line.
Algebra liniowa
2 min czytania
How to find the determinant of a 2×2 or 3×3 matrix
How to find the determinant of a matrix: the ad − bc rule for 2×2, cofactor expansion for 3×3, a shortcut for triangular matrices, and what a zero determinant tells you.
Matematyka na co dzień
2 min czytania
How to calculate percentages: the three types of problem
How to solve percentage problems of all three kinds — a percentage of an amount, one number as a percentage of another, and percentage change — with checked examples.
1 min czytania
Ratios and proportions
Simplify ratios, share amounts in a ratio, and solve proportions — the maths behind recipes, maps and scale models.
Jak uczyć się matematyki z tymi lekcjami
- Wybierz temat poniżej albo go wyszukaj.
- Przeczytaj krótkie wyjaśnienie, potem prześledź rozwiązane przykłady linijka po linijce — spróbuj każdego kroku, zanim go przeczytasz.
- Przejrzyj typowe błędy, żeby wiedzieć, na co uważać.
- Ćwicz na podlinkowanych pytaniach. Zaloguj się, jeśli chcesz zapisywać postępy.
Jak wygląda prosty przykład z matematyki
Rozwiąż .
- Odejmij 4 od obu stron: .
- Podziel obie strony przez 3: .
- Sprawdzenie: . ✓
Każda lekcja działa tak samo: jeden mały krok naraz i sprawdzony wynik.
Najczęstsze pytania
Czy jest łatwy sposób na naukę trudnych tematów?
Podziel każdy temat na małe kroki, które już znasz. Lekcje robią to za ciebie — równania kwadratowe zaczynają się np. od rozkładu na czynniki, zanim pojawi się delta.
Jak sprawdzane są rozwiązane przykłady?
Przykłady są ponownie rozwiązywane przez silnik matematyczny lub kalkulatory Sumzo w testach automatycznych, więc błędny wynik zostaje wykryty, zanim lekcja trafi do sieci.
Czy potrzebuję konta?
Nie. Wszystkie lekcje są darmowe. Konto zapisuje tylko postępy w ćwiczeniach.