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Curriculum (Poland)

Mathematics — Grade 8

Grade 8 mathematics: systems of equations, prisms and pyramids, special right triangles, probability in two-stage experiments — and a review of the entire primary school curriculum before the eighth-grade exam. Ready-to-print worksheets for every topic.

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What does a student learn in 8th grade mathematics?

In Grade 8, students solve systems of two linear equations, calculate the volume and surface area of prisms and pyramids, use the properties of 45°, 30°, and 60° triangles, and calculate probability in two-stage experiments. Grade 8 concludes primary school, so it revisits powers, percentages, algebraic expressions, and the Pythagorean theorem — material covered by the eighth-grade exam.

Important note about the curriculum: the new core curriculum (Journal of Laws 2026, item 378) will cover Grade 8 starting from the 2030/2031 school year — it is phased in year by year. Until then, Grade 8 follows the 2017 curriculum; the scope of topics on this page is similar in both curricula. The curriculum for Grades VII–VIII is block-based — it defines what a student should know by the end of primary school. Assigning a topic to a specific grade is our editorial choice based on a typical teaching schedule; in the "Curriculum Scope" section, you can see the entire VII–VIII block with quotations from the regulation.

Page status: powers and roots, practical percentages, proportionality, signed rational numbers, algebraic expressions, equations and systems of equations, the Pythagorean theorem, circles, prisms and pyramids, and probability are ready. Geometric constructions and exam-style practice questions are being added successively.

Curriculum scope

  1. Curriculum point MAT.VII-VIII.1 · Teaching content · checked against the act

    Numbers

    our translation · original wording (PL): Liczby

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, treści nauczania kl. VII i VIII, dział 1

  2. Curriculum point MAT.VII-VIII.1.1 · Teaching content · checked against the act

    carries out calculations and reasoning in which they use percentage notation, including calculating:

    our translation · original wording (PL): prowadzi obliczenia i rozumowania, w których korzysta z zapisu procentowego, w tym oblicza:

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 1

  3. Curriculum point MAT.VII-VIII.1.2 · Teaching content · checked against the act

    applies percentage calculations to solve problems in a practical context, including financial, such as calculating: prices after an increase or before their increase or decrease, interest on an annual deposit, loan costs, tax amounts – economic and financial module

    our translation · original wording (PL): stosuje obliczenia procentowe do rozwiązywania problemów w kontekście praktycznym, w tym finansowym, takie jak obliczanie: cen po podwyżce lub przed jej podwyżką albo obniżką, odsetek dla rocznej lokaty, kosztów kredytu, wysokości podatków – moduł ekonomiczno-finansowy

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 2

    Calculating percentage increases and decreases (we teach in grade 7-8) · Calculating interest on an annual deposit (we teach in grade 7-8)

  4. Curriculum point MAT.VII-VIII.1.3 · Teaching content · checked against the act

    calculates powers of rational numbers with natural exponents

    our translation · original wording (PL): oblicza potęgi liczb wymiernych o wykładnikach naturalnych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 3

    Calculating powers of integers and fractions (we teach in grade 7-8)

  5. Curriculum point MAT.VII-VIII.1.4 · Teaching content · checked against the act

    converts powers with a negative integer exponent into corresponding powers with natural exponents

    our translation · original wording (PL): zamienia potęgi o wykładniku całkowitym ujemnym na odpowiednie potęgi o wykładnikach naturalnych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 4

    Powers with a negative exponent (we teach in grade 7-8)

  6. Curriculum point MAT.VII-VIII.1.5 · Teaching content · checked against the act

    applies the properties of operations on powers with natural bases and integer exponents, including:

    our translation · original wording (PL): stosuje własności działań na potęgach o podstawach naturalnych i wykładnikach całkowitych, w tym:

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 5

    Operations on powers and the missing exponent (we teach in grade 7-8)

  7. Curriculum point MAT.VII-VIII.1.6 · Teaching content · checked against the act

    writes numbers in scientific notation and compares numbers written in this way

    our translation · original wording (PL): zapisuje liczby w notacji wykładniczej i porównuje liczby zapisane w ten sposób

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 6

    Writing and comparing numbers in scientific notation (we teach in grade 7-8)

  8. Curriculum point MAT.VII-VIII.1.7 · Teaching content · checked against the act

    calculates the values of square and cube roots of numbers that are, respectively, squares or cubes of rational numbers, and calculates the approximate values of roots of other numbers using a calculator

    our translation · original wording (PL): oblicza wartości pierwiastków kwadratowych i sześciennych z liczb, które są odpowiednio kwadratami lub sześcianami liczb wymiernych, a przybliżone wartości pierwiastków z innych liczb oblicza za pomocą kalkulatora

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 7

    Calculating square and cube roots (we teach in grade 7-8)

  9. Curriculum point MAT.VII-VIII.1.8 · Teaching content · checked against the act

    estimates the value of a simple arithmetic expression containing roots

    our translation · original wording (PL): szacuje wartość prostego wyrażenia arytmetycznego zawierającego pierwiastki

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 8

    Estimating the value of roots (we teach in grade 7-8)

  10. Curriculum point MAT.VII-VIII.1.9 · Teaching content · checked against the act

    applies the properties of operations on roots for square and cube roots, including:

    our translation · original wording (PL): stosuje własności działań na pierwiastkach dla pierwiastków kwadratowych i sześciennych, w tym:

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 9

    Taking a factor out of a radical (we teach in grade 7-8)

  11. Curriculum point MAT.VII-VIII.1.10 · Teaching content · checked against the act

    recognizes directly proportional relationships, applies them to determine the value of one quantity based on another, and uses proportional division into two or three parts in a practical context

    our translation · original wording (PL): rozpoznaje zależności wprost proporcjonalne, stosuje je do wyznaczania wartości jednej wielkości na podstawie drugiej oraz wykorzystuje podział proporcjonalny na dwie lub trzy części w kontekście praktycznym

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 10

    Dividing a quantity in a given ratio (we teach in grade 7-8)

  12. Curriculum point MAT.VII-VIII.1.11 · Teaching content · checked against the act

    applies the multiplication rule to count pairs of elements in situations requiring two choices

    our translation · original wording (PL): stosuje regułę mnożenia do zliczania par elementów w sytuacjach wymagających dwukrotnego wyboru

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 11

  13. Curriculum point MAT.VII-VIII.1.12 · Teaching content · checked against the act

    uses a calculator and computer applications to perform calculations when solving problems set in a practical context.

    our translation · original wording (PL): używa kalkulatora i aplikacji komputerowych do wykonywania obliczeń przy rozwiązywaniu problemów osadzonych w kontekście praktycznym.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 12

  14. Curriculum point MAT.VII-VIII.1.1.a · Teaching content · checked against the act

    what percentage of one number another number is

    our translation · original wording (PL): jakim procentem jednej liczby jest druga liczba

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 1 lit. a

    Calculating what percentage one number is of another (we teach in grade 7-8)

  15. Curriculum point MAT.VII-VIII.1.5.a · Teaching content · checked against the act

    multiplies and divides powers with the same base

    our translation · original wording (PL): mnoży i dzieli potęgi o takich samych podstawach

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 5 lit. a

    Operations on powers and the missing exponent (we teach in grade 7-8)

  16. Curriculum point MAT.VII-VIII.1.9.a · Teaching content · checked against the act

    calculates the roots of the product and quotient of two numbers

    our translation · original wording (PL): oblicza pierwiastki z iloczynu i ilorazu dwóch liczb

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 9 lit. a

    Taking a factor out of a radical (we teach in grade 7-8)

  17. Curriculum point MAT.VII-VIII.1.1.b · Teaching content · checked against the act

    a given percentage of a given number

    our translation · original wording (PL): podany procent danej liczby

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 1 lit. b

  18. Curriculum point MAT.VII-VIII.1.5.b · Teaching content · checked against the act

    multiplies and divides powers with the same exponents

    our translation · original wording (PL): mnoży i dzieli potęgi o takich samych wykładnikach

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 5 lit. b

    Operations on powers and the missing exponent (we teach in grade 7-8)

  19. Curriculum point MAT.VII-VIII.1.9.b · Teaching content · checked against the act

    multiplies and divides roots of the same degree

    our translation · original wording (PL): mnoży i dzieli pierwiastki tego samego stopnia

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 9 lit. b

    Taking a factor out of a radical (we teach in grade 7-8)

  20. Curriculum point MAT.VII-VIII.1.1.c · Teaching content · checked against the act

    a number based on a given percentage of it

    our translation · original wording (PL): liczbę na podstawie danego jej procentu

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 1 lit. c

    Finding a number from its percentage (we teach in grade 7-8)

  21. Curriculum point MAT.VII-VIII.1.5.c · Teaching content · checked against the act

    raises a power to a power

    our translation · original wording (PL): podnosi potęgę do potęgi

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 5 lit. c

    Operations on powers and the missing exponent (we teach in grade 7-8)

  22. Curriculum point MAT.VII-VIII.1.9.c · Teaching content · checked against the act

    takes a number out of the radical sign and moves a number under the radical sign

    our translation · original wording (PL): wyłącza liczbę przed znak pierwiastka i włącza liczbę pod znak pierwiastka

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 1 pkt 9 lit. c

    Taking a factor out of a radical (we teach in grade 7-8)

  23. Curriculum point MAT.VII-VIII.2 · Teaching content · checked against the act

    Algebra

    our translation · original wording (PL): Algebra

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, treści nauczania kl. VII i VIII, dział 2

  24. Curriculum point MAT.VII-VIII.2.1 · Teaching content · checked against the act

    creates algebraic expressions, also expressions in two variables, describing relationships between quantities presented in a problem, and uses them to solve problems, including those set in a practical context

    our translation · original wording (PL): tworzy wyrażenia algebraiczne, także wyrażenia dwóch zmiennych, opisujące związki między wielkościami przedstawionymi w zadaniu oraz używa ich do rozwiązywania zadań, w tym osadzonych w kontekście praktycznym

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 1

  25. Curriculum point MAT.VII-VIII.2.2 · Teaching content · checked against the act

    calculates the numerical value of an algebraic expression that is a sum of at most three monomials for given integer values of variables

    our translation · original wording (PL): oblicza wartość liczbową wyrażenia algebraicznego będącego sumą co najwyżej trzech jednomianów dla podanych całkowitych wartości zmiennych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 2

    Calculating the value of an algebraic expression (we teach in grade 7-8)

  26. Curriculum point MAT.VII-VIII.2.3 · Teaching content · checked against the act

    simplifies monomials and combines like terms in algebraic sums

    our translation · original wording (PL): porządkuje jednomiany i redukuje wyrazy podobne w sumach algebraicznych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 3

    Ordering and combining like terms (we teach in grade 7-8)

  27. Curriculum point MAT.VII-VIII.2.4 · Teaching content · checked against the act

    adds and subtracts algebraic sums consisting of several monomials with integer coefficients

    our translation · original wording (PL): dodaje i odejmuje sumy algebraiczne składające się z kilku jednomianów o współczynnikach całkowitych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 4

    Ordering and combining like terms (we teach in grade 7-8)

  28. Curriculum point MAT.VII-VIII.2.5 · Teaching content · checked against the act

    multiplies a monomial by an algebraic sum of at most three monomials with integer coefficients

    our translation · original wording (PL): mnoży jednomian przez sumę algebraiczną co najwyżej trzech jednomianów o współczynnikach całkowitych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 5

    Multiplying through parentheses and factoring out (we teach in grade 7-8)

  29. Curriculum point MAT.VII-VIII.2.6 · Teaching content · checked against the act

    factors out a monomial from an algebraic sum consisting of two monomials with integer coefficients

    our translation · original wording (PL): wyłącza poza nawias jednomian z sumy algebraicznej złożonej z dwóch jednomianów o współczynnikach całkowitych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 6

    Multiplying through parentheses and factoring out (we teach in grade 7-8)

  30. Curriculum point MAT.VII-VIII.2.7 · Teaching content · checked against the act

    multiplies two binomials with integer coefficients

    our translation · original wording (PL): mnoży przez siebie dwa dwumiany o współczynnikach całkowitych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 7

    Multiplying two binomials (we teach in grade 7-8)

  31. Curriculum point MAT.VII-VIII.2.8 · Teaching content · checked against the act

    rearranges simple formulae to determine a given quantity, including in geometric and physical formulae

    our translation · original wording (PL): przekształca proste wzory, aby wyznaczyć zadaną wielkość, w tym we wzorach geometrycznych i fizycznych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 8

  32. Curriculum point MAT.VII-VIII.2.9 · Teaching content · checked against the act

    checks whether a given integer is a solution to an equation (of the first, second, or third degree) with one unknown

    our translation · original wording (PL): sprawdza, czy dana liczba całkowita jest rozwiązaniem równania (stopnia pierwszego, drugiego lub trzeciego) z jedną niewiadomą

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 9

    Checking if a number is a solution to an equation (we teach in grade 7-8)

  33. Curriculum point MAT.VII-VIII.2.10 · Teaching content · checked against the act

    checks whether a given rational number is a solution to a first-degree equation

    our translation · original wording (PL): sprawdza, czy dana liczba wymierna jest rozwiązaniem równania stopnia pierwszego

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 10

    Checking if a number is a solution to an equation (we teach in grade 7-8)

  34. Curriculum point MAT.VII-VIII.2.11 · Teaching content · checked against the act

    solves first-degree equations, as well as equations that can be easily reduced to a first-degree equation

    our translation · original wording (PL): rozwiązuje równania pierwszego stopnia, a także równania, które dają się w prosty sposób sprowadzić do równania pierwszego stopnia

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 11

    Solving two-step equations (ax + b = c) (we teach in grade 7-8) · Solving equations with the unknown on both sides (we teach in grade 7-8)

  35. Curriculum point MAT.VII-VIII.2.12 · Teaching content · checked against the act

    solves word problems using linear equations with one unknown or two linear equations with two unknowns that can be easily reduced to one linear equation.

    our translation · original wording (PL): rozwiązuje zadania tekstowe za pomocą równań pierwszego stopnia z jedną niewiadomą lub dwóch równań pierwszego stopnia z dwiema niewiadomymi, które dają się w prosty sposób sprowadzić do jednego równania pierwszego stopnia.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 2 pkt 12

    Solving systems of two equations (we teach in grade 8)

  36. Curriculum point MAT.VII-VIII.3 · Teaching content · checked against the act

    Shapes

    our translation · original wording (PL): Figury

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, treści nauczania kl. VII i VIII, dział 3

  37. Curriculum point MAT.VII-VIII.3.1 · Teaching content · checked against the act

    recognises in a drawing and draws the perpendicular bisector of a line segment and the bisector of an angle, uses their properties

    our translation · original wording (PL): rozpoznaje na rysunku i rysuje symetralną odcinka oraz dwusieczną kąta, posługuje się ich własnościami

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 1

  38. Curriculum point MAT.VII-VIII.3.2 · Teaching content · checked against the act

    recognises and draws axially symmetric and centrally symmetric figures, identifies their axes of symmetry and centre of symmetry

    our translation · original wording (PL): rozpoznaje i rysuje figury osiowosymetryczne i środkowosymetryczne, wskazuje ich osie symetrii i środek symetrii

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 2

  39. Curriculum point MAT.VII-VIII.3.3 · Teaching content · checked against the act

    calculates the measure of an interior angle of a regular polygon

    our translation · original wording (PL): oblicza miarę kąta wewnętrznego wielokąta foremnego

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 3

  40. Curriculum point MAT.VII-VIII.3.4 · Teaching content · checked against the act

    recognises right-angled triangles in various geometric situations, applies the Pythagorean theorem

    our translation · original wording (PL): dostrzega trójkąty prostokątne w różnych sytuacjach geometrycznych, stosuje twierdzenie Pitagorasa

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 4

    Applying the Pythagorean Theorem (we teach in grade 7-8)

  41. Curriculum point MAT.VII-VIII.3.5 · Teaching content · checked against the act

    uses the properties of special right triangles, that is, right triangles with acute angles of 30° and 60° and isosceles right triangles

    our translation · original wording (PL): korzysta z własności trójkątów prostokątnych szczególnych, to jest trójkątów prostokątnych o kątach ostrych 30° i 60° oraz trójkątów prostokątnych równoramiennych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 5

    Right triangles with 30°, 60°, and 45° angles (we teach in grade 8)

  42. Curriculum point MAT.VII-VIII.3.6 · Teaching content · checked against the act

    applies triangle congruence criteria

    our translation · original wording (PL): stosuje cechy przystawania trójkątów

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 6

  43. Curriculum point MAT.VII-VIII.3.7 · Teaching content · checked against the act

    calculates the circumference of a circle and the area of a circle

    our translation · original wording (PL): oblicza długość okręgu i pole koła

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 7

    Calculating the circumference and area of a circle (we teach in grade 7-8)

  44. Curriculum point MAT.VII-VIII.3.8 · Teaching content · checked against the act

    divides polygons into parts or recognizes figures as parts of another figure in order to calculate, compare, or estimate areas and to identify other relationships

    our translation · original wording (PL): dzieli wielokąty na części lub dostrzega figury jako części innej figury w celu obliczania, porównywania lub szacowania pól oraz dostrzegania innych zależności

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 8

  45. Curriculum point MAT.VII-VIII.3.9 · Teaching content · checked against the act

    solves problems concerning real-world objects that can be represented using a geometric model, also in a situation requiring necessary simplifications

    our translation · original wording (PL): rozwiązuje problemy dotyczące rzeczywistych obiektów, które dają się przedstawić za pomocą modelu geometrycznego, również w sytuacji wymagającej koniecznych uproszczeń

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 9

    Applying the Pythagorean Theorem (we teach in grade 7-8) · Surface area and volume of a right prism (we teach in grade 8) · Calculating the circumference and area of a circle (we teach in grade 7-8)

  46. Curriculum point MAT.VII-VIII.3.10 · Teaching content · checked against the act

    draws polygons in a coordinate system with vertices at lattice points

    our translation · original wording (PL): rysuje w układzie współrzędnych wielokąty o wierzchołkach w punktach kratowych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 10

  47. Curriculum point MAT.VII-VIII.3.11 · Teaching content · checked against the act

    calculates the areas and perimeters of triangles and quadrilaterals with vertices at lattice points

    our translation · original wording (PL): oblicza pola i obwody trójkątów i czworokątów o wierzchołkach w punktach kratowych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 11

  48. Curriculum point MAT.VII-VIII.3.12 · Teaching content · checked against the act

    identifies right and regular prisms and regular pyramids

    our translation · original wording (PL): rozpoznaje graniastosłupy proste i prawidłowe oraz ostrosłupy prawidłowe

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 12

  49. Curriculum point MAT.VII-VIII.3.13 · Teaching content · checked against the act

    indicates the height of the lateral faces and the height of a pyramid

    our translation · original wording (PL): wskazuje wysokość ścian bocznych i wysokość ostrosłupa

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 13

  50. Curriculum point MAT.VII-VIII.3.14 · Teaching content · checked against the act

    calculates the surface areas and volumes of right prisms and regular pyramids.

    our translation · original wording (PL): oblicza pola powierzchni i objętości graniastosłupów prostych oraz ostrosłupów prawidłowych.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 3 pkt 14

    Surface area and volume of a right prism (we teach in grade 8) · Surface area and volume of a regular pyramid (we teach in grade 8)

  51. Curriculum point MAT.VII-VIII.4 · Teaching content · checked against the act

    Data and random events

    our translation · original wording (PL): Dane i zdarzenia losowe

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, treści nauczania kl. VII i VIII, dział 4

  52. Curriculum point MAT.VII-VIII.4.1 · Teaching content · checked against the act

    obtains and selects data from various sources, organizes them and evaluates their reliability

    our translation · original wording (PL): pozyskuje i selekcjonuje dane z różnych źródeł, porządkuje je i ocenia ich wiarygodność

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 4 pkt 1

  53. Curriculum point MAT.VII-VIII.4.2 · Teaching content · checked against the act

    selects a method of presenting data that facilitates their analysis, using appropriate computer applications

    our translation · original wording (PL): dobiera sposób przedstawienia danych ułatwiający ich analizę, posługując się odpowiednimi aplikacjami komputerowymi

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 4 pkt 2

  54. Curriculum point MAT.VII-VIII.4.3 · Teaching content · checked against the act

    interprets the relationship between two quantities described by a graph, for example the change of a certain quantity over time, including: distance travelled, salary levels, cost of living, inflation – economic and financial module

    our translation · original wording (PL): interpretuje zależność między dwiema wielkościami opisaną za pomocą wykresu, przykładowo zmianę pewnej wielkości w czasie, w tym: przebyta droga, wysokość wynagrodzeń, koszty życia, inflacja – moduł ekonomiczno-finansowy

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 4 pkt 3

  55. Curriculum point MAT.VII-VIII.4.4 · Teaching content · checked against the act

    uses the arithmetic mean and the median in a practical context to compare data sets, including financial data – economic and financial module

    our translation · original wording (PL): stosuje średnią arytmetyczną i medianę w kontekście praktycznym do porównania zestawów danych, w tym danych finansowych – moduł ekonomiczno-finansowy

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 4 pkt 4

    Determining the median of a dataset (we teach in grade 7-8)

  56. Curriculum point MAT.VII-VIII.4.5 · Teaching content · checked against the act

    critically interprets data, including financial data, draws conclusions from them and justifies them, referring to the properties of the data set

    our translation · original wording (PL): krytycznie interpretuje dane, w tym dane finansowe, wyciąga z nich wnioski i uzasadnia je, odwołując się do własności zestawu danych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 4 pkt 5

  57. Curriculum point MAT.VII-VIII.4.6 · Teaching content · checked against the act

    conducts simple random experiments consisting, for example, of tossing a coin or rolling a die once or twice and drawing a ball from a set of balls, analyses random experiments and determines the probability of the considered events.

    our translation · original wording (PL): przeprowadza proste doświadczenia losowe polegające przykładowo na jednokrotnym lub dwukrotnym rzucie monetą lub kostką do gry oraz na losowaniu kuli spośród zestawu kul, analizuje doświadczenia losowe i określa, jakie jest prawdopodobieństwo rozważanych zdarzeń.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 4 pkt 6

    Calculating probability in simple situations (we teach in grade 7-8) · Probability in Two-Stage Experiments (we teach in grade 8)

  58. Curriculum point MAT.VII-VIII.5 · Teaching content · checked against the act

    Mathematical thinking

    our translation · original wording (PL): Myślenie matematyczne

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, treści nauczania kl. VII i VIII, dział 5

  59. Curriculum point MAT.VII-VIII.5.1 · Teaching content · checked against the act

    applies general methods of solving mathematical problems, including:

    our translation · original wording (PL): stosuje ogólne metody rozwiązywania problemów matematycznych, w tym:

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 1

  60. Curriculum point MAT.VII-VIII.5.2 · Teaching content · checked against the act

    explains their way of thinking when solving a mathematical problem by:

    our translation · original wording (PL): wyjaśnia swój sposób myślenia przy rozwiązywaniu problemu matematycznego przez:

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 2

  61. Curriculum point MAT.VII-VIII.5.3 · Teaching content · checked against the act

    conducts mathematical reasoning, thereby demonstrating the ability to:

    our translation · original wording (PL): prowadzi rozumowania matematyczne, wykazując przy tym umiejętność:

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 3

  62. Curriculum point MAT.VII-VIII.5.1.a · Teaching content · checked against the act

    records the relationships between the data in the problem in a way that facilitates the analysis of the problem

    our translation · original wording (PL): zapisuje związki między danymi występującymi w zadaniu w sposób ułatwiający analizę problemu

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 1 lit. a

  63. Curriculum point MAT.VII-VIII.5.2.a · Teaching content · checked against the act

    providing mathematical arguments justifying successive steps of reasoning

    our translation · original wording (PL): podawanie argumentów matematycznych uzasadniających kolejne kroki rozumowania

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 2 lit. a

  64. Curriculum point MAT.VII-VIII.5.3.a · Teaching content · checked against the act

    distinguishing a theorem from its converse

    our translation · original wording (PL): odróżniania twierdzenia od twierdzenia odwrotnego

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 3 lit. a

  65. Curriculum point MAT.VII-VIII.5.1.b · Teaching content · checked against the act

    analyses their solution to a problem and – if possible – simplifies the solution

    our translation · original wording (PL): analizuje swoje rozwiązanie problemu i – jeśli to możliwe – upraszcza rozwiązanie

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 1 lit. b

  66. Curriculum point MAT.VII-VIII.5.2.b · Teaching content · checked against the act

    constructing a justification referring only to necessary arguments

    our translation · original wording (PL): budowanie uzasadnienia odwołującego się tylko do niezbędnych argumentów

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 2 lit. b

  67. Curriculum point MAT.VII-VIII.5.3.b · Teaching content · checked against the act

    using uncomplicated, one-step indirect reasoning

    our translation · original wording (PL): stosowania nieskomplikowanych, jednokrokowych rozumowań nie wprost

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 3 lit. b

  68. Curriculum point MAT.VII-VIII.5.1.c · Teaching content · checked against the act

    poses additional questions arising from the obtained solution

    our translation · original wording (PL): stawia dodatkowe pytania wynikające z otrzymanego rozwiązania

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 1 lit. c

  69. Curriculum point MAT.VII-VIII.5.2.c · Teaching content · checked against the act

    formulating conclusions resulting from reasoning

    our translation · original wording (PL): formułowanie wniosków wynikających z rozumowania

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 2 lit. c

  70. Curriculum point MAT.VII-VIII.5.3.c · Teaching content · checked against the act

    generalizing observed special cases

    our translation · original wording (PL): uogólniania zaobserwowanych przypadków szczególnych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 3 lit. c

  71. Curriculum point MAT.VII-VIII.5.1.d · Teaching content · checked against the act

    notices similarities between methods for solving various mathematical problems

    our translation · original wording (PL): dostrzega podobieństwa między metodami rozwiązywania różnych problemów matematycznych

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 1 lit. d

  72. Curriculum point MAT.VII-VIII.5.3.d · Teaching content · checked against the act

    justifying, e.g. by providing a counterexample, that a given property does not hold.

    our translation · original wording (PL): uzasadniania, np. przez podanie kontrprzykładu, że dana własność nie zachodzi.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, kl. VII i VIII, dział 5 pkt 3 lit. d

  73. Curriculum point MAT.VII-VIII.WO.1 · General aims · checked against the act

    Using mathematical tools – understanding and using mathematical concepts and properties also in a new situation, recognizing relationships between properties as well as differences and similarities between mathematical objects.

    our translation · original wording (PL): Korzystanie z narzędzi matematycznych – rozumienie i używanie pojęć oraz własności matematycznych także w nowej sytuacji, dostrzeganie związków między własnościami oraz różnic i podobieństw między obiektami matematycznymi.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, cele kształcenia — wymagania ogólne, kl. VII-VIII, pkt 1

  74. Curriculum point MAT.VII-VIII.WO.2 · General aims · checked against the act

    Performing arithmetic and algebraic calculations, striving for an optimal method of calculation, using a calculator or computational applications when justified by the complexity of the calculation or the purpose of the task.

    our translation · original wording (PL): Wykonywanie obliczeń arytmetycznych i algebraicznych, dążenie do optymalnego sposobu rachowania, używanie kalkulatora lub aplikacji obliczeniowych, gdy jest to uzasadnione złożonością rachunku lub celem zadania.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, cele kształcenia — wymagania ogólne, kl. VII-VIII, pkt 2

  75. Curriculum point MAT.VII-VIII.WO.3 · General aims · checked against the act

    Drawing conclusions from information provided by means of text, tables, diagrams, graphs, etc., using mathematical concepts, tools, and arguments to critically interpret data, including recognizing unjustified conclusions.

    our translation · original wording (PL): Wyciąganie wniosków z informacji podanych za pomocą tekstu, tabel, diagramów, wykresów itp., używanie pojęć, narzędzi i argumentów matematycznych do krytycznego interpretowania danych, w tym do rozpoznawania nieuprawnionych wniosków.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, cele kształcenia — wymagania ogólne, kl. VII-VIII, pkt 3

  76. Curriculum point MAT.VII-VIII.WO.4 · General aims · checked against the act

    Using adequate mathematical tools to describe, analyze and solve problems from various fields, including everyday life.

    our translation · original wording (PL): Stosowanie adekwatnych narzędzi matematycznych do opisu, analizy i rozwiązywania problemów z różnych dziedzin, w tym z życia codziennego.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, cele kształcenia — wymagania ogólne, kl. VII-VIII, pkt 4

  77. Curriculum point MAT.VII-VIII.WO.5 · General aims · checked against the act

    Building multi-step strategies for solving a mathematical problem, recognizing correct solutions to a problem, analyzing the solution in order to verify its correctness or simplify the line of reasoning.

    our translation · original wording (PL): Budowanie kilkuetapowych strategii rozwiązania problemu matematycznego, dostrzeganie poprawnych rozwiązań problemu, analizowanie rozwiązania w celu sprawdzenia jego poprawności lub uproszczenia sposobu rozumowania.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, cele kształcenia — wymagania ogólne, kl. VII-VIII, pkt 5

  78. Curriculum point MAT.VII-VIII.WO.6 · General aims · checked against the act

    Conducting reasoning and justifying its correctness, formulating conclusions resulting from premises, selecting arguments by providing examples and counterexamples, verifying reasoning given by others.

    our translation · original wording (PL): Prowadzenie rozumowań i uzasadnianie ich poprawności, formułowanie wniosków wynikających z przesłanek, dobieranie argumentów przez podawanie przykładów i kontrprzykładów, weryfikowanie rozumowań podanych przez innych.

    Dz.U. 2026 poz. 378, zal. nr 2, Matematyka, cele kształcenia — wymagania ogólne, kl. VII-VIII, pkt 6

Skills step by step

Applying the Pythagorean Theorem

The student recognizes right triangles, checks the relationships between their sides, and calculates the missing leg or hypotenuse using the Pythagorean theorem.

Curriculum point MAT.VII-VIII.3.4 recognises right-angled triangles in various geometric situations, applies the Pythagorean theorem (our translation)
Curriculum point MAT.VII-VIII.3.9 solves problems concerning real-world objects that can be represented using a geometric model, also in a situation requiring necessary simplifications (our translation)

In grade 8, students use the relationship between the side lengths of a right triangle to find unknown measurements. Based on two known sides, they can calculate the third, and they also use the converse theorem to determine whether a given triangle has a right angle.

A common mistake is mechanically adding the squares of the given numbers without checking which segment lies opposite the right angle. Students sometimes add the squares of the sides even when the problem requires finding a leg (instead of subtracting the square of the known leg from the square of the hypotenuse), or they forget to take the square root of the final sum or difference.

In practice, worksheet exercises include three types of problems:

  • Calculate the hypotenuse – squaring the lengths of both legs, adding them, and taking the square root of the result,
  • Calculate a leg – subtracting the square of the known leg from the square of the hypotenuse,
  • Is the triangle a right triangle? – checking whether the sum of the squares of the two shorter sides is equal to the square of the longest side.
Create a worksheet for this skill

Task types: Find a leg of a right triangle · Find the hypotenuse · Hypotenuse — choose the answer · Is the triangle right-angled?

Right triangles with 30°, 60°, and 45° angles

The student calculates the side lengths in right triangles with angles of 30° and 60°, as well as 45°, using relationships involving the diagonal of a square and the height of an equilateral triangle.

Curriculum point MAT.VII-VIII.3.5 uses the properties of special right triangles, that is, right triangles with acute angles of 30° and 60° and isosceles right triangles (our translation)

In eighth grade, the student determines the missing side lengths in right triangles with acute angles of 30° and 60°, as well as in isosceles right triangles (with 45° angles). Given the measure of just one side, they can calculate the other two using constant geometric relationships and expressing the results using square roots.

The most common mistake is confusing the assignment of sides in a 30°-60° triangle—students often place the longer leg opposite the smaller angle. Coefficients are also sometimes swapped: using the square root of 2 instead of 3 or vice versa, which stems from confusing the properties of half a square with half an equilateral triangle.

In practice, problems involve the direct application of these rules in typical geometric configurations:

  • 30°-60° triangle — calculating the lengths of the legs or the hypotenuse based on one known side;
  • Height of an equilateral triangle — determining the height given the side, or calculating the side when the height is given;
  • Diagonal of a square — relating the side of a square to its diagonal as the hypotenuse of a 45° triangle.
Create a worksheet for this skill

Task types: Diagonal of a square · Height of an equilateral triangle · The 30-60-90 triangle

Calculating probability in simple situations

The student analyzes simple random situations and calculates the probability of an event by comparing the number of favorable outcomes to the number of all possible outcomes.

Curriculum point MAT.VII-VIII.4.6 conducts simple random experiments consisting, for example, of tossing a coin or rolling a die once or twice and drawing a ball from a set of balls, analyses random experiments and determines the probability of the considered events. (our translation)

In grade 8, the student determines the number of all possible outcomes in a single random experiment and identifies those that satisfy a given condition. Based on this, they write the probability of an event as a fraction, examining situations with equally likely outcomes.

A typical mistake is placing the number of remaining outcomes in the denominator instead of all possible outcomes. For example, with 3 white balls and 5 black balls in a box, a student determines the chance of drawing a white ball as 3/5 instead of 3/8. Another common error is incorrectly counting numbers that meet given criteria, such as overlooking the fact that the number 1 is not a prime number.

In practice, tasks take the form of simple experiments:

  • Rolling a die — determining the probability of obtaining a specific outcome, e.g., rolling a number greater than 4 or divisible by 2.
  • Drawing a ball from a box — calculating the chance of drawing a ball of a given color when the contents of the container are known.
  • Drawing a card with a number — identifying the probability of drawing a number that satisfies a specific condition from a prepared pool of cards.
Create a worksheet for this skill

Task types: Drawing a ball from a box · Rolling a die · Drawing a numbered card

Calculating percentage increases and decreases

The student calculates the new value or price after increasing or decreasing the initial amount by a given percentage.

Curriculum point MAT.VII-VIII.1.2 applies percentage calculations to solve problems in a practical context, including financial, such as calculating: prices after an increase or before their increase or decrease, interest on an annual deposit, loan costs, tax amounts – economic and financial module (our translation)

In eighth grade, a student efficiently determines a new amount after a change by a given percentage. In practical financial situations, this means being able to calculate the size of the increase or decrease itself and connect it to the starting price—by adding the calculated portion to the initial amount or subtracting it.

The most common mistake is treating a percentage like a regular number and subtracting it directly from the price (for example, subtracting the number 20 instead of 20% of the given amount). Students also sometimes struggle to correctly identify which initial value the percentage should be calculated from.

In practice, the exercises are based on tasks from the templates Increase by a percentage and Decrease by a percentage. The student is presented with a specific amount and information about a percentage change, and calculates the final result—for example, the price of shoes after a 15% discount or a new rate after a 10% increase.

Create a worksheet for this skill

Task types: Decrease by a percent · Increase by a percent · Percent increase — choose the answer

Multiplying through parentheses and factoring out

The student multiplies a monomial by a sum of up to three terms with integer coefficients and factors out a common monomial from a sum of two terms.

Curriculum point MAT.VII-VIII.2.5 multiplies a monomial by an algebraic sum of at most three monomials with integer coefficients (our translation)
Curriculum point MAT.VII-VIII.2.6 factors out a monomial from an algebraic sum consisting of two monomials with integer coefficients (our translation)

In Grade 8, the student consolidates transforming algebraic expressions with integer coefficients. They perform two related operations: multiplying a monomial by an algebraic sum consisting of up to three terms, and performing the inverse operation, that is, factoring out a monomial from a sum consisting of exactly two monomials.

The most common mistake is losing signs when multiplying by a negative monomial and omitting the remaining terms of the sum—the student multiplies only the first term in parentheses and mindlessly copies the rest. On the other hand, when factoring out a common factor, difficulty arises when one of the terms of the sum is identical to the monomial being factored out; students often forget to write the number 1 inside the parentheses in such cases.

Exercises from the Multiplying a monomial by a sum template require writing the given product in the form of a simplified algebraic sum. Tasks from the Factoring out a common factor template involve finding the common numerical divisor and common variables for two terms and writing the expression in the form of a product.

Create a worksheet for this skill

Task types: Expand brackets · Factor out a common factor

Calculating square and cube roots

The student calculates the exact values of square and cube roots of numbers that are squares or cubes of rational numbers.

Curriculum point MAT.VII-VIII.1.7 calculates the values of square and cube roots of numbers that are, respectively, squares or cubes of rational numbers, and calculates the approximate values of roots of other numbers using a calculator (our translation)

In Grade 8, students fluently find the exact values of square roots and cube roots of numbers that are, respectively, squares or cubes of rational numbers. They understand taking roots as the inverse operation of exponentiation—they know what number they are looking for so that raising it to the second or third power yields the value under the radical.

A common mistake at this stage is confusing root extraction with division. Students often give an answer of 8 instead of 4 for the square root of 16 (by dividing by 2) or an answer of 9 instead of 3 for the cube root of 27 (by dividing by 3). Overlooking the degree of the root also occurs, especially when exercises require alternating between calculating square and cube roots.

Practice materials for this skill include the templates Calculate the square root and Calculate the cube root. The tasks involve directly evaluating the radical in integer or fractional form, which also serves as a basis for review before the eighth-grade exam.

Create a worksheet for this skill

Task types: Find the cube root · Find the square root

Calculating powers of integers and fractions

The student calculates powers of integers and fractions with natural exponents, preparing for review before the eighth-grade exam.

Curriculum point MAT.VII-VIII.1.3 calculates powers of rational numbers with natural exponents (our translation)

In eighth grade, the student efficiently evaluates powers where the base is an integer (positive or negative) or a common or decimal fraction, and the exponent is a natural number. They understand that this notation means multiplying the same rational number by itself multiple times.

The most common mistake is confusing exponentiation with multiplying the base by the exponent (for example, calculating 3 to the power of 3 as 9 instead of 27). When working with fractions, a common problem is raising only the numerator to the power while ignoring the denominator, and in the case of negative numbers – incorrectly determining the sign of the result, especially when distinguishing between even and odd exponents.

In practice, the student tackles tasks from the templates Calculate the power of an integer and Calculate the power of a fraction. These involve directly computing the result of operations such as squaring or cubing a negative number, as well as evaluating the power of a common fraction written in parentheses.

Create a worksheet for this skill

Task types: Evaluate a power of a fraction · Evaluate a power of an integer · Value of a power — choose the answer

Calculating the circumference and area of a circle

The student calculates the circumference and the area of a circle based on a given radius or diameter using the number pi.

Curriculum point MAT.VII-VIII.3.7 calculates the circumference of a circle and the area of a circle (our translation)
Curriculum point MAT.VII-VIII.3.9 solves problems concerning real-world objects that can be represented using a geometric model, also in a situation requiring necessary simplifications (our translation)

In grade 8, students calculate the circumference and area of a circle. They apply formulas relating these quantities to the radius or diameter of the figure, expressing results accurately using the number π or calculating with its approximations.

A common issue is confusing the formula for the circumference with the formula for the area—students often square the radius when determining the circumference or multiply the radius by two instead of squaring it when calculating the area. Another frequent error is substituting the given diameter directly into the formulas without first dividing it in half, which particularly distorts the result for the area of a circle.

The exercises in the materials Circumference of a Circle and Area of a Circle include both straightforward determinations of circumference and area based on a given radius or diameter, as well as inverse operations—calculating circle dimensions when its area or circumference is known.

Create a worksheet for this skill

Task types: Area of a circle · Circumference of a circle

Probability in Two-Stage Experiments

The student calculates the probability of events in two-stage experiments, such as rolling a die twice or drawing balls with replacement.

Curriculum point MAT.VII-VIII.4.6 conducts simple random experiments consisting, for example, of tossing a coin or rolling a die once or twice and drawing a ball from a set of balls, analyses random experiments and determines the probability of the considered events. (our translation)

In Grade 8, the student analyzes the course of two-step random experiments. They can determine the total number of possible outcomes and the number of favorable outcomes for a given event by listing pairs of numbers or organizing them in a table. Based on this, they calculate the probability of the outcome of interest.

A common mistake is treating different orders as the same outcome or omitting some of the possibilities. For example, when rolling two dice, students often consider the pairs (2, 5) and (5, 2) to be the exact same outcome, thereby incorrectly determining the number of favorable outcomes and underestimating the total number of possible combinations (believing there are 21 instead of 36).

In practice, the student encounters two types of problems:

  • Sum of dots on two dice – involves calculating the chance of rolling a specific sum (e.g., equal to 7 or greater than 9) when rolling two six-sided dice;
  • Two draws with replacement – requires determining the probability of drawing balls with specific properties, where the drawn ball is returned to the set before the second draw.
Create a worksheet for this skill

Task types: Sum of two dice · Two draws with replacement

Surface area and volume of a right prism

An eighth-grader calculates the total surface area and volume of right prisms, including solids with a triangular base.

Curriculum point MAT.VII-VIII.3.14 calculates the surface areas and volumes of right prisms and regular pyramids. (our translation)
Curriculum point MAT.VII-VIII.3.9 solves problems concerning real-world objects that can be represented using a geometric model, also in a situation requiring necessary simplifications (our translation)

In Grade 8, students calculate the surface area and volume of a right prism, focusing in this part of the material on solids with a triangular base. They independently determine the area of the triangular base, sum the areas of the rectangular lateral faces, and multiply the base area by the height of the entire solid to obtain its volume.

The most common difficulty is confusing the height of the triangle in the base with the height of the prism itself. Students also tend to forget to include two bases when calculating the total surface area and add the area of the triangle only once.

In practice, exercises from the templates Surface area of a triangular prism and Volume of a triangular prism involve reading dimensions from a drawing of the solid or a verbal description, calculating the area of the base triangle, and then correctly determining the lateral surface area or the volume of the entire figure.

Create a worksheet for this skill

Task types: Volume of a triangular prism · Surface area of a triangular prism

Multiplying two binomials

The student learns to multiply two binomials with integer coefficients, ordering the resulting expression and combining like terms.

Curriculum point MAT.VII-VIII.2.7 multiplies two binomials with integer coefficients (our translation)

In Grade 8, students multiply two binomials that contain only integer coefficients. The task involves correctly multiplying each term in the first set of parentheses by each term in the second set of parentheses, followed by combining like terms.

The most common mistake is overlooking sign rules with negative numbers—especially when there is subtraction inside the parentheses and the student forgets that the minus sign belongs to that term. Another common error is multiplying only the first term by the first and the second by the second, which leads to omitting the middle terms.

In practice, exercises are based on worksheets from the templates Multiplying Binomials (x + a)(x + b), where the variable has a coefficient of 1, and Multiplying Binomials, where integer coefficients can also appear directly in front of the variables.

Create a worksheet for this skill

Task types: Multiply (x + a)(x + b) · Multiply two binomials

Surface area and volume of a regular pyramid

An eighth-grader calculates the total surface area and volume of regular pyramids using the properties of their bases and lateral faces.

Curriculum point MAT.VII-VIII.3.14 calculates the surface areas and volumes of right prisms and regular pyramids. (our translation)

In Grade 8, students calculate the total surface area and volume of regular pyramids. They identify the components of the solid—base area, lateral surface area, and the height of the entire solid—and combine them in the appropriate geometric formulas.

The most common difficulty is confusing the height of the pyramid with the slant height of its lateral face. Students also often forget the fraction one-third in the volume formula, calculating it as they would for a prism, or they omit the base area when determining the total surface area.

In practice, exercises are based on the templates Surface Area of a Pyramid and Volume of a Pyramid. The tasks require determining the areas of the individual triangular faces and the polygonal base, or substituting the given dimensions directly into the formula for the volume of the solid.

Create a worksheet for this skill

Task types: Surface area of a square pyramid · Volume of a square pyramid

Ordering and combining like terms

The student arranges monomials and adds and subtracts like terms with integer coefficients, simplifying expressions with one or two variables.

Curriculum point MAT.VII-VIII.2.3 simplifies monomials and combines like terms in algebraic sums (our translation)
Curriculum point MAT.VII-VIII.2.4 adds and subtracts algebraic sums consisting of several monomials with integer coefficients (our translation)

In grade 8, students efficiently rearrange algebraic sums and combine like terms. They add and subtract monomials with integer coefficients, reducing the expression to its simplest form. They perform operations on expressions containing one variable or two different variables.

The most common difficulty is losing the minus sign directly preceding a combined term when rearranging the components. Incorrectly combining unlike terms also occurs—for example, attempting to add monomials with different variables (such as 3x and 2y) or adding a constant to the coefficient of a variable.

In practice, the exercises are based on the templates Combining like terms and Combining like terms with two variables. The task involves identifying terms with the same variables in a given expression, summing their numerical coefficients, and writing the result in simplified, ordered form.

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Task types: Combine like terms · Combine like terms in two variables

Finding a number from its percentage

An eighth-grader can efficiently calculate the whole sought quantity when they know the value corresponding to a given percentage of it.

Curriculum point MAT.VII-VIII.1.1.c a number based on a given percentage of it (our translation)

In grade 8, a student is able to find the whole quantity (that is, 100%) knowing only a part of it and the corresponding percentage. To find the unknown number, they convert the percentage to a common fraction or decimal and perform division, set up a simple proportion, or first calculate the value of 1% of the given quantity.

The most common mistake at this stage is confusing this type of problem with finding the percentage of a number. Instead of dividing the given value by the fraction, students instinctively multiply the two numbers. As a result, instead of a larger, original whole, they get a result much smaller than the given part, failing to notice that the outcome makes no sense.

Problems from the template Number from a given percentage require finding an unknown in situations such as: "Find a number whose 15% is 45" or in simple word problems where it is necessary to determine, for example, the original price of an item after a price increase or the full length of a route based on a completed stage.

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Calculating what percentage one number is of another

The student calculates what part of the whole a given value represents and writes this result as a percentage.

Curriculum point MAT.VII-VIII.1.1.a what percentage of one number another number is (our translation)

In grade 8, students are able to compare two numbers and determine what percentage one is of the other. They write the ratio of these values as a common fraction or decimal and then convert it into a percentage by multiplying by 100%.

The most common mistake is confusing the base value (the whole, which should be in the denominator) with the number being compared. Out of habit, students often divide the larger number by the smaller one instead of checking the problem statement to see which value the question refers to.

In tasks from the What percentage is a number? template, eighth-graders encounter questions testing this skill directly (for example, what percentage of 50 is 15) as well as in a simple practical context, such as calculating the percentage of students present or the covered portion of a route.

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Determining the median of a dataset

The student orders a given set of numbers and determines its median, that is, the middle value or the average of the two middle numbers.

Curriculum point MAT.VII-VIII.4.4 uses the arithmetic mean and the median in a practical context to compare data sets, including financial data – economic and financial module (our translation)

In grade 8, the student efficiently finds the median of a given set of numerical data. The first step is always to arrange all values in ascending order. If the set contains an odd number of elements, the student identifies the value that is exactly in the middle. When there is an even number of data points, they calculate the arithmetic mean of the two middle numbers.

The most common mistake is identifying the middle number directly from the problem statement, without first sorting the numbers from smallest to largest. Students also confuse the rule for finding the median of an even number of data points and, instead of calculating the mean of the two middle values, randomly choose one of them.

Tasks from the Find the median template take the form of a simple list of numbers (for example, measurement results, grades, or amounts of money) for which the student must correctly determine and enter the middle value.

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Powers with a negative exponent

An eighth-grader learns to convert powers with negative integer exponents into corresponding powers with natural exponents.

Curriculum point MAT.VII-VIII.1.4 converts powers with a negative integer exponent into corresponding powers with natural exponents (our translation)

In the eighth grade, students master the rule of exponentiation with a negative exponent. According to the curriculum, they convert powers with negative integer exponents into corresponding powers with natural exponents. This involves using the reciprocal operation: the student replaces the base of the power with its reciprocal and transforms the negative exponent into its corresponding natural number.

A typical mistake involves confusing a negative exponent with the sign of the entire power. Students often add a minus sign in front of the number or multiply the base by the exponent (for example, assuming that 3 to the power of -2 is -6 or -9). This stems from the habit of associating a minus sign with subtraction or negative numbers rather than with taking the reciprocal of a number.

In exercises from the template Power with a negative exponent, students encounter tasks that require the direct transformation of powers of integers and fractions (for example, writing an expression as a power with a positive exponent and giving the final result as a common fraction or decimal).

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Operations on powers and the missing exponent

The student applies the rules of multiplying and dividing powers with natural bases and integer exponents to determine the missing exponent in an equation.

Curriculum point MAT.VII-VIII.1.5 applies the properties of operations on powers with natural bases and integer exponents, including: (our translation)
Curriculum point MAT.VII-VIII.1.5.a multiplies and divides powers with the same base (our translation)
Curriculum point MAT.VII-VIII.1.5.b multiplies and divides powers with the same exponents (our translation)
Curriculum point MAT.VII-VIII.1.5.c raises a power to a power (our translation)

In grade 8, the student proficiently uses the properties of operations on powers with natural bases and integer exponents (including negative exponents and zero). They use the rules for adding, subtracting, and multiplying exponents when multiplying and dividing powers with the same bases, as well as when raising a power to a power.

The most common mistake is mechanically confusing operations—multiplying exponents instead of adding them when finding the product of powers with the same bases. Negative exponents present an additional difficulty: when dividing powers, students often forget the rule for subtracting a negative number, which leads to an incorrect sign in the result.

In tasks from the template Missing Exponent — Operations on Powers, the student analyzes an equation in which an empty box appears in place of one of the exponents. Their goal is to simplify the expression and enter the correct integer that satisfies the given equality.

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Dividing a quantity in a given ratio

The student divides a given quantity into two or three parts in a given ratio and applies this skill in practical word problems.

Curriculum point MAT.VII-VIII.1.10 recognizes directly proportional relationships, applies them to determine the value of one quantity based on another, and uses proportional division into two or three parts in a practical context (our translation)

In Grade 8, students solve problems requiring the division of a given quantity—such as an amount of money, length, or mass—into two or three parts according to a given ratio. They determine the value of one share in the division and then calculate the quantities corresponding to each individual share.

A typical mistake is treating the numbers in the ratio as fractions of the total quantity rather than as its component parts. Students often divide the initial quantity directly by the terms of the ratio instead of first finding the total number of parts and dividing the initial quantity by that sum.

In problems involving Dividing in a given ratio, students encounter tasks set in practical contexts, such as sharing profits in a ratio of 2 : 3 or preparing mortar with ingredient proportions of 1 : 2 : 4. The task requires setting up an appropriate equation or summing the parts and providing the specific quantities for each component of the division.

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Taking a factor out of a radical

The student learns to simplify square and cube roots by factoring the radicand and extracting the appropriate factor.

Curriculum point MAT.VII-VIII.1.9 applies the properties of operations on roots for square and cube roots, including: (our translation)
Curriculum point MAT.VII-VIII.1.9.a calculates the roots of the product and quotient of two numbers (our translation)
Curriculum point MAT.VII-VIII.1.9.b multiplies and divides roots of the same degree (our translation)
Curriculum point MAT.VII-VIII.1.9.c takes a number out of the radical sign and moves a number under the radical sign (our translation)

In Grade 8, students master simplifying square and cube roots. They use the properties of operations to write the number under the radical sign as a product and take out a factor whose root of the given degree can be extracted.

A typical mistake is stopping at incomplete simplification—for example, rewriting the square root of 72 into a form with 18 under the radical instead of bringing it to its simplest form. Another common issue with cube roots is mechanically looking for squares of numbers instead of their cubes.

Exercises practicing this skill appear in the form of the instruction Take the factor out of the radical. The student receives a single expression with a square or cube root and transforms it into the product of an integer and a radical with a smaller radicand.

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Estimating the value of roots

The student learns to determine the approximate value of a root by identifying the adjacent integers between which the given number lies.

Curriculum point MAT.VII-VIII.1.8 estimates the value of a simple arithmetic expression containing roots (our translation)

In Grade 8, students can estimate the value of a root and a simple arithmetic expression containing roots without using a calculator. To do this, they use their knowledge of squares and cubes of natural numbers, determining between which two consecutive integers the given value lies.

A typical mistake at this stage is confusing finding roots with division—for example, assuming that the square root of 10 is 5, instead of noticing that it lies between 3 and 4 (since 3² = 9 and 4² = 16). Students also sometimes try to guess decimal values instead of relying on the nearest known powers.

In practice, in tasks from the Estimate the Root series, students fill in the missing integers bounding a given root in an inequality or determine which integer is closest to the given value.

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Writing and comparing numbers in scientific notation

The student writes numbers in scientific notation and compares quantities presented in this form.

Curriculum point MAT.VII-VIII.1.6 writes numbers in scientific notation and compares numbers written in this way (our translation)

In Grade 8, the student efficiently converts numbers from standard decimal notation to scientific notation and vice versa. They write numbers as a product in which the first factor is in the range from 1 to 10 (excluding 10), and the second is a power of 10. Based on the exponent and the value of the first factor, they can also quickly compare two numbers.

A common mistake is leaving the first factor in an incorrect form—for example, writing 45 · 10³ instead of the correct 4.5 · 10⁴. Students also tend to mechanically count the zeros in a number instead of checking how many places the decimal point actually moved.

The exercises in the Scientific Notation set involve converting given numbers to scientific notation, reading values written using powers of 10, and ordering a set of numbers from least to greatest.

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Solving equations with the unknown on both sides

The student organizes and solves first-degree equations in which the letter representing the unknown appears on both the left and right sides of the equals sign.

Curriculum point MAT.VII-VIII.2.11 solves first-degree equations, as well as equations that can be easily reduced to a first-degree equation (our translation)

In Grade 8, students solve first-degree equations with one unknown located on both sides of the equals sign. The student simplifies the equation: groups terms containing the unknown on one side (usually the left) and constants on the other, then combines like terms and determines the unknown value.

A typical difficulty at this stage is changing signs when moving numbers or algebraic expressions to the other side of the equation. Students often mechanically copy the plus or minus sign instead of changing it to its opposite. Errors also appear in the final step when both sides must be divided by a negative coefficient of the unknown.

In practice, within the Equation with an unknown on both sides template, the task takes the form of a given linear equation (e.g., 5x - 3 = 2x + 9). The student writes out the successive transformation steps, performing operations on both sides until reaching the final result.

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Checking if a number is a solution to an equation

The student can check whether a given integer or rational number satisfies an equation with one unknown by substituting it in place of the letter and calculating the values of both sides.

Curriculum point MAT.VII-VIII.2.10 checks whether a given rational number is a solution to a first-degree equation (our translation)
Curriculum point MAT.VII-VIII.2.9 checks whether a given integer is a solution to an equation (of the first, second, or third degree) with one unknown (our translation)

In Grade 8, a student can determine whether a given number satisfies an equation with one unknown. In the case of first-degree equations, they check any rational numbers, including common and decimal fractions. For second- and third-degree equations, they test integers. This skill consists of substituting the given value for the unknown and separately calculating the values of the left-hand and right-hand sides of the equation to check if they are equal.

The most common mistakes occur in calculations involving negative numbers and exponentiation. Students confuse the notation (-2)² with -2² or lose minus signs during multiplication. Another frequent difficulty is the unnecessary attempt to algebraically rearrange and solve the entire equation (especially second- or third-degree ones), instead of simply checking the given number using arithmetic calculation.

In practice, in exercises from the template Is the number a solution to the equation?, an eighth-grader is given an equation and a specific number to check. After substituting it for the variable and performing the operations, the student compares the results of the left-hand and right-hand sides and then unambiguously indicates the answer: yes or no.

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Solving two-step equations (ax + b = c)

The student solves first-degree equations in the form ax + b = c by sequentially transposing the constant term and dividing by the coefficient of the unknown.

Curriculum point MAT.VII-VIII.2.11 solves first-degree equations, as well as equations that can be easily reduced to a first-degree equation (our translation)

In Grade 8, the student efficiently solves linear equations written in the form ax + b = c. The task is completed in two structured steps: first, isolating the term with the unknown by adding or subtracting the number b on both sides, and then dividing both sides by the coefficient a to determine the value of the unknown number.

The most common mistake is dropping or confusing signs when moving the constant term to the other side of the equation (for example, adding a number instead of subtracting it). Students also sometimes struggle with the order of operations—attempting to divide by the coefficient of the unknown before first eliminating the constant term, which unnecessarily complicates the calculations.

Exercises from the Equation ax + b = c template require finding the value of the unknown in simple algebraic equations, such as 3x + 5 = 26 or those involving negative numbers, e.g., -2x + 7 = -5. The student writes out the successive steps of transformation, leading the equation to the final result.

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Task types: Solution of an equation — choose the answer · Solve ax + b = c

Calculating the value of an algebraic expression

The student substitutes integers for letters in a short algebraic expression (up to three monomials) and calculates its final result.

Curriculum point MAT.VII-VIII.2.2 calculates the numerical value of an algebraic expression that is a sum of at most three monomials for given integer values of variables (our translation)

In grade 8, the student efficiently calculates the numerical value of an algebraic expression consisting of at most three monomials. In place of the letters, they substitute the given integers — both positive and negative, or zero — and then correctly perform arithmetic operations according to the order of operations.

The most common mistakes stem from carelessness when working with negative numbers. Students forget about parentheses when raising a negative number substituted for a variable to a power, or lose a minus sign when subtracting a monomial with a negative value. Violations of the order of operations also occur, for example, adding numbers before performing the multiplication written in a monomial.

In the tasks from the "Value of an algebraic expression" worksheet, the student receives a short expression and values of variables, for example: calculate the value of the expression 3a - 2b + 4 for a = -2 and b = 5. The task consists of accurately substituting the numbers in place of the symbols and carrying out the calculations to a single numerical result.

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Solving systems of two equations

The student solves word problems by describing the situation using two equations with two unknowns and reducing them to a single first-degree equation.

Curriculum point MAT.VII-VIII.2.12 solves word problems using linear equations with one unknown or two linear equations with two unknowns that can be easily reduced to one linear equation. (our translation)

In eighth grade, students solve word problems using two linear equations with two unknowns. According to the curriculum, they master systems that can be easily reduced to a single equation with one unknown — for example, by isolating one variable and substituting it into the other equation.

A common mistake occurs at the stage of substituting the isolated expression. When a minus sign or a numerical coefficient precedes the variable, students often forget to place parentheses around the substituted binomial, which leads to an incorrect sign in subsequent terms and distorts the entire result.

In practice, using the System of two equations template, students encounter word problems (e.g., involving the prices of reduced-fare and standard tickets or the number of coins of different denominations). Their task is to define both unknowns, set up two related equations, and determine the correct numerical values.

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Calculating interest on an annual deposit

The student calculates the profit from a one-year bank deposit and the total amount of capital increased by interest at a given interest rate.

Curriculum point MAT.VII-VIII.1.2 applies percentage calculations to solve problems in a practical context, including financial, such as calculating: prices after an increase or before their increase or decrease, interest on an annual deposit, loan costs, tax amounts – economic and financial module (our translation)

In Grade 8, students apply percentage calculations in a practical financial context. Based on the deposited principal and the given annual interest rate, they are able to calculate the amount of interest earned after one year, as well as determine the final savings balance after adding the profit to the deposited sum.

A typical mistake is confusing the interest amount itself with the final amount paid out by the bank at the end of the savings period. Students sometimes treat the calculated interest as the total funds in the account or attempt to add the percentage value directly to the sum of money (e.g., adding 5 PLN instead of 5% of the principal).

Tasks from the Interest on a Deposit template present realistic financial situations. A typical task requires calculating how much interest an initial deposit of a specific amount (e.g., 3,000 PLN) will earn at an annual interest rate (e.g., 6%), and what total sum the client will receive after one year of the term deposit.

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Task templates to print

Every task type, grouped by skill. Clicking one opens the worksheet generator with it already selected.

Applying the Pythagorean Theorem

Right triangles with 30°, 60°, and 45° angles

Calculating probability in simple situations

Calculating percentage increases and decreases

Multiplying through parentheses and factoring out

Calculating square and cube roots

Calculating powers of integers and fractions

Calculating the circumference and area of a circle

Probability in Two-Stage Experiments

Surface area and volume of a right prism

Multiplying two binomials

Surface area and volume of a regular pyramid

Ordering and combining like terms

Finding a number from its percentage

Calculating what percentage one number is of another

Determining the median of a dataset

Powers with a negative exponent

Operations on powers and the missing exponent

Dividing a quantity in a given ratio

Taking a factor out of a radical

Estimating the value of roots

Writing and comparing numbers in scientific notation

Solving equations with the unknown on both sides

Checking if a number is a solution to an equation

Solving two-step equations (ax + b = c)

Calculating the value of an algebraic expression

Solving systems of two equations

Calculating interest on an annual deposit

Other task types

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