English
Exam (Poland)

Eighth-grade mathematics exam — printable mock exam

Preparation for the eighth-grade mathematics exam: what the CKE paper looks like, how long it lasts, and how it is scored — and above all, a one-click printable mock exam: 21 tasks in the CKE paper layout, with scoring, an answer key, and error diagnosis.

Build a practice exam

What does the eighth-grade mathematics exam look like?

The mathematics exam lasts 125 minutes (students with accommodations may receive up to 40 additional minutes). The paper contains 20–21 tasks, for which you can score 30 points:

  • 14–15 closed-ended tasks — 1 point each: single-choice, true–false, matching; this accounts for about half of the score,
  • 5–6 open-ended tasks — 2 or 3 points each; the entire line of reasoning is evaluated, and partial credit can also be awarded for an incomplete solution.

In the exam paper, closed-ended tasks come first, followed by open-ended ones. You cannot "fail" the exam — the score is reported as a percentage and on a percentile scale. Details and sample papers: information guides of the Central Examination Board.

Mock exam with one click

The “Create printable mock exam” button compiles an exam paper in the CKE format: 15 closed-ended questions (single choice of four options A–D and yes/no) and 6 open-ended questions, with point values for each question, the total score, and instructions in the header. The answer key includes a points column.

Every incorrect answer in a closed-ended question is a named error — beneath the answer key, you can see who “multiplied the base by the exponent” and who “added the legs instead of using the Pythagorean theorem”. The teacher immediately knows what to review. Each click generates a different set and different numbers, and each section of the paper can later be modified.

What does the exam cover?

The exam tests the requirements of the core curriculum from the entire primary school, with an emphasis on grades 7–8: powers and roots, percentages in practice, proportionality, rational numbers, algebraic expressions and equations, the Pythagorean theorem, 2D and 3D shapes, probability, and data interpretation.

Important note on the core curriculum: the new curriculum (Dz.U. 2026 item 378) will cover grade 8 from the 2030/2031 school year — until then, eighth-graders follow the 2017 curriculum, and the scope of exam topics is similar in both curricula. You can also find the material for grades 7 and 8 on the pages for grade 7 and grade 8.

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 price increases and decreases by a percentage (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 numbers with a natural exponent (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

  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

    Scientific notation and comparing numbers (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 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

  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

    Factoring out terms from under the radical sign (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) · Calculating the missing value from a proportion (we teach in grade 5-6)

  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 of one number another number is (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

    Factoring out terms from under the radical sign (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

    Calculating a percentage of a given number (we teach in grade 6)

  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

    Factoring out terms from under the radical sign (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

    Calculating a whole number from a given 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

    Factoring out terms from under the radical sign (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

    Evaluating 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

    Combining like terms in expressions (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

    Combining like terms in expressions (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 brackets 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 brackets 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 whether 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 whether 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 linear equations (we teach in grade 7-8) · Solving equations with variables 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

    Systems of two equations in word problems (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

    Pythagorean theorem in right-angled triangles (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 angles 30°, 60°, and 45° (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 the 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

    Pythagorean theorem in right-angled triangles (we teach in grade 7-8) · Surface area and volume of a triangular prism (we teach in grade 8) · Calculating the circumference and the 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 triangular prism (we teach in grade 8) · Calculating the 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

    Arithmetic Mean in Everyday Situations (we teach in grade 5-6)

  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 the probability of drawing a ball (we teach in grade 7-8) · Probability in two tosses or draws (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

Pythagorean theorem in right-angled triangles

An eighth-grader is able to calculate the length of the hypotenuse or a leg, and determine whether a triangle with given side lengths is a right-angled triangle.

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)

An eighth-grader recognizes right-angled triangles within geometric figures and correctly applies the relationship between the squares of the lengths of their sides. Based on information from a diagram or the problem text, the student sets up the appropriate equation, squares the lengths, and finds the missing side by taking the square root at the end.

The most common mistake is confusing the legs with the hypotenuse when writing down the formula. Students often mechanically add the squares of the two given numbers, even if one of them is the triangle's longest side, instead of subtracting the square of a leg from the square of the hypotenuse. Another frequent error is skipping the exponentiation entirely and simply adding the lengths of the segments.

In exam tasks, the student performs direct calculations in formats such as Calculate the hypotenuse and Calculate the leg. They also encounter multiple-choice questions (Hypotenuse — choose the answer) as well as problems testing the converse theorem: Is the triangle right-angled?, where they check whether the sum of the squares of the two shorter sides equals 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?

Calculating price increases and decreases by a percentage

The student can calculate the new price of an item after an increase or decrease by a given percentage in practical context problems.

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)

As part of the preparation for the eighth-grade exam, the student applies percentage calculations in everyday life situations. At this stage, they are able to determine the new value of a given quantity or the price of an item after increasing or decreasing it by a specific percentage, efficiently combining finding a fraction of a number with addition or subtraction.

A typical mistake on the exam is stopping after calculating only the amount of the change and treating it as the final result. Students also frequently confuse addition with subtraction or, in a rush, select the answer that contains only the value of the increase instead of the new price.

On the exam paper, this skill appears in both open-ended and multiple-choice tasks. Tasks based on the templates Increase by a percentage and Decrease by a percentage require independently calculating and entering the new amount, whereas the format Percentage increase — choose the answer tests the ability to efficiently identify the correct final price from the given options A–D.

Create a worksheet for this skill

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

Calculating powers of numbers with a natural exponent

An eighth-grader is able to efficiently calculate the value of a power of a rational number, in particular an integer, when the exponent is a natural number.

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

At this stage, the student raises rational numbers—including positive and negative integers—to natural powers. They understand the exponential notation as repeated multiplication of the same number by itself and can determine its exact numerical result.

A common mistake on the exam is confusing exponentiation with simply multiplying the base by the exponent (for example, assuming that 3 to the power of 3 is 9 instead of 27). The signs of negative numbers also cause difficulty: students often lose the minus sign or forget that a power with an even exponent yields a positive result only when the negative number is enclosed in parentheses.

On the mock test, the assessment of this skill takes the form of open-ended tasks requiring the result to be entered, such as calculate the power of an integer, or multiple-choice questions in the format of value of a power — choose the answer, where the student identifies the correct result from the given options.

Create a worksheet for this skill

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

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

The student calculates the side lengths of right-angled triangles with 30° and 60° angles and determines the diagonal of a square as well as the sides of an isosceles right-angled 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 8th grade, students use the relationships between the side lengths in right-angled triangles with acute angles of 30° and 60°, as well as in isosceles right-angled triangles (with angles of 45°, 45°, and 90°). Given the length of just one side, they are able to calculate the measures of the other two, relying on the properties of half an equilateral triangle and the formula for the diagonal of a square.

The most common mistake is confusing the assignment of sides in a 30°-60°-90° triangle. Students often place the shortest side opposite the 60° angle instead of the 30° angle, or assign the factor with the square root of 3 to the hypotenuse. Another challenge is finding the side length of a square when the given diagonal is an integer, which requires correctly rationalizing the denominator.

In exam preparation tasks, this skill appears in the templates 30°-60°-90° Triangle and Diagonal of a Square. In practice, students encounter diagram-based problems where, based on a given angle and one side, they must calculate the area, perimeter of a figure, or a missing segment in a composite figure.

Create a worksheet for this skill

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

Calculating the circumference and the area of a circle

The student calculates the circumference of a circle and the area of a disk 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 eighth grade, the student determines the circumference and the area of a circle using the appropriate mathematical formulas with the constant π. Based on a given radius or diameter length, they calculate the circumference and area of a circle, and can also determine the radius when the circumference or area is known.

A common problem is confusing the two formulas—for example, doubling the radius instead of squaring it when calculating the area of a circle. Students also often make the mistake of substituting the entire diameter into the formulas instead of the radius, which leads to incorrect results.

Exam preparation tasks are based on the templates Circumference of a circle and Area of a circle. They appear on the exam paper as multiple-choice or open-ended tasks in which the student writes the result in exact form (with the symbol π) or performs simple computational transformations.

Create a worksheet for this skill

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

Probability in two tosses or draws

The student calculates the probability of events in rolling a die twice and in a two-stage drawing with replacement, determining the number of all possible and favorable 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, students analyze simple two-stage experiments. They can correctly determine the number of all possible outcomes (for example, 36 equally likely pairs when rolling a standard six-sided die twice), find the number of favorable outcomes, and express the required probability as a fraction.

The most common mistake is ignoring the order of outcomes across consecutive stages. Students often treat a pair of outcomes, such as (2, 5) and (5, 2), as one and the same event, which leads to undercounting the number of favorable outcomes. Another typical error is adding the number of die faces (6 + 6 = 12) instead of multiplying them (6 · 6 = 36) when determining all possible outcomes.

On the exam, tasks testing this skill take the form of problems in which the student examines:

  • the sum of the numbers rolled on two dice — for example, calculating the probability of rolling a sum equal to 8 or a sum greater than 10,
  • two draws with replacement — where the drawn item is returned to the set before the second turn, so the conditions and the number of balls remain identical in both stages.
Create a worksheet for this skill

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

Surface area and volume of a triangular prism

The student calculates the total surface area and volume of a right triangular prism based on the given dimensions.

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 eighth grade, the student is able to determine the total surface area and volume of a right prism, including a triangular prism. They correctly select the formulas: they calculate the volume as the product of the base area and the height of the solid, and the surface area as the sum of the areas of the two triangular bases and the three rectangular lateral faces.

A common mistake on the exam is confusing the height of the base triangle with the height of the entire solid. Students also often forget that the solid has two bases and include only one triangle when calculating the total surface area. Another frequent error is assuming that all lateral faces are identical, whereas their dimensions differ if the base is not an equilateral triangle.

Exam preparation tasks include two specific types of exercises: surface area of a triangular prism and volume of a triangular prism. The student solves problems with net diagrams or 3D drawings of the solid, as well as word problems in which they first calculate the area of the triangular base on their own, and then the total volume or surface area.

Create a worksheet for this skill

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

Solving two-step linear equations

The student solves linear equations in the form ax + b = c by successively moving 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 eighth grade, the student efficiently solves linear equations in one unknown of the form ax + b = c. The task requires performing two ordered operations: first, adding or subtracting the constant term on both sides, and then dividing both sides of the equation by the coefficient of the unknown to determine a single x.

The most common problem is reversing the proper order of steps or making errors when working with positive and negative signs. Students tend to divide by the coefficient a before subtracting the term b, which unnecessarily complicates fraction calculations, or they forget to change the sign of a number when moving it to the other side of the equals sign.

In the practice test, working on this skill is based on tasks where students must determine the number satisfying the notation Equation ax + b = c, or on multiple-choice questions (template Equation solution — choose the answer), where the student identifies the correct result from options A, B, C, and D.

Create a worksheet for this skill

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

Calculating a whole number from a given percentage

The student is able to determine the initial quantity or number when they know what fraction of it is represented by a given percentage value.

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

In grade 8, a student can determine the whole (i.e., 100%) knowing a specific part of it expressed as a percentage. They efficiently convert a percentage into a common fraction or a decimal and perform the appropriate division, or set up and solve a simple equation with an unknown representing the sought-after value.

A typical mistake on the exam is confusing the direction of the calculations: instead of dividing the known value by the percentage, the student mindlessly multiplies the given number by that percentage. As a result, they calculate a percentage of a given number instead of reconstructing the initial whole (for example, instead of finding an amount of which 20% is 40 PLN, they mistakenly calculate 20% of 40).

On the exam paper, tasks within the Finding a number from its percentage template most often take the form of word problems set in everyday contexts. For example, the student must calculate the initial price of a product before a discount, the total length of a trip route, or the total number of survey participants based on data about one of its subsets.

Create a worksheet for this skill

Dividing common fractions

The student divides common fractions with single- and two-digit denominators, efficiently converting division into multiplication by the reciprocal of the second fraction.

Curriculum point MAT.IV-VI.1.27 adds, subtracts, multiplies and divides common fractions with one- and two-digit denominators (our translation)

As part of revision for the eighth-grade exam, the student divides common fractions with one- and two-digit denominators. They efficiently transform the operation by multiplying the first fraction by the reciprocal of the second, and also simplify fractions before calculating the final result.

The most common mistake is inverting the wrong number—the first fraction instead of the second—or attempting to directly divide the numerator by the numerator and the denominator by the denominator. It also happens that the student inverts the divisor, but forgets to change the division sign to a multiplication sign.

A task of the type Divide a fraction by a fraction tests this skill directly: the student is given the quotient of two common fractions to calculate and reduce the result to its simplest form, that is, an irreducible fraction.

Create a worksheet for this skill

Multiplying brackets and factoring out

The student multiplies a monomial by an algebraic sum with integer coefficients and factors out a common factor.

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, a student skillfully performs algebraic transformations: multiplies a monomial by a sum consisting of at most three monomials and factors out a monomial from a sum of two terms. These operations are performed on expressions with integer coefficients.

A common mistake is multiplying only the first term in parentheses by the monomial and rewriting the rest unchanged (for example, writing 2(x + 4) = 2x + 4 instead of 2x + 8). Students also often lose a minus sign when there is a negative number in front of the parentheses, or they forget to add the exponents when multiplying identical variables.

On the exam paper, this skill appears in problems from the template Multiplying a monomial by a sum, where one must correctly expand parentheses and combine like terms to simplify the entire expression to its simplest form.

Create a worksheet for this skill

Calculating what percentage of one number another number is

The student is able to determine what percentage of one quantity another quantity is by writing the appropriate fraction and converting it into a percentage.

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

An eighth-grader is able to determine what percentage of a given number another quantity is. In practice, this means comparing two numbers by writing their quotient as a fraction and then converting it to a percentage — for example, by expanding the fraction to a denominator of 100 or by multiplying it by 100%.

The most common mistake on the exam is confusing the numerator with the denominator, resulting from misidentifying the base of reference. The student swaps the number representing the whole with the number representing its part, thereby obtaining the reciprocal of the intended ratio.

In problems of the type What percentage is a number?, the questions are most often set in practical contexts. A student encounters them, for instance, when calculating what percentage of all students in a class are boys, or what fraction of the entire route a cyclist has covered.

Create a worksheet for this skill

Calculating square roots

The student calculates exact values of square roots of rational numbers that are squares of other 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, the student efficiently finds the values of square roots of numbers that are squares of rational numbers. This means proficiently finding the non-negative number that, when squared, gives the radicand — with respect to integers, as well as common and decimal fractions.

A common mistake at this stage is confusing square roots with dividing by two (for example, assuming that the square root of 16 is 8). Students also struggle to correctly determine the number of decimal places when taking square roots of decimals, which leads to incorrect results, such as assigning a square root of 0.2 to the number 0.4.

Tasks from the template Calculate the square root involve directly determining the numerical value of a root. On the mock exam, this skill appears on its own in arithmetic problems or serves as an initial step for further calculations, for example, when finding the side lengths of plane figures based on their areas.

Create a worksheet for this skill

Calculating the area of a trapezoid

The student is able to correctly calculate the area of a trapezoid based on the lengths of its bases and its height, efficiently applying the geometric formula.

Curriculum point MAT.IV-VI.4.14 calculates the area of a triangle, square, rectangle, rhombus, parallelogram, trapezoid and the areas of figures that can be built from them (our translation)

An eighth-grader preparing for the exam is able to determine the area of a trapezoid by correctly identifying the pair of parallel bases and the height perpendicular to them in the figure. This skill, established in grades 4–6, requires proficiency in performing arithmetic operations and applying the area formula or decomposing the polygon into simpler shapes — a rectangle and triangles.

The most common mistake is confusing the height of a trapezoid with the length of its slanted leg, especially when the figure is not a right trapezoid. Students also tend to have difficulties with the correct order of operations: they forget to place the sum of the bases in parentheses before multiplying by the height, or they forget to divide the final result by two.

On the exam paper, tasks following the Area of a trapezoid template appear as geometric drawings with given dimensions or as figures placed on a square grid, from which the student reads the lengths of the segments independently. A task may also require calculating a missing base or height when the area of the figure is already given.

Create a worksheet for this skill

Calculating the area of a triangle

The student calculates the area of a triangle based on the length of the base and the height dropped to it, using the appropriate geometric formula.

Curriculum point MAT.IV-VI.4.14 calculates the area of a triangle, square, rectangle, rhombus, parallelogram, trapezoid and the areas of figures that can be built from them (our translation)

As part of preparation for the eighth-grade exam, the student determines the area of a triangle using the relationship between the length of a side and the height drawn to that side or its extension. This is a skill introduced in grades 4–6, which on the exam paper is used to solve plane geometry problems—both standalone tasks and those requiring calculating the areas of composite figures.

A typical mistake made by eighth-graders is omitting the division by 2, which leads to giving the area of a parallelogram instead of a triangle. Students also struggle to correctly associate a side with its corresponding height: they multiply segments that are not perpendicular to each other, or confuse the length of a triangle's side with its height (especially in obtuse-angled triangles, where the height lies outside the figure).

In tasks from the Area of a triangle template on the mock exam paper, the student works with an auxiliary drawing showing dimensions or reads the data from a grid. The instructions require identifying the correct line segments, substituting them into the formula, and correctly performing the multiplication and division.

Create a worksheet for this skill

Arithmetic Mean in Everyday Situations

The student calculates the arithmetic mean of everyday numerical data, such as expenses, and draws practical conclusions from it.

Curriculum point MAT.IV-VI.5.3 calculates the arithmetic mean when describing everyday phenomena (our translation)
Curriculum point MAT.IV-VI.5.4 interprets data on expenses, calculates their average value, and formulates conclusions regarding the cost-effectiveness of different solutions – economic-financial module. (our translation)
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, a student can sum collected values and divide the obtained result by the number of data points to determine their arithmetic mean. They use this tool to describe everyday phenomena and analyze expenses and finances, which allows them to compare different data sets and evaluate the cost-effectiveness of various solutions.

A common calculation error is dividing the sum by the incorrect number of components—for example, omitting values equal to zero from the denominator, even though they represent a legitimate element of the data set. Students also often forget the order of operations when writing the calculation as a single expression, which leads to dividing only the last added term by the number of elements instead of the entire sum.

On the exam paper, in tasks based on the Average in everyday life template, the student works with data presented in a short text or a simple table. The task most often involves calculating an average cost, an average score, or comparing two sets of data and formulating a concise conclusion.

Create a worksheet for this skill

Calculating the missing value from a proportion

The student can identify a directly proportional relationship and calculate a missing value, for example, the cost or weight for a different number of items.

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)
Curriculum point MAT.IV-VI.6.3.d uses proportional reasoning, also as a method facilitating multiplication and division. (our translation)

In Grade 8, students use the properties of directly proportional quantities to find an unknown value based on related data. Using multiplication and division, they can reduce quantities to a common unit (for example, by calculating the price per single item) or apply the equality of products to find the required number.

A common mistake is using additive reasoning instead of proportional reasoning. Instead of multiplying and dividing, students try to add or subtract the difference between known values—for example, assuming that because the number of items increased by 2, the total cost will also increase by 2 PLN, rather than by the price of two items.

On the exam paper, this skill appears in tasks such as How much will it be for a different number of items?. In these problems, students are given information about the price, weight, or volume of a set of products and must determine what that quantity will be when the number of purchased items changes.

Create a worksheet for this skill

Calculating distance, speed, and time

The student calculates the distance traveled by an object given its speed and time, correctly applying and reconciling the units km/h and m/s.

Curriculum point MAT.IV-VI.2.12 in a practical context calculates: distance given speed and time, speed given distance and time, time given distance and speed, and uses the units of speed km/h and m/s (our translation)

The student solves problems set in a practical context by determining the distance traveled based on the given speed and travel time. They efficiently use basic units of speed, such as km/h and m/s, ensuring unit consistency before proceeding with multiplication.

The most common mistake at this stage is directly multiplying quantities without first standardizing the units — for example, multiplying speed in km/h by time given in minutes. The student then forgets to convert minutes into a fraction of an hour, leading to an incorrect and unrealistic result.

In exam preparation tasks, such as “What distance will they cover?”, the student analyzes a concise description of the movement of a pedestrian, cyclist, or car. The task consists of correctly reading the speed and time data, performing any necessary unit conversions, and calculating the final distance.

Create a worksheet for this skill

Operations on Powers and the Missing Exponent

The student applies the rules of exponents with natural bases and integer exponents to simplify expressions and determine an unknown exponent.

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)

An eighth-grader is able to transform expressions containing powers with natural number bases and integer exponents (positive, negative, or equal to zero). In doing so, they use the rules for multiplying and dividing powers with the same base and raising a power to a power, reducing complex expressions to their simplest form.

The most common mistake is confusing the rules and multiplying exponents instead of adding them when multiplying powers with the same base. Subtracting negative exponents during division also causes trouble, where it is easy to make an error with signs.

In the practice exam, the task in this area consists of filling in the missing exponent in an equation. The student must combine several powers with the same base using the appropriate operations to determine what number should be in the exponent of the result.

Create a worksheet for this skill

Dividing a quantity in a given ratio

The student divides a given quantity into two or three parts in a given ratio and calculates their values in practical situations.

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)

An eighth-grader is able to divide a given quantity into two or three parts according to a given numerical ratio. They understand that a ratio specifies the proportions between fragments of a whole, which is why they first determine the value of one base part and then calculate the sizes of individual allocations in a practical context.

A typical mistake involves dividing the entire quantity by the number of participants in the division (for example, by 2 or 3) instead of by the sum of the numbers forming the ratio (e.g., by 5 in a division with a ratio of 2 : 3). Students also sometimes struggle to maintain the order of the data, which leads to assigning the larger share to the person or quantity that was meant to receive fewer units.

In tasks of the Division in a Given Ratio type, the student tackles practical problems, such as sharing profits from joint work, the proportions of ingredients in cooking recipes, or cutting a line segment into fragments in a given ratio (e.g., 1 : 4 or 2 : 3 : 5). The worksheet requires them to sum the shares and calculate the specific value of one or all of the parts.

Create a worksheet for this skill

Factoring out terms from under the radical sign

The student transforms square and cube roots by correctly taking a number out of the radical sign.

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 the eighth grade, a student preparing for the exam efficiently simplifies square and cube roots. They decompose the radicand (the number under the radical sign) into a product of factors, at least one of which is a perfect square or cube, and then evaluate that root and write the value in front of the radical.

A common problem is incompletely factoring out a term, meaning choosing too small a divisor (for example, writing 2√12 instead of 4√3). Students also confuse the rules for different radical degrees: with cube roots, they tend to look for pairs of identical factors instead of triplets, or they divide the radicand by the degree of the root instead of properly calculating its root.

On the exam paper, the task tests the instruction Take the factor outside the radical. An eighth-grader is required to transform a single square or cube root into the product of an integer and a radical, which is also an essential step in further comparing numbers and adding radical expressions.

Create a worksheet for this skill

Scientific notation and comparing numbers

The student writes numbers in scientific notation and compares values 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)

An eighth-grader preparing for the exam is able to write numbers in scientific notation and compare quantities given in this form. They efficiently convert numbers from decimal notation into the product of a number in the range from 1 to 10 (excluding 10) and an appropriate power of 10, as well as determine greater-than and less-than relationships between them.

A typical mistake results from ignoring the condition for the first factor—students often accept notations where the first number is less than 1 or greater than or equal to 10 (for example, writing 25 multiplied by a power of ten instead of 2.5). When comparing, it is also common to rely solely on the exponent of the power of ten without paying attention to the value of the initial factor.

On the exam paper, tasks from the Scientific notation template most often involve converting a large or small quantity into correct scientific notation or ordering several given numbers from least to greatest.

Create a worksheet for this skill

Multiplying Negative Numbers and Sign Rules

The student correctly determines the sign of the result and performs straightforward multiplication of negative numbers in exam tasks.

Curriculum point MAT.IV-VI.1.32 performs uncomplicated calculations involving rational numbers (our translation)

An eighth-grader performs simple calculations on rational numbers, skillfully applying sign rules when multiplying negative numbers. They know the rule stating that the product of numbers with the same signs is positive, and with different signs — negative, and can determine the sign of the result without hesitation before performing the actual multiplication.

The most common problem is the mechanical application of the rule "two minuses make a plus" in inappropriate places, for example when adding negative numbers (the expression -3 - 5 is sometimes mistakenly assumed to yield a positive result). Students also sometimes have difficulty correctly counting the minuses when multiplying three or more factors, forgetting that an odd number of negative signs always produces a negative result.

On the exam paper, tasks based on the Multiplication of negative numbers template test computational fluency in simple arithmetic operations. They require calculating the direct result of multiplication or identifying the correct value in closed-ended tasks, where proper sign manipulation determines the choice of the correct answer.

Create a worksheet for this skill

Multiplying two binomials

The student multiplies two binomials with integer coefficients and writes the result as a simplified algebraic sum.

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

In grade 8, a student can multiply two binomial expressions containing only integers (both positive and negative). The task involves systematically multiplying each term in the first set of parentheses by each term in the second set of parentheses, followed by arranging and combining like terms.

The most common difficulties arise from confusing signs when multiplying negative numbers (for example, overlooking that the product of two negative numbers is positive). Another typical mistake is multiplying only the first terms together and the last terms together, omitting the "cross" terms.

On the exam paper, tasks based on the Multiplying binomials template involve simplifying an expression given as the product of two sets of parentheses—for example, (2x - 3)(x + 4)—and reducing it to the simplest form of an algebraic sum, or selecting the correct answer from the given options.

Create a worksheet for this skill

Calculating the probability of drawing a ball

The student is able to determine the probability of drawing a ball of a specified color from a set of balls in a box and write the result as a fraction.

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)

An eighth-grader is able to analyze a simple random experiment and calculate the probability of a specified event. In practice, this means determining the total number of all possible outcomes (the total number of balls in a container) and the number of favorable outcomes (balls meeting the given condition, e.g., of a specific color), and then expressing this relationship as a fraction.

A typical mistake is placing the number of remaining objects in the denominator of the fraction instead of their total sum—for example, dividing the number of white balls by the number of black balls instead of by all the balls in the box. Students also make mistakes when the problem requires accounting for adding or removing a ball, forgetting to update the total number of all elements.

On the exam paper, this skill is tested in problems from the template Drawing a ball from a box. The problem describes a container with balls of various colors and requires calculating the probability of drawing a ball with a specific characteristic or determining how many balls need to be added or removed to achieve the desired probability.

Create a worksheet for this skill

Calculating the volume of a regular pyramid

The student calculates the volume of a regular pyramid using the base area and the height of the solid according to the formula.

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

In the eighth grade, an exam-preparing student is able to calculate the volume of a regular pyramid. The task requires efficiently determining the area of the base polygon and relating it to the height of the solid using the appropriate formula.

The most common mistake is omitting the fraction one-third in the formula for the volume of a pyramid, which makes the calculated result three times too large—corresponding to the volume of a prism with the same dimensions. Students can also be careless when reading the given data, confusing the height of the entire solid with the slant height of its lateral face.

A task of the Volume of a pyramid type on the exam paper is based on a reference diagram or a verbal description of the solid. Based on the given edges or heights, the eighth grader must calculate the base area, substitute the data into the formula, and perform the arithmetic operations correctly.

Create a worksheet for this skill

Solving equations with variables on both sides

The student solves first-degree equations in which the variable appears on both sides of the equals sign, efficiently rearranging terms and determining the result.

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 eighth grade, the student fluently transforms and solves linear equations where the unknown appears on both sides of the equals sign. They can reduce the equation to a simpler form by moving terms with the unknown to one side and known numbers to the other using addition or subtraction on both sides.

A typical mistake at this stage is failing to change the sign to its opposite when moving a term to the other side of the equation. Students also often drop minus signs when combining like terms or divide both sides by the wrong number (e.g., subtracting the coefficient instead of dividing by it).

On the exam paper, a task based on the template Equation with an unknown on both sides requires finding the number that satisfies a given algebraic condition. The task may be in a closed-ended format—where the student selects the correct value from the given options—or open-ended, where a complete sequence of transformations leading to the solution must be written down.

Create a worksheet for this skill

Checking whether a number is a solution to an equation

The student checks whether a given integer or rational number satisfies a given equation by substituting it for the unknown 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 eighth grade, a student is able to check the correctness of a solution without having to transform the entire equation. They check whether a given integer satisfies an equation with one unknown of the first, second, or third degree, and in the case of any rational numbers (e.g., fractions), they verify first-degree equations. It is enough to substitute the specified value in place of the variable and calculate the values of the left and right sides separately.

The most common mistake involves losing minus signs when substituting negative numbers—especially during exponentiation or subtraction—as well as following an incorrect order of operations. Students also often unnecessarily attempt to solve the entire equation from scratch (which is difficult for second or third-degree equations), instead of simply performing the calculations with the given numbers.

On a practice test, a task based on the template Is the number a solution to the equation? usually takes the form of a closed-ended question where an equation and a specific number are provided. The student's task is to calculate both sides and clearly evaluate whether the obtained results are equal.

Create a worksheet for this skill

Evaluating an algebraic expression

The student calculates the numerical value of an algebraic expression consisting of up to three monomials, substituting the given integers for the letters.

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)

As part of the preparation for the eighth-grade exam, the student is able to substitute given integers for variables and calculate the final result. Operations are performed on algebraic expressions that are the sum of at most three monomials, such as 2a − 3b + 5 or x² − 4x.

The most common source of mistakes is ignoring parentheses when substituting negative numbers. Students often confuse the notation (−2)² with −2² or drop the minus sign when subtracting a negative value (for example, writing 5 − 3 instead of 5 − (−3)). Violations of the order of operations also occur, especially multiplying before performing exponentiation.

In tasks of the type Value of an algebraic expression, the student receives a specific expression and integer values of the variables specified in the problem statement. Their task is to correctly perform the arithmetic calculations step by step and indicate the correct answer in a closed-ended question or write down the result.

Create a worksheet for this skill

Combining like terms in expressions

The student simplifies monomials and adds and subtracts like terms with integer coefficients, reducing algebraic expressions to their simplest form.

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 eighth grade, a student efficiently simplifies monomials and adds and subtracts algebraic sums consisting of several terms with integer coefficients. They can recognize monomials with an identical variable part and combine them into a single term by performing operations on their numerical coefficients.

The most common mistake is losing the minus sign in front of a given term while simplifying a sum. Students also often combine terms that are not like terms—for example, adding monomials containing different letters, confusing variables with different exponents, or forgetting that a standalone minus sign before a letter represents a coefficient equal to -1.

Tasks from the template Combining like terms involve simplifying a given algebraic expression and writing it in its simplest form. On a practice exam, students encounter this topic as multiple-choice questions or as an essential step in solving equations as well as word problems and geometry open-ended tasks.

Create a worksheet for this skill

Systems of two equations in word problems

The student solves word problems by writing the relationships between two unknowns as a simple system of equations and reducing it to a single equation with one unknown.

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, the student solves word problems using systems of two linear equations with two unknowns. At this stage, systems that can be easily reduced to a single equation with one unknown are required—most often by immediately expressing one quantity and substituting it into the second expression.

A typical problem on the exam is losing signs (especially minus signs) when transforming expressions in parentheses, as well as failing to account for all the conditions of the problem when setting up the second equation. Students also often stop after determining just one of the unknowns, forgetting to complete the calculations for the second unknown quantity.

A problem based on the System of two equations template usually takes the form of an open-ended question with a practical context, such as ticket prices, the number of coins, or the ages of two people. The eighth-grader must correctly define the unknowns, set up the system of relations, write down a clear calculation process, and provide the final answer.

Create a worksheet for this skill

Calculating interest on an annual deposit

The student calculates the interest earned on a one-year bank deposit and the final account balance at a given annual 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)

An eighth-grader can calculate the profit from capital deposited in a bank for a period of one year. In practice, this means skillfully applying percentage calculations: the student converts the given annual interest rate into a fraction and multiplies it by the initial deposit amount, determining the amount of interest due.

The most common mistake in this type of problem is confusing the amount of interest alone with the total amount paid out when the deposit matures. Students often give only the accrued profit as the answer, forgetting to add the deposited principal, or vice versa—they provide the total account balance even though the question asked only for the value of the added interest.

Tasks from the Interest on a deposit template describe a realistic savings scenario: the student is given an initial amount (principal) and the annual interest rate. Their task is to correctly calculate the annual profit or determine the total amount the customer will have after one year.

Create a worksheet for this skill

Calculating a percentage of a given number

An eighth-grader is able to calculate a given percentage of a number and identify the correct result in an exam task.

Curriculum point MAT.IV-VI.1.33 interprets 100 % of a given quantity as a whole, 50 % – as a half, 25 % – as one quarter, 10 % – as one tenth, 1 % – as one hundredth part of this quantity (our translation)
Curriculum point MAT.IV-VI.1.34 calculates a percentage of a given quantity in cases set in a practical context (our translation)
Curriculum point MAT.VII-VIII.1.1.b a given percentage of a given number (our translation)

In the eighth grade, the student proficiently calculates a given percentage of a number. The task involves converting the percentage into a common fraction or a decimal and multiplying it by the given numerical value, which allows for quickly determining the desired quantity.

A typical error results from incorrectly writing the percentage as a fraction. Students often confuse single-digit values with tens—for example, writing 4% as 0.4 instead of 0.04, which leads to overestimating the final result tenfold.

In the practice exam paper, this skill is tested in a task of the type Percentage of a number — choose the answer. The student performs the necessary calculations in rough notes and then selects one correct option from the given variants A, B, C, or D.

Create a worksheet for this skill

Task templates to print

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

Pythagorean theorem in right-angled triangles

Calculating price increases and decreases by a percentage

Calculating powers of numbers with a natural exponent

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

Calculating the circumference and the area of a circle

Probability in two tosses or draws

Surface area and volume of a triangular prism

Solving two-step linear equations

Calculating a whole number from a given percentage

Dividing common fractions

Multiplying brackets and factoring out

Calculating what percentage of one number another number is

Calculating square roots

Calculating the area of a trapezoid

Calculating the area of a triangle

Arithmetic Mean in Everyday Situations

Calculating the missing value from a proportion

Calculating distance, speed, and time

Operations on Powers and the Missing Exponent

Dividing a quantity in a given ratio

Factoring out terms from under the radical sign

Scientific notation and comparing numbers

Multiplying Negative Numbers and Sign Rules

Multiplying two binomials

Calculating the probability of drawing a ball

Calculating the volume of a regular pyramid

Solving equations with variables on both sides

Checking whether a number is a solution to an equation

Evaluating an algebraic expression

Combining like terms in expressions

Systems of two equations in word problems

Calculating interest on an annual deposit

Calculating a percentage of a given number

Build a practice exam

Common questions

How long does the eighth-grade mathematics exam last?

125 minutes. Students who have been granted accommodations may write for up to 40 minutes longer.

How many points can you score and how many tasks are there in the exam paper?

30 points for 20–21 tasks: 14–15 closed-ended worth 1 point each and 5–6 open-ended worth 2 or 3 points.

Can you fail the eighth-grade exam?

No. The exam does not have a passing threshold — the result is given as a percentage and on a percentile scale, and secondary schools take it into account during admissions.

How to print a practice exam?

Click “Create printable mock exam”. The generator will automatically assemble the exam paper in the CKE layout — you can immediately download a PDF with the tasks, answer key, and solutions, or modify selected sections.