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Curriculum (Poland)

Mathematics — Grade 7

Grade 7 Mathematics: powers and roots, scientific notation, percentages in practice, proportional division, algebraic expressions, linear equations, the Pythagorean theorem, circles, and introductory probability. Ready-to-print worksheet templates for every topic.

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What does a student learn in 7th grade mathematics?

In grade 7, students perform operations with powers and roots, estimate roots, and write large and small numbers in scientific notation. They calculate price increases, discounts, and interest, and divide quantities in a given ratio. They simplify algebraic expressions, multiply algebraic sums, and solve linear equations. In geometry, they learn the Pythagorean theorem, the circumference and area of a circle, and in probability theory — events in simple experiments.

Important note regarding the curriculum: the new core curriculum (Journal of Laws 2026, item 378) will apply to grade 7 starting from the 2029/2030 school year — it is being introduced cohort by cohort. Until then, grade 7 follows the 2017 curriculum; the scope of topics on this page is similar in both curricula. The curriculum for grades VII–VIII is organized in blocks — it defines what a student should know by the end of primary school. Assigning a topic to a specific grade is our decision based on a typical course schedule; in the "Curriculum Scope" section, you can see the entire VII–VIII block with quotations from the ministerial 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, as well as probability are ready. Geometric constructions and eighth-grade exam style tasks are being added consecutively.

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 prices after a percentage increase and decrease (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

    Exponentiation 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 negative exponents (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 missing exponents (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 roots 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 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

    Proportional division into two or three parts (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 missing exponents (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 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

  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 missing exponents (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 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

    Finding a number given a percentage of it (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 missing exponents (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 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 algebraic expressions (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 through brackets and factoring out common terms (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 brackets and factoring out common terms (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 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 satisfies 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 satisfies 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 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

  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 and 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

  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

    Pythagorean theorem and right-angled triangles (we teach in grade 7-8) · 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

  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 random draws (we teach in grade 7-8)

  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 and right-angled triangles

The student recognizes a right triangle and calculates the length of the missing side – a leg or the 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 7, students learn and apply the relationship between the sides of a right triangle: equating the sum of the squares of the lengths of the legs to the square of the length of the hypotenuse. Based on this, they can calculate the length of the missing side when the other two are known, and verify whether a triangle with three given side lengths is a right triangle.

A typical mistake is mechanically adding the squares of the given numbers without determining which side is the hypotenuse opposite the right angle. As a result, students add the squares of a leg and the hypotenuse together instead of subtracting when finding a missing leg. Another common error is forgetting to take the square root of the final result.

In practice, students solve problems based on three templates:

  • Calculate the hypotenuse – square the lengths of the legs, add them, and take the square root of the result;
  • Calculate a leg – subtract the square of the known leg from the square of the hypotenuse to find the unknown side;
  • Is the triangle a right triangle? – check whether the square of the longest side is equal to the sum of the squares of the two shorter sides.
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 probability in simple random draws

The student learns to determine the chances of random events and write probability as a fraction during simple experiments, such as rolling a die or drawing a ball.

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 7, students take their first steps in probability theory. They analyze simple random events: determine the total number of possible outcomes and the number of favorable outcomes for a given event, and express the probability as a common fraction or a decimal.

A common mistake is comparing the number of favorable outcomes to unfavorable outcomes instead of to all possible outcomes. For example, when there are 2 white balls and 3 black balls in a container, children often describe the chance of drawing a white ball as 2 to 3 and write the fraction 2/3 instead of the correct 2/5.

Practice exercises use clear sampling models:

  • Rolling a die – the student determines the chance of rolling a specific number of dots, for example an even number or a number less than 5;
  • Drawing a ball from a box – calculates the probability of drawing a ball of a specified color from a closed set;
  • Drawing a slip of paper with a number – determines the chance of drawing a slip with a number satisfying a given condition (e.g., divisible by 3 from a set of numbers from 1 to 15).
Create a worksheet for this skill

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

Calculating prices after a percentage increase and decrease

The student learns to calculate a new price or value after an increase or decrease by a given percentage in everyday financial situations.

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 7, students apply percentage calculations in a practical context, determining a new value after a price change. At this stage, the student learns to calculate the amount of an increase or decrease based on the initial price, and then add it to or subtract it from the base value to determine the final purchase cost.

A common mistake at this level is treating a percentage like a regular amount and subtracting it directly from the price (for example, writing the calculation 80 zł − 15% as 80 − 15 = 65 zł). Remembering that a percentage always refers to a specific value and that one must first calculate a fraction of the given number can be challenging.

Exercises practicing this skill are based on the templates Increase by a percentage and Decrease by a percentage. The student solves simple arithmetic problems where they are given the initial price along with information about the change (e.g., an item costing 120 zł is discounted by 20%) and determines the new, final price after the discount or markup.

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

Multiplying through brackets and factoring out common terms

The student learns to multiply a monomial by a sum in parentheses and to factor out a common factor in expressions with integer coefficients.

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 7, students perform basic algebraic transformations on integers. They master multiplying a monomial by a sum consisting of at most three monomials, as well as the reverse operation: factoring out a common monomial from a sum of two monomials.

The most common mistake when multiplying through parentheses is multiplying only the first term and leaving the rest unchanged, or making sign errors when multiplying by a negative number. In turn, when factoring out, difficulty arises when one of the terms is identical to the factored-out factor—students often omit it or write zero instead of the number 1.

The templates Multiplying a Monomial by a Sum and Factoring Out a Common Factor are designed for regular practice. The exercises involve expanding the product and simplifying expressions with variables, or finding the greatest common divisor of the numbers along with the common variable part and writing the expression in product form.

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Task types: Expand brackets · Factor out a common factor

Calculating square roots and cube roots

The student calculates the exact values of square and cube roots of rational numbers and determines their approximations using a calculator.

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 7, students learn to determine the exact values of square roots and cube roots of numbers that are squares or cubes of rational numbers, respectively. This involves working fluently with integers, common fractions, and decimals. In the case of numbers that do not have an exact rational root, students use a calculator to find their approximate value.

A common mistake at this stage is confusing finding a root with dividing by the index of the root—for example, treating a square root as dividing by 2 (mistakenly assuming that the square root of 16 is 8 instead of 4) or a cube root as dividing by 3. Students also sometimes struggle to determine the correct sign of the result when taking the cube root of a negative number.

In practice, exercises are based on the templates Calculate the square root and Calculate the cube root. The tasks involve directly finding the number that, when raised to the second or third power respectively, yields the value under the radical symbol.

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

Exponentiation of Integers and Fractions

The student calculates the values of powers with natural exponents for integers, common fractions, and decimals.

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

In Grade 7, students learn to raise rational numbers—including positive and negative integers as well as fractions—to powers with natural exponents (such as the second, third, or fourth power). They understand exponentiation as repeated multiplication of the same number by itself and efficiently determine the sign of the result depending on whether the exponent is even or odd.

A common mistake at this stage is confusing exponentiation with simple multiplication of the base by the exponent (for example, calculating 3² = 6 instead of 9). When raising common fractions to a power, students often raise only the numerator to the power while leaving the denominator unchanged, which leads to an incorrect result.

In practice, students encounter tasks based on two templates:

  • Calculate the power of an integer — involving determining the value of powers of positive and negative numbers,
  • Calculate the power of a fraction — requiring correctly raising a common fraction or a decimal to a power.
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

Adding and subtracting signed decimals

The student proficiently adds and subtracts positive and negative decimals, correctly determining the sign of the result.

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

In Grade 7, the student performs straightforward calculations with rational numbers, focusing on adding and subtracting signed decimals—both positive and negative numbers.

A common mistake at this stage is confusing the rules for determining the sign of the result. Students often get lost when subtracting a negative number (where two minus signs turn into addition) or mistakenly assume that the sum of two negative numbers yields a positive result. Keeping track of the sign while accurately calculating the decimal parts also poses a difficulty.

Practical exercises use task templates: Adding signed decimals and Subtracting signed decimals. The tasks involve directly calculating the values of single arithmetic expressions, such as –2.4 + 1.1 or 0.5 – (–1.3).

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Task types: Add signed decimals · Subtract signed decimals

Multiplying and dividing negative numbers

The student performs simple calculations on rational numbers, correctly applying the rules of signs when multiplying and dividing negative numbers.

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

At this stage, the student performs simple calculations with rational numbers, focusing on mastering the rules for determining the sign of the result. They efficiently multiply and divide positive and negative numbers, separating the decision about the final sign from the calculation of the numerical values itself.

The most common mistake is transferring the rules of multiplication to addition. Students remember the rule that "two minuses make a plus" and incorrectly apply it when adding negative numbers. It is also common to drop the minus sign when multiplying or dividing one negative number and one positive number.

Tasks practicing this skill are based on the templates Multiplying negative numbers and Dividing negative numbers. They take the form of direct arithmetic operations in which the student calculates products and quotients of pairs of numbers with the same or opposite signs.

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Task types: Divide signed numbers · Multiply signed numbers

Calculating the circumference and area of a circle

The student calculates the circumference and area of a circle based on a given radius or diameter, correctly 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 7, students learn how to determine the circumference of a circle and the area of a circle. They use formulas relating these quantities to the radius or diameter of the figure and the number π. They write the result in exact form (with the symbol π) or calculate an approximate value by substituting, for example, 3.14 for π.

The most common difficulty is confusing the formulas: students often double the radius instead of squaring it when calculating area. Another typical mistake is directly substituting the diameter given in the problem into the formula instead of the circle's radius.

The exercises in the templates Circumference of a Circle and Area of a Circle involve straightforward calculations of circumference or area based on the radius or diameter, as well as inverse operations—finding the radius of the figure when its circumference or surface area is already known.

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

Multiplying Binomials

The student learns to multiply two binomials with integer coefficients and write the result in standard form.

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

In Grade 7, students multiply two binomials involving only integer coefficients (both positive and negative). They master the rule of multiplying each term in the first set of parentheses by each term in the second set of parentheses, starting with simpler forms such as (x + a)(x + b).

A common mistake is omitting the middle terms—students often multiply only the first terms together and the second terms together, giving an incorrect result (for example, writing (x + 2)(x + 3) as x² + 6 instead of accounting for all four products). Another frequent issue is losing signs, especially when subtraction appears in a binomial and two negative numbers must be multiplied together.

In practice, the tasks are based on the templates Multiplying binomials (x + a)(x + b) and Multiplying binomials. The student receives the product of two expressions in parentheses, expands it into the sum of individual products, takes care of the proper plus and minus signs, and finally combines like terms.

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

Combining like terms in expressions

The student learns to arrange monomials and add and subtract like terms with integer coefficients in 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 7, the student learns to simplify algebraic sums by efficiently identifying monomials with the same letters. The child organizes written monomials and adds and subtracts like terms with integer coefficients, working with expressions that have one variable or two different variables.

The most common mistake is losing the minus sign in front of a given monomial and combining terms that differ in their variables or exponents. Children also tend to struggle with monomials that do not have a visible number in front, forgetting that writing x means a coefficient of 1, and -x corresponds to the number -1.

In practice, the student practices this skill using the templates Combining like terms and Combining like terms with two variables. The tasks consist of transforming an expanded expression into its simplest possible form by adding the coefficients of identical letters and separately adding the constant terms.

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

Finding a number given a percentage of it

The student learns to reconstruct the whole quantity, knowing only the value of its part and the corresponding percentage.

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

In Grade 7, the student can determine the entire initial value (100%) given information about a part of it and its corresponding percentage. To solve the problem, they convert the percentage to a fraction and divide, use the proportion method, or use an intermediate step: first determining what 1% of the given quantity is, and then multiplying the result by 100.

A common mistake is confusing this operation with finding the percentage of a given number. Instead of dividing the value by the percentage, the student mechanically multiplies the given quantities—for example, when asked for a number whose 20% is 40, they answer 8 (calculating 20% of 40) instead of the correct 200. Another frequent difficulty is forgetting to convert the percentage to a fraction before dividing.

In tasks from the Number from a given percentage template, the student tackles problems such as “Find the number whose 15% is 45” or simple real-world word problems where they must reconstruct an initial salary, the total distance of a trip, or the total capacity of a tank based on information about a completed part.

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Calculating what percentage one number is of another

The student learns to compare two quantities and express their ratio as a percentage, determining what part of the whole a given number represents.

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

In Grade 7, students learn to compare two values and express their relationship as a percentage. The solution involves writing the quotient of two numbers as a fraction (where the part being examined goes in the numerator, and the reference quantity goes in the denominator), and then converting this fraction into a percentage—either by expanding it to a denominator of 100 or by multiplying by 100%.

The most common mistake at this stage is swapping the numerator and the denominator. Students tend to automatically divide the larger number by the smaller one to avoid a fraction less than 1. For example, when answering what percentage of 40 the number 8 is, they might divide 40 by 8 instead of correctly setting up the fraction 8/40 and converting it to 20%.

In exercises based on the template What percentage is a number?, students encounter tasks requiring direct calculation of the relationship between two numbers (e.g., determining what percentage of a whole a given part represents) as well as simple real-world scenarios, such as calculating the percentage of people present in a group or points scored on a test.

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Determining the median of a dataset

The student organizes a set of numerical data and determines its median, also in the context of simple financial data.

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 7, students learn how to determine the median of a given set of numerical data, including in practical and financial contexts (e.g., related to prices or savings). To identify this middle value, students arrange the numbers in non-decreasing order: for an odd number of elements, they identify the number located exactly in the middle, and for an even number, they calculate the arithmetic mean of the two middle values.

A common mistake at this stage is looking for the middle number directly in an unsorted list, without first arranging the numbers from smallest to largest. Students also sometimes struggle when a data set has an even number of values—instead of calculating the mean of the pair of middle numbers, they provide both at once or choose just one of them.

Tasks from the Find the median template involve reading a set of numbers, ordering it, and correctly determining the middle value.

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Powers with negative exponents

The student learns to convert powers with a negative integer exponent into corresponding powers with a natural exponent, using the reciprocal of the base.

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

In 7th grade, students master transforming powers with negative integer exponents. The task consists of converting such an expression into a power with a natural exponent—by finding the reciprocal of the base and dropping the minus sign in the exponent.

A common mistake at this stage is treating the minus sign in the exponent as a negative sign for the entire result or confusing exponentiation with multiplication. Students often assign a negative value to the result instead of taking the reciprocal of the base.

Tasks based on the Power with a negative exponent template require students to convert the given numerical expression into a form with a natural exponent and then calculate its final value.

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Operations on powers and missing exponents

The student applies the rules of multiplication and division of powers with natural bases and integer exponents to determine missing numbers in equations.

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 7, students learn to efficiently use the exponent rules with natural number bases and integer exponents. At this stage, students combine the rules for multiplying and dividing powers with the same base with their knowledge of negative numbers, transforming expressions without the need to tediously multiply out large values.

The most common mistake is confusing adding exponents with multiplying them when multiplying powers with the same base. Negative exponents present an additional challenge—when dividing powers, students often forget that subtracting a negative number results in addition, which leads to an incorrect sign in the final result.

In practice, using the worksheets Missing Exponent — Operations on Powers, students solve equations with a single blank space. The task involves analyzing the operation (e.g., the product or quotient of powers) and entering an integer in the exponent so that the left side of the equation equals the right side.

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Proportional division into two or three parts

The student learns to divide a given quantity into two or three parts according to a given ratio, solving practical everyday 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 7, the student learns to divide a given quantity into two or three parts in a given numerical ratio (for example, 2 : 3 or 1 : 2 : 4). They apply this skill in a practical context, determining specific values for each part based on their mutual relationship.

A common mistake is dividing the original number by the individual terms of the ratio instead of by their sum. For example, a student might try to divide an amount in the ratio 2 : 3 by dividing it directly by 2 or 3, instead of first determining that the whole consists of 5 equal parts and calculating the value of one such part.

In practice, using the Division in a Given Ratio template, the student tackles word problems in which they divide, among other things, sums of money, masses of ingredients, or lengths of segments, calculating the exact size of each separated part.

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Taking a factor out of the radical sign

The student learns to simplify square and cube roots by factoring the radicand and taking factors out of the radical.

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 7th grade, students learn how to simplify square roots and cube roots. Using the properties of operations on radicals, they factor the number under the radical sign into a product of factors, one of which is a perfect square (e.g., 4, 9, 16, 25) or a perfect cube (e.g., 8, 27), and then take its root and write the result in front of the radical sign.

A common mistake is copying the identified factor in front of the radical sign without taking its root—for example, writing that √12 is 4√3 instead of 2√3. Students also sometimes confuse the index of the root, such as looking for perfect squares inside cube roots.

Exercises from the Take factor out of the radical template involve simplifying a given radical to its simplest form, such as changing √50 to 5√2 or ∛24 to 2∛3.

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Estimating the value of roots

The student learns to determine the approximate value of a root and indicate between which two consecutive integers it lies.

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

In 7th grade, students develop the skill of estimating the value of square roots that do not result in an integer. The student does not need to calculate the exact decimal expansion – they learn to bound a given root from below and above using known squares of natural numbers, for example, by determining that the value of √20 lies between 4 and 5.

A typical mistake at this stage is confusing finding the square root with dividing by two. Students then try to estimate the value of √50 as close to 25, instead of referring to the nearest perfect squares (49 and 64) and noticing that the result is just over 7.

In practice, using worksheets from the Estimate the root template, students tackle tasks where they enter two consecutive integers bounding a given root or indicate which of them the number in question is closer to.

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Writing and comparing numbers in scientific notation

The student learns to write numbers in scientific notation and compare 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 7, students master writing numbers in scientific notation, which represents a number as the product of a value from 1 to 10 (excluding 10) and a power of 10. At this stage, students learn to convert numbers into this form, read them, and correctly compare pairs or sequences of quantities written in this way.

A common issue is overlooking the condition regarding the first factor. Students often write expressions such as 45 · 10³ instead of 4.5 · 10⁴, mistakenly considering any product with a power of ten to be scientific notation. Another typical error when comparing is looking solely at the first number rather than the exponent—a student might judge 7.2 · 10³ to be greater than 1.5 · 10⁵ simply because 7.2 is greater than 1.5.

Exercises from the Scientific notation template cover converting numbers from standard notation to scientific notation and vice versa. Students also practice comparing quantities: identifying the larger number, inserting the correct relational sign between two expressions, or ordering a set of numbers from least to greatest.

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Solving equations with variables on both sides

The student learns to solve linear 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 7, students master solving linear equations where the same unknown appears on both sides. The student learns to reduce such an expression to a simpler form by adding or subtracting terms with the unknown and constants from both sides, ultimately gathering the unknowns on one side and the numerical values on the other.

A typical problem at this stage involves sign errors (plus and minus) when performing operations on both sides of the equation. Students often forget to change the sign when moving a term or confuse the order of operations, and they also attempt to add numbers to terms with an unknown (e.g., treating the expression 3x + 5 as 8x).

In practice, using the Equation with an unknown on both sides template, the student is given a linear equation (e.g., 5x + 3 = 2x + 12). The task consists of performing step-by-step transformations, finding the value of the unknown, and providing the final numerical result.

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Checking whether a number satisfies an equation

The student checks whether a given integer or rational number satisfies the equation by substituting it in place of the unknown and comparing 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 7, students learn to verify 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), as well as whether a given rational number — including a common fraction or a decimal — is a solution to a first-degree equation.

A typical error results from carelessness with the order of operations and sign rules, especially when substituting negative numbers. Students often forget parentheses when raising a negative number to a power (confusing, for example, (-3)² with -3²) or mix up signs when multiplying by a negative. It is also common for a student to calculate only one side of the equation instead of separately evaluating the left-hand and right-hand sides to check if the equality LHS = RHS holds.

Tasks based on the template Is the number a solution to the equation? involve substituting the given number into the written formula and performing the calculations. The student writes down the calculations for the left and right sides and, on this basis, formulates a clear answer as to whether the tested number actually satisfies the given condition.

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

The student solves first-degree equations of the form ax + b = c by sequentially performing subtraction or addition, followed by division 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 7th grade, students learn to solve for the unknown in a linear equation of the form ax + b = c. They master a two-step procedure: first, adding or subtracting the constant term on both sides, and second, dividing both sides of the equation by the coefficient of the unknown.

A typical mistake at this stage is reversing the order of transformations—attempting to divide by the coefficient a before eliminating the number b, which often results in mistakenly omitting the division of the constant term. Students also tend to struggle with signs, confusing inverse operations (e.g., adding instead of subtracting) when eliminating numbers from the left side of the equation.

Exercises from the Equation ax + b = c template contain a specific numerical equation, for example 3x + 5 = 26 or 4x - 7 = 13. The student's task is to perform and write down the transformations step by step and provide the final value of the unknown.

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

Evaluating algebraic expressions

The student substitutes the given integers for the letters in an expression consisting of at most 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)

At this stage of learning, the student substitutes letters in an expression with specific integers—positive, negative, or zero. In accordance with the requirements for grades 7–8, the complexity of such expressions is limited to the sum of at most three monomials, which means performing substitutions in formulas with a simple structure (for example, with one or two variables).

A common mistake occurs when substituting negative numbers. Students often forget to enclose the substituted negative number in parentheses, confusing notations such as (-3)² with -3², or dropping minus signs when subtracting a negative number or multiplying it by a negative coefficient of a monomial.

In tasks based on the "Value of an algebraic expression" template, the student receives a ready-made expression and a set of numbers assigned to individual letters. The task consists of correctly replacing the symbols with integers, performing exponentiation, multiplication, and addition in the correct order, and writing down the final result.

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Calculating interest on an annual deposit

The student learns to calculate the amount of interest added after a year of saving and the total sum of money in the account at a given deposit 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 7, the student uses percentage calculations in a practical financial context, calculating the return on an annual bank deposit. The student calculates a percentage of the deposited amount (initial principal) to determine the amount of interest earned and the final savings balance after a full year.

The most common mistake is confusing the amount of interest with the total balance in the account—when asked for the savings balance after a year, the student provides only the calculated profit, forgetting to add the deposited principal. Another frequent error occurs when converting the interest rate into a decimal, for example, treating 4% as 0.4 instead of 0.04.

In exercises from the Interest on a deposit template, the student analyzes specific financial situations: given the amount deposited in the bank and the annual interest rate, they calculate the amount the bank will add to the principal after twelve months.

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Task templates to print

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

Pythagorean theorem and right-angled triangles

Calculating probability in simple random draws

Calculating prices after a percentage increase and decrease

Multiplying through brackets and factoring out common terms

Calculating square roots and cube roots

Exponentiation of Integers and Fractions

Adding and subtracting signed decimals

Multiplying and dividing negative numbers

Calculating the circumference and area of a circle

Multiplying Binomials

Combining like terms in expressions

Finding a number given a percentage of it

Calculating what percentage one number is of another

Determining the median of a dataset

Powers with negative exponents

Operations on powers and missing exponents

Proportional division into two or three parts

Taking a factor out of the radical sign

Estimating the value of roots

Writing and comparing numbers in scientific notation

Solving equations with variables on both sides

Checking whether a number satisfies an equation

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

Evaluating algebraic expressions

Calculating interest on an annual deposit

Other task types

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