English
Curriculum (Poland)

Mathematics — Grade 10

Mathematics in Grade 2 of high school and technical secondary school: quadratic function, quadratic equations and inequalities, polynomials, exponential function, trigonometry, and plane geometry — angles in a circle, similarity, the law of sines and the law of cosines. Ready-to-print exercises for each topic.

Create a worksheet for this curriculum

What does a student learn in mathematics in the 2nd grade of high school?

In Grade 2, the core of the curriculum is the quadratic function: vertex form, vertex, maximum and minimum values on an interval, quadratic equations and inequalities, and Viète's formulas. The student divides polynomials and solves polynomial and rational equations, learns about the exponential function and graph transformations. In trigonometry, they calculate function values in a right-angled triangle and use identities, and in plane geometry, they apply Thales's theorem, inscribed and central angles, similarity, as well as the law of sines and the law of cosines.

Important note about the core curriculum: the curriculum for general and technical secondary schools is formulated in blocks — it describes what the student knows by the end of secondary school, separately for the basic and extended levels. Assigning a topic to a specific grade is our decision based on a typical syllabus distribution; in the "Curriculum Scope" section, you can see the curriculum items with quotes.

Page status: the exercises for this grade are ready in non-contextual form (the instruction and formula only); problems with diagrams, proofs, and word problems will be added in subsequent steps.

Curriculum scope

  1. Curriculum point MAT.LO.1 · Teaching content · checked against the act

    Real numbers

    our translation · original wording (PL): Liczby rzeczywiste

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 1

  2. Curriculum point MAT.LO.1.P.1 · Teaching content · checked against the act

    performs operations (addition, subtraction, multiplication, division, exponentiation, taking roots, taking logarithms) in the set of real numbers

    our translation · original wording (PL): wykonuje działania (dodawanie, odejmowanie, mnożenie, dzielenie, potęgowanie, pierwiastkowanie, logarytmowanie) w zbiorze liczb rzeczywistych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 1

  3. Curriculum point MAT.LO.1.P.2 · Teaching content · checked against the act

    carries out simple proofs concerning the divisibility of integers and remainders of division, e.g.:

    our translation · original wording (PL): przeprowadza proste dowody dotyczące podzielności liczb całkowitych i reszt z dzielenia, np.:

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 2

  4. Curriculum point MAT.LO.1.P.3 · Teaching content · checked against the act

    applies the properties of roots of any degree, including odd-degree roots of negative numbers

    our translation · original wording (PL): stosuje własności pierwiastków dowolnego stopnia, w tym pierwiastków stopnia nieparzystego z liczb ujemnych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 3

  5. Curriculum point MAT.LO.1.P.4 · Teaching content · checked against the act

    applies the relationship between root extraction and exponentiation and the laws of operations on powers and roots

    our translation · original wording (PL): stosuje związek pierwiastkowania z potęgowaniem oraz prawa działań na potęgach i pierwiastkach

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 4

  6. Curriculum point MAT.LO.1.P.5 · Teaching content · checked against the act

    applies the monotonicity of exponentiation, in particular the properties: if x < y and a > 1, then ax < ay, whereas when x < y and 0 < a < 1, then ax > ay

    our translation · original wording (PL): stosuje monotoniczność potęgowania, w szczególności własności: jeśli x < y oraz a > 1, to ax < ay, zaś gdy x < y i 0 < a < 1, to ax > ay

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 5

  7. Curriculum point MAT.LO.1.P.6 · Teaching content · checked against the act

    uses the concept of a numerical interval, marks intervals on the number line

    our translation · original wording (PL): posługuje się pojęciem przedziału liczbowego, zaznacza przedziały na osi liczbowej

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 6

  8. Curriculum point MAT.LO.1.P.7 · Teaching content · checked against the act

    uses the geometric and algebraic interpretation of absolute value, solves equations of the type: |x + 4| = 5

    our translation · original wording (PL): stosuje interpretację geometryczną i algebraiczną wartości bezwzględnej, rozwiązuje równania typu: |x + 4| = 5

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 7

    Equations and inequalities with absolute value (we teach in grade 10)

  9. Curriculum point MAT.LO.1.P.8 · Teaching content · checked against the act

    uses properties of powers and roots in practical situations, including calculating compound interest, returns on deposits and loan costs

    our translation · original wording (PL): wykorzystuje własności potęgowania i pierwiastkowania w sytuacjach praktycznych, w tym do obliczania procentów składanych, zysków z lokat i kosztów kredytów

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 8

  10. Curriculum point MAT.LO.1.P.9 · Teaching content · checked against the act

    applies the relationship between logarithms and exponentiation, uses the formulas for the logarithm of a product, the logarithm of a quotient, and the logarithm of a power.

    our translation · original wording (PL): stosuje związek logarytmowania z potęgowaniem, posługuje się wzorami na logarytm iloczynu, logarytm ilorazu i logarytm potęgi.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 9

  11. Curriculum point MAT.LO.1.P.2.a · Teaching content · checked against the act

    proof of the divisibility of the product of four consecutive natural numbers by 24

    our translation · original wording (PL): dowód podzielności przez 24 iloczynu czterech kolejnych liczb naturalnych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 2 lit. a

  12. Curriculum point MAT.LO.1.P.2.b · Teaching content · checked against the act

    proof of the property: if a number leaves a remainder of 3 when divided by 4, then it is not the square of an integer

    our translation · original wording (PL): dowód własności: jeśli liczba przy dzieleniu przez 4 daje resztę 3, to nie jest kwadratem liczby całkowitej

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 1, zakres podstawowy, pkt 2 lit. b

  13. Curriculum point MAT.LO.2 · Teaching content · checked against the act

    Algebraic expressions

    our translation · original wording (PL): Wyrażenia algebraiczne

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 2

  14. Curriculum point MAT.LO.2.P.1 · Teaching content · checked against the act

    applies short multiplication formulas for: (a + b)2, (a − b)2, a2 − b2

    our translation · original wording (PL): stosuje wzory skróconego mnożenia na: (a + b)2, (a − b)2, a2 − b2

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 2, zakres podstawowy, pkt 1

  15. Curriculum point MAT.LO.2.P.2 · Teaching content · checked against the act

    adds, subtracts and multiplies polynomials in one and several variables

    our translation · original wording (PL): dodaje, odejmuje i mnoży wielomiany jednej i wielu zmiennych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 2, zakres podstawowy, pkt 2

  16. Curriculum point MAT.LO.2.P.3 · Teaching content · checked against the act

    factors out a monomial from an algebraic sum

    our translation · original wording (PL): wyłącza poza nawias jednomian z sumy algebraicznej

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 2, zakres podstawowy, pkt 3

  17. Curriculum point MAT.LO.2.P.4 · Teaching content · checked against the act

    multiplies and divides rational expressions.

    our translation · original wording (PL): mnoży i dzieli wyrażenia wymierne.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 2, zakres podstawowy, pkt 4

  18. Curriculum point MAT.LO.3 · Teaching content · checked against the act

    Equations and inequalities

    our translation · original wording (PL): Równania i nierówności

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 3

  19. Curriculum point MAT.LO.3.P.1 · Teaching content

    transforms equations and inequalities in an equivalent manner, including e.g. equivalently transforms a rational equation (example in the act)

    our translation · original wording (PL): przekształca równania i nierówności w sposób równoważny, w tym np. przekształca równoważnie równanie wymierne (przykład w akcie)

    our summary of the point · Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 3, zakres podstawowy, pkt 1

    Solving rational and radical equations (we teach in grade 10)

  20. Curriculum point MAT.LO.3.P.2 · Teaching content · checked against the act

    interprets inconsistent and identity linear equations and inequalities

    our translation · original wording (PL): interpretuje równania i nierówności liniowe sprzeczne oraz tożsamościowe

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 3, zakres podstawowy, pkt 2

  21. Curriculum point MAT.LO.3.P.3 · Teaching content · checked against the act

    solves linear inequalities in one unknown

    our translation · original wording (PL): rozwiązuje nierówności liniowe z jedną niewiadomą

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 3, zakres podstawowy, pkt 3

  22. Curriculum point MAT.LO.3.P.4 · Teaching content · checked against the act

    solves quadratic equations and inequalities

    our translation · original wording (PL): rozwiązuje równania i nierówności kwadratowe

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 3, zakres podstawowy, pkt 4

    Solving quadratic inequalities (we teach in grade 10) · Solving quadratic equations (we teach in grade 10)

  23. Curriculum point MAT.LO.3.P.5 · Teaching content · checked against the act

    solves polynomial equations of the form W(x) = 0 for polynomials reduced to factored form.

    our translation · original wording (PL): rozwiązuje równania wielomianowe postaci W(x) = 0 dla wielomianów doprowadzonych do postaci iloczynowej.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 3, zakres podstawowy, pkt 5

    Solving polynomial equations (we teach in grade 10)

  24. Curriculum point MAT.LO.4 · Teaching content · checked against the act

    Systems of equations

    our translation · original wording (PL): Układy równań

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 4

  25. Curriculum point MAT.LO.4.P.1 · Teaching content · checked against the act

    solves systems of linear equations with two unknowns, gives a geometric interpretation of determinate, indeterminate, and inconsistent systems

    our translation · original wording (PL): rozwiązuje układy równań liniowych z dwiema niewiadomymi, podaje interpretację geometryczną układów oznaczonych, nieoznaczonych i sprzecznych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 4, zakres podstawowy, pkt 1

  26. Curriculum point MAT.LO.4.P.2 · Teaching content · checked against the act

    uses systems of equations to solve word problems.

    our translation · original wording (PL): stosuje układy równań do rozwiązywania zadań tekstowych.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 4, zakres podstawowy, pkt 2

  27. Curriculum point MAT.LO.5 · Teaching content · checked against the act

    Functions

    our translation · original wording (PL): Funkcje

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 5

  28. Curriculum point MAT.LO.5.P.1 · Teaching content · checked against the act

    defines functions as a unique assignment using a verbal description, a table, a graph, a formula (also with different formulas on different intervals)

    our translation · original wording (PL): określa funkcje jako jednoznaczne przyporządkowanie za pomocą opisu słownego, tabeli, wykresu, wzoru (także różnymi wzorami na różnych przedziałach)

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 1

  29. Curriculum point MAT.LO.5.P.2 · Teaching content · checked against the act

    calculates the value of a function given by an algebraic formula

    our translation · original wording (PL): oblicza wartość funkcji zadanej wzorem algebraicznym

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 2

    Calculating the value of a quadratic function (we teach in grade 10)

  30. Curriculum point MAT.LO.5.P.3 · Teaching content · checked against the act

    reads and interprets the values of functions defined by means of tables, graphs, formulas, etc., also in situations of multiple use of the same source of information or several sources simultaneously

    our translation · original wording (PL): odczytuje i interpretuje wartości funkcji określonych za pomocą tabel, wykresów, wzorów itp., również w sytuacjach wielokrotnego użycia tego samego źródła informacji lub kilku źródeł jednocześnie

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 3

  31. Curriculum point MAT.LO.5.P.4 · Teaching content · checked against the act

    reads from the graph of a function: the domain, range, zeros, intervals of monotonicity, intervals in which the function takes values greater (not less) or less (not greater) than a given number, the maximum and minimum values of the function (if they exist) in a given closed interval and the arguments for which the maximum and minimum values are attained by the function

    our translation · original wording (PL): odczytuje z wykresu funkcji: dziedzinę, zbiór wartości, miejsca zerowe, przedziały monotoniczności, przedziały, w których funkcja przyjmuje wartości większe (nie mniejsze) lub mniejsze (nie większe) od danej liczby, największe i najmniejsze wartości funkcji (o ile istnieją) w danym przedziale domkniętym oraz argumenty, dla których wartości największe i najmniejsze są przez funkcję przyjmowane

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 4

  32. Curriculum point MAT.LO.5.P.5 · Teaching content · checked against the act

    interprets the coefficients appearing in the formula of a linear function

    our translation · original wording (PL): interpretuje współczynniki występujące we wzorze funkcji liniowej

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 5

  33. Curriculum point MAT.LO.5.P.6 · Teaching content · checked against the act

    determines the formula of a linear function based on information about its graph or about its properties

    our translation · original wording (PL): wyznacza wzór funkcji liniowej na podstawie informacji o jej wykresie lub o jej własnościach

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 6

  34. Curriculum point MAT.LO.5.P.7 · Teaching content · checked against the act

    sketches the graph of a quadratic function given by a formula

    our translation · original wording (PL): szkicuje wykres funkcji kwadratowej zadanej wzorem

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 7

  35. Curriculum point MAT.LO.5.P.8 · Teaching content · checked against the act

    interprets the coefficients appearing in the formula of a quadratic function in general, vertex, and factored form (if it exists)

    our translation · original wording (PL): interpretuje współczynniki występujące we wzorze funkcji kwadratowej w postaci ogólnej, kanonicznej i iloczynowej (jeśli istnieje)

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 8

    Vertex of a parabola and the vertex form of a function (we teach in grade 10)

  36. Curriculum point MAT.LO.5.P.9 · Teaching content · checked against the act

    determines the formula of a quadratic function based on information about this function or about its graph

    our translation · original wording (PL): wyznacza wzór funkcji kwadratowej na podstawie informacji o tej funkcji lub o jej wykresie

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 9

  37. Curriculum point MAT.LO.5.P.10 · Teaching content · checked against the act

    determines the maximum and minimum value of a quadratic function on a closed interval

    our translation · original wording (PL): wyznacza największą i najmniejszą wartość funkcji kwadratowej w przedziale domkniętym

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 10

    Minimum and maximum value of a function on an interval (we teach in grade 10)

  38. Curriculum point MAT.LO.5.P.11 · Teaching content · checked against the act

    uses the properties of linear and quadratic functions to interpret geometric, physical, etc. problems, also set in a practical context

    our translation · original wording (PL): wykorzystuje własności funkcji liniowej i kwadratowej do interpretacji zagadnień geometrycznych, fizycznych itp., także osadzonych w kontekście praktycznym

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 11

    Optimization problems: maximum area (we teach in grade 10) · Minimum and maximum value of a function on an interval (we teach in grade 10)

  39. Curriculum point MAT.LO.5.P.12 · Teaching content

    based on the graph of the function y = f(x), sketches the graphs of the functions y = f(x − a), y = f(x) + b

    our translation · original wording (PL): na podstawie wykresu funkcji y = f(x) szkicuje wykresy funkcji y = f(x − a), y = f(x) + b

    our summary of the point · Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 12

    Translation of a function graph (we teach in grade 10)

  40. Curriculum point MAT.LO.5.P.13 · Teaching content

    uses the function f(x) = a/x, including its graph, to describe and interpret problems related to inversely proportional quantities, also in practical applications

    our translation · original wording (PL): posługuje się funkcją f(x) = a/x, w tym jej wykresem, do opisu i interpretacji zagadnień związanych z wielkościami odwrotnie proporcjonalnymi, również w zastosowaniach praktycznych

    our summary of the point · Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 13

    Graph and properties of the function f(x) = a/x (we teach in grade 10)

  41. Curriculum point MAT.LO.5.P.14 · Teaching content · checked against the act

    uses exponential and logarithmic functions, including their graphs, to describe and interpret issues related to practical applications.

    our translation · original wording (PL): posługuje się funkcjami wykładniczą i logarytmiczną, w tym ich wykresami, do opisu i interpretacji zagadnień związanych z zastosowaniami praktycznymi.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 5, zakres podstawowy, pkt 14

    Calculating the value of an exponential function (we teach in grade 10) · Solving exponential equations with the same base (we teach in grade 10)

  42. Curriculum point MAT.LO.6 · Teaching content · checked against the act

    Sequences

    our translation · original wording (PL): Ciągi

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 6

  43. Curriculum point MAT.LO.6.P.1 · Teaching content · checked against the act

    calculates the terms of a sequence defined by a general formula

    our translation · original wording (PL): oblicza wyrazy ciągu określonego wzorem ogólnym

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 1

  44. Curriculum point MAT.LO.6.P.2 · Teaching content · checked against the act

    calculates the initial terms of sequences defined recursively

    our translation · original wording (PL): oblicza początkowe wyrazy ciągów określonych rekurencyjnie

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 2

  45. Curriculum point MAT.LO.6.P.3 · Teaching content · checked against the act

    in simple cases examines whether a sequence is increasing or decreasing

    our translation · original wording (PL): w prostych przypadkach bada, czy ciąg jest rosnący, czy malejący

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 3

  46. Curriculum point MAT.LO.6.P.4 · Teaching content · checked against the act

    checks whether a given sequence is arithmetic or geometric

    our translation · original wording (PL): sprawdza, czy dany ciąg jest arytmetyczny lub geometryczny

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 4

  47. Curriculum point MAT.LO.6.P.5 · Teaching content · checked against the act

    uses the formula for the n-th term and for the sum of the first n terms of an arithmetic sequence

    our translation · original wording (PL): stosuje wzór na n-ty wyraz i na sumę n początkowych wyrazów ciągu arytmetycznego

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 5

  48. Curriculum point MAT.LO.6.P.6 · Teaching content · checked against the act

    applies the formula for the n-th term and for the sum of the first n terms of a geometric sequence

    our translation · original wording (PL): stosuje wzór na n-ty wyraz i na sumę n początkowych wyrazów ciągu geometrycznego

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 6

  49. Curriculum point MAT.LO.6.P.7 · Teaching content · checked against the act

    uses the properties of sequences, including arithmetic and geometric, to solve problems, also set in a practical context.

    our translation · original wording (PL): wykorzystuje własności ciągów, w tym arytmetycznych i geometrycznych, do rozwiązywania zadań, również osadzonych w kontekście praktycznym.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 6, zakres podstawowy, pkt 7

  50. Curriculum point MAT.LO.7 · Teaching content · checked against the act

    Trigonometry

    our translation · original wording (PL): Trygonometria

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 7

  51. Curriculum point MAT.LO.7.P.1 · Teaching content · checked against the act

    uses the definitions of the functions: sine, cosine and tangent for angles from 0° to 180°, in particular determines the values of trigonometric functions for angles of 30°, 45°, 60°

    our translation · original wording (PL): wykorzystuje definicje funkcji: sinus, cosinus i tangens dla kątów od 0° do 180°, w szczególności wyznacza wartości funkcji trygonometrycznych dla kątów 30°, 45°, 60°

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 7, zakres podstawowy, pkt 1

    Exact values of trigonometric functions (we teach in grade 10)

  52. Curriculum point MAT.LO.7.P.2 · Teaching content

    uses the formulas sin²α + cos²α = 1, tan α = sin α / cos α

    our translation · original wording (PL): korzysta z wzorów sin²α + cos²α = 1, tg α = sin α / cos α

    our summary of the point · Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 7, zakres podstawowy, pkt 2

    Calculating cosine and tangent from a known sine (we teach in grade 10)

  53. Curriculum point MAT.LO.7.P.3 · Teaching content

    applies the law of cosines and the formula for the area of a triangle P = ½ · a · b · sin γ

    our translation · original wording (PL): stosuje twierdzenie cosinusów oraz wzór na pole trójkąta P = ½ · a · b · sin γ

    our summary of the point · Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 7, zakres podstawowy, pkt 3

    Laws of Sines and Cosines and the Area of a Triangle (we teach in grade 10)

  54. Curriculum point MAT.LO.7.P.4 · Teaching content · checked against the act

    calculates the angles of a right-angled triangle and the lengths of its sides given appropriate data (solves right-angled triangles, including using trigonometric functions).

    our translation · original wording (PL): oblicza kąty trójkąta prostokątnego i długości jego boków przy odpowiednich danych (rozwiązuje trójkąty prostokątne, w tym z wykorzystaniem funkcji trygonometrycznych).

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 7, zakres podstawowy, pkt 4

    Identifying legs relative to an angle (we teach in grade 10)

  55. Curriculum point MAT.LO.8 · Teaching content · checked against the act

    Plane geometry

    our translation · original wording (PL): Planimetria

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 8

  56. Curriculum point MAT.LO.8.P.1 · Teaching content · checked against the act

    determines radii and diameters of circles, lengths of chords of circles and tangent segments, including using the Pythagorean theorem

    our translation · original wording (PL): wyznacza promienie i średnice okręgów, długości cięciw okręgów oraz odcinków stycznych, w tym z wykorzystaniem twierdzenia Pitagorasa

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 1

    Calculating the length of a chord and a tangent segment (we teach in grade 10)

  57. Curriculum point MAT.LO.8.P.2 · Teaching content · checked against the act

    recognizes acute, right, and obtuse triangles given the lengths of the sides (among other things, applies the converse of the Pythagorean theorem and the law of cosines); applies the theorem: in a triangle, the longer side lies opposite the greater interior angle

    our translation · original wording (PL): rozpoznaje trójkąty ostrokątne, prostokątne i rozwartokątne przy danych długościach boków (m.in. stosuje twierdzenie odwrotne do twierdzenia Pitagorasa i twierdzenie cosinusów); stosuje twierdzenie: w trójkącie naprzeciw większego kąta wewnętrznego leży dłuższy bok

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 2

    Determining the type of triangle from side lengths (we teach in grade 10)

  58. Curriculum point MAT.LO.8.P.3 · Teaching content · checked against the act

    recognises regular polygons and uses their basic properties

    our translation · original wording (PL): rozpoznaje wielokąty foremne i korzysta z ich podstawowych własności

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 3

  59. Curriculum point MAT.LO.8.P.4 · Teaching content · checked against the act

    uses the properties of angles and diagonals in rectangles, parallelograms, rhombuses, and trapezoids

    our translation · original wording (PL): korzysta z własności kątów i przekątnych w prostokątach, równoległobokach, rombach i trapezach

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 4

  60. Curriculum point MAT.LO.8.P.5 · Teaching content · checked against the act

    applies properties of inscribed and central angles

    our translation · original wording (PL): stosuje własności kątów wpisanych i środkowych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 5

    Inscribed and central angle in a circle (we teach in grade 10)

  61. Curriculum point MAT.LO.8.P.6 · Teaching content · checked against the act

    applies formulas for the area of a sector of a circle and the length of an arc of a circle

    our translation · original wording (PL): stosuje wzory na pole wycinka koła i długość łuku okręgu

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 6

    Area of a circular sector and arc length (we teach in grade 10)

  62. Curriculum point MAT.LO.8.P.7 · Teaching content · checked against the act

    applies Thales' theorem

    our translation · original wording (PL): stosuje twierdzenie Talesa

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 7

    Calculating segment lengths using Thales's theorem (we teach in grade 10)

  63. Curriculum point MAT.LO.8.P.8 · Teaching content · checked against the act

    uses triangle similarity criteria

    our translation · original wording (PL): korzysta z cech podobieństwa trójkątów

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 8

  64. Curriculum point MAT.LO.8.P.9 · Teaching content · checked against the act

    uses relationships between perimeters and between areas of similar figures

    our translation · original wording (PL): wykorzystuje zależności między obwodami oraz między polami figur podobnych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 9

    Areas and perimeters of similar figures (we teach in grade 10)

  65. Curriculum point MAT.LO.8.P.10 · Teaching content · checked against the act

    identifies basic special points in a triangle: the center of the inscribed circle of a triangle, the center of the circumscribed circle of a triangle, the orthocenter, the centroid, and uses their properties

    our translation · original wording (PL): wskazuje podstawowe punkty szczególne w trójkącie: środek okręgu wpisanego w trójkąt, środek okręgu opisanego na trójkącie, ortocentrum, środek ciężkości oraz korzysta z ich własności

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 10

    Inscribed and circumscribed circles of a right triangle (we teach in grade 10)

  66. Curriculum point MAT.LO.8.P.11 · Teaching content · checked against the act

    carries out geometric proofs

    our translation · original wording (PL): przeprowadza dowody geometryczne

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 11

  67. Curriculum point MAT.LO.8.P.12 · Teaching content · checked against the act

    applies trigonometric functions to determine the lengths of segments in plane figures and to calculate the areas of figures.

    our translation · original wording (PL): stosuje funkcje trygonometryczne do wyznaczania długości odcinków w figurach płaskich oraz obliczania pól figur.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 8, zakres podstawowy, pkt 12

    Calculating the area of a parallelogram using sine (we teach in grade 10)

  68. Curriculum point MAT.LO.9 · Teaching content · checked against the act

    Analytic geometry on the Cartesian plane

    our translation · original wording (PL): Geometria analityczna na płaszczyźnie kartezjańskiej

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 9

  69. Curriculum point MAT.LO.9.P.1 · Teaching content · checked against the act

    identifies the relative position of lines on a plane based on their equations, including finding the common point of two lines, if it exists

    our translation · original wording (PL): rozpoznaje wzajemne położenie prostych na płaszczyźnie na podstawie ich równań, w tym znajduje wspólny punkt dwóch prostych, jeśli taki istnieje

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 9, zakres podstawowy, pkt 1

  70. Curriculum point MAT.LO.9.P.2 · Teaching content · checked against the act

    uses equations of lines in the plane, in slope-intercept and general form, including determining the equation of a line with given properties (such as, e.g., passing through two given points, a known slope, parallelism to another line)

    our translation · original wording (PL): posługuje się równaniami prostych na płaszczyźnie, w postaci kierunkowej i ogólnej, w tym wyznacza równanie prostej o zadanych własnościach (takich, jak np. przechodzenie przez dwa dane punkty, znany współczynnik kierunkowy, równoległość do innej prostej)

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 9, zakres podstawowy, pkt 2

  71. Curriculum point MAT.LO.9.P.3 · Teaching content · checked against the act

    calculates the distance between two points in a coordinate system

    our translation · original wording (PL): oblicza odległość dwóch punktów w układzie współrzędnych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 9, zakres podstawowy, pkt 3

  72. Curriculum point MAT.LO.9.P.4 · Teaching content

    uses the equation of a circle (x − a)² + (y − b)² = r²

    our translation · original wording (PL): posługuje się równaniem okręgu (x − a)² + (y − b)² = r²

    our summary of the point · Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 9, zakres podstawowy, pkt 4

  73. Curriculum point MAT.LO.9.P.5 · Teaching content · checked against the act

    determines the images of circles and polygons in axial symmetries with respect to the axes of the coordinate system, central symmetry (with the center at the origin of the coordinate system).

    our translation · original wording (PL): wyznacza obrazy okręgów i wielokątów w symetriach osiowych względem osi układu współrzędnych, symetrii środkowej (o środku w początku układu współrzędnych).

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 9, zakres podstawowy, pkt 5

  74. Curriculum point MAT.LO.10 · Teaching content · checked against the act

    Solid geometry

    our translation · original wording (PL): Stereometria

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 10

  75. Curriculum point MAT.LO.10.P.1 · Teaching content · checked against the act

    recognizes the relative positions of lines in space, in particular perpendicular non-intersecting lines

    our translation · original wording (PL): rozpoznaje wzajemne położenie prostych w przestrzeni, w szczególności proste prostopadłe nieprzecinające się

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 10, zakres podstawowy, pkt 1

  76. Curriculum point MAT.LO.10.P.2 · Teaching content · checked against the act

    uses the concept of the angle between a line and a plane and the concept of a dihedral angle between half-planes

    our translation · original wording (PL): posługuje się pojęciem kąta między prostą a płaszczyzną oraz pojęciem kąta dwuściennego między półpłaszczyznami

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 10, zakres podstawowy, pkt 2

  77. Curriculum point MAT.LO.10.P.3 · Teaching content · checked against the act

    identifies in prisms and pyramids angles between line segments (e.g. edges, edges and diagonals) and angles between faces, calculates the measures of these angles

    our translation · original wording (PL): rozpoznaje w graniastosłupach i ostrosłupach kąty między odcinkami (np. krawędziami, krawędziami i przekątnymi) oraz kąty między ścianami, oblicza miary tych kątów

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 10, zakres podstawowy, pkt 3

  78. Curriculum point MAT.LO.10.P.4 · Teaching content · checked against the act

    identifies in cylinders and cones the angle between line segments and the angle between line segments and planes (e.g. the apex angle of a cone, the angle between the generatrix and the base), calculates the measures of these angles

    our translation · original wording (PL): rozpoznaje w walcach i w stożkach kąt między odcinkami oraz kąt między odcinkami i płaszczyznami (np. kąt rozwarcia stożka, kąt między tworzącą a podstawą), oblicza miary tych kątów

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 10, zakres podstawowy, pkt 4

  79. Curriculum point MAT.LO.10.P.5 · Teaching content · checked against the act

    calculates the volumes and surface areas of prisms, pyramids, a cylinder, a cone, and a sphere, also using trigonometry

    our translation · original wording (PL): oblicza objętości i pola powierzchni graniastosłupów, ostrosłupów, walca, stożka i kuli, również z wykorzystaniem trygonometrii

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 10, zakres podstawowy, pkt 5

  80. Curriculum point MAT.LO.10.P.6 · Teaching content · checked against the act

    uses the relationship between the volumes of similar solids.

    our translation · original wording (PL): wykorzystuje zależność między objętościami brył podobnych.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 10, zakres podstawowy, pkt 6

  81. Curriculum point MAT.LO.11 · Teaching content · checked against the act

    Combinatorics

    our translation · original wording (PL): Kombinatoryka

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 11

  82. Curriculum point MAT.LO.11.P.1 · Teaching content · checked against the act

    counts objects in simple combinatorial situations

    our translation · original wording (PL): zlicza obiekty w prostych sytuacjach kombinatorycznych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 11, zakres podstawowy, pkt 1

  83. Curriculum point MAT.LO.11.P.2 · Teaching content · checked against the act

    counts objects, using the rules of multiplication and addition (also in combination) for any number of actions, e.g.:

    our translation · original wording (PL): zlicza obiekty, stosując reguły mnożenia i dodawania (także łącznie) dla dowolnej liczby czynności, np.:

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 11, zakres podstawowy, pkt 2

  84. Curriculum point MAT.LO.11.P.2.a · Teaching content · checked against the act

    calculating how many four-digit odd positive integers there are such that in their decimal representation there is exactly one digit 1 and exactly one digit 2

    our translation · original wording (PL): obliczenie, ile jest czterocyfrowych nieparzystych liczb całkowitych dodatnich takich, że w ich zapisie dziesiętnym występuje dokładnie jedna cyfra 1 i dokładnie jedna cyfra 2

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 11, zakres podstawowy, pkt 2 lit. a

  85. Curriculum point MAT.LO.11.P.2.b · Teaching content · checked against the act

    calculating how many four-digit even positive integers there are such that exactly one digit 0 and exactly one digit 1 appear in their decimal representation.

    our translation · original wording (PL): obliczenie, ile jest czterocyfrowych parzystych liczb całkowitych dodatnich takich, że w ich zapisie dziesiętnym występuje dokładnie jedna cyfra 0 i dokładnie jedna cyfra 1.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 11, zakres podstawowy, pkt 2 lit. b

  86. Curriculum point MAT.LO.12 · Teaching content · checked against the act

    Probability and statistics

    our translation · original wording (PL): Rachunek prawdopodobieństwa i statystyka

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 12

  87. Curriculum point MAT.LO.12.P.1 · Teaching content · checked against the act

    calculates probability in the classical model

    our translation · original wording (PL): oblicza prawdopodobieństwo w modelu klasycznym

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 12, zakres podstawowy, pkt 1

  88. Curriculum point MAT.LO.12.P.2 · Teaching content · checked against the act

    calculates the arithmetic mean and the weighted mean, finds the median and the mode.

    our translation · original wording (PL): oblicza średnią arytmetyczną i średnią ważoną, znajduje medianę i dominantę.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 12, zakres podstawowy, pkt 2

  89. Curriculum point MAT.LO.13 · Teaching content · checked against the act

    Optimization and differential calculus

    our translation · original wording (PL): Optymalizacja i rachunek różniczkowy

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, treści nauczania, dział 13

  90. Curriculum point MAT.LO.13.P.1 · Teaching content · checked against the act

    solves optimization problems in situations that can be described by a quadratic function.

    our translation · original wording (PL): rozwiązuje zadania optymalizacyjne w sytuacjach dających się opisać funkcją kwadratową.

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, dział 13, zakres podstawowy, pkt 1

    Optimization problems: maximum area (we teach in grade 10)

  91. Curriculum point MAT.LO.WO.1 · General aims · checked against the act

    Computational fluency

    our translation · original wording (PL): Sprawność rachunkowa

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele kształcenia, I

  92. Curriculum point MAT.LO.WO.1.1 · General aims · checked against the act

    Performing calculations on real numbers, also using a calculator, applying the laws of mathematical operations when transforming algebraic expressions, and using these skills when solving problems in real-world and theoretical contexts

    our translation · original wording (PL): Wykonywanie obliczeń na liczbach rzeczywistych, także przy użyciu kalkulatora, stosowanie praw działań matematycznych przy przekształcaniu wyrażeń algebraicznych oraz wykorzystywanie tych umiejętności przy rozwiązywaniu problemów w kontekstach rzeczywistych i teoretycznych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, I

  93. Curriculum point MAT.LO.WO.2 · General aims · checked against the act

    Using and creating information

    our translation · original wording (PL): Wykorzystanie i tworzenie informacji

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele kształcenia, II

  94. Curriculum point MAT.LO.WO.2.1 · General aims · checked against the act

    Interpreting and working with information presented in text, both mathematical and popular science, as well as in the form of graphs, diagrams, tables

    our translation · original wording (PL): Interpretowanie i operowanie informacjami przedstawionymi w tekście, zarówno matematycznym, jak i popularnonaukowym, a także w formie wykresów, diagramów, tabel

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, II pkt 1

  95. Curriculum point MAT.LO.WO.2.2 · General aims · checked against the act

    Using mathematical language to create mathematical texts, including describing reasoning carried out and justifying conclusions, as well as presenting data

    our translation · original wording (PL): Używanie języka matematycznego do tworzenia tekstów matematycznych, w tym do opisu prowadzonych rozumowań i uzasadniania wniosków, a także do przedstawiania danych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, II pkt 2

  96. Curriculum point MAT.LO.WO.3 · General aims · checked against the act

    Using and interpreting representations

    our translation · original wording (PL): Wykorzystanie i interpretowanie reprezentacji

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele kształcenia, III

  97. Curriculum point MAT.LO.WO.3.1 · General aims · checked against the act

    Using mathematical objects and operating on them, interpreting mathematical concepts

    our translation · original wording (PL): Stosowanie obiektów matematycznych i operowanie nimi, interpretowanie pojęć matematycznych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, III pkt 1

  98. Curriculum point MAT.LO.WO.3.2 · General aims · checked against the act

    Selecting and creating mathematical models when solving practical and theoretical problems

    our translation · original wording (PL): Dobieranie i tworzenie modeli matematycznych przy rozwiązywaniu problemów praktycznych i teoretycznych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, III pkt 2

  99. Curriculum point MAT.LO.WO.3.3 · General aims · checked against the act

    Creating auxiliary mathematical objects based on existing ones, in order to carry out argumentation or solve a problem

    our translation · original wording (PL): Tworzenie pomocniczych obiektów matematycznych na podstawie istniejących, w celu przeprowadzenia argumentacji lub rozwiązania problemu

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, III pkt 3

  100. Curriculum point MAT.LO.WO.3.4 · General aims · checked against the act

    Indicating the necessity or possibility of modifying a mathematical model in cases requiring special reservations, additional assumptions, consideration of specific conditions

    our translation · original wording (PL): Wskazywanie konieczności lub możliwości modyfikacji modelu matematycznego w przypadkach wymagających specjalnych zastrzeżeń, dodatkowych założeń, rozważenia szczególnych uwarunkowań

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, III pkt 4

  101. Curriculum point MAT.LO.WO.4 · General aims · checked against the act

    Reasoning and argumentation

    our translation · original wording (PL): Rozumowanie i argumentacja

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele kształcenia, IV

  102. Curriculum point MAT.LO.WO.4.1 · General aims · checked against the act

    Conducting reasoning, including multi-step reasoning, providing arguments justifying the correctness of reasoning, distinguishing a proof from an example

    our translation · original wording (PL): Przeprowadzanie rozumowań, także kilkuetapowych, podawanie argumentów uzasadniających poprawność rozumowania, odróżnianie dowodu od przykładu

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, IV pkt 1

  103. Curriculum point MAT.LO.WO.4.2 · General aims · checked against the act

    Recognizing regularities, similarities and analogies, formulating conclusions based on them and justifying their correctness

    our translation · original wording (PL): Dostrzeganie regularności, podobieństw oraz analogii, formułowanie wniosków na ich podstawie i uzasadnianie ich poprawności

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, IV pkt 2

  104. Curriculum point MAT.LO.WO.4.3 · General aims · checked against the act

    Selecting arguments to justify the correctness of solving problems, constructing a chain of arguments guaranteeing the correctness of the solution and effectiveness in seeking solutions to a problem

    our translation · original wording (PL): Dobieranie argumentów do uzasadnienia poprawności rozwiązywania problemów, tworzenie ciągu argumentów gwarantujących poprawność rozwiązania i skuteczność w poszukiwaniu rozwiązań zagadnienia

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, IV pkt 3

  105. Curriculum point MAT.LO.WO.4.4 · General aims · checked against the act

    Applying and creating strategies when solving problems, also in non-routine situations

    our translation · original wording (PL): Stosowanie i tworzenie strategii przy rozwiązywaniu zadań, również w sytuacjach nietypowych

    Dz.U. 2018 poz. 467 w brzmieniu Dz.U. 2024 poz. 1019, matematyka, cele, IV pkt 4

Skills step by step

Viète's formulas: sum and product of roots

The student determines the sum, product, and sum of the reciprocals of the roots of a quadratic equation directly from the coefficients, without the need to calculate the discriminant and the solutions themselves.

In the second year of high school and technical school, students determine the sum and product of the roots of a quadratic trinomial directly from its numerical coefficients. At this stage, they also learn to transform algebraic expressions, for example, reducing the sum of the reciprocals of the roots to a fraction composed of their sum and product.

A common issue is dropping the minus sign in the formula for the sum of the roots (confusing the expression -b/a with b/a). Students also often forget to divide by the coefficient a when the leading coefficient is different from 1, which completely distorts the result.

The exercises in this section include specific calculation tasks:

  • Sum of roots using Vieta's formulas and product of roots using Vieta's formulas – directly reading and calculating values from the coefficients of a quadratic equation,
  • Sum of the reciprocals of roots – finding a common denominator for the expression and applying both formulas at once.
Create a worksheet for this skill

Task types: Product of roots (Vieta) · Sum of reciprocals of roots · Sum of roots (Vieta)

Laws of Sines and Cosines and the Area of a Triangle

The student calculates unknown sides and angles in any triangle using the Law of Sines and the Law of Cosines, and determines the area of a triangle using the sine of an angle.

Curriculum point MAT.LO.7.P.3 applies the law of cosines and the formula for the area of a triangle P = ½ · a · b · sin γ (our translation)

In the 2nd grade of high school and technical secondary school, students solve general triangles that do not have to be right-angled. They apply the law of sines and the law of cosines to determine missing side lengths or angle measures based on given data. They can also efficiently calculate the area of a triangle using the formula that connects the lengths of two sides with the sine of the included angle.

A typical difficulty when applying the law of cosines is a calculation error in the order of operations or failing to take into account the negative sign of the cosine for an obtuse angle. When calculating the area of a triangle, on the other hand, students often choose a random angle from the diagram instead of the one strictly included between the two known sides.

Printable worksheets in this area cover three specific groups of exercises:

  • The Law of Cosines — calculating the length of the third side given two sides and the angle between them, or determining the cosine of an angle when all three sides are given,
  • The Law of Sines — finding missing sides and angles based on the proportion between a side and the sine of its opposite angle,
  • Area of a triangle using sine — calculating the area of the figure without having to determine its height first.
Create a worksheet for this skill

Task types: Area of a triangle with sine · Law of cosines · Law of sines

Calculating the value of an exponential function

The student calculates the values of an exponential function for given arguments and recognizes when such a function is decreasing.

Curriculum point MAT.LO.5.P.14 uses exponential and logarithmic functions, including their graphs, to describe and interpret issues related to practical applications. (our translation)

In Grade 2 of secondary school, students determine the values of an exponential function for specific arguments. They understand the relationship between the base of a power and the behavior of the function, allowing them to identify when an exponential function is decreasing and when it is increasing.

The most common difficulties arise with negative or fractional exponents, especially when the base of the function is a fraction. Students sometimes confuse taking the reciprocal of a number with changing its sign, or they incorrectly assess monotonicity, forgetting that a base between zero and one means the function is decreasing.

Exercises from the Value of an exponential function template require substituting a given number for the variable and performing the appropriate operations on powers. In turn, exercises from the Decreasing exponential function template involve analyzing the base of the power in the formula and determining whether the function decreases as the arguments increase.

Create a worksheet for this skill

Task types: Decreasing exponential function · Value of an exponential function

Dividing a polynomial by a binomial x − a

The student divides a polynomial by a binomial of the form x − a and determines the quotient and the remainder of such division.

In the second year of high school and technical school, students divide polynomials by a linear binomial of the form x − a. During calculations, they determine the quotient polynomial and find the remainder, checking whether the polynomial is divisible completely or with a numerical remainder.

The most common problem with polynomial long division is losing signs when subtracting successive terms, especially when the subtracted term has a negative coefficient. Students also often make mistakes when the dividend polynomial does not have all consecutive powers — forgetting to insert a zero in place of a missing degree leads to misaligning like terms.

In practice, students encounter two types of problems:

  • Polynomial division by a binomial — involves performing the full division step by step and writing the result as a quotient and remainder;
  • Remainder of polynomial division — requires determining only the remainder from division by x − a for a given polynomial.
Create a worksheet for this skill

Task types: Remainder of polynomial division · Dividing a polynomial by a binomial

Solving quadratic inequalities

The student solves quadratic inequalities written in general and factored form and correctly determines the solution set in the form of numerical intervals.

Curriculum point MAT.LO.3.P.4 solves quadratic equations and inequalities (our translation)

In the second grade of high school and technical school, students learn to determine the sets of numbers that satisfy quadratic inequalities. They independently calculate the roots of the corresponding quadratic function or read them from factored form, and then sketch a rough graph of the parabola, taking into account the direction of its opening, to correctly read off the interval of positive or negative values.

A common issue at this stage is incorrectly determining the direction of the parabola's opening when the leading coefficient is negative, which leads to reading the wrong interval. Students also confuse closed and open brackets, failing to pay attention to whether the inequality is strict (<, > signs) or non-strict (≤, ≥ signs).

Practice exercises cover two basic formats:

  • Quadratic inequality in factored form — the student directly reads the roots from the factors, e.g., from the form (x - 3)(x + 2) ≥ 0, and determines the interval on the number line;
  • Quadratic inequality — an example in standard form, e.g., x² - 4x + 3 < 0, which first requires factoring the expression or calculating the discriminant (delta) to find the x-intercepts.
Create a worksheet for this skill

Task types: Quadratic inequality · Quadratic inequality in factored form

Equations and inequalities with absolute value

The student learns to determine numbers that satisfy equations and solution intervals of inequalities involving absolute value.

Curriculum point MAT.LO.1.P.7 uses the geometric and algebraic interpretation of absolute value, solves equations of the type: |x + 4| = 5 (our translation)

In the 2nd grade of high school and technical secondary school, the student solves algebraic problems in which an expression containing an unknown is inside an absolute value. They efficiently break the problem down into cases or use a geometric interpretation on the number line, finding specific numbers that satisfy the condition or giving the solution set in the form of an interval.

A typical problem is confusing logical connectives when writing inequalities. Students often use the union of intervals instead of their intersection (or vice versa), which stems from confusing the conditions for "less than" and "greater than" relations. Another common mistake is failing to reverse the inequality sign when considering the case with a negative number.

Work on this topic includes the templates Absolute value equation and Absolute value inequality. In the problems, the student transforms the given algebraic expression, eliminates the absolute value symbol, and writes the final solution as a set of numbers or numerical intervals.

Create a worksheet for this skill

Task types: Absolute value equation · Absolute value inequality

Solving polynomial equations

The student solves polynomial equations written in factored form and biquadratic equations using an auxiliary variable.

Curriculum point MAT.LO.3.P.5 solves polynomial equations of the form W(x) = 0 for polynomials reduced to factored form. (our translation)

In the second grade of secondary school, students master finding solutions to polynomial equations of degree higher than two in specific, accessible setups. They work with expressions written in factored form, using the fact that a product equals zero when at least one factor is zero. They also solve biquadratic equations (fourth-degree equations with only even powers) by reducing them to quadratic equations through substituting a new variable for the square of the unknown.

The most common mistake with biquadratic equations is stopping the calculations after finding the values of the auxiliary variable and forgetting to return to the original unknown. Students also sometimes attempt to take the square root of negative values of the auxiliary variable in the set of real numbers. On the other hand, with equations in factored form, students often reflexively expand the parentheses, which unnecessarily complicates the problem, instead of immediately setting each factor equal to zero.

Tasks testing this skill are based on two patterns:

  • Polynomial equation in factored form — requires breaking down the product of several factors into simpler linear or quadratic equations and providing the full set of solutions.
  • Biquadratic equation — requires a change of variables, calculating the discriminant of the quadratic equation (delta), and then finding the final values of the unknown.
Create a worksheet for this skill

Task types: Polynomial equation in factored form · Biquadratic equation

Solving rational and radical equations

The student learns to determine the domain and correctly solve equations containing an unknown in the denominator of a fraction or under a radical sign.

Curriculum point MAT.LO.3.P.1 transforms equations and inequalities in an equivalent manner, including e.g. equivalently transforms a rational equation (example in the act) (our translation)

In the 2nd grade of high school and technical secondary school, students solve algebraic equations in which the unknown appears under a radical sign or in the denominator of a fractional expression. A key element at this stage is determining the domain, which means identifying the numbers for which the expression is mathematically valid—in particular, excluding values that make the denominator zero and taking into account the conditions for radicals.

The most common mistake is mechanically transforming the equation without checking the initial conditions. Students often forget about the domain or that squaring both sides of an equation can lead to extraneous solutions, resulting in providing a number as the correct answer that does not actually satisfy the original equation.

In practice, the exercises are based on the Rational Equation and Radical Equation templates. In these exercises, students encounter problems involving simplifying fractional expressions, cross-multiplying, or isolating the radical on one side of the equation, followed by verifying the obtained results against the domain.

Create a worksheet for this skill

Task types: Radical equation · Rational equation

Vertex of a parabola and the vertex form of a function

The student learns to determine the coordinates of the vertex of a parabola and to write and read the formula of a quadratic function in vertex form.

Curriculum point MAT.LO.5.P.8 interprets the coefficients appearing in the formula of a quadratic function in general, vertex, and factored form (if it exists) (our translation)

A student in the 2nd grade of high school and technical secondary school masters determining the coordinates of the vertex of a parabola and working with the vertex form of a quadratic function. They efficiently read the vertex coordinates from the formula in the form y = a(x - p)² + q, and also convert the standard form of a function into vertex form.

The most common difficulty is correctly reading the x-coordinate of the vertex due to the minus sign in the formula. Students confuse the sign of the number inside the parentheses: with the expression (x - 4)², they give the x-coordinate as -4 instead of 4, and with the expression (x + 3)², they forget that this coordinate is -3.

In practice, tasks from the templates Vertex of a parabola and Vertex form of a quadratic function involve reading the coordinates of the vertex directly from a given formula, calculating them from standard form, or writing the formula of a function with a specified vertex on their own.

Create a worksheet for this skill

Task types: Vertex of a parabola · Vertex form of a quadratic function

Minimum and maximum value of a function on an interval

The student learns to determine the maximum and minimum values of a quadratic function in a given closed interval by checking the vertex of the parabola and its endpoints.

Curriculum point MAT.LO.5.P.10 determines the maximum and minimum value of a quadratic function on a closed interval (our translation)
Curriculum point MAT.LO.5.P.11 uses the properties of linear and quadratic functions to interpret geometric, physical, etc. problems, also set in a practical context (our translation)

In grade 2 of high school and technical school, students learn to determine the minimum and maximum values of a quadratic function on a given closed interval. They analyze the function formula, calculate the coordinate of the parabola's vertex, and check whether it lies within the specified interval, then compare the values attained at its endpoints and at the vertex.

The most common mistake is mechanically calculating the function's values solely at the endpoints of the interval while ignoring the vertex of the parabola, where the function may reach its actual extremum. The opposite mistake also occurs: students automatically give the value at the vertex, forgetting to check whether its coordinate actually falls within the given interval.

Tasks in this area correspond to the templates Maximum value on an interval and Minimum value on an interval. The student receives the formula of a quadratic function and a specific numerical interval, and their goal is to identify a single number representing the minimum or maximum value of the function within these bounds.

Create a worksheet for this skill

Task types: Greatest value on an interval · Least value on an interval

Identifying legs relative to an angle

The student identifies the leg opposite a given acute angle and the leg adjacent to this angle in a right triangle.

Curriculum point MAT.LO.7.P.4 calculates the angles of a right-angled triangle and the lengths of its sides given appropriate data (solves right-angled triangles, including using trigonometric functions). (our translation)

In the 2nd year of high school and technical school, students master the correct identification of the sides in a right-angled triangle in relation to a chosen acute angle. They precisely distinguish which leg forms the given angle (is adjacent to it) and which one lies on the opposite side, which is an essential step toward defining trigonometric functions.

A typical mistake is confusing the leg adjacent to an angle with the leg opposite to it. This difficulty increases especially when the triangle is drawn in an unusual orientation (rotated) and the student tries to judge the sides based on their vertical or horizontal layout on the page, instead of checking their position relative to the vertex of the angle.

The tasks in this section are visual and involve analyzing the drawing of a right-angled triangle with a marked angle. Students solve exercises such as:

  • Leg adjacent to the angle — finding and naming the side adjacent to the indicated angle,
  • Leg opposite the angle — identifying the side lying opposite the given acute angle.
Create a worksheet for this skill

Task types: Leg adjacent to an angle · Leg opposite an angle

Calculating cosine and tangent from a known sine

The student determines the value of the cosine and tangent of an angle using knowledge of the sine and trigonometric identities.

Curriculum point MAT.LO.7.P.2 uses the formulas sin²α + cos²α = 1, tan α = sin α / cos α (our translation)

In the second year of high school and technical secondary school, students apply fundamental trigonometric identities: the relationship sin²α + cos²α = 1 (the Pythagorean trigonometric identity) and the definition of the tangent as the quotient tan α = sin α / cos α. Given the numerical value of the sine of an angle (often written as a common fraction), they can efficiently calculate the corresponding cosine value and then find the tangent.

Typical issues include calculation errors when transforming the equation: incorrectly squaring fractions (for example, forgetting to square the denominator) or difficulties with simplifying and taking the square root of the resulting value. When determining the tangent, students also make mistakes when dividing by a fraction, forgetting to multiply by its reciprocal.

The exercises in this section are based on the following patterns:

  • Cosine from sine — involves substituting the known sine value into the Pythagorean trigonometric identity, subtracting it from one, and taking the square root of the result;
  • Tangent from sine — the student first determines the cosine, and in the next step calculates the value of the tangent by dividing the given sine by the calculated cosine.
Create a worksheet for this skill

Task types: Cosine from sine · Tangent from sine

Area of a circular sector and arc length

The student calculates the length of an arc of a circle and the area of a sector of a circle based on a given radius and the measure of the central angle.

Curriculum point MAT.LO.8.P.6 applies formulas for the area of a sector of a circle and the length of an arc of a circle (our translation)

In grade 2 of high school and technical secondary school, students determine the measures of fragments of a circle and its boundary. Based on a given radius and central angle measure expressed in degrees, they calculate what fraction of the entire circumference is made up by an arc and what fraction of the entire circle's area is occupied by a given sector.

A typical mistake at this stage is confusing formulas: squaring the radius when calculating arc length (instead of using the circumference formula) or using twice the radius instead of squaring it when calculating sector area. Students also often forget the constant π during transformations or make arithmetic errors when simplifying the fraction containing the angle measure over 360 degrees.

Exercises practicing this skill come in two forms:

  • Arc length – the student calculates the length of a specified part of the circle for a given radius and central angle.
  • Area of a sector of a circle – the exercise consists of calculating the area of a circular sector with given parameters.
Create a worksheet for this skill

Task types: Arc length · Area of a sector

Inscribed and central angle in a circle

The student calculates the measure of an inscribed angle based on a central angle subtended by the same arc and determines the measure of a central angle when knowing the measure of the corresponding inscribed angle.

Curriculum point MAT.LO.8.P.5 applies properties of inscribed and central angles (our translation)

In grade 2 of high school and technical secondary school, the student uses a fundamental planimetric relationship in a circle: the measure of a central angle is twice the measure of an inscribed angle subtended by the same arc. In practice, this means efficiently performing simple calculations – multiplying or dividing a given angle measure in degrees by 2.

The most common mistake is confusing the direction of this relationship and dividing by two where multiplication was required (or vice versa), leading to an illogical result where the inscribed angle becomes larger than the central angle. Students also sometimes struggle to verify on a diagram whether both angles definitely subtend the exact same arc of the circle.

Tasks in this area are based on two straightforward calculation patterns:

  • Central angle from an inscribed angle – based on a given measure of an inscribed angle (e.g., 35°), the student determines the measure of the central angle by multiplying the value by 2;
  • Inscribed angle from a central angle – the student is given the measure of a central angle (e.g., 110°) and calculates the corresponding inscribed angle by dividing this value by two.
Create a worksheet for this skill

Task types: Central angle from the inscribed angle · Inscribed angle from the central angle

Calculating the length of a chord and a tangent segment

The student calculates the lengths of chords and tangent segments to a circle using the properties of the radius, angles, and right triangles.

Curriculum point MAT.LO.8.P.1 determines radii and diameters of circles, lengths of chords of circles and tangent segments, including using the Pythagorean theorem (our translation)

Students in grade 2 of high school and technical school determine the lengths of chords and tangent segments to a circle. In geometric calculations, they combine data on the radius of the circle and distances between points, utilizing right triangles inscribed in the configuration of the circle and lines.

A common problem is overlooking the fact that a radius drawn to the point of tangency is always perpendicular to the tangent, which prevents correctly setting up the Pythagorean theorem equation. In problems involving chords, students also make mistakes by forgetting that a segment drawn from the center of the circle perpendicular to the chord bisects it into two equal parts.

The exercises are based on two types of problems:

  • Chord length — the student calculates the length of a chord given the radius of the circle and its distance from the center, or determines the missing radius given the chord,
  • Tangent segment — the problem requires determining the distance from an external point to the point of tangency with the circle, given the distance from the center of the circle and its radius.
Create a worksheet for this skill

Task types: Length of a chord · Length of a tangent segment

Inscribed and circumscribed circles of a right triangle

The student calculates the length of the radius of the inscribed circle and the circumscribed circle of a right-angled triangle based on the lengths of its sides.

Curriculum point MAT.LO.8.P.10 identifies basic special points in a triangle: the center of the inscribed circle of a triangle, the center of the circumscribed circle of a triangle, the orthocenter, the centroid, and uses their properties (our translation)

In Grade 2, the student uses the properties of a right triangle to determine the radii of its associated circles. They notice and apply a key geometric relationship: the hypotenuse of a right triangle is also the diameter of its circumscribed circle, which allows them to immediately determine the radius as half its length. At the same time, the student calculates the radius of the inscribed circle by connecting the lengths of the legs and the hypotenuse with the appropriate algebraic formula or by relating the area of the triangle to its perimeter.

The most common problem is mechanically confusing the formulas for the circumscribed and inscribed circle radii, as well as making mistakes when distinguishing the legs from the hypotenuse. Difficulties also arise when only two sides of the triangle are given in the problem—the student then forgets to first calculate the missing side using the Pythagorean theorem.

Tasks testing this skill appear in two distinct variants: Radius of a circle circumscribed about a right triangle and Radius of a circle inscribed in a right triangle. In the exercises, the student is given the side lengths (or two of them) and identifies or calculates the exact value of the desired radius.

Create a worksheet for this skill

Task types: Circumradius of a right triangle · Inradius of a right triangle

Optimization problems: maximum area

The student learns to determine the dimensions of a rectangle or an enclosure that yield the largest possible area, using the properties of the vertex of a parabola in a quadratic function.

Curriculum point MAT.LO.13.P.1 solves optimization problems in situations that can be described by a quadratic function. (our translation)
Curriculum point MAT.LO.5.P.11 uses the properties of linear and quadratic functions to interpret geometric, physical, etc. problems, also set in a practical context (our translation)

In the 2nd grade of high school and technical school, students can describe a geometric problem using a quadratic function and determine its maximum value. At this stage, they translate the problem statement into an equation describing the area of a figure as a function of the length of one of its sides, and then—using the coordinates of the parabola's vertex—identify the optimal dimensions.

A common mistake is failing to take into account the structural conditions of the problem, for example, when the plot borders a wall and does not require fencing on all sides. Students then reflexively apply the standard formula for the perimeter of a four-sided rectangle instead of summing only the three fenced segments, which leads to an incorrect function formula and a wrong final result.

In practice, students encounter word problems set in a real-world context, such as:

  • Rectangle with the maximum area – determining the dimensions of a plot given a predetermined total fence length;
  • Enclosure against a wall with the maximum area – planning a rectangular space utilizing an existing building wall, where fencing is placed on only three sides.
Create a worksheet for this skill

Task types: Pen against a wall with the greatest area · Rectangle with the greatest area

Solving exponential equations with the same base

The student can reduce both sides of an exponential equation to the same base and equate their exponents to determine the unknown.

Curriculum point MAT.LO.5.P.14 uses exponential and logarithmic functions, including their graphs, to describe and interpret issues related to practical applications. (our translation)

In the second year of high school and technical secondary school, students learn how to solve equations where the unknown is in the exponent. This skill involves transforming both sides of the equation to obtain powers with the same base (e.g., expressing numbers as powers of two, three, or fractions). When the bases are identical, the student equates the exponents and determines the value of the unknown variable.

A common problem at this stage is making mistakes when applying the laws of exponents—especially confusing the multiplication of powers with raising a power to a power (e.g., adding exponents instead of multiplying them). Students also sometimes attempt to prematurely "drop" the bases before reducing both sides of the equation to single powers with the exact same base.

In practice, problems from the Exponential Equation template require solving an equality with exponential expressions on both sides. The student reduces the numbers to a common base, sets up a simple linear equation from the resulting exponents, and calculates the final result.

Create a worksheet for this skill

Coefficients in the binomial expansion

The student learns to determine specific numerical coefficients of powers of a variable in the binomial expansion using the binomial theorem.

The student applies the binomial theorem to determine specific numerical coefficients in the expansion of a binomial power. Instead of repeatedly multiplying out the parentheses, they can directly identify and calculate the numerical value of the coefficient of a chosen power of the variable using the binomial coefficient.

A common mistake is failing to account for the coefficients of the variable itself inside the parentheses. Students focus on calculating the value of the binomial coefficient, forgetting that the entire term is raised to the power — for example, when raising the expression 2x to a power, both the number 2 and the variable must be raised to it. Correctly handling minus signs when subtracting terms inside the parentheses also poses a challenge.

In problems from the template Coefficient in a binomial expansion, the student is given a power of a binomial and determines the coefficient of the specified power of the variable without having to expand all the other terms.

Create a worksheet for this skill

Solving polynomial inequalities

The student solves polynomial inequalities by finding their zeros and reading the appropriate intervals from the number line.

In the second year of secondary school, students progress from quadratic inequalities to higher-degree polynomial inequalities. They factor the polynomial, determine its roots, and then plot them on the number line and sketch a simplified graph to determine for which numbers the expression takes positive or negative values.

The most common mistake is failing to take into account the multiplicity of the roots when sketching the graph on the number line. Students tend to mechanically cross the axis at every root, forgetting that for roots of even multiplicity, the graph "bounces off" the axis, and the sign of the polynomial does not change at that point.

A typical task from the Polynomial inequality template involves finding all numbers that satisfy the given inequality (often already written in factored form or requiring simple grouping of terms) and correctly expressing the solution as an interval or a union of intervals.

Create a worksheet for this skill

Solving quadratic equations

The student learns to determine the solutions of quadratic equations using the formulas for the roots and the discriminant delta, or by factoring the expression.

Curriculum point MAT.LO.3.P.4 solves quadratic equations and inequalities (our translation)

In the 2nd grade of high school and technical secondary school, students solve quadratic equations in standard and incomplete forms. Based on the numerical coefficients, they calculate the discriminant of the equation (delta), determine the number of real roots, and then precisely determine their values using standard formulas or algebraic transformations.

The most common mistake is sign errors when calculating delta, especially when coefficient c or b is a negative number. In incomplete equations, students often divide both sides by the unknown, which leads to the irreversible loss of one of the solutions (usually zero), instead of factoring out the common factor.

Exercises from the Quadratic Equation template involve bringing the given equation into standard form and finding all solutions belonging to the set of real numbers, or correctly justifying that no such solutions exist.

Create a worksheet for this skill

Translation of a function graph

The student learns to sketch graphs of functions resulting from translations along the coordinate axes and to determine their new formulas.

Curriculum point MAT.LO.5.P.12 based on the graph of the function y = f(x), sketches the graphs of the functions y = f(x − a), y = f(x) + b (our translation)

In Grade 2 of high school and technical secondary school, students master transforming function graphs by translating them along the axes of the coordinate system. They can shift the graph of a base function vertically (by a given number of units up or down) and horizontally (left or right), and link this translation to a change in the function's formula.

The most common difficulty is the correct interpretation of horizontal shifts. Students frequently confuse directions along the horizontal axis and shift the graph of the function f(x - p) to the left instead of to the right, instinctively associating the minus sign with the negative direction of the axis.

Tasks from the template Translation of a Function Graph involve sketching the transformed graph based on a given formula or determining from a drawing by how many units and in which direction the original graph was shifted, and then writing down the newly formed formula.

Create a worksheet for this skill

Finding the intersection points of a line and a parabola

The student learns to determine the points of intersection of a line and a parabola by algebraically solving a system of a linear equation and a quadratic equation.

In Grade 2, the student combines analytic geometry with algebra, solving systems composed of a linear and a quadratic equation. They independently express one variable from the equation of the line and substitute it into the equation of the parabola, reducing the problem to a quadratic equation with one unknown, which they solve using the discriminant (delta).

A typical mistake at this stage is stopping after finding the unknown x. Students correctly find the roots of the quadratic equation, but forget to calculate the corresponding y values, thereby failing to provide the full coordinates of the point. Sign errors also frequently occur when moving all terms to one side of the equation.

Exercises from the template Intersection points of a parabola and a line involve algebraically determining pairs of numbers (coordinates of points) where the graphs of the given functions intersect. The student checks whether the line intersects the parabola at two points, touches it at a single point of tangency, or whether the graphs have no points in common.

Create a worksheet for this skill

Calculating the value of a quadratic function

The student learns to calculate the value of a quadratic function for a given argument, correctly substituting numbers into the formula and performing the operations.

Curriculum point MAT.LO.5.P.2 calculates the value of a function given by an algebraic formula (our translation)

At this stage, the student is able to find the value of a quadratic function given in standard form for a specified argument. The task requires substituting the given number for the variable x, squaring it, multiplying by the coefficients, and adding and subtracting the resulting terms.

The most common mistake is confusing the order of operations and incorrectly raising negative numbers to a power. Students often forget parentheses with a negative argument and write, for example, -32 = 9 instead of (-3)2 = 9. It also happens that they first multiply the argument by the coefficient of the leading term and only then square the product.

In tasks from the Value of a Quadratic Function template, the student is given the formula of a function and a specific number and is tasked with calculating the result, for example, determining the value of the function f(x) = 2x2 - 5x + 1 for x = -2.

Create a worksheet for this skill

Graph and properties of the function f(x) = a/x

The student draws the graph of the function f(x) = a/x, determines its domain, and reads the basic properties of the hyperbola depending on the sign of the coefficient a.

Curriculum point MAT.LO.5.P.13 uses the function f(x) = a/x, including its graph, to describe and interpret problems related to inversely proportional quantities, also in practical applications (our translation)

At this stage, the student works with the basic form of inverse proportionality, that is, the function f(x) = a/x for a fixed number a ≠ 0. They are able to determine the domain of the function (taking into account the condition x ≠ 0), prepare a table of values for selected positive and negative numbers, and then sketch the graph — a hyperbola. Based on the formula or the graph, they determine the range, monotonicity on intervals, and identify the quadrants of the coordinate system in which the branches of the graph lie.

A typical mistake is mechanically connecting both branches of the hyperbola into a single continuous line that intersects the y-axis, which stems from ignoring the fact that zero does not belong to the domain of the function. Students also have difficulty correctly determining monotonicity: they mistakenly describe the function as decreasing over its entire domain instead of in two separate intervals, and they also confuse the position of the graph's branches when the coefficient a is negative.

In tasks from the template Function f(x) = a/x, the student most often completes a table and sketches a graph for a specific numerical parameter, determines the formula of the function based on the coordinates of a point lying on the hyperbola, or reads from the graph the arguments for which the function takes specific values.

Create a worksheet for this skill

Exact values of trigonometric functions

The student determines the exact values of trigonometric functions for standard angles, using reduction formulas and the properties of functions in the respective quadrants.

Curriculum point MAT.LO.7.P.1 uses the definitions of the functions: sine, cosine and tangent for angles from 0° to 180°, in particular determines the values of trigonometric functions for angles of 30°, 45°, 60° (our translation)

In the 2nd grade of high school and technical secondary school, students learn to determine the exact values of sine, cosine, or tangent for obtuse angles and angles greater than 90°. Using reduction formulas, they reduce calculations for such angles to the known values of acute angles (primarily 30°, 45°, and 60°).

The most common problem in these calculations is determining the correct sign of the result (plus or minus). Students often confuse the quadrant in which the given angle lies, or they determine the sign based on the function obtained after reduction instead of the original function.

A typical exercise from the template Exact value of a trigonometric function involves giving the exact number (written as a fraction or radical, without decimal approximations) for a given angle, such as 120°, 225°, or 300°.

Create a worksheet for this skill

Calculating the area of a parallelogram using sine

The student calculates the area of a parallelogram using the lengths of two adjacent sides and the sine of the angle between them.

Curriculum point MAT.LO.8.P.12 applies trigonometric functions to determine the lengths of segments in plane figures and to calculate the areas of figures. (our translation)

In the 2nd grade of secondary school, students combine their knowledge of plane figures with trigonometry. They can find the area of a parallelogram using the formula based on the lengths of two adjacent sides and the sine of the angle between them: P = a · b · sin(α).

A common mistake at this stage is substituting the angle measure directly into the formula instead of using its sine value (for example, multiplying by 60 instead of sin 60°). Students also tend to struggle with obtuse parallelograms, forgetting that the sine of an obtuse angle is equal to the sine of the acute angle that supplements it to 180°.

Problems from the Area of a Parallelogram with Sine template require calculating the area of the figure given the lengths of the sides and a known angle measure (most often with typical values such as 30°, 45°, 60°, or 120°) or determining a missing side when the area of the parallelogram is known.

Create a worksheet for this skill

Areas and perimeters of similar figures

The student learns to determine the area or perimeter of a similar figure using the scale of similarity and its square.

Curriculum point MAT.LO.8.P.9 uses relationships between perimeters and between areas of similar figures (our translation)

In grade 2 of high school and technical school, the student uses relationships between similar figures to calculate their dimensions. They skillfully use the scale factor k, applying the rule that the ratio of the perimeters of the figures is equal to k, and the ratio of their areas is k².

The most common mistake is directly multiplying the area of the figure by the scale factor k instead of its square. Students often mechanically apply the rule regarding side lengths and perimeters to surface area, which leads to incorrect results.

In a typical exercise from the template Area of a similar figure, the student knows the area of one of the figures and the scale factor (or the lengths of corresponding sides) and, on this basis, determines the area of the other, enlarged or reduced figure.

Create a worksheet for this skill

Calculating segment lengths using Thales's theorem

The student uses Thales's theorem to calculate missing lengths of segments on lines intersected by parallel lines.

Curriculum point MAT.LO.8.P.7 applies Thales' theorem (our translation)

In grade 2 of secondary school, students apply Thales's theorem to plane geometry problems. Based on the relationships between segments intercepted on lines by parallel lines, they set up the appropriate proportions and determine the unknown lengths.

The most common mistake is setting up the segments in the proportion incorrectly, especially when the problem involves the lengths of segments lying on the parallel lines. Students often confuse the segments intercepted on the arms of the angle with the entire arms, comparing the section lying between the parallels instead of the segment measured from the vertex of the angle.

Problems from the Thales's Theorem template feature a diagram showing an angle or intersecting lines cut by parallel lines. Given the numerical lengths of some of the segments, the student sets up a proportion equation and calculates the missing value.

Create a worksheet for this skill

Determining the type of triangle from side lengths

The student checks whether a triangle can be formed from the given side lengths and determines whether it is acute, right, or obtuse.

Curriculum point MAT.LO.8.P.2 recognizes acute, right, and obtuse triangles given the lengths of the sides (among other things, applies the converse of the Pythagorean theorem and the law of cosines); applies the theorem: in a triangle, the longer side lies opposite the greater interior angle (our translation)

Based on three numbers describing side lengths, the student can examine the geometry of the figure: first, they verify the triangle inequality, and then determine whether the triangle is acute, right, or obtuse. In grade 2 of high school and technical school, they do this by comparing the sum of the squares of the two shorter sides with the square of the longest side, using the generalization of the Pythagorean theorem.

A typical mistake is skipping the first step, which is the condition for the existence of a triangle (the sum of the two shorter sides must be greater than the longest side). Students jump straight to squaring the numbers and draw incorrect conclusions for sets of numbers that cannot form a triangle at all. Another common error is confusing the direction of the inequality—assuming that when the sum of the squares of the shorter sides is less than the square of the longest side, the triangle is acute instead of obtuse.

In exercises from the template Type of triangle from side lengths, the student receives a set of three specific lengths (integers or roots, e.g., 4, 6, 8) and is tasked with classifying it into the appropriate type of triangle or showing that a triangle with such dimensions does not exist.

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.

Viète's formulas: sum and product of roots

Laws of Sines and Cosines and the Area of a Triangle

Calculating the value of an exponential function

Dividing a polynomial by a binomial x − a

Solving quadratic inequalities

Equations and inequalities with absolute value

Solving polynomial equations

Solving rational and radical equations

Vertex of a parabola and the vertex form of a function

Minimum and maximum value of a function on an interval

Identifying legs relative to an angle

Calculating cosine and tangent from a known sine

Area of a circular sector and arc length

Inscribed and central angle in a circle

Calculating the length of a chord and a tangent segment

Inscribed and circumscribed circles of a right triangle

Optimization problems: maximum area

Solving exponential equations with the same base

Coefficients in the binomial expansion

Solving polynomial inequalities

Solving quadratic equations

Translation of a function graph

Finding the intersection points of a line and a parabola

Calculating the value of a quadratic function

Graph and properties of the function f(x) = a/x

Exact values of trigonometric functions

Calculating the area of a parallelogram using sine

Areas and perimeters of similar figures

Calculating segment lengths using Thales's theorem

Determining the type of triangle from side lengths

Other task types

Create a worksheet for this curriculum