English
Curriculum (Poland)

Mathematics — Grade 11

Mathematics for grade 3 of high school and technical secondary school: arithmetic and geometric sequences, trigonometry with radian measure, analytic geometry — lines, circles, vectors — combinatorics and probability. Ready-to-print exercises for every topic.

Create a worksheet for this curriculum

What does a student learn in mathematics in the 3rd year of high school?

In grade 3, the student studies sequences: determining terms, sums of arithmetic and geometric sequences, and in the extended curriculum, limits and geometric series. They learn about radian measure, periodic functions, and trigonometric equations. In analytic geometry, they determine equations of lines and circles, intersection points of a line and a circle, tangents, and vector operations. In combinatorics and probability theory, they calculate events using the multiplication rule, Newton's binomial theorem, and classical probability.

Important note on the core curriculum: the curriculum for general secondary and technical secondary schools is formulated in blocks — it describes what a 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; in the “Curriculum scope” section, you can see the curriculum points with quotes.

Page status: the exercises for this grade are ready in a context-free format (instruction and formula only); exercises with diagrams, proofs, and story 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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

    Calculating terms of a sequence from the general formula and recurrence (we teach in grade 11)

  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

    Calculating terms of a sequence from the general formula and recurrence (we teach in grade 11)

  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

    Calculating the nth Term and Sum of an Arithmetic Sequence (we teach in grade 11) · Calculating the terms and sum of a geometric sequence (we teach in grade 11)

  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

    Calculating the nth Term and Sum of an Arithmetic Sequence (we teach in grade 11)

  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

    Calculating the terms and sum of a geometric sequence (we teach in grade 11)

  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

    Calculating the nth Term and Sum of an Arithmetic Sequence (we teach in grade 11) · Calculating the terms and sum of a geometric sequence (we teach in grade 11)

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

  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

    Equations of lines: perpendicularity and point of intersection (we teach in grade 11)

  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

    Equations of lines: perpendicularity and point of intersection (we teach in grade 11)

  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

    Calculating the distance between points on a plane (we teach in grade 11)

  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

    Determining the center and radius of a circle from its equation (we teach in grade 11)

  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

    Counting possibilities: multiplication rule and permutations (we teach in grade 11)

  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

    Counting possibilities: multiplication rule and permutations (we teach in grade 11)

  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

    Counting possibilities: multiplication rule and permutations (we teach in grade 11)

  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

    Counting possibilities: multiplication rule and permutations (we teach in grade 11)

  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

    Probability in simple experiments (we teach in grade 11) · Probability in sampling without replacement (we teach in grade 11) · Calculating the probability of the union of events (we teach in grade 11)

  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

  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

Counting possibilities: multiplication rule and permutations

The student calculates the number of possible arrangements and choices using the rules of multiplication and addition, permutations, and variations in problems involving digits or the distribution of prizes.

Curriculum point MAT.LO.11.P.1 counts objects in simple combinatorial situations (our translation)
Curriculum point MAT.LO.11.P.2 counts objects, using the rules of multiplication and addition (also in combination) for any number of actions, e.g.: (our translation)
Curriculum point MAT.LO.11.P.2.a 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)
Curriculum point MAT.LO.11.P.2.b 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)

A student in the 3rd grade of secondary school analyzes combinatorial situations and determines the number of possible outcomes without the need to list all cases. They apply the multiplication and addition rules in multi-stage processes, calculate the number of permutations of a finite set when arranging objects in a specific order, and use variations when selecting a specific number of elements and assigning them to specific positions.

A common mistake is failing to take boundary conditions into account at individual stages of selection. For example, students forget that the first digit of a formed number cannot be zero, or they fail to consider that parity depends solely on the last digit. If a given problem requires a specific digit to appear only once, errors stem from an improper separation of stages: instead of first determining the position of the distinguished digit, the student thoughtlessly multiplies the available possibilities.

Exercises practicing this skill are based on models of sampling and ordering. The student solves problems that involve finding the number of ways to arrange people in a queue (permutations), distribute distinct trophies or places (variations — award distribution), and form even or odd numbers of a given number of digits where a specified digit appears exactly once.

Create a worksheet for this skill

Task types: Permutations — standing in a queue · Arrangements — awarding prizes · Even numbers containing a digit once · Odd numbers containing a digit once

Probability in simple experiments

The student calculates the probability of an event occurring in a single trial, such as rolling a die or drawing a ball from a box, writing the result as a fraction.

Curriculum point MAT.LO.12.P.1 calculates probability in the classical model (our translation)

At this stage, the student determines classical probability in single random experiments with a finite number of equally likely outcomes. They can correctly count all possible outcomes of a given experiment, identify those that satisfy a given condition (favorable outcomes), and then represent the chance of the event occurring as the quotient of these two values.

A common mistake is incorrectly determining the number of favorable outcomes. Students often overlook certain possibilities when checking properties of numbers (for example, forgetting that the number 2 is a prime number) or confuse the number of favorable outcomes with the total number of elements in the set. Sometimes, probability is also incorrectly identified with just the count of matching outcomes instead of being written as a fraction.

In practice, the tasks involve analyzing a specific random situation:

  • Rolling a die – for example, calculating the chance of rolling a prime number or a number greater than a given value,
  • Drawing a ball from a box – calculating the probability of drawing a ball of a specific color given the count of balls of each type,
  • Drawing a card with a number – determining the chance of drawing a card with a number that meets a specific condition, e.g., divisible by 3.
Create a worksheet for this skill

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

Equations of lines: perpendicularity and point of intersection

The student determines the point of intersection of two lines, finds the equation of a perpendicular line passing through a point, and reads the slope from the general form.

Curriculum point MAT.LO.9.P.1 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)
Curriculum point MAT.LO.9.P.2 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)

As part of analytic geometry, the student converts the general equation of a line to slope-intercept form in order to accurately determine its slope. Based on this, they can determine the condition for perpendicularity and write the equation of a perpendicular line passing through a specified point with known coordinates. They also algebraically determine the coordinates of the intersection point of two lines in a plane.

The most common problem when finding a perpendicular line is confusing opposite numbers with reciprocals — students often take only the reciprocal of the slope (e.g., changing 3 to 1/3 instead of -1/3). Another difficulty is correctly dividing by the coefficient of the variable y when converting the general equation, especially when this coefficient is negative.

Exercises testing this skill include typical computational tasks:

  • Slope from the general equation — converting the equation of a line and reading its slope;
  • Perpendicular line through a point — finding the new equation of a line that satisfies the perpendicularity condition and passes through a given point;
  • Point of intersection of lines — determining the pair of numbers that is the common solution to the equations of two lines.
Create a worksheet for this skill

Task types: Slope from the general form · Intersection of two lines · Perpendicular line through a point

Calculating the distance between points on a plane

The student calculates the distance between two points in a coordinate system using the geometric relationship from the Pythagorean theorem.

Curriculum point MAT.LO.9.P.3 calculates the distance between two points in a coordinate system (our translation)

Within analytic geometry, the student determines the distance between two points with given numerical coordinates. The line segment connecting these points is treated as the hypotenuse of a right-angled triangle, whose legs result directly from the differences between the coordinates on the axes. On this basis, the student applies the Pythagorean theorem to calculate the exact length of the segment.

A typical difficulty at this stage involves arithmetic errors when subtracting negative numbers while determining the lengths of the legs (e.g., subtracting a negative coordinate from a positive one). Students also tend to forget to take the square root at the very end of the calculations, leaving the square of the distance as the final result.

In practice, in tasks such as Distance between points using the Pythagorean theorem and Distance between points, the student is given the coordinates of two points and calculates the distance between them. The tasks may require illustrating an auxiliary triangle in the coordinate system or directly calculating the length of the segment, potentially factoring out terms from under the radical sign.

Create a worksheet for this skill

Task types: Distance between two points · Distance between points with Pythagoras

Calculating the nth Term and Sum of an Arithmetic Sequence

The student determines any term of an arithmetic sequence based on the general formula and calculates the sum of a specified number of its initial terms.

Curriculum point MAT.LO.6.P.4 checks whether a given sequence is arithmetic or geometric (our translation)
Curriculum point MAT.LO.6.P.5 uses the formula for the n-th term and for the sum of the first n terms of an arithmetic sequence (our translation)
Curriculum point MAT.LO.6.P.7 uses the properties of sequences, including arithmetic and geometric, to solve problems, also set in a practical context. (our translation)

In grade 3 of high school and technical secondary school, the student proficiently uses the properties of an arithmetic sequence. They can determine the value of any specified term based on the first term and the common difference of the sequence, as well as calculate the sum of a given number of consecutive initial terms.

A common issue at this stage is confusing the term number with its actual value (that is, the index n with the term an). Mistakes also occur with decreasing sequences, where the common difference is a negative number, leading to sign errors when substituting values into the sum formula.

Exercises practicing this skill appear in two clear forms:

  • Arithmetic sequence — nth term: the student receives the initial sequence data and calculates a specific, distant term (e.g., the tenth or twentieth).
  • Arithmetic sequence — sum of terms: the task consists of summing a specified number of the first terms of a given sequence using the appropriate formula.
Create a worksheet for this skill

Task types: Arithmetic sequence — nth term · Arithmetic sequence — sum of terms

Calculating the terms and sum of a geometric sequence

The student determines any term of a geometric sequence and calculates the sum of its initial terms based on the given data.

Curriculum point MAT.LO.6.P.4 checks whether a given sequence is arithmetic or geometric (our translation)
Curriculum point MAT.LO.6.P.6 applies the formula for the n-th term and for the sum of the first n terms of a geometric sequence (our translation)
Curriculum point MAT.LO.6.P.7 uses the properties of sequences, including arithmetic and geometric, to solve problems, also set in a practical context. (our translation)

In the 3rd grade of high school and technical secondary school, students master the rules governing geometric sequences. Based on the first term and the common ratio, they can determine the general formula and calculate the value of any n-th term. They also use the formula for the sum of the first n terms of a sequence, skillfully combining multiplication by a constant ratio with exponentiation.

A common issue is confusing the common ratio of a geometric sequence with the common difference of an arithmetic sequence—students reflexively add a constant value instead of multiplying successive terms. Difficulties also arise with the order of operations and raising fractions and negative numbers to powers, which leads to sign errors when evaluating the power in the sum formula.

Printable worksheets include the following templates:

  • Geometric sequence — nth term: calculating a specific term or the common ratio based on initial data;
  • Geometric sequence — sum of terms: determining the sum of a given number of initial terms of a sequence.
Create a worksheet for this skill

Task types: Geometric sequence — nth term · Geometric sequence — sum of terms

Calculating terms of a sequence from the general formula and recurrence

The student calculates specific terms of a number sequence based on a given general formula or recurrence relation.

Curriculum point MAT.LO.6.P.1 calculates the terms of a sequence defined by a general formula (our translation)
Curriculum point MAT.LO.6.P.2 calculates the initial terms of sequences defined recursively (our translation)

In Grade 3 of high school and technical secondary school, the student efficiently determines the values of specified terms of a numerical sequence. They are able to substitute the position number n into the general formula to directly calculate the result, as well as determine subsequent numbers in the sequence based on a recursive formula, using their knowledge of the previous terms.

A common difficulty is confusing a term's index with its value — for example, mistaking the notation for the previous term for subtracting one from the value of the current term. With recursive sequences, computational discipline can also be a challenge: a minor arithmetic error made at the beginning of calculations cascades through all subsequent terms.

The exercises are based on two types of tasks:

  • A term of a sequence from the general formula — the student receives an explicit formula for the sequence and calculates a specific value (e.g., the fifth or tenth term) by substituting the appropriate natural number for n;
  • A term of a recursive sequence — given the initial term and the rule for generating subsequent steps, the student gradually calculates consecutive elements of the sequence until reaching the desired term.
Create a worksheet for this skill

Task types: Term of a sequence from its formula · Term of a recursive sequence

Calculating and applying the binomial coefficient

The student learns to calculate the value of the binomial coefficient using factorials and apply it to determine the number of ways to choose a group of people or elements.

In grade 3, students learn about the binomial coefficient and learn how to efficiently calculate its numerical value. They master expanding and simplifying fractions containing factorials, which allows for quick calculations without the need to multiply large numbers. They use this notation to describe situations where a subset of elements is chosen without regard to their order.

A typical mistake at this stage is confusing situations where order matters with situations where only the composition of the group is important. As a result, students often forget to divide the product by the factorial of the number of chosen elements. Arithmetic errors also frequently occur when simplifying the denominator of the fraction.

In practice, exercises for this skill include two types of problems:

  • Value of the binomial coefficient — purely arithmetic calculation of the value of a given coefficient by simplifying products of numbers;
  • Selecting a delegation — word problems involving calculating how many ways a smaller group (e.g., a representation of several people) can be chosen from a larger set of people or objects.
Create a worksheet for this skill

Task types: Value of a binomial coefficient · Choosing a delegation

Converting degrees to radians and radians to degrees

The student efficiently converts an angle measure given in degrees to radians and converts a measure expressed in radians to degrees.

In Grade 3, the student learns radian measure as an alternative way of expressing the size of an angle. At this stage, they master converting fluently between degrees and radians, based on the fundamental relationship that an angle of 180° corresponds to π radians. They use proportions and multiplication by the appropriate fraction to determine the exact value of the angle in the new unit.

A typical problem is reversing the conversion formula. Students often confuse when to multiply by π/180° and when by 180°/π, which leads to an incorrect result with a double π or its absence. Correctly reducing the resulting common fractions to simplest form can also be challenging.

The exercises are based on two types of tasks: in Degrees to radians exercises, the student converts a given angle (e.g., 45° or 120°) into a fraction with the symbol π. In Radians to degrees exercises, they perform the reverse operation, replacing π with 180° and calculating the final measure in degrees.

Create a worksheet for this skill

Task types: Degrees to radians · Radians to degrees

Determining the period and amplitude of a sinusoid

The student learns to determine the fundamental period and the maximum value of the sine function based on its formula or graph.

In grade 3 of high school and technical school, students analyze graphs and formulas of trigonometric functions using radian measure. As part of this skill, they determine two key parameters of a sinusoid: its period (the length of the repeating cycle) and amplitude, which is the deviation from the midline related to the maximum value reached.

A common mistake is confusing the effect of individual coefficients in the function formula on its shape. Students often equate the coefficient of the argument directly with the period, instead of calculating it using the formula that accounts for the 2π periodicity, or confuse vertical stretching of the graph with horizontal compression.

Printable worksheets in this area cover two typical exercise formats:

  • Period of a sinusoid — the student calculates or reads the cycle length of a function with a transformed argument;
  • Maximum value of a sinusoid — the student identifies the maximum value the function attains by analyzing its amplitude and vertical shift.
Create a worksheet for this skill

Task types: Maximum of a sine curve · Period of a sine curve

Quadrilaterals inscribed in a circle and circumscribed about a circle

The student calculates angle measures in a quadrilateral inscribed in a circle and side lengths in a quadrilateral circumscribed about a circle, applying relationships between opposite elements of the figure.

A student in the 3rd grade of secondary school can determine missing quantities in quadrilaterals related to a circle. In the case of a figure inscribed in a circle, they use the rule that the sum of the measures of opposite angles is 180°. When a quadrilateral is circumscribed about a circle, the student applies the property stating that the sums of the lengths of opposite pairs of sides are equal.

The most common problem is confusing these two properties or applying them to adjacent elements of the figure instead of opposite ones. Students mistakenly sum angles lying along one side to 180° or equate the sums of adjacent sides, which leads to setting up an incorrect equation.

In practice, the tasks involve reading data from a diagram or the problem statement and calculating the missing value. In the template Angle in an inscribed quadrilateral, the student determines the unknown angle measure, and in the exercise Side of a quadrilateral circumscribed about a circle, they calculate the length of the fourth side based on the three given segments.

Create a worksheet for this skill

Task types: Angle in a cyclic quadrilateral · Side of a tangential quadrilateral

Determining the center and radius of a circle from its equation

The student learns to read the coordinates of the center and the length of the radius of a circle directly from its standard form equation.

Curriculum point MAT.LO.9.P.4 uses the equation of a circle (x − a)² + (y − b)² = r² (our translation)

At this stage, the student analyzes the equation of a circle and determines two key parameters from it: the coordinates of the center of the circle and the length of its radius. They effectively connect the algebraic representation of the relationship between variables with its geometric meaning in the coordinate system.

A typical issue is confusing signs when determining the coordinates of the center. The formula involves subtraction, so a plus sign inside the parentheses indicates a negative coordinate, which students often forget. Another frequent mistake is equating the right-hand side of the equation directly with the radius instead of its square—for example, entering a radius of 25 instead of 5.

The tasks take the form of direct analysis of the given equation. In the exercise Center of a circle from an equation, the student's task is to correctly provide the coordinate pair of the center point, while in the task Radius of a circle from an equation, they determine the correct length of the radius by taking the square root of the number on the right-hand side of the equals sign.

Create a worksheet for this skill

Task types: Centre of a circle from its equation · Radius of a circle from its equation

Length and operations on vectors

The student calculates the length of a vector in a plane based on its coordinates and determines the coordinates of vectors resulting from addition, subtraction, and multiplication by a number.

In the 3rd grade of secondary school, students work with vectors in a coordinate system. They efficiently determine the length of a vector using the Pythagorean theorem for its coordinates, and perform basic operations: adding and subtracting vectors, as well as multiplying them by real numbers to form their linear combinations.

A common mistake is losing minus signs when squaring negative coordinates while calculating the length of a vector. For example, students write -32 = -9 instead of (-3)2 = 9, which leads to incorrect results under the square root. Another difficulty is correctly subtracting vectors when coordinates contain negative numbers, where it is easy to make an error when resolving a double negative.

Problems in this topic appear in two main forms:

  • Vector length – the student is given the coordinates of a vector and calculates its numerical value, often simplifying radical expressions;
  • Vector combination – the student determines the coordinates of a new vector described by an algebraic formula, for example 2a - 3b, by operating directly on the coordinates of the given vectors.
Create a worksheet for this skill

Task types: Length of a vector · Linear combination of vectors

Calculating the sum of a geometric series

The student checks the convergence condition and calculates the sum of an infinite sequence of numbers using the formula for the sum of a geometric series.

In Grade 3 of upper secondary school, students learn the concept of a geometric series, which is the sum of infinitely many consecutive terms of a geometric sequence. They learn to evaluate when such a sum yields a finite number at all—checking whether the common ratio of the sequence satisfies the condition |q| < 1—and then calculate its exact value using a formula based on the first term and the common ratio.

A typical issue is the mechanical application of the formula without verifying whether the series converges. Students attempt to calculate the sum even when the common ratio is, for example, 2 or -3, which leads to incorrect conclusions. Another frequent stumbling block involves calculation errors when the common ratio is negative, where subtracting a negative number appears in the denominator (1 - q).

In tasks from the Sum of a geometric series template, students most often receive a sequence of fractions or integers (e.g., 1/2, 1/4, 1/8...) written as an infinite sum, with instructions to determine the common ratio, verify whether it can be summed, and find the final numerical result.

Create a worksheet for this skill

Calculating the probability of the union of events

The student calculates the probability of at least one of two events occurring, using the formula for the probability of the union of events and taking into account their intersection.

Curriculum point MAT.LO.12.P.1 calculates probability in the classical model (our translation)

A student in the 3rd grade of high school and technical secondary school determines the probability that at least one of two random events will occur. To do this, they use the relation connecting the probability of the union of events with the probabilities of the individual events and their intersection: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

The most common mistake is mechanically adding the values of P(A) and P(B) without subtracting the intersection P(A ∩ B). Students forget to check whether the events can occur simultaneously, which leads to double-counting the same outcomes and often results in an erroneous probability greater than 1.

Exercises from the Probability of the union of events template require calculating the missing probability based on numerical data provided directly or embedded in word problems about drawing numbers, picking cards, or rolling dice, where the student must independently determine the individual probabilities and the intersection of both situations.

Create a worksheet for this skill

Probability in sampling without replacement

The student calculates the probability of random events, applying the rules of combinatorics to multi-stage experiments, such as drawing balls from an urn without replacement.

Curriculum point MAT.LO.12.P.1 calculates probability in the classical model (our translation)

The student combines the classical definition of probability with combinatorial methods. At this stage, they can correctly determine the total number of possible outcomes of an experiment and the number of favorable outcomes in situations where objects are selected in batches or sequentially without replacement.

A common mistake is failing to take into account that the number of available elements in the set decreases after each step. Students tend to treat subsequent stages as if they were drawing with replacement, or they struggle to decide whether the order of drawing the objects matters when counting favorable combinations.

Problems practicing this skill use the model of drawing balls without replacement. The student analyzes the contents of a container with balls of different colors and calculates the probability of drawing a specific set—for example, drawing exactly two balls of one color or balls of different colors.

Create a worksheet for this skill

Calculating limits of rational sequences

The student learns to determine the limit of a sequence defined by a fractional (rational) expression by examining the behavior of its terms as the index approaches infinity.

In the 3rd grade of high school and technical school, students practice determining the limit of a sequence whose terms are given by a fraction with polynomials in the variable n in the numerator and denominator. The task involves determining what value the successive terms of the sequence approach as n becomes arbitrarily large. The student uses the technique of factoring out the highest power of n from both the numerator and the denominator, which allows the fraction to be brought into a readable form.

A common mistake is attempting to directly substitute infinity into the formula and perform regular arithmetic operations on it, which leads to incorrect conclusions in the presence of indeterminate forms. Students also sometimes struggle with correctly dividing each term of the sum by the factored-out power or forget that fractions with n in the denominator approach zero as the index grows.

Tasks from the Limit of a rational sequence template involve calculating the limit of a specific algebraic fraction. The student transforms the given sequence formula, reduces powers, and writes down the final numerical value of the limit, or determines that the sequence diverges to infinity.

Create a worksheet for this skill

Finding the composition of two functions

The student learns to create a new function by substituting the formula of one function into the argument of another and to calculate the values of such a composition.

At this stage, the student is able to find the formula of a composite function by applying the operation of composing two functions, such as f(g(x)) or g(f(x)). They understand that the value returned by the inner function becomes the input argument for the outer function, and on this basis, they can determine the final algebraic form of the new function.

A common mistake is confusing the order of composition and treating the operation f(g(x)) the same as g(f(x)), whereas function composition is generally not commutative. Students also frequently forget to enclose the substituted expression in parentheses, which leads to sign errors or losing powers when replacing the variable x with the entire formula of the inner function.

Problems from the Function Composition template most often require finding the general formula for the composition of two given algebraic functions or calculating a specific numerical value for a designated argument, for example, the value of the expression f(g(3)).

Create a worksheet for this skill

Solving simple trigonometric equations

The student solves simple trigonometric equations, determining angles in radians or degrees, taking into account their periodicity.

In Grade 3, the student solves basic equations with a single trigonometric function, such as sine, cosine, or tangent (for example sin x = 1/2 or tan x = 1). They read angle values using radian measure expressed in terms of π or degree measure, and write down complete sets of solutions, taking into account the periodicity of the functions.

A typical mistake at this stage is giving only one angle instead of two within a single period—for example, specifying the angle π/6 for a sine equal to 1/2 and missing the value 5π/6. Students also often confuse the periods of functions, incorrectly adding + 2kπ to the tangent instead of + kπ, or completely omitting the integer parameter symbol k.

Problems from the template Trigonometric equation most often require transforming the given expression into basic form, and then determining all matching solutions in a given interval (e.g., from 0 to 2π) or providing a general formula for all numbers satisfying the equation.

Create a worksheet for this skill

Determining the points of intersection of a line and a circle

The student learns to determine the coordinates of the points of intersection or tangency between a line and a circle and to determine the number of their common points.

In coordinate geometry, the student combines algebra with geometry to determine the relative position of a line and a circle. By solving a system of a linear equation and a quadratic equation, they can determine whether the line intersects the circle at two points, is tangent to it, or has no points in common with it, as well as calculate the exact coordinates of these points in the coordinate system.

A common issue involves calculation errors when substituting the expressed variable from the equation of the line into the equation of the circle. Students make mistakes when applying algebraic expansion formulas (especially with minus signs) and often stop after finding only the x-coordinates, forgetting to calculate the corresponding y-values.

Typical problems from the Intersection points of a circle and a line template provide the equation of a line and the equation of a circle in algebraic form. The student's task is to solve the system of equations, determine the number of solutions, and give the pairs of numbers representing the coordinates of the intersection points.

Create a worksheet for this skill

Finding the equation of a tangent line to a circle at a point

The student determines the equation of a tangent line to a circle at a given point on that circle, using methods of analytic geometry in a coordinate system.

As part of analytic geometry in Grade 3, the student is able to determine the equation of a tangent line to a circle at a point lying on it. The task is based on linking the equation of a circle with the properties of straight lines: the student determines the coordinates of the circle's center, finds the line containing the radius connecting the center to the point of tangency, and then constructs the equation of the line perpendicular to this radius.

A common mistake with this topic is an error in determining the slope of the perpendicular line (for example, changing the sign without taking the reciprocal) or misreading the coordinates of the circle's center from its equation, especially when negative signs appear in the formula. Another issue arises when the tangent line is vertical — the student then tries to force it into slope-intercept form instead of the form x = c.

In practice, in tasks of the type Tangent to a circle at a point, the student receives the algebraic equation of a circle and the coordinates of the point of tangency. Their goal is to perform the calculations leading to the final equation of the tangent line in slope-intercept or general form.

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.

Counting possibilities: multiplication rule and permutations

Probability in simple experiments

Equations of lines: perpendicularity and point of intersection

Calculating the distance between points on a plane

Calculating the nth Term and Sum of an Arithmetic Sequence

Calculating the terms and sum of a geometric sequence

Calculating terms of a sequence from the general formula and recurrence

Calculating and applying the binomial coefficient

Converting degrees to radians and radians to degrees

Determining the period and amplitude of a sinusoid

Quadrilaterals inscribed in a circle and circumscribed about a circle

Determining the center and radius of a circle from its equation

Length and operations on vectors

Calculating the sum of a geometric series

Calculating the probability of the union of events

Probability in sampling without replacement

Calculating limits of rational sequences

Finding the composition of two functions

Solving simple trigonometric equations

Determining the points of intersection of a line and a circle

Finding the equation of a tangent line to a circle at a point

Other task types

Create a worksheet for this curriculum