Mathematics
high school and technical college

Learning objectives - general requirements

  1. Calculation skills.
    1. Performing calculations on real numbers, also using a calculator, applying the laws of mathematical operations when manipulating algebraic expressions, and using these skills to solve problems in real and theoretical contexts.
  2. Use and creation of information.
    1. Interpreting and manipulating information presented in the text, both mathematical and popular science, as well as in the form of charts, diagrams, tables.
    2. Using mathematical language to create mathematical texts, including description of reasoning and justification of conclusions, as well as to present data.
  3. Use and interpretation of representation.
    1. Applying and operating mathematical objects, interpreting mathematical concepts.
    2. Selection and creation of mathematical models for solving practical and theoretical problems.
    3. Creating auxiliary mathematical objects on the basis of existing, to conduct arguments or solve a problem.
    4. Indicating the necessity or the possibility of modifying the mathematical model in cases requiring special reservations, additional assumptions, consideration of specific conditions.
  4. Reasoning and argumentation.
    1. Carrying out reasoning, including several stages, providing arguments justifying the correctness of reasoning, distinguishing evidence from an example.
    2. Seeing regularities, similarities and analogies, formulating conclusions based on them and justifying their correctness.
    3. Choosing arguments to justify the correctness of problem solving, creating a series of arguments that guarantee correctness of the solution and effectiveness in finding solutions to the problem.
    4. Applying and creating strategies when solving tasks, also in unusual situations.

Teaching contents - detailed requirements

I. Real numbers

Standard level. A pupil:
  1. performs operations (addition, subtraction, multiplication, division, exponentiation, extracting roots, logarithm) in a set of real numbers;
  2. carries simple proofs about the divisibility of integers and division remainders, no more difficult than:
    (a) proof that the product of four consecutive natural numbers is divisible by $24$ of ;
    (b) proof that if a number when dividing by $5$ gives the remainder of $3$, then its third power when dividing by $5$ gives the remainder of $2$;
  3. applies properties of roots of any degree, including odd degree roots of negative numbers;
  4. applies the relationship between roots and exponents, and laws of exponents and roots;
  5. applies monotonicity properties of exponentiation, in particular: if $x < y$ and $a>1$, then $a^x<a^y$ , and if $x < y$ and $0<a<1$, then $a^x>a^y$ ;
  6. uses the concept of real interval, marks intervals on a number line ;
  7. applies geometrical and algebraic interpretation of absolute value, solves equations and inequalities of the type: $\left|x + 4\right| = 5$, $\left|x - 2\right| < 3$, $\left|x+3\right| \geq 4$;
  8. uses properties of exponent and root properties in practical situations, including calculating compound interest, investment returns, and loan costs;
  9. uses the relation between logarithms and exponents, uses formulas for a logarithm of the product, a logarithm of the quotient and a logarithm of a power .

A pupil meets the requirements specified for the standard level, and also
  1. 1R. applies the formula to replace the base of the logarithm.

II. Algebraic expressions.

Standard level. A pupil:
  1. uses short multiplication formulas: $(a+b)^2$, $(a-b)^2$, $a^2-b^2$, $(a+b)^3$, $(a-b)^3$, $a^3-b^3$, $a^n-b^n$;
  2. adds, subtracts and multiplies one and many variables polynomials;
  3. takes a common monomial out of an algebraic sum;
  4. factorizes polynomials by grouping terms, in no more difficult cases than factorizing the polynomial $W(x)=2x^3-\sqrt{3}x^2+4x-2\sqrt{3}$;
  5. finds integer rootsof a polynomial with integer coefficients;
  6. dzieli z resztą wielomian jednej zmiennej $W(x)$ przez dwumian postaci $x-a$;
  7. adds and subtracts rational expressions, in cases not more difficult than: $\frac{1}{x+1}-\frac{1}{x}$, $\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}$, $\frac{x+1}{x+2}+\frac{x-1}{x+1}$.

A pupil meets the requirements specified for the standard level, and also
  1. 1R. finds integer and rational polynomial roots with integer coefficients
  2. 2R. uses the basic properties of the Pascal triangle and the following properties of the binomial coefficient: $\binom{n}{0}=1$, $\binom{n}{1}=n$, $\binom{n}{n-1}=n$, $\binom{n}{k}=\binom{n}{n-k}$, $\binom{n}{k}+\binom{n}{k+1}=\binom{n+1}{k+1}$;
  3. 3R. uses formulas: $a^3+b^3$, $(a+b)^n$ and $(a-b)^n$.

III. Equations and inequalities.

Standard level. A pupil:
  1. transforms equations and inequalities in an equivalent manner;
  2. interprets equations and inequalities that are identities or contradictions;
  3. solves linear inequalities with one unknown;
  4. solves quadratic equations and inequalities;
  5. solves compound equations that lead to a quadratic equation, in particular to an equation of the form $ax^4+bx^2+c=0$;
  6. solves polynomial equations of the form $W (x) = 0$ for polynomials given in factorized form or polynomials that can be factorized by grouping terms;
  7. solves rational equations of the form $\frac{V (x)}{W (x)} = 0$, where polynomials $V(x)$ and $W(x)$ are written in a factorized form.

A pupil meets the requirements specified for the standard level, and also:
  1. III. 1R. solves polynomial inequalities such as: $W(x)> 0$, $W(x)\geq 0$, $W(x)<0$ for polynomials given in a factorized form or polynomials that can be factorized by taking out a common factor or by grouping terms;
  2. III. 2R. solves rational equations and inequalities, no more difficult than $\frac{x+1}{x(x-1)}+\frac{1}{x+1} \geq \frac{2x}{(x-1)(x+1)}$;
  3. III. 3R. uses Vièta's formulas for quadratic equations;
  4. III. 4R. solves equations and inequalities with an absolute value of no more difficult than: $\left|x + 2\right|+ 3\left|x -1\right| = 13$, $\left|x + 2\right|+ 3\left|x -1\right| < 11$;
  5. III. 5R. analyzes linear equations and inequalities with parameters and quadratic equations and inequalities with parameters, in particular, determines the number of solutions depending on parameters, gives the conditions under which solutions have the desired property, and determines solutions depending on parameters.

IV. Systems of equations.

Standard level. A pupil:
  1. solves systems of linear equations with two unknowns, presents geometric interpretation of conssistent independent, consistent dependent and inconsistent systems of equations;
  2. uses equation systems to solve word problems;
  3. uses substitution method to solve systems of equations, one of which is linear and the other is quadratic: $\begin{cases} ax + by = e \\ x^2 + y^2 + cx + dy = f \end{cases}$  or   $\begin{cases} ax + by = e \\ y = cx^2 + dx + f \end{cases}$ .

A pupil meets the requirements specified for the standard level, and also
  1. IV. 1R. solves systems of quadratic equations of the form $\begin{cases} x^2 + y^2 + ax + by = c \\ x^2 + y^2 + cx + dy = f \end{cases}$  .

V. Functions.

Standard level. A pupil:
  1. defines functions as an unambigues assignment using a verbal description, table, graph, formula (also with different patterns at different intervals);
  2. evaluates output of a function based on given input and algebraic formula of a function;
  3. reads and interprets the values of functions defined by tables, charts, formulas etc., also in case of multiple use of the same source of information or several sources at the same time;
  4. reads from the graph of functions: domain, range, zeros, monotonicity intervals, intervals in which the function reaches values larger (or not smaller) or smaller (or not larger) than a given number, the largest and smallest values of a function (if any) in a given closed interval and inputs for which the function has the largest and smallest values;
  5. interprets the coefficients in a formula of the linear function;
  6. determines the formula of a linear function based on information about its graph or its properties;
  7. sketches a graph of a quadratic function given by a formula;
  8. interprets the coefficients appearing in the formula of the quadratic function in general, canonical and product form (if it exists);
  9. determines a formula for a quadratic function based on information about the function or its graph;
  10. determines the largest and smallest values of the quadratic function in a closed interval;
  11. uses the properties of linear and quadratic functions to interpret geometric, physical, etc. issues, also embedded in a practical context;
  12. based on the graph of a function $y=f(x)$ sketches graphs of functions $y=f(x-a)$, $y=f(x)+b$, $y=-f(x)$, $y=f(-x)$;
  13. uses the function $f(x)=\frac{a}{x}$, including its graph, to describe and interpret issues related to inversely proportional quantities, also in practical applications;
  14. uses exponential and logarithmic functions, including their graphs, to describe and interpret issues related to practical applications.

A pupil meets the requirements specified for the standard level, and also
  1. V. 1R. based on the graph of the function $y=f(x)$ draws a graph of the function $y=\left| f(x) \right|$;
  2. V. 2R. uses composition of functions;
  3. V. 3R. proves the monotonicity of the function given by the formula, as in the example: prove that the function $f(x)=\frac{x-1}{x+2}$   is monotonic in the interval $(-\infty , -2)$.

VI. Sequences.

Standard level. A pupil:
  1. evaluates terms of a sequence defined by a closed formula;
  2. evaluates a few initial terms of a sequence given by recursion, like in following examples:
    a) $\begin{cases} a_1 = 0,001 \\ a_{n+1}=a_{n}+\frac{1}{2}a_{n}(1-a_{n}) \end{cases}$  ,    b) $\begin{cases} a_1 = 1 \\ a_2 = 1 \\a_{n+2}=a_{n+1}+a_{n} \end{cases}$  .
  3. in simple cases, examines whether a given sequence is increasing or decreasing;
  4. checks if the given sequence is arithmetic or geometric;
  5. uses a formula for $n$-th term and the sum of $n$ initial terms of the arithmetic sequence;
  6. uses a formula for $n$-th term and the sum of $n$ initial terms of the geometric sequence;
  7. uses properties of sequences, including arithmetic and geometric ones, to solve tasks, also embedded in a practical context.

A pupil meets the requirements specified for the standard level, and also
  1. VI. 1R. ; calculates limits of sequences using limits of the sequences such as $ \ frac {1} {n} $, $ \ sqrt [n] {a} $ & nbsp; and theorems on the limits of sum, difference, product and quotient of convergent sequences, as well as theorems on three sequences;
  2. VI. 2R. recognizes converging geometrical series and calculates their sum.

VII. Trigonometry.

Standard level. A pupil:
  1. uses the definitions of sine, cosine and tangent for angles from $0^\circ $ to $ 180^\circ$, in particular, sets the values of trigonometric functions for angles $30^\circ$, $45^\circ$, $60^\circ$;
  2. finds approximate values of trigonometric functions using tables or a calculator;
  3. finds the approximate angle size using trigonometric tables or a calculator if the value of the trigonometric function is given;
  4. uses trigonometric identities $\sin^2\alpha + \cos^2 \alpha =1$; tg$\alpha=\frac{sin\alpha}{cos\alpha}$;
  5. uses theorems of sines and cosines and the formula $P=\frac{1}{2}absin\gamma$ to calculate the are of a triangle;
  6. calculates angle sizes of a triangle and lengths of its sides with the appropriate data (solves triangles).

A pupil meets the requirements specified for the standard level, and also
  1. VII. 1R. uses an arc measure, converts an angle measure from degrees to radianse, and vice versa;
  2. VII. 2R. uses graphs of trigonometric functions: sine, cosine, tangent;
  3. VII. 3R. uses periodicity of trigonometric functions;
  4. VII. 4R. uses reduction formulas for trigonometric functions
  5. VII. 5R. uses formulas for sine, cosine and tangent of sum and difference of angles, as well as for trigonometric functions of doubled angles;
  6. VII. 6R. solves trigonometric equations and inequalities of no more difficulty than in the examples: $4\cos2x\cos5x = 2\cos7x+1$, $2\sin^2x \leq 1$.

VIII. Planimetry

Standard level. A pupil:
  1. determines radii and diameters of circles, lengths of chords of circles and tangent segments, including using Pythagoras' theorem;
  2. recognizes acute, right and obtuse triangles of given side lengths (using, if necessary, the inverse theorem to Pythagoras' theorem and cosine theorem); uses the statement: in the triangle opposite the larger internal angle lies the longer side;
  3. recognizes regular polygons and uses their basic properties;
  4. uses the properties of angles and diagonals in rectangles, parallelograms, rhombuses and trapeziums;
  5. applies properties of inscribed and central angles;
  6. applies formulas of the area of the circle sector and the length of the arc of a circle;
  7. applies theorems: Thales's theorem, inverse to Thales theorem, theorem about angle bisector in a triangle and the angle between the tangent and the chord;
  8. uses the similarity rules for triangles;
  9. uses relationships between areas of similar figures;
  10. indicates special points in a triangle: incenter, circumcenter, orthocenter, centroid, and uses their properties;
  11. uses trigonometric functions to determine lengths of segments in a plane figure and to calculate the area of a figure;
  12. carries out geometric proofs.

A pupil meets the requirements specified for the standard level, and also
  1. VIII.1R. applies the properties of quadrangles inscribed in a circle and described on a circle.

IX. Analytical geometry on the Cartesian plane.

Standard level. A pupil:
  1. recognizes the relative position of lines on a plane based on their equations, including finding a common point of two lines, if one exists;
  2. recognizes the relative position of lines on a plane based on their equations, including finding a common point of two lines, if one exists;
  3. calculates the distance between two points in a coordinate system;
  4. uses the circle equation $(x-a)^2+(y-b)^2=r^2$;
  5. calculates the distance of a point from a straight line;
  6. finds common points of the straight line and the circle as well as the straight line and the parabola being a graph of the quadratic function;
  7. determines the images of circles and polygons in axial symmetries about axes of the coordinate system, central symmetry (with a center at the origin).

A pupil meets the requirements specified for the standard level, and also
  1. IX. 1R. uses the equation of a circle in general form;
  2. IX. 2R. finds common points of two circles;
  3. IX. 3R. knows the concept of a vector and calculates its coordinates and length, adds vectors and multiplies the vector by a number, both of these actions are performed both analytically and geometrically.

X. Stereometry.

Standard level. A pupil:
  1. recognizes the relative position of lines in space, in particular perpendicular lines that do not intersect;
  2. uses the concept of the angle between the straight line and the plane and the concept of the dihedral angle between the half planes;
  3. recognizes angles between segments (e.g. edges, edges and diagonals) and angles between faces in prisms and pyramids, calculates measures of these angles;
  4. recognizes in cylinders and cones the angle between segments and the angle between segments and planes (e.g. cone opening angle, slant angle), calculates measures of these angles;
  5. determines what figure is a given cross-section of a given cuboid;
  6. calculates the volume and surface area of prisms, pyramids, cylinder, cone and sphere, also using trigonometry and theorems known;
  7. uses the relationship between the volumes of similar solids.

A pupil meets the requirements specified for the standard level, and also
  1. X. 1R. knows and applies the theorem about a line perpendicular to the plane and about three perpendiculars;
  2. X. 2R. determines the cross-sections of the cube and normal pyramids and calculates their fields, also using trigonometry.

XI. Combinatorics.

Standard level. A pupil:
  1. counts objects in simple combinatorial situations;
  2. counts objects using multiplication and addition rules (also together) for any number of actions in situations not more difficult than:
    a) finding the number of four-digit odd positive integers such that exactly one digit 1 and exactly one digit 2 appear in their decimal notation,
    b) finding the number of positive four-digit even integers such that exactly one digit 0 and exactly one digit 1 appear in their decimal notation;

A pupil meets the requirements specified for the standard level, and also
  1. XI. 1R. calculates the number of possible situations that meet certain criteria, using the rule of multiplication and addition (also together) and formulas for the number of: permutations, combinations and variations, also in cases requiring consideration of a complex model of counting elements;
  2. XI. 2R. uses the binomial coefficient and its properties in solving combinatorial problems.

XII. Probability and statistics.

Standard level. A pupil:
  1. calculates the probability in a classic model;
  2. uses the centile scale;
  3. calculates the arithmetic and weighted average, finds the median and dominant;
  4. calculates the standard deviation of a data set (also in the case of appropriately grouped data), interprets this parameter for empirical data;
  5. calculates the expected value, e.g. when determining the amount of winnings in simple games of chance and lotteries.

A pupil meets the requirements specified for the standard level, and also
  1. XII. 1R. calculates the conditional probability and uses the Bayes formula, practically applies the total probability theorem;
  2. XII. 2R. uses the Bernoulli scheme.

XIII. Optimization and calculus

Standard level. A pupil:
  1. A pupil solves optimization tasks in situations that can be described by a quadratic function.

A pupil meets the requirements specified for the standard level, and also
  1. XIII. 1R. calculates function boundaries (including one-sided);
  2. XIII. 2R. uses the Darboux property to justify the existence of a function zero and to find the approximate value of a zero;
  3. XIII. 3R. applies the definition of a derivative of function, gives a geometric and physical interpretation of the derivative;
  4. XIII. 4R. calculates the derivative of a power function with a real exponent and calculates the derivative using theorems on the derivative of sum, difference, product, quotient and composite function;
  5. XIII. 5R. uses a derivative to study the monotonicity of a function;
  6. XIII.6R. solves optimization problems using a derivative.

Conditions and manner of implementation.

Correlation. Because of the usefulness of mathematics and its applications in school teaching Physics, computer science, geography and chemistry it is advised to implement the teaching content specified in sections: I point 9 (logarithms) and, if possible, V point 14, V point 1 (concept of function) and V point 5 (linear functions) in the first half of the first year, and the content of teaching specified in sections: V point 11 (quadratic functions) and V point 13 (inverse proportionality) no later than the end of first grade. The content of teaching specified in section VI point 2 (calculating the initial words of the recursively specified strings) can be performed in correlation with the same problem of the core curriculum in computer science.

Mathematical symbols. A pupils should use commonly accepted symbols for numerical sets, in particular: for integers $\mathbb{Z}$, for rational numbers - $\mathbb{Q}$, for real numbers - $\mathbb{R}$. The symbol $C$ for the set of integers can lead to confusion and should be avoided.

Intervals. The A pupil should use the intervals to describe the set of solutions of an inequality. It is worth emphasizing that the most important thing about the answer is its correctness. For example, resolving the inequality $ x^2-9x+20>0$ can be credited to any of the following ways:
Logarithm applications. When teaching logarithms, it is worth highlighting their applications. in explaining natural phenomena. In nature, processes whose logarithmic function describes are common. This happens when in a certain period of time a given quantity always increases (or decreases) with a constant fold. The following sample problems illustrate the use of logarithms.

Vertical form. When dealing with square polynomials it should be emphasized vertical form of a quadratic function and the resulting properties. It should be noted that the formulas for the roots of quadratic aquation and the coordinates of the top of the parabola are only conclusions from it. It is worth emphasizing that many issues associated with the quadratic function can be solved directly from the the vertex form, without mechanical application of formulas. In particular, the vertical form allows you to find the smallest or largest value of a quadratic function, as well as the axis of symmetry of its plot.

Composite functions and inverse functions. The definition of a composite function appears only in the advanced level, but in the standard level a A pupil is expected to be able to use data from several sources simultaneously. However, this does not require any formal introduction of composition or inverse function.

Equivalent transformations . When solving equations and inequalities, it should be noted that instead the method of equivalent transformations, you can use the inference method (ancient analysis method). After determining the potential set of solutions, it is checked which of the determined values are the solutions. In many situations, it is not worth demanding equivalent transformations when the inference method leads to quick results. In addition, A pupils should know that the legitimate method of proof is equivalent transformation of the thesis.

Applications of algebra. A prerequisite for successful math teaching process is efficient using algebraic expressions. Algebraic methods can often be used in geometric situations and vice versa - geometric illustration allows a better understanding of algebraic issues.

Sequences This issue should be discussed so that A pupils realize that there are others besides arithmetic and geometric sequences. Similarly, it should be emphasized that apart from non-decreasing, growing, non-growing, decreasing and constant sequences, there are also ones that are not monotonic. It is worth noting that some sequences describe the dynamics of processes occurring in nature or society. For example, given in section VI point 2 lit. and the string describes the spread of the rumor ($a_n$ indicates how many people have heard of the rumor). A similar model can be used to describe the spread of the epidemic.

Planimetry Solving classic geometric problems is an effective way to shape mathematical awareness. As a result, A pupils who solve construction problems acquire skills in solving geometric problems of various types, for example, A pupils can easily acquire the properties of circles inscribed in a triangle or quadrangle, if they can construct these figures. Teaching geometric constructions can be carried out in a classic way, using a ruler and a compass, or you can use specialized computer programs, such as GeoGebra.

Stereometry. Spatial imagination is particularly developed during the implementation of teaching content from stereometry. Using solid models, as well as the ability to draw their projections, will greatly facilitate the determination of different sizes in solids. Cross-section analysis of a tetrahedron and a cube can be very informative; particularly valuable is the answer to the question: what a cross-section can be. Experience teaches that, for example, the question of the existence of a cube cross-section, which is a trapezium but not isosceles, can cause trouble for many A pupils.

Binomial expansion. It is important to emphasize the importance of the binomial coefficient $\binom{n}{k}$ in combinatorics when teaching the formula for $(a + b)^n$. It is also worth to write it in the form $\binom{n}{k}=\frac{n(n-1)\cdot ... \cdot(n-k + 1)}{1 \cdot 2 \cdot ... \cdot (k-1) \cdot k}$, because in this form its interpretation is more visible and easier to calculate for small $k$.

Probability. In the future, A pupils will deal with issues related to randomness that occur in various areas of life and science, for example, when analyzing surveys, issues in economics and financial market research or in natural and social sciences. It is worth mentioning the paradoxes in the theory of probability, which show typical errors in reasoning and discuss some of them. It is also worth conducting experiments with A pupils, e.g. an experiment in which A pupils save a long string of heads and tails without tossing-up coins, and then save the string of heads and heads resulting from random coin tosses. Misconceptions about randomness usually suggest that there should not be long sequence of tails (or heads), when in reality such long sequence of tails (or heads) occur. Discussing the basic expected value does not require the introduction of the concept of random variable. It is advisable to use an intuitive understanding of the expected value of profit or to determine the number of objects that meet certain properties. In this way, the A pupil has the opportunity to see the relationship of probability with everyday life, also has the chance to shape the ability to avoid risky behaviors, e.g. in financial decisions
In the advanced level, it is important to make A pupils aware that the theory of probability is not limited to the classical scheme and the combinatorics used there. A good illustration are examples of using the Bernoulli scheme for a large number of attempts.

Proofs. Idependent carrying out proofs by A pupils develops skills such as logical thinking, precise expression of thoughts and the ability to solve complex problems. Command allows you to improve your ability to choose the right arguments and construct the right reasoning. One of the methods to develop the skill of proving is to analyze the evidence of the theorems learned. In this way, you can teach what a properly conducted piece of evidence should look like. Being able to formulate correct reasoning and justifications is also important outside of mathematics. Below is a list of statements whose evidence the A pupil should know.

Theorems, proofs - standard level.:

  1. The existence of infinitely many primes.
  2. Proof of irrationality of numbers: $\sqrt{2}$ , $\log_2{5}$ itp.
  3. Formulas for zeros of the quadratic trinomial.
  4. Basic properties of powers (with integer and rational exponents) and logarithms.
  5. Theorem about division with the remainder of the polynomial by a binomial of the form $x-a$ together with recursive formulas for the quotient and remainder coefficients (Horner's algorithm) - proof can be carried out in a special case, e.g. for a fourth-degree polynomial.
  6. Closed formulas for $n$-th term and the sum of $n$ initial terms of the arithmetic and geometric sequence.
  7. Theorem on angles in a circle:
    1) Central angle is twice any inscribed angle subtended by the same arc;
    2) Two angles inscribed in the same circle are congruent if and only if they are subtended by the arc of the same length.
  8. Theorem of segments in a right triangle:
    If a segment $CD$ is the height of a right triangle $ABC$ with the right angle $ACB$, then $\left|AD\right| \cdot\left|BD\right|=\left|CD\right|^2$, $\left|AC\right|^2= \left|AB\right|\cdot\left|AD\right|$ oraz $\left|BC\right|^2= \left|AB\right|\cdot\left|BD\right|$.
  9. Triangle angle bisector theorem:
    If a line $CD$ is the angle bisector of the angle $ACB$ in a triangle $ABC$ and the point $D$ lies on the side $AB$, then $\frac{|AD|}{|BD|}=\frac{|AC|}{|BC|}$.
  10. The formula for the area of a triangle $P=\frac{1}{2}ab \sin \gamma$ .
  11. Sine theorem.
  12. Cosine theorem and the theorem inverse to Pythagoras's theorem.