Mathematics
high school and technical college

From 2025, the Matura examination in Formula 2023 will be conducted on the basis of the requirements specified in the general and specific requirements of the general education core curriculum from 2024.

Core curriculum for general education from 2024 [CKE pdf pl]

Learning objectives - general requirements

Teaching contents - detailed requirements

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.

• the inequality is satisfied by numbers $x$ that are less than 4 or greater than $5$;
• all numbers $x$ less than $4$ and all numbers $x$ greater than $5$ satisfy the inequality;
• $x<4$ or $x>5$;
• $x\in (-\infty, 4)$ or $x\in (5, \infty)$;
• $ x\in (-\infty, 4) \cup (5, \infty)$.

• Problem 1. The Richter scale is used to determine the strength of earthquakes. This force is described by the formula $R=\log \frac{A}{A_0}$, where $A$ is the quake amplitude expressed in centimeters, $A_0=10^{-4}$ cm is a constant, called the reference amplitude. On May 5, 2014, a $6.2$ rich magnitude earthquake occurred in Thailand. Calculate the earthquake amplitude of the land in Thailand.
• Problem 2. The patient took a dose of $100$ mg of the drug. The mass of this drug remaining in the body after time $t$ is determined by the relationship $M(t)=a\cdot b^t$. After five hours, the body removes $30$% of the drug. Calculate how much medicine will remain in the patient's body after a day.

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.