Continued fractions in mathematics are mainly known due to the need for a more detailed presentation of rational and irrational numbers. Continued fractions of rational numbers are finite, while for presentation of irrational numbers, it is necessary to introduce infinite continued fractions. The theoretical part of the master's thesis will, in addition to the presentation of the basic concepts of continued fractions, convergence, and irrationality of a continued fraction, also include a more detailed discussion of the number e, which can be easily expanded into an infinite continued fraction in order to demonstrate its irrationality. We are able to represent irrational numbers in a unique way with an infinite continued fraction, with its finite approximations representing an excellent rational approximation of the number, also called convergents. Hereinafter, we will address periodic continued fractions, and show that an infinite simple continued fraction is a quadratic irrational, if and only if it is periodic. This theoretical content, together with the presentation of the basic algorithm of the computation of continued fractions, will be an introduction to the empirical part. The empirical part will represent the project by means of which we will present continued fractions as an enrichment of the content for teaching mathematics in elementary and secondary schools. We will show how to use different learning approaches in teaching continued fractions to gifted students.
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