Improved pixel estimates of measures of plane sets as methodological approach to introducing the concept of «area of a figure» in geometry course. Part 1

The paper formulates an approach to obtaining a pixel estimate for a measure of a plane set (area of a figure) with an estimate of its error. The approach is presented by the authors as a modern alternative methodical approach to the introduction of the concept of “Area of a figure” in the systematic course of the geometry of a basic common school. The approach is consistent with the cognitive patterns of modern “IT-oriented” students and successfully combines deep set-theoretical foundations and simple and visual geometric images. The developed approach can be realized in the form of laboratory works in mathematics, either in “manual” mode (on squired paper) or by using appropriate computer procedures (on the monitor screen). The approach is simple in explanation and application, and is successfully used by the authors in teaching modern IT-oriented students-mathematicians at a pedagogical university and college students in the framework of coordinated courses of mathematics and informatics. The first part of the work analyzes the approaches of textbooks on the geometry of various groups of authors to the implementation of the concept of “Area of a figure” and associated with it; the basics are developed and tested of building improved pixel estimates of areas of plane figures. It is shown that the statistical method can be “experimentally” clearly confirmed the validity of the key formula for the further presentation: $S=a^2$ for the area of square.