{"id":39103,"date":"2014-08-27T21:59:29","date_gmt":"2014-08-27T18:59:29","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lvi-2014\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"matematika-i-informatika-knizhka-4-godina-lvi-2014","status":"publish","type":"post","link":"https:\/\/strategies.azbuki.bg\/en\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lvi-2014\/","title":{"rendered":"Mathematics and Informatics, Number 4\/2014, Volume 57"},"content":{"rendered":"<p><strong><a name=\"top\">&nbsp;<\/a>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 <\/strong><\/p>\n<p><strong>EDUCATIONAL TECHNOLOGIES \/&nbsp; \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<\/strong> <\/p>\n<p><a href=\"#art01\">\u0418\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u043c\u0435\u0442\u0430\u043f\u0440\u0435\u0434\u043c\u0435\u0442\u043d\u043e\u0441\u0442\u0438 [Informatics and Mathematics in the Context of Interdisciplinarity] \/ <\/a><br \/><em>\u0421. \u0410. \u0411\u0435\u0448\u0435\u043d\u043a\u043e\u0432, \u0410. \u0425. \u0414\u0437\u0430\u043c\u044b\u0445\u043e\u0432 \/ Sergei Beshenkov, Alibek Dzamiho<\/em>v &#8211; \u0441\u0442\u0440. 335<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art02\">\u041a\u0440\u0430\u0442\u043a\u043e \u0440\u044a\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e \u0437\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u0437\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0430 Wolfram Mathematica [A Short Manual for the Computer Algebra System Wolfram Mathematica] <\/a><br \/><em>\u0422\u0438\u0445\u043e\u043c\u0438\u0440 \u0418\u0432\u0430\u043d\u043e\u0432 \/ Tihomir Ivanov<\/em> &#8211; \u0441\u0442\u0440.&nbsp;343<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art03\">\u041a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u043e \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0430\u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430: \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043d\u0430 \u041a\u043e\u0441\u043d\u0438\u0442\u0430 \u0432 \u0415\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f [Computer-Generated Mathematics: Kosnita Products in Euclidean Geometry] <\/a><em>\/<br \/>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432 \/ Sava Grozdev, Deko Dekov<\/em> &#8211; \u0441\u0442\u0440. 355<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art05\">\u041d\u044f\u043a\u043e\u043b\u043a\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u0435\u0434\u0438\u043d \u0432\u0438\u0434 \u043a\u0440\u0438\u0432\u0438, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430 \u041d\u0430\u0433\u0435\u043b [Some Properties of a Type of Curves, Generated by a Nagel Point] \/ <\/a><br \/><em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Sava Grozdev, Veselin Nenkov<\/em> &#8211; \u0441\u0442\u0440. 384<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><strong>EDUCATIONAL RESEARCH \/ \u041d\u0410\u0423\u0427\u041d\u041e\u2013\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u0410\u0422\u0418\u0418 <\/strong><\/p>\n<p><a href=\"#art06\">Honeycomb \u2013 a Genius Creation of Nature \/ <\/a><br \/><em>Risto Mal\u010deski<\/em> &#8211; \u0441\u0442\u0440. 402<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art07\">Three Solutions of a Problem with Four Circles \/ <\/a><br \/><em>\u0160efket Arslanagi\u0107<\/em> &#8211; \u0441\u0442\u0440. 408<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art08\">\u041c\u043e\u0442\u0438\u0432\u0430\u0446\u0438\u044f \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438 \u0447\u0440\u0435\u0437 \u043f\u0440\u0435\u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u043e\u0432\u043a\u0430 \u043d\u0430 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0442\u0430 [Motivation in Problem Solving by Task Preformulation] \/ <\/a><br \/><em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430 \/ Sava Grozdev, Diana Stefanova<\/em> \u2013 p. 416<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><a href=\"#art09\">\u0423\u0440\u043e\u043a \u0437\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0437\u0430\u0434\u0430\u0447\u0438 \u043f\u043e \u0438\u043a\u043e\u043d\u043e\u043c\u0438\u043a\u0430 [A Class Lesson for the Application of Functions to Problems in Economics] \/<\/a><br \/><em>\u041f\u0435\u0442\u044f \u0421\u044f\u0440\u043e\u0432\u0430 \/ Petya Syarova<\/em> &#8211; \u0441\u0442\u0440. 422<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><strong>CONTEST PROBLEMS \/ \u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong> <\/p>\n<p>\u041a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430 \u0431\u0440\u043e\u044f [Contest Problems of the Issue] &#8211; \u0441\u0442\u0440. 431<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u0431\u0440\u043e\u0439 5, 2013 [Solutions of the Contest Problems from Issue 5, 2013] &#8211; \u0441\u0442\u0440. 432<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">&nbsp;<a href=\"https:\/\/strategies.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p><strong>READ IN THE LATEST ISSUES OF \u201cAZ BUKI\u201c JOURNALS \/&nbsp; <\/strong><br \/><strong>\u0412 \u041d\u041e\u0412\u0418\u0422\u0415 \u0411\u0420\u041e\u0415\u0412\u0415 \u041d\u0410 \u0421\u041f\u0418\u0421\u0410\u041d\u0418\u042f\u0422\u0410 \u041d\u0410 \u0418\u0417\u0414\u0410\u0422\u0415\u041b\u0421\u0422\u0412\u041e \u201e\u0410\u0417 \u0411\u0423\u041a\u0418\u201c \u0427\u0415\u0422\u0415\u0422\u0415<\/strong> &#8211; \u0441\u0442\u0440. 435<\/p>\n<p><strong>GUIDE FOR AUTHORS \/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> &#8211; \u0441\u0442\u0440. 437<\/p>\n<hr \/>\n<p><strong><a name=\"art01\">&nbsp;<\/a>\u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0422\u0418\u041a\u0410 \u0418 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410 \u0412 \u041a\u041e\u041d\u0422\u0415\u041a\u0421\u0422\u0415 \u041c\u0415\u0422\u0410\u041f\u0420\u0415\u0414\u041c\u0415\u0422\u041d\u041e\u0421\u0422\u0418 <\/strong><\/p>\n<p><strong>1\u0421. \u0410. \u0411\u0435\u0448\u0435\u043d\u043a\u043e\u0432, 2\u0410. \u0425. \u0414\u0437\u0430\u043c\u044b\u0445\u043e\u0432 1\u0424\u0413\u041d\u0423 <\/strong><em><br \/>\u00ab\u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u0438 \u043c\u0435\u0442\u043e\u0434\u043e\u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u044f\u00bb \u2013 \u0420\u0410\u041e <br \/>2\u041a\u0430\u0440\u0430\u0447\u0430\u0435\u0432\u043e-\u0427\u0435\u0440\u043a\u0435\u0441\u0441\u043a\u0438\u0439 \u0433\u043e\u0441\u0443\u0434\u0430\u0440\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442<\/em><\/p>\n<p><strong>Abstract<\/strong>. \u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u043e\u0431\u0441\u0443\u0436\u0434\u0430\u044e\u0442\u0441\u044f \u043d\u0435\u0441\u043a\u043e\u043b\u044c\u043a\u043e \u043f\u043e\u0434\u0445\u043e\u0434\u043e\u0432 \u043a \u0441\u043e\u0432\u043c\u0435\u0449\u0435\u043d\u0438\u044e \u0441\u043e\u0434\u0435\u0440\u0436\u0430\u043d\u0438\u044f \u0443\u0447\u0435\u0431\u043d\u044b\u0445 \u0434\u0438\u0441\u0446\u0438\u043f\u043b\u0438\u043d \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0438 \u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438. <\/p>\n<p><em>Keywords<\/em>: mathematics, informatics, interdisciplinary approach, contents<\/p>\n<p><strong>INFORMATICS AND MATHEMATICS IN THE CONTEXT OF INTERDISCIPLINARITY<\/strong><\/p>\n<p><strong>Abstract<\/strong>. The present paper considers several approaches to a common content of Informatics and Mathematics teaching disciplines.<\/p>\n<p><strong>Prof. Sergei Beshenkov, DSc<\/strong><br \/>Laboratory on Didactics of Informatics <br \/>Institute for Content and Teaching <br \/>Methods Russian Academy of Education <br \/>Moscow, Russia<\/p>\n<p><strong>Dr. Alibek Dzamihov, Assoc. Prof.<\/strong> <br \/>Department of Mathematics and Methodology of Teaching<br \/>Karachay-Cherkessian State University Cherkessk, Russia<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art02\">&nbsp;<\/a>\u041a\u0420\u0410\u0422\u041a\u041e \u0420\u042a\u041a\u041e\u0412\u041e\u0414\u0421\u0422\u0412\u041e&nbsp; \u0417\u0410 \u0421\u0418\u0421\u0422\u0415\u041c\u0410\u0422\u0410 \u0417\u0410 \u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u0410 \u0410\u041b\u0413\u0415\u0411\u0420\u0410 WOLFRAM MATHEMATICA <\/strong><\/p>\n<p><strong>\u0422\u0438\u0445\u043e\u043c\u0438\u0440 \u0418\u0432\u0430\u043d\u043e\u0432,<\/strong> <br \/><em>\u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430, \u0411\u0410\u041d<\/em><\/p>\n<p><strong>Abstract<\/strong>. \u041d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u043f\u044a\u0440\u0432\u0430 \u0432 \u043f\u043e\u0440\u0435\u0434\u0438\u0446\u0430\u0442\u0430 \u201e\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043c\u043e\u0434\u0435\u043b\u0438 \u043d\u0430 \u0440\u0435\u0430\u043b\u043d\u0438 \u043f\u0440\u043e\u0446\u0435\u0441\u0438 \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0438\u0442\u0435 \u0437\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0430 \u0437\u0430 \u0442\u044f\u0445\u043d\u043e\u0442\u043e \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u201c \u0438 \u0438\u043c\u0430 \u043f\u043e-\u0441\u043a\u043e\u0440\u043e \u0443\u0432\u043e\u0434\u0435\u043d \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440. \u041e\u0441\u043d\u043e\u0432\u043d\u0430\u0442\u0430 \u043d\u0438 \u0446\u0435\u043b \u0435 \u0434\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438 \u0435\u0434\u043d\u0430 \u043e\u0442 \u043d\u0430\u0439-\u043f\u043e\u043f\u0443\u043b\u044f\u0440- \u043d\u0438\u0442\u0435 \u0441\u0438\u0441\u0442\u0435\u043c\u0438 \u0437\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0430 \u2013 Wolfram Mathematica. \u041f\u043e\u043a\u0430\u0437\u0430\u043d\u0438 \u0441\u0430 \u043d\u044f\u043a\u043e\u0438 \u0431\u0430\u0437\u043e\u0432\u0438 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430, \u043a\u043e\u0438\u0442\u043e \u0431\u0438\u0445\u0430 \u0431\u0438\u043b\u0438 \u043f\u043e\u043b\u0435\u0437\u043d\u0438 \u0432 \u0441\u0440\u0435\u0434\u043d\u0438\u044f \u043a\u0443\u0440\u0441 \u043d\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435, \u0430 \u0437\u0430\u043f\u043e\u0437\u043d\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u0441 \u0442\u044f\u0445 \u0449\u0435 \u043f\u043e\u0434\u0433\u043e\u0442\u0432\u0438 \u0447\u0438\u0442\u0430\u0442\u0435\u043b\u044f \u0437\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u0432\u0430\u0449\u0438\u0442\u0435 \u0441\u0442\u0430\u0442\u0438\u0438, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u043c\u0430- \u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0442\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435. \u0412\u043a\u043b\u044e\u0447\u0435\u043d\u0438 \u0441\u0430 \u0438 \u043d\u044f\u043a\u043e\u0438 \u043f\u0440\u043e\u0441\u0442\u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u0438 \u0437\u0430 \u0441\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u043d\u0430 \u0440\u0430\u0431\u043e\u0442\u0430, \u043a\u043e\u0438\u0442\u043e \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0432\u0430\u0442 \u0443\u0441\u0432\u043e\u044f\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043f\u044a\u0440\u0432\u043e\u043d\u0430\u0447\u0430\u043b\u043d\u0438 \u0443\u043c\u0435\u043d\u0438\u044f \u0437\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441\u044a\u0441 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430. <\/p>\n<p><em>Keywords<\/em>: mathematical modeling, computer algebra systems, education, Wolfram Mathematica<\/p>\n<p><strong>A SHORT MANUAL FOR THE COMPUTER ALGEBRA SYSTEM WOLFRAM MATHEMATICA<\/strong><\/p>\n<p><strong>Abstract.<\/strong> This is the first (introductory) article in the series \u201cMathematical models of real processes and applications of computer algebra systems to their study\u201d. We present one of the most popular computer algebra systems\u2014Wolfram Mathematica. Some of its basic capabilities that could be useful for the high school education in mathematics are shown. Studying the latter is a necessary background that will prepare the reader for the following articles, dealing with mathematical modeling. Also, a few simple problems are included to develop some basic skills for working with the system.<\/p>\n<p><strong>Mr. Tihomir Ivanov<\/strong> <br \/>Institute of Mathematics and Informatics <br \/>Bulgarian Academy of Sciences<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art03\">&nbsp;<\/a>\u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u041e \u0413\u0415\u041d\u0415\u0420\u0418\u0420\u0410\u041d\u0410 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410: \u041f\u0420\u041e\u0418\u0417\u0412\u0415\u0414\u0415\u041d\u0418\u042f \u041d\u0410 \u041a\u041e\u0421\u041d\u0418\u0422\u0410 \u0412 \u0415\u0412\u041a\u041b\u0418\u0414\u041e\u0412\u0410\u0422\u0410 \u0413\u0415\u041e\u041c\u0415\u0422\u0420\u0418\u042f <\/strong><\/p>\n<p><strong>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432,&nbsp; \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432<\/strong> <br \/><em>Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n<p><strong>Abstract<\/strong>. \u041f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u041a\u043e\u0441\u043d\u0438\u0442\u0430 \u043d\u0430 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u0432 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430 \u0441\u0430 \u0432\u044a\u0432\u0435\u0434\u0435\u043d\u0438 \u043f\u0440\u0435\u0437 2011 \u0433. \u043e\u0442 Randy Hutson. \u0412 \u0435\u043d\u0446\u0438\u043a\u043b\u043e\u043f\u0435\u0434\u0438\u0438\u0442\u0435 \u043d\u0430 Weisstein \u0438 Kimberling \u0441\u0430 \u0434\u0430\u0434\u0435\u043d\u0438 \u0442\u0430\u0431\u043b\u0438\u0446\u0438 \u0441 \u0442\u0435\u043e\u0440\u0435\u043c\u0438 \u0437\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043d\u0430 \u041a\u043e\u0441\u043d\u0438\u0442\u0430. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u043f\u043e\u043a\u0430\u0437\u0432\u0430 \u043a\u0430\u043a \u0442\u0435\u0437\u0438 \u0442\u0430\u0431\u043b\u0438\u0446\u0438 \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u0440\u0430\u0437\u0448\u0438\u0440\u0435\u043d\u0438 \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 70 \u0442\u0435\u043e\u0440\u0435\u043c\u0438 \u0437\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u043d\u0430 \u041a\u043e\u0441\u043d\u0438\u0442\u0430, \u043e\u0442\u043a\u0440\u0438\u0442\u0438 \u043e\u0442 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c. <\/p>\n<p><em>Keywords<\/em>: computer-generated mathematics, Kosnita product, Euclidean geometry, Discoverer.<\/p>\n<p><strong>COMPUTER-GENERATED MATHEMATICS: KOSNITA PRODUCTS IN EUCLIDEAN GEOMETRY<\/strong><\/p>\n<p><strong>Abstract.<\/strong>&nbsp; In 2011 Randy Hutson introduced the notion of Kosnita products. In the encyclopedias of Weisstein and Kimberling some tables&nbsp; are given with theorems on Kosnita products.&nbsp; It is shown in the paper how these tables could be extended by mens of the computer program \u201cDiscoverer\u201d. The paper contains 70 theorems on Kosnita products, produced by \u201cDiscoverer\u201d.<\/p>\n<p><strong>Prof. Sava Grozdev, DSc<\/strong><br \/>Institute of Mathematics and Informatics \u2013 BAS Acad.<\/p>\n<p><strong>Dr. Deko Dekov, Assoc. Prof.<\/strong><\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art04\">&nbsp;<\/a>\u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415 \u0412 \u0421\u0422\u0418\u041b EDUTAINMENT \u0421 \u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u041d\u0410 \u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u0410 \u0413\u0420\u0410\u0424\u0418\u041a\u0410 <\/strong><\/p>\n<p><strong>\u0425\u0440\u0438\u0441\u0442\u043e \u041a\u0440\u0443\u0448\u043a\u043e\u0432, \u0410\u0441\u0435\u043d \u0420\u0430\u0445\u043d\u0435\u0432, \u041c\u0430\u0440\u0438\u0430\u043d\u0430 \u041a\u0440\u0443\u0448\u043a\u043e\u0432\u0430<\/strong> <br \/><em>Plovdiv University \u2018\u2019Paisii Hilendarski\u2019<\/em><\/p>\n<p><strong>Abstract<\/strong>. \u0412 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0442\u0435 \u0433\u043e\u0434\u0438\u043d\u0438 \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438 \u0435\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0438 \u0434\u043e\u043a\u0430\u0437\u0432\u0430\u0442 \u043e\u0433\u0440\u043e\u043c\u043d\u0430- \u0442\u0430 \u043f\u043e\u043b\u0437\u0430 \u043e\u0442 \u0430\u043a\u0442\u0438\u0432\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 (active learning). \u0422\u043e \u0432\u043a\u043b\u044e\u0447\u0432\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u043f\u043e\u0434\u0445\u043e\u0434\u0438, \u043e\u0431\u0435\u0434\u0438\u043d\u0435\u043d\u0438 \u043e\u0442 \u0435\u0434\u043d\u043e \u2013 \u043f\u043e\u0441\u0442\u0430\u0432\u044f\u043d\u0435\u0442\u043e \u043d\u0430 \u043e\u0431\u0443\u0447\u0430\u0435\u043c\u0438\u044f \u043a\u0430\u0442\u043e \u0430\u043a\u0442\u0438\u0432\u043d\u0430 \u0435\u0434\u0438\u043d\u0438\u0446\u0430 \u0432 \u0446\u0435\u043d\u0442\u044a\u0440\u0430 \u043d\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u043d\u0438\u044f \u043f\u0440\u043e\u0446\u0435\u0441. \u041f\u0440\u0438 \u043d\u0435\u0433\u043e \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u044a\u0442 \u043d\u0435 \u0435 \u043f\u0430\u0441\u0438\u0432\u0435\u043d \u0440\u0435\u0433\u0438\u0441\u0442\u0440\u0430\u0442\u043e\u0440 \u043d\u0430 \u043b\u0435\u043a\u0442\u043e\u0440- \u0441\u043a\u0438\u0442\u0435 \u0438\u0434\u0435\u0438, \u0430 \u0430\u043a\u0442\u0438\u0432\u0435\u043d \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u043a \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e, \u043a\u043e\u0435\u0442\u043e \u0441\u0435 \u043f\u0440\u0435\u0432\u0440\u044a\u0449\u0430 \u0432 \u043e\u0431\u043c\u0435\u043d \u043d\u0430 \u0438\u0434\u0435\u0438. \u0411\u0443\u0440\u043d\u043e\u0442\u043e \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435 \u043d\u0430 \u0418\u041a\u0422, \u0438\u043d\u0442\u0435\u0440\u043d\u0435\u0442 \u0438 \u0441\u043e\u0446\u0438\u0430\u043b\u043d\u0438\u0442\u0435 \u043c\u0440\u0435\u0436\u0438 \u0434\u043e\u043f\u0440\u0438\u043d\u0435\u0441\u0435 \u0438\u0437\u043a\u043b\u044e\u0447\u0438\u0442\u0435\u043b- \u043d\u043e \u0437\u0430 \u0440\u0430\u0437\u0448\u0438\u0440\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0438\u0442\u0435 \u0437\u0430 \u0430\u043a\u0442\u0438\u0432\u043d\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f\u0442 \u043f\u043e\u0434\u0445\u043e\u0434\u0438 \u0437\u0430 \u0430\u043a\u0442\u0438\u0432\u043d\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0432 \u043e\u0431\u043b\u0430\u0441\u0442\u0442\u0430 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438- \u043a\u0430\u0442\u0430 \u0438 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0438\u0442\u0435 \u043d\u0430\u0443\u043a\u0438. \u041e\u043f\u0438\u0441\u0430\u043d\u0438 \u0441\u0430 \u0430\u0432\u0442\u043e\u0440\u0441\u043a\u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u0437\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0433\u0440\u0430\u0444\u0438- \u043a\u0430, \u043f\u043e\u0434\u043f\u043e\u043c\u0430\u0433\u0430\u0449\u0438 \u0430\u043a\u0442\u0438\u0432\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0432 \u0441\u0442\u0438\u043b edutainment. \u0422\u044f\u0445\u043d\u043e\u0442\u043e \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435 \u0435 \u0441\u044a\u043e\u0431\u0440\u0430\u0437\u0435\u043d\u043e \u0441 \u043d\u0438\u0432\u043e\u0442\u043e \u043d\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438\u0442\u0435 \u0438 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0432\u0430 \u0434\u0430 \u0441\u0435 \u0443\u0441\u0432\u043e\u044f\u0432\u0430\u0442 \u0437\u043d\u0430\u043d\u0438\u044f \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430- \u0442\u0438\u043a\u0430, \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430, \u043e\u0431\u0435\u043a\u0442\u043d\u043e \u043e\u0440\u0438\u0435\u043d\u0442\u0438\u0440\u0430\u043d\u043e \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u0438 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0433\u0440\u0430\u0444\u0438\u043a\u0430. <\/p>\n<p><em>Keywords<\/em>: active learning, edutainment, teaching methodology, object-oriented programming, computer graphics<\/p>\n<p><strong>EDUTAINMENT LEARNING STYLE USING COMPUTER GRAPHICS<\/strong><\/p>\n<p><strong>Abstract.<\/strong> In recent years pedagogical experiments have proven the immense ben- efits of active learning. Active learning includes different approaches with one thing in common \u2013 placing the learner as an active unit in the centre of the educational process. The student is not only a registrar of lecturer\u2019s ideas but an active participant in the exchange of ideas. The rapid development of ICT, the Internet and the social networks have led to a huge expand of the possibilities of active learning. An approach to active learning in mathematics and computer science is presented in the paper. Authoring tools are described for computer graphics, supporting edutainment as an active learning method. Using these tools students acquire knowledge in math- ematics, informatics, object-oriented programming and computer graphics.<\/p>\n<p><strong>Dr. Hristo Krushkov, Assoc. Prof.,<\/strong> <\/p>\n<p><strong>Prof. Dr. Asen Rahnev,<\/strong> <\/p>\n<p><strong>Dr. Mariana Krushkova, Assist. Prof.<\/strong> <br \/>Department of Software Engineering <br \/>Faculty of Mathematics and Informatics <br \/>University of Plovdiv<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art05\">&nbsp;<\/a>\u041d\u042f\u041a\u041e\u041b\u041a\u041e \u0421\u0412\u041e\u0419\u0421\u0422\u0412\u0410 \u041d\u0410 \u0415\u0414\u0418\u041d \u0412\u0418\u0414 \u041a\u0420\u0418\u0412\u0418, \u041f\u041e\u0420\u041e\u0414\u0415\u041d\u0418 \u041e\u0422 \u0422\u041e\u0427\u041a\u0410 \u041d\u0410 \u041d\u0410\u0413\u0415\u041b<\/strong><\/p>\n<p><strong> 1\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, 2\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/strong> <br \/><em>1\u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u2013 \u0411\u0410\u041d<\/em><br \/><em> 2\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u043a\u043e\u043b\u0435\u0436 \u2013 \u041b\u043e\u0432\u0435\u0447<\/em> <\/p>\n<p><strong>Abstract<\/strong>. \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u043e\u043f\u0438\u0441\u0430\u043d\u043e \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435 \u043d\u0430 \u0435\u0434\u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u043e\u0442 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430\u0442\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043f\u0440\u0435\u0437 2013 \u0433. <\/p>\n<p><em>Keywords<\/em>: triangle, conic, Ceva circle, Ceva curve, Feuerbach configuration, Nagel point, GSP.<\/p>\n<p><strong>SOME PROPERTIES OF A TYPE OF CURVES, GENERATED BY A NAGEL POINT <\/strong><\/p>\n<p><strong>Abstract<\/strong>. The paper considers a generalization of a geometric problem from the International Mathematical Olympiad in 2013. <\/p>\n<p><strong>Prof. Sava Grozdev, DSc<\/strong><br \/> Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p><strong>Dr. Veselin Nenkov, Assoc. Prof.<\/strong> <br \/>Technical College Lovech<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art06\">&nbsp;<\/a>HONEYCOMB \u2013 A GENIUS&nbsp; CREATION OF NATURE<\/strong><\/p>\n<p><strong>Risto Mal\u010deski<\/strong> <br \/><em>FON University, Skopje<\/em><\/p>\n<p><strong>Abstract<\/strong>. The paper considers how bees construct the cells of the beehive. It turns out that the constructions are subjected to strict mathematical laws, which makes them surprising masterpieces of nature. <\/p>\n<p><em>Keywords<\/em>: bee, cell, beehive, regular polygon, area.<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art07\">&nbsp;<\/a>THREE SOLUTIONS OF A PROBLEM&nbsp; WITH FOUR CIRCLES <\/strong><\/p>\n<p><strong>\u0160efket Arslanagi\u0107 University of Sarajevo<\/strong><\/p>\n<p><strong>Abstract<\/strong>. The paper considers three solutions of an interesting problem with four circles. <\/p>\n<p><em>Keywords<\/em>: solution, circles, geometry<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art08\">&nbsp;<\/a>\u041c\u041e\u0422\u0418\u0412\u0410\u0426\u0418\u042f \u041f\u0420\u0418 \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0418 \u0427\u0420\u0415\u0417 \u041f\u0420\u0415\u0424\u041e\u0420\u041c\u0423\u041b\u0418\u0420\u041e\u0412\u041a\u0410 \u041d\u0410 \u0423\u0421\u041b\u041e\u0412\u0418\u042f\u0422\u0410 <\/strong><\/p>\n<p><strong>1\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, <\/strong><br \/><strong>2\u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430<\/strong> <br \/><em>1\u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u2013 \u0411\u0410\u041d <\/em><br \/><em>2\u041e\u0441\u043d\u043e\u0432\u043d\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u201e\u041d. \u0412\u0430\u043f\u0446\u0430\u0440\u043e\u0432\u201c \u2013 \u0433\u0440. \u0410\u0441\u0435\u043d\u043e\u0432\u0433\u0440\u0430\u0434 &nbsp;<\/em><\/p>\n<p><strong>Abstract<\/strong>. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0442\u0430 \u0437\u0430 \u0437\u0430\u043e\u0441\u0442\u0440\u044f\u043d\u0435 \u043d\u0430 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438 \u0447\u0440\u0435\u0437 \u043f\u0440\u0435\u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0442\u0430 \u043a\u0430\u0442\u043e \u0437\u0430\u0434\u0430\u0447\u0438-\u0432\u044a\u043f\u0440\u043e\u0441\u0438. <\/p>\n<p><em>Keywords<\/em>: problem, pre-formulation, problem-question, solution&nbsp;<\/p>\n<p><strong>MOTIVATION IN PROBLEM SOLVING BY TASK PREFORMULATION&nbsp; <\/strong><\/p>\n<p><strong>Abstract.<\/strong> The paper considers a possibility to intensify students\u2019 attention in problem solving by a task pre-formulation to problem-questions.<\/p>\n<p><strong>Prof. Sava Grozdev, DSc<\/strong> <br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p><strong>Dr. Diana Stefanova<\/strong> <br \/>Primary School \u201cNikola Vaptsarov\u201d<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art09\">&nbsp;<\/a>\u0423\u0420\u041e\u041a \u0417\u0410 \u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u041d\u0410 \u0424\u0423\u041d\u041a\u0426\u0418\u0418 \u0412 \u0417\u0410\u0414\u0410\u0427\u0418 \u041f\u041e \u0418\u041a\u041e\u041d\u041e\u041c\u0418\u041a\u0410 <\/strong><\/p>\n<p><strong>\u041f\u0435\u0442\u044f \u0421\u044f\u0440\u043e\u0432\u0430<\/strong> <br \/><em>\u0421\u041e\u0423 \u201e\u0412\u0430\u0441\u0438\u043b \u041b\u0435\u0432\u0441\u043a\u0438\u201c \u2013 \u042f\u043c\u0431\u043e\u043b<\/em><\/p>\n<p><strong>Abstract<\/strong>. \u0421\u043f\u043e\u0434\u0435\u043b\u0435\u043d e \u0443\u0447\u0438\u0442\u0435\u043b\u0441\u043a\u0438 \u043e\u043f\u0438\u0442 \u0437\u0430 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0437\u043d\u0430\u043d\u0438\u044f \u0437\u0430 \u043b\u0438\u043d\u0435\u0439\u043d\u0430\u0442\u0430 \u0444\u0443\u043d\u043a\u0446\u0438\u044f \u043a\u044a\u043c \u0448\u0438\u0440\u043e\u043a\u043e \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0438 \u043f\u043e\u043d\u044f\u0442\u0438\u044f \u043e\u0442 \u0438\u043a\u043e\u043d\u043e\u043c\u0438\u043a\u0430\u0442\u0430 \u0438 \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u043f\u0430\u0437\u0430\u0440\u0430. \u0414\u0430\u0434\u0435\u043d\u0438 \u0441\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0438 \u0437\u0430 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0438 \u043d\u0430 \u0434\u0440\u0443\u0433\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438.<\/p>\n<p> <em>Keywords<\/em>: linear function, straight line, demand, suply, market equilibrium<\/p>\n<p><strong>A CLASS LESSON FOR THE APPLICATION OF FUNCTIONS TO PROBLEMS IN ECONOMICS<\/strong><\/p>\n<p><strong>Abstract. T<\/strong>his article shares teacher\u2019s experience in applying the linear function knowledge to common concepts in the field of Economics and market trends. Some examples for using other functions are also given. <\/p>\n<p><strong>Mrs. Petya Syarova Teacher<\/strong><\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>","protected":false},"excerpt":{"rendered":"<p>&nbsp;CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 EDUCATIONAL TECHNOLOGIES \/&nbsp; \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418 \u0418\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u043c\u0435\u0442\u0430\u043f\u0440\u0435\u0434\u043c\u0435\u0442\u043d\u043e\u0441\u0442\u0438 [Informatics and Mathematics in the Context of Interdisciplinarity] \/ \u0421. \u0410. \u0411\u0435\u0448\u0435\u043d\u043a\u043e\u0432, \u0410. \u0425. \u0414\u0437\u0430\u043c\u044b\u0445\u043e\u0432 \/ Sergei Beshenkov, Alibek Dzamihov &#8211; \u0441\u0442\u0440. 335 &nbsp; &nbsp; \u041a\u0440\u0430\u0442\u043a\u043e \u0440\u044a\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e \u0437\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u0437\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0430 Wolfram Mathematica [A Short Manual for the Computer Algebra [&hellip;]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[2882],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>&quot;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&quot;, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LVII, 2014 - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/newspaper.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lvi-2014\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"&quot;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&quot;, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LVII, 2014 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"&nbsp;CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 EDUCATIONAL TECHNOLOGIES \/&nbsp; \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418 \u0418\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0435 \u043c\u0435\u0442\u0430\u043f\u0440\u0435\u0434\u043c\u0435\u0442\u043d\u043e\u0441\u0442\u0438 [Informatics and Mathematics in the Context of Interdisciplinarity] \/ \u0421. \u0410. \u0411\u0435\u0448\u0435\u043d\u043a\u043e\u0432, \u0410. \u0425. \u0414\u0437\u0430\u043c\u044b\u0445\u043e\u0432 \/ Sergei Beshenkov, Alibek Dzamihov &#8211; \u0441\u0442\u0440. 335 &nbsp; &nbsp; \u041a\u0440\u0430\u0442\u043a\u043e \u0440\u044a\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e \u0437\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u0437\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0430 Wolfram Mathematica [A Short Manual for the Computer Algebra [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/newspaper.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lvi-2014\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Azbuki55\/\" \/>\n<meta property=\"article:published_time\" content=\"2014-08-27T18:59:29+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-24T14:33:15+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" 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