{"id":38136,"date":"2012-11-21T10:24:19","date_gmt":"2012-11-21T07:24:19","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-6-godina-lv-2012\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"matematika-i-informatika-knizhka-6-godina-lv-2012-2","status":"publish","type":"post","link":"https:\/\/pedagogy.azbuki.bg\/en\/matematics\/matharticles\/matematika-i-informatika-knizhka-6-godina-lv-2012-2\/","title":{"rendered":"Mathematics and Informatics, Number 6\/2012, Volume 55"},"content":{"rendered":"

<\/a>\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS<\/strong><\/p>\n

LeMATH Communiqu\u00e9 - Learning Mathematics with New Means of Communication - p. 487<\/p>\n

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\u0414\u0432\u0435 \u0434\u0438\u0434\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0441\u0442\u044a\u043b\u0431\u0438 <\/a>
[Two Didactical Ladders] \/ <\/a>
\u0421 . \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0421. \u0414\u043e\u0439\u0447\u0435\u0432 \/ S. Doichev<\/em> \u2013 p. 489<\/p>\n

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\u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043d\u0430 \u041f\u043e\u043d\u0441\u0435\u043b\u0435 \u0437\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438 <\/a>
[Poncelet Theorem for Quadrilaterals] \/ <\/a>
\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov<\/em> \u2013 p. 517<\/p>\n

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\u041d\u044f\u043a\u043e\u0438 \u043d\u0435\u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u0438 \u043c\u0435\u0442\u043e\u0434\u0438 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u0438 \u0441\u0438\u0441\u0442\u0435\u043c\u0438 <\/a>
[Some Non-Standard Methods in Problem Solving of Equations, Inequalities and Systems] \/ <\/a>
\u0417. \u0417\u0430\u043f\u0440\u044f\u043d\u043e\u0432 \/ Z. Zapryankov, \u041d. \u0420\u0430\u0439\u043a\u043e\u0432 \/ N. Raikov<\/em> \u2013 p. 526<\/p>\n

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\u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0440\u0438\u0446\u0438 \u0432 \u0443\u0447\u0438\u043b\u0438\u0449\u043d\u0438\u044f \u043a\u0443\u0440\u0441 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 <\/a>
[Using Matrices in School Educational Course] \/ <\/a>
\u0411. \u041a\u0438\u0440\u0438\u043b\u043e\u0432\u0430 \/ B. Kirilova<\/em> \u2013 p. 536<\/p>\n

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\u041f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438, \u0442\u0430\u0431\u043b\u0438\u0446\u0438 \u0438 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0438 \u0433\u0440\u0430\u0444\u0438\u043a\u0438 <\/a>
[Parameters, Tables and Dynamic Charts] \/<\/a>
\u0412. \u0411\u0438\u0437\u043e\u0432\u0430-\u041b\u0430\u043b\u0435\u0432\u0430 \/ V. Bizova-Laleva<\/em> \u2013 p. 549<\/p>\n

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\u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u043c\u0435\u0441\u0442\u0430, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \u0441\u043f\u0440\u0435\u0433\u043d\u0430\u0442\u0438 \u043f\u0440\u0430\u0432\u0438 \u0438 \u0447\u0435\u0432\u0438\u0430\u043d\u0438 <\/a>
[Loci, That are Generated by Conjugated Lines and Cevians] \/ <\/a>
\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov<\/em> \u2013 p. 562<\/p>\n

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\u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 Visual Basic for Applications \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 <\/a>
[Using Visual Basic for Applications in Solving Practical Problems in Information Technology Training] \/ <\/a>
\u0410. \u0420\u0430\u0445\u043d\u0435\u0432 \/ A. Rahnev, \u041a. \u0413\u044a\u0440\u043e\u0432 \/ K. Garov, \u0421. \u0410\u043d\u0435\u0432\u0430 \/ S. Aneva – <\/em>\u2013 p. 578<\/p>\n

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\u0418\u0437\u0432\u043b\u0438\u0447\u0430\u043d\u0435 \u043d\u0430 \u043e\u0431\u0435\u043a\u0442\u0438\u0432\u043d\u0438 \u0437\u043d\u0430\u043d\u0438\u044f \u043e\u0442 \u0438\u043d\u0442\u0435\u0440\u043d\u0435\u0442 <\/a>
[Extraction of Objective Knowledge from Internet] \/ <\/a>
\u0418. \u041f\u0435\u043d\u0435\u0432 \/ I. Penev, \u041f. \u041f\u0435\u043d\u0435\u0432 \/ P. Penev<\/em> \u2013 p. 603<\/p>\n

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\u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0432\u044a\u043d\u0448\u043d\u043e\u0442\u043e \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0437\u0430 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0438 \u043a\u043b\u0430\u0441 \u0437\u0430 2012 \u0433\u043e\u0434\u0438\u043d\u0430 \u0432 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f <\/a>
[External Evaluation Analysis of Mathematics at Fourth Grade for Year 2012 in Bulgaria] \/ <\/a>
\u041c. \u0417\u0430\u043c\u0444\u0438\u0440\u043e\u0432 \/ M. Zamfirov<\/em> \u2013 p. 603<\/p>\n

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\u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 [CONTEST PROBLEMS]<\/strong>
The column is led by Dr. Svetlozar Doychev and Dr. Veselin Nenkov<\/em><\/p>\n

\u041a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430 \u0431\u0440\u043e\u044f – \u0441\u0442\u0440. 623<\/p>\n

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\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0431\u0440\u043e\u0439 1, 2012 – \u0441\u0442\u0440. 624<\/p>\n

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In Memoriam \u2013 \u0434-\u0440 \u0421\u0432\u0435\u0442\u043b\u043e\u0437\u0430\u0440 \u0414\u043e\u0439\u0447\u0435\u0432 – \u0441\u0442\u0440. 628<\/p>\n

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\u0423\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415 \/ GUIDE FOR AUTHORS – <\/strong>\u2013 p. 631<\/p>\n

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Annual contents<\/strong> \u2013 p. 637<\/p>\n

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\n

<\/a>\u0414\u0412\u0415 \u0414\u0418\u0414\u0410\u041a\u0422\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u042a\u041b\u0411\u0418<\/strong><\/p>\n


\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0421\u0432\u0435\u0442\u043b\u043e\u0437\u0430\u0440 \u0414\u043e\u0439\u0447\u0435\u0432<\/em><\/p>\n

Abstract.<\/strong> \u0412 \u0441\u0440\u0435\u0434\u0430\u0442\u0430 \u043d\u0430 2003 \u0433. \u043e\u0441\u0435\u043c \u0434\u044a\u0440\u0436\u0430\u0432\u0438 \u043e\u0442 \u0415\u0432\u0440\u043e\u043f\u0435\u0439\u0441\u043a\u0438\u044f \u0441\u044a\u044e\u0437 \u2013 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f, \u0412\u0435\u043b\u0438\u043a\u043e\u0431\u0440\u0438\u0442\u0430\u043d\u0438\u044f (\u043f\u043e-\u043a\u044a\u0441\u043d\u043e \u0437\u0430\u043c\u0435\u0441\u0442\u0435\u043d\u0430 \u043e\u0442 \u0423\u043d\u0433\u0430\u0440\u0438\u044f), \u0413\u0435\u0440\u043c\u0430\u043d\u0438\u044f, \u0413\u044a\u0440\u0446\u0438\u044f, \u0418\u0442\u0430\u043b\u0438\u044f, \u041a\u0438\u043f\u044a\u0440, \u0420\u0443\u043c\u044a\u043d\u0438\u044f \u0438 \u0427\u0435\u0445\u0438\u044f, \u0434\u0430\u0434\u043e\u0445\u0430 \u0441\u0442\u0430\u0440\u0442 \u043d\u0430 \u0448\u0438\u0440\u043e\u043a\u043e \u043c\u0430\u0449\u0430\u0431\u043d\u0438\u044f \u0435\u0432\u0440\u043e\u043f\u0435\u0439\u0441\u043a\u0438 \u043f\u0440\u043e\u0435\u043a\u0442 \u201cMATHEU: \u043e\u0442\u043a\u0440\u0438\u0432\u0430\u043d\u0435, \u043c\u043e\u0442\u0438\u0432\u0438\u0440\u0430\u043d\u0435 \u0438 \u043f\u043e\u0434\u043a\u0440\u0435\u043f\u0430 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0442\u0435 \u0442\u0430\u043b\u0430\u043d\u0442\u0438 \u0432 \u0435\u0432\u0440\u043e\u043f\u0435\u0439\u0441\u043a\u0438\u0442\u0435 \u0443\u0447\u0438\u043b\u0438\u0449\u0430\u201d. \u0415\u0434\u043d\u0430 \u043e\u0442 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0442\u0430 \u0431\u0435\u0448\u0435 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0438 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438 \u0441 \u043d\u0430\u0440\u0430\u0441\u0442\u0432\u0430\u0449\u0430 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442, \u043d\u0430\u0440\u0435\u0447\u0435\u043d\u0438 \u0441\u0442\u044a\u043b\u0431\u0438, \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u043a\u043e\u0438\u0442\u043e \u0434\u0430 \u0441\u0435 \u043e\u0441\u044a\u0449\u0435\u0441\u0442\u0432\u044f\u0432\u0430\u0442 \u043f\u043e\u0441\u0442\u0430\u0432\u0435\u043d\u0438\u0442\u0435 \u0446\u0435\u043b\u0438. \u041d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0434\u0432\u0435 \u0442\u0430\u043a\u0438\u0432\u0430 \u0441\u0442\u044a\u043b\u0431\u0438.<\/p>\n

Keywords:<\/em> problem solving, didactical ladder, mathematical talent, identification, motivation.<\/p>\n

TWO DIDACTICAL LADDERS<\/strong><\/p>\n

Abstract.<\/strong> In the middle of 2003 eight countries from the European Union \u2013 Bulgaria, United Kingdom (later substituted by Hungary), Germany, Greece, Italy, Cyprus, Romania and Czech Republic, started the large scale European project \u201cMATHEU: identification, motivation and support of mathematical talents in European schools\u201d. One of the main tasks of the project was the elaboration of systems of problems with increasing difficulty, named ladders, to realize the defined goals. The present paper is dedicated to such two ladders. <\/p>\n

Sava Grozdev
<\/strong>Professor, Doctor in Mathematics, DSc in Pedagogy<\/em>
Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n


Svetlozar Doichev
<\/strong>Teacher in Mathematics, PhD in Pedagogy<\/em>
Mathematical High School, Stara Zagora<\/em><\/div>\n
\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/div>\n
\n
\n

<\/a>\u0422\u0415\u041e\u0420\u0415\u041c\u0410 \u041d\u0410 \u041f\u041e\u041d\u0421\u0415\u041b\u0415 \u0417\u0410 \u0427\u0415\u0422\u0418\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u0426\u0418<\/strong><\/p>\n


Sava Grozdev, Veselin Nenkov
<\/em>\u041d\u0430 \u043d\u0430\u0448\u0438\u044f \u043a\u043e\u043b\u0435\u0433\u0430 \u0438 \u043f\u0440\u0438\u044f\u0442\u0435\u043b \u0421\u0432\u0435\u0442\u043b\u044c\u043e \u0414\u043e\u0439\u0447\u0435\u0432<\/strong><\/em><\/p>\n

Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0435\u0442\u044e\u0434\u0438 \u0432\u044a\u0440\u0445\u0443 \u0442\u0435\u043e\u0440\u0435\u043c\u0430\u0442\u0430 \u043d\u0430 \u041f\u043e\u043d\u0441\u0435\u043b\u0435 \u0437\u0430 \u043f\u043e\u0440\u0430\u0436\u0434\u0430\u043d\u0435 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438, \u0432\u043f\u0438\u0441\u0430\u043d\u0438 \u0432 \u0435\u0434\u043d\u0430 \u043a\u0440\u0438\u0432\u0430 \u0438 \u043e\u043f\u0438\u0441\u0430\u043d\u0438 \u043e\u043a\u043e\u043b\u043e \u0434\u0440\u0443\u0433\u0430.<\/p>\n

Keywords:<\/em> inscribed quadrilateral, circumscribed quadrilateral, conic.<\/p>\n

\n

PONCELET THEOREM FOR QUADRILATERALS<\/strong><\/p>\n


Abstract.<\/strong> The paper is dedicated to some studies on Poncelet theorem for the generation of quadrilaterals that are inscribed in a conic and circumscribed to another conic. <\/p>\n

\n

Sava Grozdev<\/strong>
Professor, Doctor in Mathematics, DSc in Pedagogy<\/em>
Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n


Veselin Nenkov
<\/strong>Doctor in Mathematics
<\/em>Technical College Lovech<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n

<\/a>\u041d\u042f\u041a\u041e\u0418 \u041d\u0415\u0421\u0422\u0410\u041d\u0414\u0410\u0420\u0422\u041d\u0418 \u041c\u0415\u0422\u041e\u0414\u0418 \u0417\u0410 \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 <\/strong>\u041d\u0410 \u0423\u0420\u0410\u0412\u041d\u0415\u041d\u0418\u042f, \u041d\u0415\u0420\u0410\u0412\u0415\u041d\u0421\u0422\u0412\u0410 \u0418 \u0421\u0418\u0421\u0422\u0415\u041c\u0418<\/strong><\/p>\n


\u0417\u0430\u043f\u0440\u044f\u043d \u0417\u0430\u043f\u0440\u044f\u043d\u043e\u0432, \u041d\u0438\u043a\u043e\u043b\u0430\u0439 \u0420\u0430\u0439\u043a\u043e\u0432<\/em><\/p>\n

Abstract.<\/strong> \u0412 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u0442\u0430 \u043b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0430 \u0441\u0430 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0438 \u043d\u0435\u043c\u0430\u043b\u043a\u043e \u043d\u0435\u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u0438 \u0438\u0434\u0435\u0438, \u043f\u043e\u0434\u0445\u043e\u0434\u0438 \u0438 \u043c\u0435\u0442\u043e\u0434\u0438 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u043e-\u0442\u0440\u0443\u0434\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f, \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u0438 \u0441\u0438\u0441\u0442\u0435\u043c\u0438. \u0417\u0430 \u0443\u0441\u043f\u0435\u0448\u043d\u043e\u0442\u043e \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0442\u0430\u043a\u044a\u0432 \u0432\u0438\u0434 \u0437\u0430\u0434\u0430\u0447\u0438 \u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e \u0434\u0430 \u0441\u0435 \u043e\u0432\u043b\u0430\u0434\u0435\u044f\u0442 \u0441\u044a\u0449\u0435\u0441\u0442\u0432\u0443\u0432\u0430\u0449\u0438\u0442\u0435 \u043c\u0435\u0442\u043e\u0434\u0438 \u0438 \u0434\u0430 \u0441\u0435 \u0437\u0430\u0442\u0432\u044a\u0440\u0434\u044f\u0442 \u0443\u043c\u0435\u043d\u0438\u044f\u0442\u0430 \u0434\u0430 \u0441\u0435 \u043f\u0440\u0430\u0432\u044f\u0442 \u0437\u0430\u0434\u044a\u043b\u0431\u043e\u0447\u0435\u043d\u0438 \u043b\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438 \u0440\u0430\u0437\u0441\u044a\u0436\u0434\u0435\u043d\u0438\u044f, \u0438\u0437\u0438\u0441\u043a\u0432\u0430\u0449\u0438 \u0441\u044a\u043e\u0431\u0440\u0430\u0437\u0438\u0442\u0435\u043b\u043d\u043e\u0441\u0442, \u0434\u043e\u0441\u0435\u0442\u043b\u0438\u0432\u043e\u0441\u0442 \u0438 \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u0438\u0432\u043d\u043e\u0441\u0442. \u0412 \u043a\u043d\u0438\u0433\u0430\u0442\u0430 (\u0417\u0430\u043f\u0440\u044f\u043d\u043e\u0432 & \u0420\u0430\u0439\u043a\u043e\u0432, 2012) \u0441\u0430 \u0440\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 11 \u043d\u0435\u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u0438 \u043c\u0435\u0442\u043e\u0434\u0430, \u043a\u0430\u0442\u043e \u0432\u0441\u0435\u043a\u0438 \u043e\u0442 \u0442\u044f\u0445 \u0435 \u0440\u0430\u0437\u044f\u0441\u043d\u0435\u043d \u0438 \u0438\u043b\u044e\u0441\u0442\u0440\u043e\u0432\u0430\u043d \u0441 \u0440\u0435\u0448\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u043c\u043d\u043e\u0433\u043e \u0437\u0430\u0434\u0430\u0447\u0438. \u041d\u0435\u0439\u043d\u0430\u0442\u0430 \u043e\u0441\u043d\u043e\u0432\u043d\u0430 \u0446\u0435\u043b \u0435 \u0434\u0430 \u043f\u043e\u0432\u0438\u0448\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u0442\u0430 \u043a\u0443\u043b\u0442\u0443\u0440\u0430 \u043d\u0430 \u0447\u0438\u0442\u0430\u0442\u0435\u043b\u044f \u0432 \u0440\u0430\u043c\u043a\u0438\u0442\u0435 \u043d\u0430 \u0443\u0447\u0438\u043b\u0438\u0449\u043d\u0438\u044f \u043a\u0443\u0440\u0441 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0434\u0430 \u0437\u0430\u0441\u0438\u043b\u0438 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0430 \u043a\u044a\u043c \u043e\u0432\u043b\u0430\u0434\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043d\u0435\u0441\u0442\u0430\u043d\u0434\u0430\u0440\u0442\u043d\u0438 \u043f\u043e\u0434\u0445\u043e\u0434\u0438 \u0438 \u043c\u0435\u0442\u043e\u0434\u0438 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438 \u0441 \u043f\u043e\u0432\u0438\u0448\u0435\u043d\u0430 \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442.<\/p>\n

\n

Keywords:<\/em> problem solving, equation, inequality, system, non-standart problem, method.<\/p>\n

\n

SOME NON-STANDARD METHODS IN PROBLEM SOLVING OF EQUATIONS, INEQUALITIES AND SYSTEMS<\/strong><\/p>\n


Abstract.<\/strong> A considerable number of non-standard ideas, approaches and methods are known in the mathematical references to solve more difficult equations, inequalities and systems. For a successful solving of such problems it is necessary to master the existing methods and to become positive in the skill of deep logic reasoning which requires quick wits, guessing and combinative ability. Eleven non-standard methods are considered in the book (Zapryankov & Raikov, 2012) and each of them is explained and illustrated by the solutions of many problems. The main goal of the book is to increase the mathematical culture of the reader in the limits of the school mathematical course and to strengthen the interest of mastering non-standard approaches and methods in problem solving of tasks with higher difficulty. <\/p>\n

\n

Zapryan Zapryanov
<\/strong>Professor, DSc in Mathematics<\/em><\/p>\n

\n

Nikolay Raikov<\/strong><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/div>\n

\n
\n

<\/a>\u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u041d\u0410 \u041c\u0410\u0422\u0420\u0418\u0426\u0418 <\/strong>\u0412 \u0423\u0427\u0418\u041b\u0418\u0429\u041d\u0418\u042f \u041a\u0423\u0420\u0421 \u041d\u0410 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415<\/strong><\/p>\n


Borislava Kirilova<\/strong><\/p>\n


Abstract.<\/strong> \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043c\u0430\u0442\u0440\u0438\u0446\u0438 \u0432 \u0443\u0447\u0438\u043b\u0438\u0449\u043d\u0438\u044f \u043a\u0443\u0440\u0441 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432\u044a\u0432 \u0432\u0440\u044a\u0437\u043a\u0430 \u043f\u0440\u0435\u0434\u0438\u043c\u043d\u043e \u0441 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u0438 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435 \u0431\u0440\u043e\u044f \u043d\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0438 \u043b\u0438\u043d\u0435\u0439\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f. \u0412 \u0435\u0434\u0438\u043d \u043e\u0442 \u043f\u0440\u0438\u043c\u0435\u0440\u0438\u0442\u0435 \u0435 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u043e \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435\u0442\u043e Microsoft Excel.<\/p>\n

\n

Keywords:<\/em> IB, assessment, matrix, determinant, system of equations, parameter, vector<\/p>\n

\n

USING MATRICES IN SCHOOL EDUCATIONAL COURSE<\/strong><\/p>\n


Abstract.<\/strong> The present paper is dedicated to the use of matrices in the school education Mathematics course mainly in connection with solving systems of linear equations and examining the number of solutions. Microsoft Excel is used in one of the examples.<\/p>\n

\n

Borislava Kirilova<\/strong>
Teacher in Mathematics, <\/em>
American College of Sofia<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n<\/div>\n
\n

<\/a><\/strong>\u041f\u0410\u0420\u0410\u041c\u0415\u0422\u0420\u0418, \u0422\u0410\u0411\u041b\u0418\u0426\u0418 <\/strong>\u0418 \u0414\u0418\u041d\u0410\u041c\u0418\u0427\u041d\u0418 \u0413\u0420\u0410\u0424\u0418\u041a\u0418<\/strong><\/p>\n


\u0412\u0430\u043d\u044f \u0411\u0438\u0437\u043e\u0432\u0430-\u041b\u0430\u043b\u0435\u0432\u0430<\/strong><\/p>\n


Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430, \u0440\u0435\u0448\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u043a\u043e\u0438\u0442\u043e \u0441\u0435 \u0441\u0432\u0435\u0436\u0434\u0430\u0442 \u0434\u043e \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0438 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430. \u0417\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u0441\u0430 \u0440\u0435\u0448\u0435\u043d\u0438 \u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u043e \u0438 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u043e. \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u0442 \u0441\u0435 \u0442\u0430\u0431\u043b\u0438\u0446\u0438 \u0437\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u0438\u0442\u0435 \u0440\u0435\u0448\u0435\u043d\u0438\u0435 \u0438 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0430\u0442\u0430 \u0441\u0440\u0435\u0434\u0430 GeoGebra \u0437\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438\u0442\u0435.<\/p>\n

\n

Keywords:<\/em> parameters, tables, dynamic software, GeoGebra.<\/p>\n

\n

PARAMETERS, TABLES AND DYNAMIC CHARTS<\/strong><\/p>\n


Abstract.<\/strong> Some parametric inequalites are considered in the paper whose solutions are reduced to quadratic ones. The problems are solved algebraically and geometrically. Tables are used for the algebraic solutions and the dynamical environment GeoGebra for the geometric ones.<\/p>\n

\n

Vania Bizova\u2013Laleva
<\/strong>National High School of Commerce<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n

<\/a>\u0413\u0415\u041e\u041c\u0415\u0422\u0420\u0418\u0427\u041d\u0418 \u041c\u0415\u0421\u0422\u0410, <\/strong>\u041f\u041e\u0420\u041e\u0414\u0415\u041d\u0418 \u041e\u0422 \u0421\u041f\u0420\u0415\u0413\u041d\u0410\u0422\u0418 \u041f\u0420\u0410\u0412\u0418 \u0418 \u0427\u0415\u0412\u0418\u0410\u041d\u0418<\/strong><\/p>\n


Sava Grozdev, Veselin Nenkov<\/strong><\/p>\n


Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u043d\u0430 \u0435\u0434\u043d\u0430 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u041c\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 2010 \u0433.<\/p>\n

\n

Keywords:<\/em> Olympiad, problem solving, generalization, the Geometer\u2019s Sketchpad. <\/p>\n

\n

LOCI, THAT ARE GENERATED BY CONJUGATED LINES AND CEVIANS<\/strong><\/p>\n


Abstract.<\/strong> The paper considers generalizations of a problem from the International Mathematical Olympiad in 2010.<\/p>\n

\n

Sava Grozdev<\/strong>
Professor, Doctor in Mathematics, DSc in Pedagogy<\/em>
Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n

Veselin Nenkov<\/strong>
Doctor in Mathematics<\/em>
Technical College Lovech<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n

<\/a>\u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u041d\u0410 VISUAL BASIC FOR <\/strong>APPLICATIONS \u041f\u0420\u0418 \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415\u0422\u041e <\/strong>\u041d\u0410 \u041f\u0420\u0410\u041a\u0422\u0418\u0427\u0415\u0421\u041a\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 \u0412 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415\u0422\u041e<\/strong>
\u041f\u041e \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0426\u0418\u041e\u041d\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<\/strong><\/p>\n


\u0410\u0441\u0435\u043d \u0440\u0430\u0445\u043d\u0435\u0432, \u041a\u043e\u0441\u0442\u0430 \u0413\u044a\u0440\u043e\u0432, \u0421\u0442\u0435\u0444\u043a\u0430 \u0410\u043d\u0435\u0432\u0430<\/strong><\/p>\n


Abstract.<\/strong> \u041d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 Visual Basic for Applications \u043f\u0440\u0438 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u043e\u0434\u0443\u043b\u0430 \u201e\u0418\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0434\u0435\u0439\u043d\u043e\u0441\u0442\u0438 \u0438 \u0434\u043e\u043a\u0443\u043c\u0435\u043d\u0442\u0438\u201c \u0432 \u043f\u0440\u043e\u0444\u0438\u043b\u0438\u0440\u0430\u043d\u0430\u0442\u0430 \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0432 \u0433\u043e\u0440\u043d\u0438\u0442\u0435 \u043a\u043b\u0430\u0441\u043e\u0432\u0435 \u043d\u0430 \u0441\u0440\u0435\u0434\u043d\u043e\u0442\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435. \u0420\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u0441\u0435 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0430 \u0437\u0430\u0434\u0430\u0447\u0430, \u0432 \u043a\u043e\u044f\u0442\u043e \u0441\u0430 \u0440\u0435\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0435\u0442\u0430\u043f\u0438 \u043d\u0430 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 Visual Basic for Applications \u0432 MS Excel. \u041c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u044a\u0442 \u0435 \u043f\u0440\u0435\u0434\u043d\u0430\u0437\u043d\u0430\u0447\u0435\u043d \u0437\u0430 \u0443\u0447\u0438\u0442\u0435\u043b\u0438 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438, \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u0449\u0438 \u0432 11. \u0438 12. \u043a\u043b\u0430\u0441 \u043d\u0430 \u0421\u041e\u0423.<\/p>\n

\n

Keywords:<\/em> Information Technology, training, Visual Basic for Applications, macro.<\/p>\n

\n

PRACTICAL PROBLEMS IN INFORMATION TECHNOLOGY TRAINING<\/strong><\/p>\n

\n

Abstract.<\/strong> This paper is dedicated to the use of Visual Basic for Applications for teaching the module \u201eIntegration of activities and documents\u201c in specialized Information Technology training in upper classes of secondary school. We consider a practical problem where the main stages of the implementation of Visual Basic for Applications are made in MS Excel. The paper is intended for IT teachers.<\/p>\n

\n

Asen Rahnev
<\/strong>Professor, PhD<\/em>
Faculty of Mathematics and Informatics<\/em><\/p>\n

Kosta Garov<\/strong>
Associated professor, PhD<\/em>
Faculty of Mathematics and Informatics<\/em><\/p>\n

 <\/p>\n

Stefka Aneva<\/strong>
Assistant professor<\/em>
Faculty of Mathematics and Informatics<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n

<\/a>\u0418\u0417\u041b\u0418\u0427\u0410\u041d\u0415 \u041d\u0410 \u041e\u0411\u0415\u041a\u0422\u0418\u0412\u041d\u0418 <\/strong>\u0417\u041d\u0410\u041d\u0418\u042f \u041e\u0422 \u0418\u041d\u0422\u0415\u0420\u041d\u0415\u0422<\/strong><\/p>\n


\u0418\u0432\u0430\u0439\u043b\u043e \u041f\u0435\u043d\u0435\u0432, \u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432<\/strong><\/p>\n


Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430 \u043f\u043e\u0434\u0445\u043e\u0434 \u0438 \u043f\u043e\u043c\u043e\u0449\u043d\u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u0437\u0430 \u043e\u0442\u0434\u0435\u043b\u044f\u043d\u0435 \u043d\u0430 \u0432\u0430\u0436\u043d\u0430\u0442\u0430 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u043f\u043e \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0443\u0432\u0430\u0449\u0430 \u043d\u0438 \u0442\u0435\u043c\u0430 \u0432 \u0418\u043d\u0442\u0435\u0440\u043d\u0435\u0442. \u041e\u043f\u0438\u0441\u0432\u0430 \u0441\u0435 \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0430\u0446\u0438\u043e\u043d\u043d\u0430 \u043e\u0431\u043e\u0431\u0449\u0430\u0432\u0430\u0449\u043e \u0442\u044a\u0440\u0441\u0435\u0449\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u201e\u0421\u043f\u0440\u0430\u0432\u043e\u0447\u043d\u0438\u043a\u201c, \u0441\u044a\u0434\u044a\u0440\u0436\u0430\u0449\u0430 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0438 \u043d\u0430 \u0438\u0437\u043a\u0443\u0441\u0442\u0432\u0435\u043d \u0438\u043d\u0442\u0435\u043b\u0435\u043a\u0442. \u0420\u0435\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d \u0435 \u043f\u0440\u043e\u0441\u0442 \u043c\u0435\u0445\u0430\u043d\u0438\u0437\u044a\u043c \u0437\u0430 \u043f\u043e\u043b\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0434\u0435\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u0438 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f. \u0418\u0437\u0433\u0440\u0430\u0436\u0434\u0430 \u0441\u0435 \u0447\u0435\u0441\u0442\u043e\u0442\u0435\u043d \u0440\u0435\u0447\u043d\u0438\u043a \u0438 \u043f\u0440\u0435\u0434\u043c\u0435\u0442\u0435\u043d \u0443\u043a\u0430\u0437\u0430\u0442\u0435\u043b. \u0417\u0430\u043f\u0430\u0437\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043a\u043b\u044e\u0447\u043e\u0432\u0438\u0442\u0435 \u0434\u0443\u043c\u0438 \u0432 \u0431\u0430\u0437\u0430 \u0437\u043d\u0430\u043d\u0438\u044f
\u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0432\u0430 \u0438\u0437\u0432\u0435\u0436\u0434\u0430\u043d\u0435 \u043d\u0430 \u0443\u043c\u043e\u0437\u0430\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u044f \u043f\u043e \u0430\u043d\u0430\u043b\u043e\u0433\u0438\u044f.<\/p>\n

Keywords:<\/em> frequency dictionary, index, knowledge base, classifier of concepts, conclusions, analogy<\/p>\n

\n

EXTRACTION OF OBJECTIVE <\/strong>KNOWLEDGE FROM INTERNET<\/strong><\/p>\n


Abstract.<\/strong> The article offers an approach and utilities for extraction of important information from Internet on a subject interesting to us. A demonstrative generalizing search system \u201eHandbook\u201c, containing elements of semantic analysis, is described. A simple technique for obtaining deductive definitions is realized. Frequency word book and an index are constructed. Storing the keywords into knowledge base allows production of
logic conclusions by analogy.<\/p>\n

Ivaylo Penev<\/strong>
Assistant<\/em>
Department of Computer Sciences and Technologies, <\/em>
Technical University of Varna<\/em><\/p>\n

Plamen Penev<\/strong>
Teacher<\/em>
\u201ePanayot Volov\u201c School<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n

<\/a>\u0410\u041d\u0410\u041b\u0418\u0417 \u041d\u0410 \u0412\u042a\u041d\u0428\u041d\u041e\u0422\u041e \u041e\u0426\u0415\u041d\u042f\u0412\u0410\u041d\u0415 \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410 \u0417\u0410 \u0427\u0415\u0422\u0412\u042a\u0420\u0422\u0418 \u041a\u041b\u0410\u0421 \u0417\u0410 2012 \u0433. \u0412 \u0411\u042a\u041b\u0413\u0410\u0420\u0418\u042f<\/strong><\/p>\n


Milen Zamfirov<\/strong><\/p>\n


Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0432\u0430 \u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b, \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d \u043d\u0430 \u0430\u043d\u0430\u043b\u0438\u0437\u0430 \u043d\u0430 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043e\u0442 \u0432\u044a\u043d\u0448\u043d\u043e\u0442\u043e \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u043e\u0441\u0442\u0438\u0436\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0438 \u043a\u043b\u0430\u0441 \u0437\u0430 2011 \u0433. \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u0442 \u0434\u0430\u043d\u043d\u0438\u0442\u0435 \u0437\u0430 2012 \u0433.
\u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0432\u043a\u043b\u044e\u0447\u0432\u0430:
\u2022 \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u043f\u0441\u0438\u0445\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438\u0442\u0435 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a\u0438 \u043d\u0430 \u0442\u0435\u0441\u0442\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430;
\u2022 \u043e\u0431\u0449 \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u043f\u043e\u0441\u0442\u0438\u0436\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435;
\u2022 \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u0438\u044f\u0442\u0430 \u0432 \u043f\u043e\u0441\u0442\u0438\u0436\u0435\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u0441\u043f\u043e\u0440\u0435\u0434 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0442\u0435\u0445\u043d\u0438 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a\u0438.<\/p>\n

\n

Keywords:<\/em> external evaluation 2012, mathematics, general analysis<\/p>\n

\n

EXTERNAL EVALUATION ANALYSIS OF MATHEMATICS AT <\/strong>FOURTH CLASS FOR YEAR 2012 IN BULGARIA<\/strong><\/p>\n


Abstract.<\/strong> This article presents analysis on external evaluation of students\u2019 achievements in mathematics school subject in fourth grade. The evaluation is conducted in 2012 with a representative number of participants (5 600 students). The number of all participants in fourth grade in mathematics in the external evaluation is 57 910 in all 28 regions of Bulgaria.
The article includes:
\u2022 analysis of psychometricla characteristics of the test for mathematics;
\u2022 general analysis of students\u2019 achievements;
\u2022 analysis of the differentces in students\u2019 achievements which correspond to their characteristics.<\/p>\n

\n

Milen Zamfirov, (corresponding author) <\/strong>PhD
<\/strong>Department of Special Education, 
<\/em>Faculty of Preschool and Primary School Education
<\/em>University of Sofia<\/em><\/p>\n

\u041d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

\n<\/div>","protected":false},"excerpt":{"rendered":"

\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS \u041a\u043e\u043c\u044e\u043d\u0438\u043a\u0435 LeMATH \u2013 \u0418\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0441 \u043d\u043e\u0432\u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u0437\u0430 \u043a\u043e\u043c\u0443\u043d\u0438\u043a\u0430\u0446\u0438\u044f – \u0441\u0442\u0440. 487 \u0414\u0432\u0435 \u0434\u0438\u0434\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0441\u0442\u044a\u043b\u0431\u0438 [Two Didactical Ladders] \/ \u0421 . \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0421. \u0414\u043e\u0439\u0447\u0435\u0432 \/ S. Doichev – \u0441\u0442\u0440. 489 \u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043d\u0430 \u041f\u043e\u043d\u0441\u0435\u043b\u0435 \u0437\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438 [Poncelet Theorem for Quadrilaterals] \/ \u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ […]<\/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":"\n"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430", \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \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:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-6-godina-lv-2012-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\""\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430", \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS \u041a\u043e\u043c\u044e\u043d\u0438\u043a\u0435 LeMATH \u2013 \u0418\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0441 \u043d\u043e\u0432\u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u0437\u0430 \u043a\u043e\u043c\u0443\u043d\u0438\u043a\u0430\u0446\u0438\u044f – \u0441\u0442\u0440. 487 \u0414\u0432\u0435 \u0434\u0438\u0434\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0441\u0442\u044a\u043b\u0431\u0438 [Two Didactical Ladders] \/ \u0421 . \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0421. \u0414\u043e\u0439\u0447\u0435\u0432 \/ S. Doichev – \u0441\u0442\u0440. 489 \u0422\u0435\u043e\u0440\u0435\u043c\u0430 \u043d\u0430 \u041f\u043e\u043d\u0441\u0435\u043b\u0435 \u0437\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438 [Poncelet Theorem for Quadrilaterals] \/ \u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. 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