{"title":"Graph solution of a system of recurrence equations","authors":"","doi":"10.57016/tm-eqwm6024","DOIUrl":"https://doi.org/10.57016/tm-eqwm6024","url":null,"abstract":"","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The relation between Pappus's and Ceva's theorem","authors":"","doi":"10.57016/tm-ftre5930","DOIUrl":"https://doi.org/10.57016/tm-ftre5930","url":null,"abstract":"","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Tabov, Asen Velchev, Rayna Alashka, Sevdalin Tsvetanov
. In the present publication, which can be considered as a continuation of the paper V. Nenkov, St. Stefanov, H. Haimov, An application of quadrilat-eral’s geometry in solving competitive mathematical problems , Synergetics and reflec-tion in mathematics education, Proceedings of the anniversary international scientific conference, Pamporovo, October 16-18, pp. 121–128, 2020, the application of the geometry of quadrilateral to the solution of exams is considered. Three examples given in the magazine “Mathematics and Informatics” have been selected, the solutions of which illustrate well the benefit of studying the recently discovered properties of convex quadrilaterals. Two solutions to the tasks are presented for comparison. The first, proposed by participants in the competition, are relatively complex and longer, and the second—based precisely on elements of the geometry of quadrilateral, are signifi-cantly simpler and shorter. These solutions are based on properties of quadrilaterals associated with some of their remarkable points.
． 在本出版物中，可视为V. Nenkov, St. Stefanov, H. Haimov论文的延续，四边形几何在解决竞争性数学问题中的应用，数学教育中的协同与反思，周年国际科学会议论文集，Pamporovo, 10月16日至18日，pp. 121-128, 2020，考虑了四边形几何在解决考试中的应用。本文选取了《数学与信息学》杂志上的三个例子，它们的解很好地说明了研究最近发现的凸四边形的性质的好处。提出了两种解决方案以供比较。第一个是由参赛者提出的，相对复杂和较长，第二个是基于四边形几何元素的，明显更简单和更短。这些解决方案是基于四边形的一些特性，这些特性与四边形的一些显著点有关。
{"title":"An application of the quadrilateral's geometry in solving competitive planimetric problems","authors":"J. Tabov, Asen Velchev, Rayna Alashka, Sevdalin Tsvetanov","doi":"10.57016/tm-gkzb9621","DOIUrl":"https://doi.org/10.57016/tm-gkzb9621","url":null,"abstract":". In the present publication, which can be considered as a continuation of the paper V. Nenkov, St. Stefanov, H. Haimov, An application of quadrilat-eral’s geometry in solving competitive mathematical problems , Synergetics and reflec-tion in mathematics education, Proceedings of the anniversary international scientific conference, Pamporovo, October 16-18, pp. 121–128, 2020, the application of the geometry of quadrilateral to the solution of exams is considered. Three examples given in the magazine “Mathematics and Informatics” have been selected, the solutions of which illustrate well the benefit of studying the recently discovered properties of convex quadrilaterals. Two solutions to the tasks are presented for comparison. The first, proposed by participants in the competition, are relatively complex and longer, and the second—based precisely on elements of the geometry of quadrilateral, are signifi-cantly simpler and shorter. These solutions are based on properties of quadrilaterals associated with some of their remarkable points.","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show how two fundamental sequential criteria for real functions, namely the sequential criterion for continuity and the sequential criterion for uniform continuity, are relaxed. Further, we derive a global continuity criterion, as an immediate consequence of the relaxed sequential criterion for continuity, and a fundamental property of uniformly continuous functions, as an application of the relaxed sequential criterion for uniform continuity.
{"title":"Relaxation of the sequential criteria for continuity and uniform continuity of a real function","authors":"Spiros Konstantogiannis","doi":"10.57016/tm-sbhr1144","DOIUrl":"https://doi.org/10.57016/tm-sbhr1144","url":null,"abstract":"We show how two fundamental sequential criteria for real functions, namely the sequential criterion for continuity and the sequential criterion for uniform continuity, are relaxed. Further, we derive a global continuity criterion, as an immediate consequence of the relaxed sequential criterion for continuity, and a fundamental property of uniformly continuous functions, as an application of the relaxed sequential criterion for uniform continuity.","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70913864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Stanojević, K. Kukić, N. Vuksanovic, Dragana Draganac
In this paper, we demonstrate how logistic equation and generalized logistic equation can be applied in teaching economics with the aim that students easily grasp the AK model. To this end, the AK model is firstly modified into three versions, after which the first and the second model are reduced to the generalized logistic equations. The purpose of this paper is to present how through applying the mathematical modifications of mentioned versions of AK models, they become more cognizable to students, which should result by more deeply understanding of well-known economic models.
{"title":"Applications of logistic and generalized logistic difference equations in economics: AK model","authors":"J. Stanojević, K. Kukić, N. Vuksanovic, Dragana Draganac","doi":"10.57016/tm-rmsh3417","DOIUrl":"https://doi.org/10.57016/tm-rmsh3417","url":null,"abstract":"In this paper, we demonstrate how logistic equation and generalized logistic equation can be applied in teaching economics with the aim that students easily grasp the AK model. To this end, the AK model is firstly modified into three versions, after which the first and the second model are reduced to the generalized logistic equations. The purpose of this paper is to present how through applying the mathematical modifications of mentioned versions of AK models, they become more cognizable to students, which should result by more deeply understanding of well-known economic models.","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70913817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Branislav Z. Popovic, Sladjana Dimitrijević, Marija Stanić, Aleksandar Milenković
In order to solve mathematical problems, students often need to make the transition from one representation of mathematical concepts in those problem formulations to another representation. In this paper we explore the influence of the representations used in the problem formulation (problems with the same mathematical background with regards to solving easier or more complex equations and determining the unknown value of the proportion) on students' success in solving those problems. On a representative sample of 584 8th grade students, we tested whether there were differences in students' success in solving mathematical problems while using symbolic, graphic, or verbal representations in the formulations of problems belonging to different level of complexity. Results of this research indicate that there is significant impact of the representations of mathematical concepts used in problem formulation on students' success. Furthermore, the level of impact of using different representations in problem formulations depends on the level of the problem complexity when it comes to students' success in solving those problems.
{"title":"Students' success in solving mathematical problems depending on different representations","authors":"Branislav Z. Popovic, Sladjana Dimitrijević, Marija Stanić, Aleksandar Milenković","doi":"10.57016/tm-bapu1403","DOIUrl":"https://doi.org/10.57016/tm-bapu1403","url":null,"abstract":"In order to solve mathematical problems, students often need to make the transition from one representation of mathematical concepts in those problem formulations to another representation. In this paper we explore the influence of the representations used in the problem formulation (problems with the same mathematical background with regards to solving easier or more complex equations and determining the unknown value of the proportion) on students' success in solving those problems. On a representative sample of 584 8th grade students, we tested whether there were differences in students' success in solving mathematical problems while using symbolic, graphic, or verbal representations in the formulations of problems belonging to different level of complexity. Results of this research indicate that there is significant impact of the representations of mathematical concepts used in problem formulation on students' success. Furthermore, the level of impact of using different representations in problem formulations depends on the level of the problem complexity when it comes to students' success in solving those problems.","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Smoothness of the signed distance function: A simple proof","authors":"","doi":"10.57016/tm-dvla5418","DOIUrl":"https://doi.org/10.57016/tm-dvla5418","url":null,"abstract":"","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Yet another elementary proof of Brauer's theorem","authors":"","doi":"10.57016/tm-kbui6835","DOIUrl":"https://doi.org/10.57016/tm-kbui6835","url":null,"abstract":"","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70913804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Parametric curves play an important role in mathematics and engineering education. The purpose of this paper is to introduce the hypotrochoid family of curves and to provide ideas and tools for teaching the topic. We give the general parametric equations of hypotrochoids and show that several special curves can be derived from these equations. To display the curves we use the Hypotrochoid Equation Tracer, which is a dynamic application made by Desmos graphic calculator. In addition we have created a LEGO robot called Spikograph, that models a Spirograph and draws different hypotrochoids providing a physical experience for students. Finally, we have compiled a student project to practice and deepen the knowledge about hypotrochoids.
{"title":"Displaying parametric curves with virtual and physical tools","authors":"Attila Körei, Szilvia Szilágyi","doi":"10.57016/tm-ehgc7743","DOIUrl":"https://doi.org/10.57016/tm-ehgc7743","url":null,"abstract":"Parametric curves play an important role in mathematics and engineering education. The purpose of this paper is to introduce the hypotrochoid family of curves and to provide ideas and tools for teaching the topic. We give the general parametric equations of hypotrochoids and show that several special curves can be derived from these equations. To display the curves we use the Hypotrochoid Equation Tracer, which is a dynamic application made by Desmos graphic calculator. In addition we have created a LEGO robot called Spikograph, that models a Spirograph and draws different hypotrochoids providing a physical experience for students. Finally, we have compiled a student project to practice and deepen the knowledge about hypotrochoids.","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"P Problem as a means to encourage students' conceptualization of fractions","authors":"","doi":"10.57016/tm-eqgl7427","DOIUrl":"https://doi.org/10.57016/tm-eqgl7427","url":null,"abstract":"","PeriodicalId":41484,"journal":{"name":"Teaching of Mathematics","volume":"33 1","pages":""},"PeriodicalIF":0.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70914224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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