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Strasburg, Ezequiel Bobsin, Fabíola Aiub Sperotto i Cinthya Maria Schneider Meneghetti. "ATIVIDADES DE TRIGONOMETRIA PARA O ENSINO FUNDAMENTAL COM O USO DO SOFTWARE GEOGEBRA". Ciência e Natura 37 (7.08.2015): 617. http://dx.doi.org/10.5902/2179460x16491.
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http://dx.doi.org/10.5902/2179460X16491This work aims to present new activities for teaching trigonometry in Elementary Education. The idea is to show how to teach the main trigonometric relations through exercises that guide students, gradually, to conclusions of these trigonometry relations. The suggested activities make use of “GeoGebra ”software, a feature that enables the construction of trigonometric circles that facilitate the understanding by students. This paper presents not only the sine relations, cosine and tangent for acute angles, as it is usually proposed by textbooks 9th year of elementary school, but also relationships with secant, cosecant and cotangent, and it is not limited to acute angles and the study of right triangles, this leaving, the student more prepared for the challenges of high school and to their professional life.
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Mateas, Victor. "Perspectives from Physics: Constraints on Our Curriculum". Mathematics Teacher: Learning and Teaching PK-12 115, nr 12 (grudzień 2022): 882–90. http://dx.doi.org/10.5951/mtlt.2021.0182.
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How trigonometry is used and portrayed differently in mathematics and physics textbooks highlights potential sources for student struggle, constraints on our trigonometry curriculum, and lessons learned when looking across STEM disciplines.
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Park, Eun young, i Oh Nam Kwon. "Comparison and Analysis of Middle School Trigonometry Textbook Tasks and Teacher Design Tasks: From the Perspective of Mathematical Modelling". Korean Association For Learner-Centered Curriculum And Instruction 23, nr 7 (15.04.2023): 817–38. http://dx.doi.org/10.22251/jlcci.2023.23.7.817.
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Objectives In this study, the triangulation task actually used in the middle school class is analyzed from the perspective of the mathematical modelling tasks.
Methods Various triangulation tasks were collected from middle school trigonometry textbooks and activity sheets produced and used by teachers, and classified based on whether connectivity and openness, which are the core characteristics of mathematical modelling tasks, were met, and cognitive activities required. Then, mathematical modelling tasks is selected from among the teacher-designed tasks, and the activities of the tasks and the interviews of the teachers in each mathematical modelling tasks are interpreted according to the mathematical modelling cycle.
Results Teacher-designed tasks were found to have higher connectivity and openness than textbook tasks, including familiar material, offering practical and specific data, or insufficient data to solve the task. On the other hand, all tasks selected as mathematical modelling tasks among teacher-designed tasks emphasized simplifying, mathematising, and working mathematically, but relatively put less emphasis on understanding, interpretation, and validation. In particular, the process after working mathematically turned out to be limited to the extent of attaching units to the mathematical results, rounding it, or reviewing whether the calculation of the result were correct. In addition, all tasks used only the concept of tangent and did not require other concepts of trigonometric ratio.
Conclusions Several examples of triangulation tasks can be found in Korean and foreign mathematical modelling studies. As such, mathematical modelling teaching/learning method is useful to teach trigonometry due to its usefulness in real life. According to the results of this paper, teachers want to design assignments rich in connectivity and openness so that students can recognize the usefulness of the concept of trigonometric ratio. Furthermore, if the various cognitive activities in the mathematical modelling cycle are evenly included, and various trigonometry tasks including various models are designed, it would lead more meaningful teaching and learning of mathematical modelling as well as trigonometry and ultimately it would help students improve problem-solving skills.
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Conway, John, i Alex Ryba. "Remembering spherical trigonometry". Mathematical Gazette 100, nr 547 (marzec 2016): 1–8. http://dx.doi.org/10.1017/mag.2016.3.
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Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.
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Mylina, Resa, Suryo Widodo i Yuni Katmingsih. "Analysis of Mathematics Textbook Problems for Class X Viewed from Creativity". International Journal of Research and Review 10, nr 5 (27.05.2023): 408–14. http://dx.doi.org/10.52403/ijrr.20230548.
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This study aims to describe the level of creativity of questions in class X high school/vocational high school mathematics textbooks, the subject matter of the trigonometry series, and comparisons based on the creativity indicators of questions. The type of research used is descriptive qualitative. To ensure the credibility of the data, the researcher uses a triangulation method. The subject of this study were Competency Test questions in Mathematics Textbooks K13 curriculum, written by Dicky Susanto, Theja Kurniawan, Savitri K. Sihombing, Eunice Salim, Marianna Magdalena Radjawane, Ummy Salmah, Ambarsari Kusuma Wardani, published by the Center for Curriculum and Books (2021). Chapter Series Sequences and Trigonometry Comparisons of 10 questions. In this study, the level of creativity in these questions will be analyzed in stages based on the creativity indicators used in solving them. The results of this study are that the percentage of questions for each level of creativity is: Very Creative 0%, Creative 80%, Quite Creative 20%, Less Creative 0%, Not Creative 0%. From these results, it was found that the Competency Test questions contained in Class X SMA/SMK Mathematics Textbooks were written by Dicky Susanto, Theja Kurniawan, Savitri K. Sihombing, Eunice Salim, Marianna Magdalena Radjawane, Ummy Salmah, Ambarsari Kusuma Wardani, published by the Central Curriculum and Bookkeeping (2021) is Creative. Keywords: [Question Creativity Level; Creative Problem Indicators; Textbook Analysis]
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Phan-Yamada, Tuyetdong, i Walter M. Yamada. "Exploroing Polar Curves with GeoGebra". Mathematics Teacher 106, nr 3 (październik 2012): 228–33. http://dx.doi.org/10.5951/mathteacher.106.3.0228.
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Most trigonometry textbooks teach the graphing of polar equations as a two-step process: (1) plot the points corresponding to values of θ such as π, π/2, π/3, π/4, π/6, and so on; and then (2) connect these points with a curve that follows the behavior of the trigonometric function in the Cartesian plane. Many students have difficulty using this method to graph general polar curves. The difficulty seems to stem from an inability to convert changes in the value of the trigonometric equation as a function of angle (abscissa vs. ordinate in Cartesian coordinates) to changes of the radius as a function of angle (r[θ] in polar coordinates). GeoGebra provides a tool to help students visualize this relationship, thus significantly improving students' ability
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Anwar, St Hartina, Muhammad Rusydi Rasyid, Suharti Suharti i Sri Sulasteri. "PENGEMBANGAN BAHAN AJAR BERBASIS MULTIPLE INTELLIGENCES UNTUK MENINGKATKAN KEMAMPUAN BERPIKIR KRITIS". Al asma : Journal of Islamic Education 2, nr 2 (23.11.2020): 170. http://dx.doi.org/10.24252/asma.v2i2.16264.
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AbstrakPenelitian ini bertujuan untuk mengembangkan bahan ajar berupa buku ajar matematika berbasis multiple intelligences pada materi trigonometri untuk meningkatkan kemampuan berpikir kritis siswa kelas X SMAN 3 Wajo dan mengetahui tingkat kevalidan, kepraktisan, dan keefetifannya. Jenis penelitian yang digunakan adalah Research & Development dengan mengacu pada model 4D yang terdiri dari Define, Design, Development dan Dissseminate. Produk yang dikembangkan berupa buku ajar berbasis multiple intelligences pada materi trigonometri yang diantaranya terdiri dari beberapa aspek multiple intelligences yaitu logis-matematis, visual-spasial, linguistik-verbal, interpersonal dan intrapersonal. Berdasarkan hasil ujicoba terbatas yang dilakukan, diperoleh bahwa (1) Buku ajar berbasis multiple intelligences dikembangkan dengan menggunakan model pengembangan 4D (define, design, development, dan disseminate), (2) Bahan ajar valid berdasarkan rata-rata penilaian hasil validasi buku ajar beserta instrumen lainnya adalah 4,17 dengan kategori sangat valid, (3) Praktis karena persentase rerata untuk respons siswa adalah 81,25% pada kategori positif dan persentase rerata respons guru adalah 92,50% pada kategori sangat positif, (4) Efektif karena rerata kemampuan guru dalam mengelola pembelajaran ialah 4,28 berada pada kategori baik, persentase rata-rata aktivitas siswa adalah 64,53% berada pada kategori baik, dan persentase ketuntasan belajar siswa berada pada kategori tinggi yaitu 80,65%. Sehingga pengembangan bahan ajar berbasis multiple intelligences pada materi trigonometri di kelas X SMAN 3 Wajo sudah memenuhi kriteria kevalidan, kepraktisan, dan keefektifan. AbstractThe study aims to develop teaching material in the form of mathematics textbook based on multiple intelligences in trigonometry material to improve critical thinking ability of class X students of SMAN 3 Wajo and to determine the level of validity, practicality, and effectiveness. The type of research used is Research and Development with reference to the 4D model which consists of Define, Design, Development, and Disseminate. The product developed is a textbook based on multiple intelligences on trigonometric material which includes several aspects of multiple intelligences, namely logical mathematical, visual-spatial, linguistic-verbal, interpersonal and intrapersonal. Based on the result of limited trials conducted, it was found that (1) Multiple intelligences based textbook were developed using the 4D development model (define, design, development, and disseminate), (2) Valid based on the average assessment of the validation results of the textbooks and other instruments is 4.17 in the very valid category, (3) Practical because the average percentage of student responses is 81.25% in the positive category and the average percentage of teacher response is 92.50% in the very positive category, (4) Effective because the teacher’s average ability in managing learning is 4.28 in the good category, the average percentage of student activity is 64.43% in the good category, and percentage of student learning completeness is in the high category, namely 80.65%. So, the development of teaching material based on multiple intelligences on trigonometric material in class X SMAN 3 Wajo has met the criteria of validity, practicality, and effectiveness.
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Brown, Alan R. "Reader Reflections". Mathematics Teacher 91, nr 4 (kwiecień 1998): 275–365. http://dx.doi.org/10.5951/mt.91.4.0275.
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After de1iving the law of sines, which is found in most trigonometry textbooks, use a visual representation to demonstrate that the relationship holds for numerical values. I have used the TI-92 calculator to accomplish this teaching strategy through demonstrations. lfTI-92s are available for classroom use, constructing and evaluating the ratios are better activities. Teachers could also use The Geometer's Sketchpad or Geometric Supposer.
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Tanu Wijaya, Tommy, Zhou Ying i Aditya Purnama. "Using Hawgent Dynamic Mathematic Software in Teaching Trigonometry". International Journal of Emerging Technologies in Learning (iJET) 15, nr 10 (1.06.2020): 215. http://dx.doi.org/10.3991/ijet.v15i10.13099.
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Learning media is a help assistant for students to understand mathematics concept. Sometimes textbooks are very boring and not interesting therefore an innovation using a technology based media is needed. This research aims to make a learning media to improve students’ ability on the topic of trigonometry. The learning process will use hawgent dynamic mathematic software as a mathematics learning media. The problems that we are going to find out through this research are 1) can developing a learning media such as hawgent dynamic mathematic software improve students’ mathematical ability? and 2) did the student show a positive response in the learning process when using hawgent dynamic mathematic software? This research will use the ADDIE method. The target of this research are 30 high school students. The instruments use for this research are the cognitive test, interview and observation. According to the research result, there is an improvement in the students’ mathematical ability for trigonometry when using hawgent dynamic mathematic software. This means that the implementation of hawgent dynamic mathematic software in the learning process is successful. Students are more active and enthusiastic in class. In conclusion, developing hawgent dynamic mathematic software can improve the students’ mathematical ability in trigonometry as students’ show positive response when the learning process is helped by hawgent.
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Lim, Woong, Ji-Won Son, Susan Gregson i Jihye Kim. "An Examination of High School Students' Online Engagement in Mathematics Problems". International Journal of Web-Based Learning and Teaching Technologies 13, nr 2 (kwiecień 2018): 1–15. http://dx.doi.org/10.4018/ijwltt.2018040101.
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This article examines high school students' engagement in a set of trigonometry problems. Students completed this task independently in an online environment with access to Internet search engines, online textbooks, and YouTube videos. The findings imply that students have the resourcefulness to solve procedure-based mathematics problems in an online environment without formal instruction. This article suggests that self-directed online learning could be more effective for solving procedure-based problems than multi-step problems. Moreover, to be successful in online learning environments, students may need training to improve keyword searching skill and their ability to utilize various online learning tools.
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Tuluk, Güler. "Knowledge of Slope Concept in Mathematics Textbooks in Undergraduate Education". Journal of Curriculum and Teaching 9, nr 3 (20.08.2020): 161. http://dx.doi.org/10.5430/jct.v9n3p161.
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This questions the prevailing approach in the presentation of the concept of slope in teaching undergraduate in Turkey. For this purpose, five mathematics textbooks, were analyzed for problems related to slope, and similarities and differences were revealed. First, the subjects related to the slope in the textbooks were categorized and examined within the context in which they were handled. Therefore, this study is a qualitative study that adopts the interpretive paradigm. These categories are discussed in the form of connectivity, exploration, and purpose using the study for the context (Rezat, 2006). Stump (1999; 2001b) and Moore-Russo et al. (2011) studies were used for cognitive development. These are determined as geometric ratio, behavioral indicator, property determiner, algebraic ratio, parametric coefficient, functional property, linear constant, real life, physical property and trigonometry. Representations of the process skills were chosen as algebraic expressions, tables, and graphics. The concept is discussed in the form of calculus. In the use of technology (Akkoyunlu, 2002; Schware & Jaramillo, 1998), BCS, Scientific and Graphing Calculators, Internet are arranged. Definition, justification, and explanation for performance are arranged. When textbooks are compared, Turkey's textbooks contains more algebraic expressions, the use of the grounds and explanations and real-life connection, contains applications that use more formulas, the description is to use less highlighted is technology, other math they do not explicitly state their links with the subject areas. In general, it was seen that translated textbooks were mostly related to real life, equipped with explanations and justifications requiring cognitive competencies, and proceeded harmoniously between the subject area's main ideas and related ideas. These books use multi-step solved problems. Turkish textbooks need to be reviewed in terms of their functional areas in terms of context, cognitive need, representations, technology, and performance.
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Holloway, Caleb. "More than Free Textbooks: Labor and Pedagogy in Implementing Open Resources in a Trigonometry Course". Journal of Humanistic Mathematics 13, nr 2 (lipiec 2023): 368–88. http://dx.doi.org/10.5642/jhummath.aska3909.
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Nurmala, R. "PENGEMBANGAN BUKU AJAR TRIGONOMETRI BERBASIS CONTEXTUAL TEACHING AND LEARNING UNTUK MAHASISWA PENDIDIKAN MATEMATIKA". Mathematic Education And Aplication Journal (META) 3, nr 1 (31.08.2021): 10–19. http://dx.doi.org/10.35334/meta.v3i1.2075.
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AbstractThis research is a research and development which aims to produce a product in the form of a trigonometric textbook based on Contextual Teaching and Learning for Mathematics Education students. The development of teaching materials in this research refers to the Thiagarajan Model. This model consists of 4 stages of development,namely Define, Design, Develop, and Disseminates. The data analysis technique used is product feasibility test analysis and product practicality test analysis. The result of the research is that the product produced in the form of trigonometric textbooks based on Contextual Teaching and Learning is feasible and practical for use for mathematics education students. This is based on the data analysis of the experts, namely the material experts, respectively, obtained a percentage of 81% with proper criteria and 84% with very feasible criteria, each presentation expert obtained 92% with very feasible criteria and 89% with very feasible criteria. Based on the results of data analysis on student responses to large-scale trials, it was found that 79% stated that trigonometric textbook based on Contextual Teaching and Learning were very good and the remaining 21% stated that Contextual Teaching and Learning-based trigonometric textbooks were good.Keywords: Development, textbook, Contextual Teaching and LearningAbstrakPenelitian pengembangan ini bertujuan untuk menghasilkan produk yaitu buku teks trigonometri berbasis Contextual Teaching and Learning yang layak dan praktis untuk mahasiswa di jurusan Pendidikan Matematika. Pengembangan produk mengacu pada model Model Thiagarajan yaitu Define, Design, Develop, dan Disseminates. Analisis yang digunakan yaitu uji kelayakan dan analisis uji kepraktisan penggunaan produk. Berdasar pada analisis data para ahli materi diperoleh skor 81% berada pada kategori layak dan 84% berada pada kategori sangat layak. Pada ahli presentasi memperoleh 92% dengan kategori sangat layak dan 89% dengan kategori layak. Adapun data respon siswa terhadap uji coba skala besar diketahui bahwa 79% menyatakan bahwa buku teks trigonometri berbasis Contextual Teaching and Learning berada pada kategori sangat baik dan 21% menyatakan buku teks trigonometri berbasis Contextual Teaching and Learning bagus.Kata Kunci: Pengembangan, buku ajar, pembelajaran kontekstual
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Schloemer, Cathy G. "Sharing Teaching Ideas: I Found Sinusoids in My Gas Bill". Mathematics Teacher 93, nr 1 (styczeń 2000): 10–12. http://dx.doi.org/10.5951/mt.93.1.0010.
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With the current emphasis on real-world mathematics applications, many recent trigonometry textbooks include examples of sinusoidal functions for which students are required to find a model. These examples typically refer to the height of a person above the ground as a function of length of time riding on a Ferris wheel or to the number of hours of daylight in a U.S. city as a function of the day of the year. Unfortunately, as real-world as these examples are, many students still reject them as artificial and contrived. Complaints of “I'll never do this in my adult life” and “Who cares how high the Ferris wheel rider is?” are common among the eleventh- and twelfth-grade students that I teach.
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Martins Lopes, Rivanaldo. "MATHEMATICAL INNOVATION". Journal of Interdisciplinary Debates 4, nr 02 (24.06.2023): 04–37. http://dx.doi.org/10.51249/jid.v4i02.1265.
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Introduction: In the teaching of trigonometry, in the subject of high school mathematics, presented according to the textbooks, bringing contents and forms of resolution with many difficulties and practicality for teaching students to find angle between the hands of a clock., has become a reality for millions of students and teachers around the world. Based on this reality, the teaching of basic education has undergone major transformations in its teaching structure with. Objective: To develop an equation to find the angle between two hands of a clock, to meet the difficulties faced by students and teachers of teaching mathematics in basic education, through innovative methodologies that will provide learning for students and contribution to the public school community of the Municipality of São Francisco/PB, Brazil. Methodology: To carry out this research, a sample was formed by a triple factorial involving: 4 schools, 5 teachers and 10 students, from the city of São Francisco/PB/Brazil. The public elementary schools chosen to be part of the research were: EEECI Dorgival Silveira, ECI Mestre Júlio Sarmento, ECI DR. Silva Mariz and E.E.E.F.M Celson Mariz, respectively. Conclusions: In general, the difficulties pointed out by teachers and students listed in this study are related to continuing education, the scarce literature that does not offer didactic material in favor of the development of new mathematical models that can help solve trigonometry problems with ease and speed on the part of students, and the lack of innovative methodologies for teaching trigonometry both in the eyes of teachers and students. Thus, there are many barriers in the teaching and learning of mathematics, requiring the development of new research that dynamically enables the teaching of this discipline
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Rudnev, Dmitrii V. "Predecessors of A. S. Shishkov at the Naval Cadet Corps". Texts and History: Journal of Philological, Historical and Cultural Texts and History Studies 3 (2020): 63–77. http://dx.doi.org/10.31860/2712-7591-2020-3-63-77.
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Despite the extensive literature on A. S. Shishkov’s linguistic views, the question of their sources has not yet been resolved. This article considers Shishkov’s views on language in the context of the cultural atmosphere in the Naval Cadet Corps in St. Petersburg. The upbringing of future marine officers was strongly influenced by the English naval science, which, apparently, left its mark on the cultural life of the Corps. In the last quarter of the 18th century, the Corps was dominated not by the Gallomania inherent in Russian high society, but by traditionalist views. An example of such traditionalism is the attitude of the Corps inspector V. V. Nikitin and his assistant P. I. Suvorov, who taught mathematics and a number of other disciplines, toward language. They put their linguistic views into practice in their two textbooks on mathematics — Euclidean elements and Two books of trigonometry where they translated all mathematical terminology into Russian. Moreover, they expounded their views in the introduction to the Euclidean elements. Based on the materials of the Russian State Naval Archive, the article traces the complex publishing history of these textbooks and their further fate. For over ten years, from the first half of the 1780s to the mid-1790s, Nikitin and Suvorov were spreading their linguistic views through their math textbooks and while teaching cadets. Shishkov, who served and taught at the Naval Cadet Corps in the 1770–1780s, was undoubtedly familiar with these views. Some features of his point of view on language suggest that it could be influenced by the linguistic views of his colleagues from the Naval Cadet Corps.
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Rahmadani, Heni, Yenita Roza i Atma Murni. "Design of Information and Technology Based Teaching Materials in Mathematics Subjects of IT High School Pekanbaru Students". Mathematics Education Journal 4, nr 1 (8.03.2020): 17. http://dx.doi.org/10.22219/mej.v4i1.11467.
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This research is based on the low mathematical representation ability of students and the lack of textbooks used by teachers in presenting teaching material. This study aims to describe the results of the development of information technology-based teaching materials to improve the mathematical representation ability of students in class X Trigonometry Function Materials that are valid, practical, and effective. The model used in this research development is the ADDIE development model, namely Analysis, Design, Development, Implementation, and Evaluation. The research instruments used in this article are validation sheets (validity aspects), teacher assessment sheets and student assessment sheets (practical aspects), and mathematical representation ability tests (effectiveness aspects). Analysis of the data used in this research is parametric statistics, the t-test with the help of SPSS 18.00 using the Independent-Sample t-Test. This research resulted in a product of teaching materials based on information and technology in valid mathematics (87.11%), practical (90.81%) and effective (0.043 <0.05).
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Kolosova, Galina I. "COLLECTION OF TEXTBOOKS AND PUBLICATIONS ON EDUCATION IN RUSSIA IN THE LIBRARY OF COUNT P.A. VALUEV". Vestnik Tomskogo gosudarstvennogo universiteta. Kul'turologiya i iskusstvovedenie, nr 41 (2021): 283–93. http://dx.doi.org/10.17223/22220836/41/25.
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The book collection of the famous Russian statesman of the XIX century Count Peter Alexandrovich Valuev (1815–1890) was acquired by the Ministry of Public Education in 1885 for a university under construction in Tomsk. This article is dedicated to the content study of this book collection, stored in the Scientific Library of TSU. The subject of the study is a collection of educational books and publications on the reform of the public education system in Russia in the 1860s. The main task is to study them in the context of state activities and personal characteristics of P.A. Valuev. The methodological basis of the research is bibliological and cultural-historical methods. The article presents the results of studying textbooks and publications of a legislative nature on education in Russia, included in this collection. In studying the collection, the entries in Valuev’s diary, made in the period from 1861 to 1868, when he headed the Ministry of Internal Affairs, were of a great help. The article provides brief information about the state activities of Valuev in the period from 1860 to 1880. On the eve of the Emancipation Reform of 1861 in Russia, the Ministry of Public Education, realizing the unsatisfactory state of educational institutions in the country, made a decision about their transformation. In 1856, they began with a revision of the university charter and the charter of gymnasiums and schools, adopted during the reign of Emperor Nicholas I. Only in the 1860s a consideration of the drafts of these charters prepared by this time had began. The first part considers publications related to the development of the new charters for educational institutions in Russia in the 1860s, which give an idea of the specific documents which Valuev worked with. “The Draft of the General Charter of Imperial Russian Universities” is presented in the collection in three different editions, published in 1862. The last draft of the charter, developed by members of scientific committee of the Ministry, was taken as the basis for the new university charter, which was published in 1863. The collection also contains “The Draft of the Charter of general education institutions”, as well as the “Charter of gymnasiums and pro-gymnasiums”, published in 1864. Numerous notes in copies belonging to Valuev, indicating his work on these projects, were analyzed. Another part of the collection – textbooks – gives an unusual touch to the owner's personality. The features of textbooks stored in the library are identified and the publication histories of some of them are revealed. Among those are copies of textbooks on trigonometry, chemistry, zoology, botany, etc., which contain the notes of his youngest son Nikolai. Being a highly educated person, Valuev appreciated good textbooks: not only he acquired them for his son, but also stored them in his library, despite the remaining traces of Nikolai's diligent work with these books during his studies. The content of the collection, somewhat unusual for Count Valuev, which is considered in this article, emphasizes the cultural and historical significance of his library.
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Fonseca, Laerte Silva da, Edmo Fernandes Carvalho, Luciano Pontes da Silva i Kleyfton Soares da Silva. "O papel das Funções Cognitivas em Praxeologias de Tipos de Tarefas Matemáticas". Jornal Internacional de Estudos em Educação Matemática 13, nr 3 (12.01.2021): 321–28. http://dx.doi.org/10.17921/2176-5634.2020v13n3p321-328.
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ResumoO objetivo principal deste trabalho é descrever uma análise sobre como as funções cognitivas atuam diretamente em organizações praxeológicas relativas a tipos de tarefas trigonométricas, relevando a complexidade neurocognitiva para realização das mesmas. A metodologia considerou uma pesquisa bibliográfica, ressaltando-se a importância de aproximar elementos da Teoria Antropológica do Didático (TAD) e da Neurociência Cognitiva. Foi verificado que para a realização de tipos de tarefas matemáticas, no campo da Trigonometria, cuja organização praxeológica esteja bem definida é possível atribuí-las aos estudantes como um convite ao desafio e, de forma gradativa, respeitando-se as condições da neurofisiologia do cérebro. Desse modo, possíveis dificuldades de estudantes frente um determinado saber matemático, pode ser atribuído ao tipo de tarefa proposta. O estudo de algumas funções cognitivas corrobora de certo modo com o estudo dessas dificuldades. Permite inclusive, compreender as implicações da falta de sentido das matemáticas escolares na vida do sujeito. Outro ponto importante a ser destacado remete à necessidade dos cursos de formação de professores de matemática oportunizar aos licenciandos conhecimentos de neurociência cognitiva, pois quanto mais se sabe sobre o funcionamento do cérebro, melhor será a escolha dos tipos de tarefas matemáticas para apresentá-las tanto nas salas de aulas, como nos livros didáticos.
Palavras-chave: Funções Cognitivas. Praxelogias. Tipos de Tarefas Matemáticas.
AbstractThe main objective of this work is to describe an analysis of how cognitive functions act directly in praxeological organizations related to types of trigonometric tasks, emphasizing the neurocognitive complexity to perform them. The methodology considered a bibliographical research, highlighting the importance of approaching elements of the Didactic Anthropological Theory and Cognitive Neuroscience. It was verified that for the accomplishment of types of mathematical tasks, in the field of Trigonometry, whose praxeological organization is well defined, it is possible to attribute them to the students as an invitation to the challenge and, in a gradual way, respecting the conditions of the neurophysiology of the brain. Thus, possible difficulties of students in the face of certain mathematical knowledge, can be attributed to the type of task proposed. The study of some cognitive functions corroborates in a way with the study of these difficulties. It even allows us to understand the implications of the lack of meaning of school mathematics in people's lives. Another important point to be highlighted is the need for training courses for mathematics teachers to give graduates the knowledge of cognitive neuroscience, since the more we know about brain functioning, the better the choice of the types of mathematical tasks to present them in classrooms, as in textbooks.
Keywords: Cognitive Functions. Praxeologies. Types of Math Tasks.
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Pambayun, Hanifa Prahastami, i Endah Retnowati. "Penerapan teknik faded examples untuk meningkatkan kemampuan pemecahan masalah materi pengayaan trigonometri SMA". Jurnal Riset Pendidikan Matematika 5, nr 1 (25.05.2018): 73. http://dx.doi.org/10.21831/jrpm.v5i1.12149.
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Penelitian ini bertujuan untuk menghasilkan dan mendeskripsikan pengembangan bahan ajar pengayaan trigonometri SMA menggunakan teknik faded examples yang berkualitas untuk meningkatkan kemampuan pemecahan masalah siswa. Kualitas bahan ajar yang dikembangkan mencakup aspek kevalidan, keefektifan, dan kepraktisan. Penelitian ini merupakan penelitian pengembangan dengan model Plomp yang mencakup tiga tahapan. (1) penelitian awal mencakup analisis kebutuhan dan analisis konteks, (2) pengembangan yang mencakup desain produk dan pembuatan produk, dan (3) evaluasi yang meliputi proses validasi dan proses implementasi. Proses pengembangan melibatkan dua ahli, satu orang guru dan 50 siswa (siswa kelas X pengayaan 1 dan X pengayaan 2) SMA IPA di Mataram. Hasil dari penelitian ini adalah bahan ajar pengayaan berupa buku guru dan buku siswa yang dikembangkan dengan menerapkan teknik faded examples. Pada buku pengayaan terdapat paket faded examples dengan jenis backward dan forward fading dimana pada akhir paket, siswa diminta untuk membuat sendiri soal sesuai dengan materi yang sedang dipelajari. Teknik ini dikembangkan oleh teori desain pembelajaran bernama Cognitive Load Theory (CLT). Hasil penelitian menunjukkan bahwa bahan ajar ini layak digunakan karena dinilai sangat baik secara isi dan penyajian oleh dosen validator, praktis oleh guru matematika dan praktis digunakan oleh siswa. Hasil ketuntasan belajar adalah sebanyak 100% siswa mencapai nilai minimal kemampuan pemecahan masalah. The application of faded examples techniques to improve student’s problem solving ability on trigonometry at high school level AbstractThis study was aimed to produce and describe the quality of the developed Trigonometric Senior High School Science Program Enrichment’s Instructional Materials using Faded-Examples Techniques to Improve Problem Solving Ability. The quality of the developed teaching materials include all aspects of validity, effectiveness, and practicality. This was a developmental research used three phases Plomp’s model which consists of: (1) preliminary studies which involved the needs and context analysis, (2) product design development, and (3) the evaluation process of product validation and implementation. The development process involves two experts as validator, one teacher, and 50 students (X pengayaan 1 and X pengayaan 2). The study results the trigonometric enrichment’s teaching materials which consists of the teacher’s textbook and the student’s textbook that was developed using the faded-examples technique. This technique based on The Cognitive Load Theory (CLT) instructional design. The results of the study showed that the quality of the developed trigonometric enrichment’s teaching materials is “very good” according to lecturer validation and “practical” according to the evaluation from the teachers and students. The results of the learning showed that 100% of the students passed the minimum grade criteria of problem solving skills.
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Ларин, С. В., i С. В. Шуманский. "The use of virtual tools of the GeoGebra environment in school trigonometry". Математический вестник Вятского государственного университета, nr 3(26) (17.04.2023): 49–53. http://dx.doi.org/10.25730/vsu.0536.22.026.
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Анимационные возможности компьютерных сред пополняют арсенал средств построения, добавляя к классическим циркулю и линейке инструменты нового типа, реализуемые на компьютерном экране. Благодаря им в статье моделируется процесс наматывания числовой прямой на единичную окружность, что превращает ее в числовую окружность, моделируется распрямление единичной окружности в отрезок. Кроме того, эти новые виртуальные инструменты положены в основу моделирования непрерывного вычерчивания графиков основных тригонометрических функций и им обратных функций. Это вносит наглядность в преподавание тригонометрии. В статье приведены фрагменты авторского учебного пособия по тригонометрии 10 класса с использованием компьютерной анимации в среде GeoGebra, содержащего в качестве приложения альбом анимационных рисунков, готовых к использованию на уроках тригонометрии. The animation capabilities of computer environments replenish the arsenal of construction tools, adding a new type of tools implemented on a computer screen to the classic compass and ruler. Thanks to them, the article models the process of winding a numerical straight line onto a unit circle, which turns it into a numerical circle, and models the straightening of a unit circle into a segment. In addition, these new virtual tools are the basis for modeling the continuous drawing of graphs of basic trigonometric functions and their inverse functions. This brings clarity to the teaching of trigonometry. The article contains fragments of the author's textbook on trigonometry of the 10th grade using computer animation in the GeoGebra environment, containing as an application an album of animated drawings ready for use in trigonometry lessons.
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Munir, Nilam Permatasari. "Pengembangan Buku Ajar Trigonometri Berbasis Konstruktivisme dengan Media E-Learning pada Prodi Tadris Matematika IAIN Palopo". Al-Khwarizmi: Jurnal Pendidikan Matematika dan Ilmu Pengetahuan Alam 6, nr 2 (30.12.2018): 167–78. http://dx.doi.org/10.24256/jpmipa.v6i2.454.
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Abstract:This study aims to determine the validity and practicality of developing a trigonometry textbook based on constructivism theory with e-learning media at Mathematics Tadris Study Program of IAIN Palopo. This research is a type of R & D (Research and Development) research by adopting the ADDIE development model which consists of five stages, namely (1) physical analysis; (2) design (3) development (4) Implementation; (5) evaluation. The target of this study were students at Mathematics Tadris study program of IAIN Palopo who programed trigonometry courses in the even semester of 2017/2018 school year. The results of this study indicate that the textbook products developed have fulfilled validity and practicality category, the average acquisition score for material expert validation is 82 on the very valid category and the average acquisition of instructional media experts is 87 on the very valid category, while the average score of practicality by ten students is 75 on the practical category.Abstrak:Penelitian ini bertujuan untuk mengetahui validitas dan paktikalitas dari pengembangkan buku ajar tigonometri berbasis teori konstruktivisme dengan media e-learning pada Pogram Studi Tadris Matematika IAIN Palopo. Penelitian ini merupakan jenis penelitian R & D (Research and Development) dengan mengadopsi model pengembangan ADDIE yang terdiri atas lima tahapan yaitu (1) tahap analisis (analyze); (2) tahap perancangan (design); (3) tahap pengembangan (development); (4) tahap implementasi (Implementation); (5) tahap evaluasi (evaluation). Sebagai sasaran dalam penelitian ini adalah mahasiswa program studi Tadris Matematika IAIN Palopo yang memprogram mata kuliah trigonometri pada semester genap tahun ajaran 2017/2018. Hasil penelitian ini menunjukkan bahwa produk buku ajar yang dikembangkan telah memenuhi kriteria kevalidan dan kepraktisan, rata-rata skor perolehan untuk validasi ahli materi adalah 82 berada dalam kategori sangat valid dan rata-rata perolehan dari ahli media pembelajaran adalah 87 berada dalam kategori sangat valid, sedangkan rata-rata skor perolehan praktikalitas oleh sepuluh orang mahasiswa adalah 75 berada dalam kategori praktis.
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Mesa, Vilma, i Bradley Goldstein. "Conceptions of Angles, Trigonometric Functions, and Inverse Trigonometric Functions in College Textbooks". International Journal of Research in Undergraduate Mathematics Education 3, nr 2 (20.10.2016): 338–54. http://dx.doi.org/10.1007/s40753-016-0042-1.
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Juita, Rahma Nia, Hari Sumardi i Hanifah Hanifah. "Analisis Level Kognitif Soal Trigonometri Buku Teks Matematika SMA/MA Kelas X Berdasarkan Taksonomi Bloom Revisi". Logaritma : Jurnal Ilmu-ilmu Pendidikan dan Sains 9, nr 02 (10.03.2022): 167–82. http://dx.doi.org/10.24952/logaritma.v9i02.4226.
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The Mathematics Textbook for 10th grade of senior high school published by the Ministry of Education and Culture is the main book used in learning at school. However, there are teachers who rarely use the problems in the book when giving exercises to students, especially on Trigonometry. Whereas, the distribution of cognitive levels on the competency test in the book is not known. This study was aimed to describe the distribution of cognitive levels of Trigonometry’s competency test questions in mathematics book for 10th grade published by the Ministry of Education and Culture based on the revised Bloom's Taxonomy. This type of research was descriptive research with documentation method. The instruments that used in this study were cognitive level classification sheet and cognitive level conformity validation sheet. The results showed that from 171 questions there was a question that was incomplete and as many as 170 questions that deserved to be analyzed obtained a distribution of cognitive level remembering (C1) as 5 questions , understanding (C2) as 44 questions , applying (C3) as 99 questions , analyzing (C4) as 15 questions , and evaluating (C5) as 7 questions , and it did not contain the cognitive level of creating (C6).
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Bonsangue, Martin V. "A Geometrical Approach to the Six Trigonometric Ratios". Mathematics Teacher 86, nr 6 (wrzesień 1993): 496–98. http://dx.doi.org/10.5951/mt.86.6.0496.
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The sine and cosine ratios are often visualized as lengths of the projections of a unit segment on the x-and y-axis. This type of pictorial approach for the other four trigonometric ratios has been presented in the problems sections of some textbooks (e.g., Wooton, Beckenbach, and Dolciani 1969, p. 82, no. 29; p. 85, no. 27; Vance 1963, p. 70, nos. 14–16). However, relationships among the trigonometric ratios are often thought of as algebraic rather than geometric. This article presents geometric interpretations of the six trigonometric ratios, as well as some trigonometric relationships derived geometrically rather than algebraically.
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Ren, Guanshen. "Match Geometric Figures with Trigonometric Identities". Mathematics Teacher 88, nr 1 (styczeń 1995): 24–25. http://dx.doi.org/10.5951/mt.88.1.0024.
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In most high school and college algebra textbooks, basic trigonometric identities are often proved by algebraic operations with other identities or presented without proof. This article presents proofs of some trigonometric identities from a geometric point of view and shows that some geometric figures beautifully match some trigonometric identities. The identities are proved with the use of geometric figmes, a process that offers a visual way to relate trigonometric properties. Studying trigonometric identities from a geometric perspective helps students make connections among various branches of mathematics and raises students' interests and appreciations in geometry. This idea was introduced in the context of a mathematics club, and the responses from students have been very positive. Some students did their own research and presented a paper at the regional Pi Mu Epsilon student conference.
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Indriantie, Nuriska, Mohamad Rif'at i Dede Suratman. "PENYUSUNAN BOOKLET SCAFFOLDING MENGGUNAKAN TEORI POLYA MATERI TRIGONOMETRI". Jurnal AlphaEuclidEdu 3, nr 1 (31.07.2022): 89. http://dx.doi.org/10.26418/ja.v3i1.53162.
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This research aims to compile a scaffolding booklet using Polya's theory of trigonometric subject. The method of this research is a research and development (R&D) method with a 4-D development model. The research subject are the student of class X MIA 1 SMAN 6 Pontianak. Data collection tools used are unstructured interviews and questionnaires. Data analysis is viewed from the results of the questionnaire assessment, how to develop booklets and booklets. From this research, it was found that the clarity of how to solve trigonometric problems in textbooks used in schools is still not clear, because the completeness of the fourth polya step, namely at the stage of re-examining, does not exist and there are still students who do not understand how to solve the problems given. The preparation of this booklet is to improve and clarify how to solve the problems in the book. The completeness of the polya steps that did not exist before came into existence and the stages of solving Polya's problems become complete in the booklet. Students who do not understand the explanations in the textbooks will at least understand enough when given an explanation of how to solve problems in the booklet.
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安, 瑞芳. "Textbook Analysis of the People’s Education “Trigonometric Function”". Advances in Education 11, nr 04 (2021): 1048–54. http://dx.doi.org/10.12677/ae.2021.114165.
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Wallace, Edward C. "Investigations Involving Involutions". Mathematics Teacher 81, nr 7 (październik 1988): 578–79. http://dx.doi.org/10.5951/mt.81.7.0578.
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Many second-year algebra textbooks include a discussion of functions and their inverses prior to introducing logarithms and the inverse trigonometric functions. Since these functions and some others can be approached neatly as inverses of more familiar functions, a good understanding of the notion of inverse allows students to understand these topics better.
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GADJIEV, D. "METHODOLOGY OF THE TEACHING OF FINITE MATHEMATICS AT THE PEDAGOGICAL UNIVERSITIES DUE TO THE NEW TRENDS IN THE EVOLUTION OF THE MATHEMATICAL SCIENCES". World of academia: Culture, Education, nr 7 (15.10.2021): 92–100. http://dx.doi.org/10.18522/2658-6983-2021-7-92-100.
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There were introduced new methods of the teaching and instruction of the following parts of the Pre-calculus: (1) Binomial Series; (2) Trigonometry; (3) Partial Fractions. The problems introduced in the article for the Pre-Calculus Course in Finite Mathematics was developed by the author. These unabridged problems are developed within the new trends in the evolutions of the novelty of the syllabi in Mathematics due to the development of the Mathematics Sciences / Theory and Applications. These new trends in the Theory and Application of Mathematics Sciences have been added new demands to the newly revised textbooks and corresponding syllabi for the Mathematics Courses taught at the Junior two years Colleges and Pedagogical Universities.These newly developed problems are reflection of the Development of Mathematical and Engineering Sciences to offer great amount of learning conclusion/sequel to those who pursue a bachelor’s degree at the universities of the pedagogical orientation. The problems presented in the article here are developed and restructured in terms of the newly developed techniques to solve the problem in Finite Mathematics and Engineering sciences. Moreover, the techniques offered in the article here are more likely to get utilized in Advanced Engineering Sciences, too, within the content of the problems, which require to obtain finite numerical solutions to the Real Phenomena Natural Problems in Engineering Sciences and Applied Problems in Mathematical Physics.The aim of this present publication is to offer new advanced techniques and instructional strategies to discuss methodology and instructional strategies of the mathematical training of the students at the Pedagogical Universities. Moreover, these new teaching techniques and strategies introduced may be extended to the engineering sciences at the technical universities, too.The results and scientific novelty of the introduced methodology and learning conclusions and sequel of the new knowledge the students at the Pedagogical universities may be benefited from are in the following list of the learning conclusions, presented in the article here. The students of the pedagogical orientation may attain the mastery skills in the following sections of the combinatorics in Finite Mathematics subject:- The n! Combination of n different terms.- Evaluate the expressions with factorials.- Identify that there are -!!( )!nrnr various of combinations of r identical terms in n variations.- Identify and evaluate the combinatorial coefficients from the Binomial Theorem.- Identify and able to build the Pascal’s triangle of the binomial coefficients.- Utilize the Binomial Theorem to expand the binomial formula for any natural powers.- Utilize the Binomial Theorem to obtain the general formula for the n-th term of binomial expansion.- Utilize the Sigma Symbols in the Binomial Theorem for the n-th terms of the binomial expansion. 19- Generate the expansion for the power of the ex, where e is the base of natural logarithmic function y = f(x) = x.Practical significance: the methods of teaching and new teaching strategies offered here in the article alongside with the application of the new trends in the development of mathematical and mathematics education sciences can be useful for prospective and currently practicing teachers of mathematics. Moreover, the materials presented here in this article can be useful for the educational professionals in their professional development plans to improve the quality in education
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梅, 方莹. "Comparative Analysis of Trigonometric Functions between Old and New High School Mathematics Textbooks". Creative Education Studies 11, nr 06 (2023): 1558–65. http://dx.doi.org/10.12677/ces.2023.116236.
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Luintel, Mahesh Chandra. "Development of Polynomial Mode Shape Functions for Continuous Shafts with Different End Conditions". Journal of the Institute of Engineering 16, nr 1 (12.04.2021): 151–61. http://dx.doi.org/10.3126/jie.v16i1.36653.
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Common methods used to determine the solutions for vibration response of continuous systems are assumed mode method, Rayleigh-Ritz method, Galerkin Method, finite element method, etc. Each of these methods requires the shape functions which satisfy the boundary conditions. Shape functions derived in most of the classical textbooks are simple trigonometric functions for some end conditions but are very complex transcendental functions for many end conditions. It is very difficult to determine the vibration response of a continuous system analytically by using such transcendental shape functions. Hence this paper presents a method to develop polynomial shape functions required to solve the vibration of continuous shafts with different end conditions. The natural frequencies obtained from the developed polynomial shape functions are compared to those obtained from the classical transcendental shape functions and found very close for the first three modes.
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Jdid, Maissam, i Fatima Al Suleiman. "A Study in the Integral of Sine and Cosine Functions". Galoitica: Journal of Mathematical Structures and Applications 2, nr 1 (2022): 33–43. http://dx.doi.org/10.54216/gjmsa.020105.
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Trigonometric functions are among the most widely used functions in many science fields, especially sine and cosine functions because they are essential for periodic functions that describe sound and light waves in different types and wavelengths. Therefore, researchers studied the integrals of sine and cosine functions in different forms of the integrating function. In this paper, we spotlighted several most important yet under-studied integrals that are poorly mentioned in Arabic and foreign textbooks and studies. In addition, we studied Integral of Sine and Cosine for n as a positive rational number and concluded that each of these integrals leads to functional series. When studying the convergence of these series using the D'Alembert ratio test, we found that these series are convergent over the entire set of real numbers. This convergence is highly useful when applying such integrals in different science fields.
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Pontes da Silva, Luciano, i Laerte Silva da Fonseca. "Analysis of textbooks of Mathematics in relation to the Disposition of Tasks in the light of Attention as a Neurocognitive Mechanism". Acta Scientiae 21, nr 4 (4.09.2019): 160–73. http://dx.doi.org/10.17648/acta.scientiae.v21iss4id3887.
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The present work has the objective of analyzing the tasks on trigonometric functions presented in three books approved in the last three PNLDs, from 2009 to 2017 regarding Levels of Attention Transfer according to Sternberg (2010). The research has a documentary nature, within its intended purpose and is divided into two sessions. The first concerns the theoretical part, concatenating three pillars: the screening of tasks according to Dante (2005); the official documents, especially the Guide of the Didactic Book and the Attention, with neurocognitive lenses like Lent (2002), Lima (2005) and psychological with Sternberg (2010) and Willingham (2011) among others. The second part brings to the fore the empirical part of the work, with the analysis of the three books, namely: "Mathematics" by Dante (2005a), "Mathematics" by Ribeiro (2010) and "New Mathematical Souza (2013). The tasks were screened at three levels of attention, where it was noticed that most of the tasks found in the analyzed books are within the second level, none appearing at the third level of attention.
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张, 金竹. "Comparative Analysis of New and Old Textbooks of High School Mathematics—Taking the Example of “Trigonometric Functions” of the A Version of Human Teaching". Creative Education Studies 11, nr 06 (2023): 1576–85. http://dx.doi.org/10.12677/ces.2023.116238.
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Singh, Rohit, Sumit Gulwani i Sriram Rajamani. "Automatically Generating Algebra Problems". Proceedings of the AAAI Conference on Artificial Intelligence 26, nr 1 (20.09.2021): 1620–28. http://dx.doi.org/10.1609/aaai.v26i1.8341.
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We propose computer-assisted techniques for helping with pedagogy in Algebra. In particular, given a proof problem p (of the form “Left-hand-side-term = Right-hand-side-term”), we show how to automatically generate problems that are similar to p. We believe that such a tool can be used by teachers in making examinations where they need to test students on problems similar to what they taught in class, and by students in generating practice problems tailored to their specific needs. Our first insight is that we can generalize p syntactically to a query Q that implicitly represents a set of problems [[Q]] (which includes p). Our second insight is that we can explore the space of problems [[Q]] automatically, use classical results from polynomial identity testing to generate only those problems in [[Q]] that are correct, and then use pruning techniques to generate only unique and interesting problems. Our third insight is that with a small amount of manual tuning on the query Q, the user can interactively guide the computer to generate problems of interest to her. We present the technical details of the above mentioned steps, and also describe a tool where these steps have been implemented. We also present an empirical evaluation on a wide variety of problems from various sub-fields of algebra including polynomials, trigonometry, calculus, determinants etc. Our tool is able to generate a rich corpus of similar problems from each given problem; while some of these similar problems were already present in the textbook, several were new!
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方, 兴发. "A Comparative Analysis of High School Mathematics Textbooks from the Perspective of Core Literacy between Human Education A Edition (2019) and Shanghai Edition (2020)—Taking Trigonometric Function as an Example". Creative Education Studies 11, nr 07 (2023): 1846–56. http://dx.doi.org/10.12677/ces.2023.117274.
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Sianturi, Iwan Andi, i Der-Ching Yang. "An Analysis of Singaporean versus Indonesian Textbooks Based on Trigonometry Content". EURASIA Journal of Mathematics, Science and Technology Education 13, nr 7 (15.06.2017). http://dx.doi.org/10.12973/eurasia.2017.00760a.
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"Sharing Teaching Ideas: Cos_x + Sin_x and the Trigonometric Sum and Difference Identities". Mathematics Teacher 95, nr 7 (październik 2002): 510–14. http://dx.doi.org/10.5951/mt.95.7.0510.
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That the cosine or sine of a sum or difference is not the sum or difference of the cosines or sines is easy to establish by counterexample. However, the actual expansions for cos (A ± B) and sin (A ± B) are not intuitively obvious for most students. Standard textbooks do cover the derivations of these expansions, but the results have a greater impact if students conjecture them first. This article describes a lesson that introduces the formal proofs of the trigonometric sum and difference identities. In this activity, students discover the expansions for the cosine and sine of a sum and difference by using technology to generate selected graphs and by analyzing patterns in these graphs. The only prerequisite is the ability to find the equation of a sinusoid when the graph of the function is given.
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"Sharing Teaching Ideas: Cos_x + Sin_x and the Trigonometric Sum and Difference Identities". Mathematics Teacher 95, nr 7 (październik 2002): 510–14. http://dx.doi.org/10.5951/mt.95.7.0510.
Pełny tekst źródłaStreszczenie:
That the cosine or sine of a sum or difference is not the sum or difference of the cosines or sines is easy to establish by counterexample. However, the actual expansions for cos (A ± B) and sin (A ± B) are not intuitively obvious for most students. Standard textbooks do cover the derivations of these expansions, but the results have a greater impact if students conjecture them first. This article describes a lesson that introduces the formal proofs of the trigonometric sum and difference identities. In this activity, students discover the expansions for the cosine and sine of a sum and difference by using technology to generate selected graphs and by analyzing patterns in these graphs. The only prerequisite is the ability to find the equation of a sinusoid when the graph of the function is given.