Pythagorean Theorem Research Articles

Development of Pythagorean Theorem Content Worksheets Based on Creative Problem Solving to Facilitate Mathematical Reasoning Skills

Mathematical reasoning ability is one of the most important parts of mathematics learning. However, the facts show that students' mathematical reasoning ability is still low, especially in the context of the Pythagorean theorem. One attempt to facilitate this is using students' worksheets based on creative problem solving (CPS). This research aims to develop valid and practical student' worksheets. It refers to the 4D development model, which includes defining, designing, developing, and disseminating. The instruments used are validation and practicality questionnaire sheets. The results of the validity assessment of students' worksheets were very valid, with an average percentage of 95,07%. The results of the practicality assessment of students' worksheets were very practical, with an average rate of 89,38%. Thus, it is in accord with the valid and practical requirements, so the worksheet is usable. This research aims to make it easier for students to build knowledge independently in mathematics learning so that mathematical reasoning abilities can be facilitated. Kemampuan penalaran matematis (KPM) menjadi salah satu aspek yang terpenting dalam pembelajaran matematika, namun fakta menunjukkan KPM peserta didik masih rendah khususnya pada konten teorema Pythagoras. Salah satu upaya untuk memfasilitasi masalah tersebut dengan menggunakan Lembar kerja peserta didik (LEMBAR KERJA SISWA) berbasis Creative Problem Solving (CPS). Tujuan penelitian ini adalah mengembangkan LEMBAR KERJA SISWA yang valid dan praktis. Penelitian pengembangan ini mengacu pada model pengembangan 4D yang meliputi define, design, develop, dan disseminate. Instrumen yang digunakan adalah angket validasi dan angket praktikalitas. Hasil analisis data angket validasi diperoleh bahwa LEMBAR KERJA SISWA dikategorikan sangat valid dengan persentase rata-rata yaitu 95,07%. Hasil analisis data angket praktikalitas diperoleh bahwa LEMBAR KERJA SISWA dikategorikan sangat praktis dengan persentase rata-rata yaitu 89,38%. Maka, LEMBAR KERJA SISWA pada penelitian ini telah memenuhi syarat valid dan praktis sehingga dapat digunakan dalam kegiatan pembelajaran. Implikasi dari penelitian ini adalah untuk mempermudah peserta didik dalam membangun pengetahuan secara mandiri pada pembelajaran matematika sehingga kemampuan penalaran matematis dapat terfasilitasi.

Proof Without Words: British Flag Theorem via the Pythagorean Theorem

Summary We visually prove the British flag theorem via the Pythagorean theorem.

A converse of De Gua’s theorem for orthocentric tetrahedra

Summary We establish a converse of De Gua’s theorem for orthocentric tetrahedra in the three-dimensional space with a proof using the converse of the Pythagorean theorem and Heron’s formula.

Understanding The Concept Of Pythagoras In Junior High School Students Using PMRI

This research aims to determine students' conceptual understanding using the PMRI learning approach with the context of cotton rope in the Pythagorean theorem material on students' conceptual understanding in class VIII SMP NEGERI 2 Pagaralam. The obstacle faced by students in studying the Pythagorean Theorem is a lack of understanding of the basic concepts in the Pythagorean theorem material. One method that can strengthen understanding of concepts in learning mathematics is Indonesian Realistic Mathematics Education (PMRI). The research method used was experimental using a control group Pre-test - Post-test design. This research involved 2 classes, namely the experimental and control classes. Data collection techniques include observation and tests, the analysis techniques used are documentation, observation and tests. Based on test data analysis, the average student score in the experimental class was 78.8, while the control class had a score of 72.50. From the research results, students' conceptual understanding was found to be better using PMRI with comparison results in the experimental class being better than in the control class.

Uma Nova Generalização dos Ternos Pitagóricos

In this work, we revisit the Pythagorean equation and its solutions with natural coordinates, called Pythagorean triples, as well as some of its well-known generalizations. Moreover, we realize a study on the solutions with natural coordinates of a quadratic equation that extends the Pythagorean equation, obtaining expressions that allow us to characterize algebraically these solutions and present a new generalization of the Pythagorean triples.

Robust block tensor PCA with F-norm projection framework

Tensor principal component analysis (TPCA), also known as Tucker decomposition, ensures that the extracted “core tensor” maximizes the variance of the sample projections. Nevertheless, this method is particularly susceptible to noise and outliers. This is due to the utilization of the squared F-norm as the distance metric. In addition, it lacks constraints on the discrepancies between the original tensors and the projected tensors. To address these issues, a novel tensor-based trigonometric projection framework is proposed using F-norm to measure projection distances. Tensor data are first processed utilizing a blocking recombination technique prior to projection, thus enhancing the representation of the data at a local spatio-temporal level. Then, we present a block TPCA with the F-norm metric (BTPCA-F) and develop an iterative greedy algorithm for solving BTPCA-F. Subsequently, regarding the F-norm projection relation as the “Pythagorean Theorem”, we provide three different objective functions, namely, the tangent, cosine and sine models. These three functions directly or indirectly achieve the two objectives of maximizing projection distances and minimizing reconstruction errors. The corresponding tangent, cosine and sine solution algorithms based on BTPCA-F (called tan-BTPCA-F, cos-BTPCA-F and sin-BTPCA-F) are presented to optimize the objective functions, respectively. The convergence and rotation invariance of these algorithms are rigorously proved theoretically and discussed in detail. Lastly, extensive experimental results illustrate that the proposed methods significantly outperform the existing TPCA and the related 2DPCA algorithms.

A New Study of Expansive Mappings and Pythagorean Theorem Relationship

A New Study of Expansive Mappings and Pythagorean Theorem Relationship

An Uncountable Number of Proofs of the Pythagorean Theorem

Summary We give an infinite number of proofs of the Pythagorean theorem. Some can be classified as ‘selfsimilar proofs’.

Five or Ten New Proofs of the Pythagorean Theorem

We present five trigonometric proofs of the Pythagorean theorem, and our method for finding proofs (Section 5) yields at least five more.

An Observer-Based View of Euclidean Geometry

An influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events forms a partially ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of a geometric product in geometric Clifford algebras.

Introducing Novel Geometric Insights and Three-Dimensional Depictions of the Pythagorean Theorem for any Triangles

Introducing Novel Geometric Insights and Three-Dimensional Depictions of the Pythagorean Theorem for any Triangles

On fuzzy inner products constructed by fuzzy numbers in linear spaces

In this paper a new definition for fuzzy inner product spaces is presented. Just as the set of real numbers can be embedded in a set of fuzzy numbers, a crisp inner product space can be considered as a special case of fuzzy inner product spaces. An example is given to demonstrate that, the new definition is a nontrivial generalization for the crisp inner product spaces. Under certain restrictions, a fuzzy inner product space can become a fuzzy normed space. Based on some elementary properties for the families of semi-inner products of endpoints, the linearly topological structure of new spaces is discussed. Moreover, the orthogonality between two vectors is considered and a fuzzy version of Pythagorean theorem is given.

A Comparative Study of Geometric Principles in the Sulba Sutras and the Pythagorean Theorem: Historical Context and Mathematical Applications

This research delves into the geometric principles detailed in the ancient Sulba Sutras and their parallels with the Pythagorean Theorem, traditionally attributed to the Greek mathematician Pythagoras. While the Pythagorean theorem is widely regarded as a cornerstone of Greek geometry, recent studies suggest that its formulation appeared centuries earlier in Indian mathematics, as evidenced in Baudhāyana’s Sulba Sutra (circa 800 BCE). These texts offer extensive insights into the geometric constructions used in Vedic rituals, including the use of Pythagorean triples and transformations between geometric shapes. The Sulba Sutras present a rich mathematical framework, emphasizing practical applications in altar construction, land measurements, and urban planning. This study aims to bridge the research gap by comparing the geometric developments in the Sulba Sutras with Greek formulations, particularly focusing on their mathematical methods, cultural contexts, and historical significance. The findings underscore the advanced state of Indian mathematics and its profound influence on subsequent mathematical traditions, offering a nuanced understanding of the parallel evolution of geometric knowledge across cultures.

Tetris Pitagórico: o jogo como ferramenta para o ensino de geometria

In this article, we introduce a creative way to approach Mathematics education, using an educational game inspired by the globally renowned Tetris game and the traditional Pythagorean theorem. Our goal is to discuss the proposed game concept titled “Pythagorean Tetris”, inspired by Pythagorean puzzles. The theoretical-methodological perspective discussed in this article focuses on the use using of recreational mathematics, specifically games, to promote more engaging Mathematics teaching. We believe this dynamic approach can enhance the quality of education by providing a more engaging learning experience for students. Through the game, we propose a perspective for Mathematics education that makes students' learning more interesting while also emphasizing the importance of accessibility and practical application of mathematical concepts.

456 Allocating distribution of pasture utilization across the grazing landscape in grazing steers equipped with virtual fencing collars

Abstract Spatial utilization of pasture landscapes by grazing animals is often heterogeneous and driven by complex environmental and physiological interactions between plants and animals. Synergistic energy pathways can be modeled using mechanistic relationships describing resource distribution, animal behavior, metabolic energy rate, and pasture utilization using biometric sensors. The objective of this study was to demonstrate the ability to measure the energy landscape by mapping fine-scale contributions to animal energy expenditure, nutrient acquisition, growth, and overall economic profitability. We utilized animal trajectories and location fixes recorded by virtual fencing collars (Vence) daily body weight (BW) using precision livestock scales (Smartscale, C-Lock Inc.) and oxygen consumption (GreenFeeds, C-Lock Inc.) on steers (n = 127), managed as part of a broader project to evaluate environmental synergies in extensive grazing systems. Steers were allocated to one of six native grass pastures assigned to either rotational or continuous grazing strategies, and one of three stocking rates (low, moderate, high) in a 2 x 3 factorial arrangement. Space utilization was quantified as the frequency of fixes located within each cell of a 5 x 5-meter Grid and the distance between grid cells and water was calculated using the Pythagorean theorem. Movement behavior was quantified using Program R to calculate step length, the distance between two temporally continuous fix points utilizing the Pythagorean Theorem, and turn angles calculated as the cosine of the pre- and pro-ceeding step lengths where larger values indicated greater deviation from a straight trajectory. A mixed linear regression model was fitted using the lmer function in the lme4 package. Space utilization decreased with increased distance from water (P < 0.05), with rotational grazing strategies increasing concentration around water sources (P < 0.05). This indicates an opportunity to utilize precision livestock technology to facilitate pasture management and increase space utilization in grazing cattle.

The Development of Canva-Assisted Scientific Approach Pythagoras Teaching Materials to Improve Students’ Mathematical Reasoning Ability on Junior High School

This research was conducted to determine the development, feasibility, effectiveness, and difficulties in the development of teaching materials on the Pythagorean Theorem using a scientific approach assisted by Canva to enhance the mathematical reasoning abilities of junior high school students. This research was conducted at SMP PGRI 10 Bandung using the Research and Development (R&D) method based on the 4D stages. The research subjects included 33 students from class IX F for limited trials, 27 students from class VIII B, and 29 students from class VIII D for extensive trials. The research instruments used in this study included interviews in the preliminary learning, student responses to the teaching materials, expert validation sheets, and 6 items of reasoning ability test questions. The material used in this research was the Pythagorean Theorem. Student reasoning ability test data were analyzed using Microsoft Excel calculations. The research results indicate that: 1) the development process of the 4D method teaching materials, including a) Define (Definition); b) Design; c) Develop; and Disseminate, have been successfully implemented according to the plan; 2) the validator calculation results obtained a percentage of 86.21%, categorized as "very valid and do not need revision"; 3) the student reasoning ability test results prove that the teaching materials are reasonably effective; 4) significant difficulties include students' lack of understanding of the material, low activity, and a lack of supportive learning facilities.

A dynamic pythagorean exponent: Applications to baseball and structural modeling in risk management

AbstractThe famous Pythagorean theorem relating a team's winning percentage over a season to it's total runs scored and surrendered has been the subject of much attention, in both popular culture and academia. In academia, a main focus has been to find the optimal value for the exponent in the formula, originally taken to be 2 in Bill James' work. The optimal value has been obtained in many published works via regression or other statistical estimation techniques, and calibrated to various sports to reflect the type of play in each sport. In this short article, we borrow from ideas in quantitative finance and risk management to estimate the distribution of these Pythagorean exponents, viewing them as random variables that can evolve over time. We carry out our analysis in a manner similar to that of implied volatility from the Black‐Scholes formula. We present the result of our analysis of 150 years of team data via the Lahman package in R, and also provide dynamic models for an evolving measure of the Pythagorean exponent. This includes suggestions for modeling credit default as well as contract valuation in sports via macro variables in connection with historically implied parameters.

PYTHAGOREAN PARABOLOGRAPH FOR REGIONS’ QUADRATURE

<p>In this paper an attempt is made to design a mathematical machine based on the Pythagorean theory of fitting an equivalent parallelogram (given angle and side) with another parallelogram. An equivalent square or rhombus is constructed given a rectangle or a parallelogram respectively. It is then proved that this mechanism can simultaneously draw a parabolic arc. The parameters of this mathematical machine are checked. In each case, an attempt is made to find the focus and the directrix of the corresponding parabola. The important help of the GeoGebra software for the initial mechanical design of the machine is highlighted, but also the important differences of this simulation from the physical construction.</p><p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0713/a.php" alt="Hit counter" /></p>

Unconventional free fall measurements of g: involving the Pythagorean Theorem and inverse tangent function

Abstract The current work presents unconventional but instructionally useful approaches to data analysis and modelling that allow for measurements of the acceleration due to gravity, g, using free-falling motion and the popular Tracker software. The two techniques detailed in this article illustrate the employability of the Pythagorean Theorem and the inverse tangent function in modelling free-falling motion that likewise, expectedly, permit for the determination of g. These two mathematical concepts are brought to relevance by strategically situating the origin of the coordinate system at an appropriate point. The models returned values of the gravitational acceleration that are comparable to that by the traditional technique, the one directly based on the vertical position-time data. Additionally, the unconventional free-fall data analyses presented in this paper suggest extra and insightful activities for the students, exercises on a notably less-automated nonlinear curve fitting on a spreadsheet (MS Excel) at most. Ultimately, by applying familiar mathematical concepts associated with a right triangle to nontraditional contexts like free-fall, this laboratory or lecture activity is perceived to spark greater interest of the students and enhance learning by unifying theory and experimental data.

Analysis of Mathematic Problem-Solving Ability in Solving Pythagoras Theorem Problems

Problem solving is about finding a solution to a problem, so learning mathematics involves solving problems that relate to everyday life. The method of research used is qualitative descriptive research aimed at describing and explaining the methods of solving problems of Pythagoras's theorem students, as seen from the analysis of student's ability to solve mathematical problems. Information obtained from the work or written answers (tests) that have been completed. In this study, 30 students participated in the 2022/2023 academic year in the eighth grade of Delitua Private High School. The highest percentage value was 98.04% for the indicator of problem understanding, while the lowest percentage was 33.34% for the re-examination indicator. Overall, the average percentage score was 69.51%, which is high. This means that the ability of the students of the eighth grade of Delitua Private High School to solve mathematical problems using the material of Pythagoras's theorem is very competent. The students' responses to the statement in the mathematical problem-solving test were effective because all the statements were answered positively by ≥ 90% of students