Mathematical communication ability is the student's ability to convey mathematical ideas through writing, symbols, and visual representations. Learning interest is one of the factors that encourages students to be more active in learning and more confident in expressing mathematical ideas. This study aims to describe students' mathematical communication abilities in solving Pythagorean Theorem problems in terms of learning interest. This research uses a qualitative descriptive approach and was conducted in class VIII of SMP Negeri 7 Mataram in the 2025/2026 academic year. The subjects of this study were 42 students, consisting of 11 students with high learning interest, 18 students with moderate learning interest, and 13 students with low learning interest. Furthermore, 8 students were selected for interviews, consisting of 3 students with high interest, 3 with moderate interest, and 2 with low interest. Data collection techniques included questionnaires, tests, and interviews. Data analysis was performed through data reduction, data presentation, and drawing conclusions. The results showed that students' mathematical communication abilities differ at each level of learning interest. Students with high interest tend to be better at expressing ideas and using symbols; students with moderate interest are fairly good but still have difficulty explaining the solution steps; whereas students with low interest have difficulty expressing ideas and explaining the problem-solving process. Thus, learning interest is not always directly proportional to students' mathematical communication abilities.

Inframammary incision placement is critical in achieving optimal aesthetic outcomes in breast augmentation, particularly in minimising visible scarring. Despite its importance, no standardised method currently exists to objectively determine the incision location. This paper presents a modified geometric model based on the Pythagorean theorem to determine the correct nipple-to-inframammary fold (IMF) distance. The model incorporates anatomical variables, such as breast tissue thickness, and implant characteristics including gel cohesiveness. The modified model provides a reproducible and individualised incision planning strategy. Soft cohesive implants necessitate a higher initial placement to anticipate postoperative tissue expansion, whereas firmer gels are placed lower to achieve immediate aesthetic ratios. This approach integrates geometry with clinical anatomy and implant biomechanics to standardise inframammary incision placement, improving aesthetic predictability and scar concealment. This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

This study investigates the impact of learner autonomy affordance on students’ learning performance and motivation within an ITS context, as well as the moderating effects of learner characteristics. Using traditional AI techniques to ensure experimental control, we developed an ITS teaching the Pythagorean Theorem to junior high school students in mainland China. Three system modes were constructed, each representing a different level of learner autonomy affordance. Through a double-blind randomized controlled trial, participants were randomly assigned to one of these three experimental conditions. We evaluated the impact of learner autonomy on learning performance and attrition and examined the potential moderating effects of learner prior knowledge and self-regulated learning abilities. Results indicated that learner autonomy affordance significantly influenced learner motivation, with this effect moderated by learners’ metacognitive self-regulation. Furthermore, learners’ intrinsic goal orientation and control of learning beliefs moderated the impact of autonomy affordance on learning performance.

Aim. The purpose of this paper is to prove theorems on the existence, nonexistence, and divisibility of power sequences. The power sequences considered in this paper consist of elements that generalize the properties of well-known Diophantine equations, such as the unsolvable equation in Fermat's Last Theorem or the equation relating the lengths of the sides of a right triangle using the Pythagorean theorem. Methodology. In this work, the methods of elementary number theory were mainly used. In this work, the methods of elementary number theory were mainly used. Results. The result of the work is completely proven theorems on sequences generalizing Diophantine equations of high degrees. Research implications. The theoretical significance of this study lies in its generalization of the concept of Diophantine equations to a set of sequences. The work is purely theoretical in nature.

The objective of this study is to show a new closed rhinoplasty surgical technique that relates nasofacial anatomy with the Pythagorean theorem and the elastic forces quantified in Hooke’s Law The Pythagorean Theorem states that in any right triangle; the square of the hypotenuse is equal to the sum of the squares of the legs. Similarly, this concept can be applied to the nasal tripods of the nose if we consider that their structure is triangular with a central vertical leg shared, or columella. The nasal wings would be the hypotenuse, and the nasolabial junction would be the triangular base. In this context, if the lower leg is shortened, the central leg lengthens, causing the lateral crura, or hypotenuse, to narrow inward. In this article, we show the developed surgical technique. For the modification to be feasible, the legs must be freed while remaining intact. Based on these principles, we have designed Two new surgical tools to release and modify the nasal triangle. The conclusion is that this new technique can be performed with very good results in closed rhinoplasty without leaving scars.

In genuine nonequilibrium systems under continuous driving, thermodynamic forces are nonconservative and cannot be described by any free energy potential. Nonetheless, we show that such systems can be associated with a derived from a large-deviation variational principle. This variational principle yields a decomposition of fluxes, forces, and entropy production into a conservative part and a nonconservative part, exemplifying an information-geometric Pythagorean theorem. The decomposition is broadly applicable—including to stochastic master equations as well as closed and open deterministic chemical reaction networks—and accessible to thermodynamic inference from short-time trajectory data. We also show that the excess entropy production obeys a thermodynamic speed limit bounding the rate of state evolution and external fluxes. We illustrate the framework on driven Markov jump processes, nonlinear chemical oscillators, and real-world metabolic networks, where we obtain tight dissipation bounds and identify futile metabolic cycles. Connections are drawn to large deviations, Onsager theory, and previous excess/housekeeping decompositions.

Abstract We construct spectral triples of one- and two-qubit states using the Hilbert-Schmidt operatorial formulation, and study the Connes spectral distances. We also construct the Dirac operator corresponding to the normal quantum trace distances. Based on the Connes spectral distances, we propose some definitions of quantum discord and coherence measure of quantum states, and explicitly calculate the coherence of one-qubit states. We also study some simple cases about two-qubit states, and the corresponding spectral distances satisfy the Pythagoras theorem. These results are significant for studies on physical relations and geometric structures of qubits and other quantum states.

Students’ mathematical reasoning ability (MRA) is an essential component of mathematics learning, particularly for solving non-routine problems. However, many junior high school students still rely on procedural strategies without sufficient conceptual understanding, resulting in weak reasoning skills, especially in geometry topics such as the Pythagorean Theorem, indicating the need for a detailed analysis of students’ mathematical reasoning ability. This study aims to analyze junior high school students’ mathematical reasoning ability on the Pythagorean Theorem based on four indicators: proposing conjectures, performing mathematical transformations, providing logical justification, and drawing conclusions. A descriptive qualitative approach was employed involving 31 eighth-grade students of SMP Negeri 1 Raha selected through purposive sampling. Data were collected using an open-ended mathematical reasoning ability test related to the Pythagorean Theorem and supported by semi-structured interviews. Data analysis was conducted using Miles and Huberman’s interactive model, consisting of data reduction, data display, and conclusion drawing and verification. The results indicate that 6% of students demonstrated high mathematical reasoning ability, 8% were categorized as moderate, and 86% were classified as low. Most students experienced difficulties in formulating conjectures, transforming contextual problems into mathematical models, providing logical justification, and drawing valid conclusions. In conclusion, students’ mathematical reasoning ability on the Pythagorean Theorem remains relatively low. Therefore, instructional strategies that emphasize conceptual understanding, reasoning processes, justification, and reflective thinking are necessary to improve students’ mathematical reasoning abilities.

Mathematical understanding is an essential goal of mathematics learning, particularly for abstract topics such as the Pythagorean Theorem, which require strong conceptual comprehension. However, mathematics instruction at the junior high school level is still largely dominated by conventional methods with limited use of interactive learning media, causing students to experience difficulties in understanding concepts. This study aims to develop an interactive Android-based learning multimedia, Pytha-Go, for the Pythagorean Theorem material for eighth-grade students at SMP Negeri 13 Tibawa. This research employed a Research and Development (R&D) approach using a modified 4D development model consisting of three stages: Define, Design, and Develop. The Define stage identified learning problems and student needs through observations and interviews. The Design stage focused on designing the structure, learning content, and interface of the multimedia. The Develop stage involved media production, expert validation, readability testing, and limited trials. Data were collected using media and material expert validation questionnaires, media readability test instruments, teacher observation sheets, and student response questionnaires. The results showed that the average scores from media and material expert validation were 3.23 and 3.17, categorized as very valid and valid. The media readability test obtained an average score of 3.34, indicating that the multimedia is feasible. Teacher observation and student response scores were 3.35 and 3.58, respectively, categorized as very practical. Based on these findings, it can be concluded that the Pytha-Go Android-based learning multimedia is feasible and practical for supporting the learning of the Pythagorean Theorem for eighth-grade students.

In real-life problems, we use square roots in natural distributions such as (the probability density function), distances and lengths in the Pythagorean theorem, and quadratic formulas in (the height of falling objects), radius of circles, harmonic movements (pendulum and springs), and standard deviation in statistics. We have observed that using fuzzy sets in real-life problems is more convenient than ordinary sets. Therefore, they are important in algebraic structures. As a result, more effort has been made to study square root structures in fuzzy sets. This paper introduces the notion of square roots fuzzy of QS-ideals on QS-algebras and some important characteristics. Some illustrative examples have been provided which prove that every SRF-BCK-ideal is an SRF QS-ideal. Also, the image and the inverse image of SRF-QS-ideals are discussed. Finally, the product of SRF-QS-ideals on QS-algebra is defined and some important properties have been proved.

The development of Problem-Based Learning (PBL)-based mathematics comics involves creating mathematics learning media that combine elements of pictures and stories to help students understand concepts and improve their critical thinking skills. The development of PBL-based math comics uses the Canva and Live Worksheets applications. This research aims to produce PBL-based math comic worksheets for students, created in Canva and Live Worksheets, to improve critical thinking skills in Phase D students. The research carried out is Research and Development (R&D), using the Analysis, Design, Development, Implementation, and Evaluation (ADDIE) model. The results of the research are PBL-based math comic worksheets to improve students' critical thinking skills, accessible on various internet-connected devices. This math comic worksheet focuses on the Pythagorean Theorem for Phase D students, presenting problems in a context-based format and using easy-to-understand language. The research was conducted at SMP IT Nurul 'Ilmi 2 Jambi City with one mathematics teacher, nine students in the small group trial, and 28 students in the large group trial. The results of the study show that the product developed underwent several revisions to become worthy of testing. The instrument used uses the Guttman scale, so that the final validation after revision reaches 100%. The validity of PBL-based math comic worksheets is 100% (very valid) according to material experts and 100% according to media design experts. The percentage of practicality of PBL-based math comic worksheets by educators is 100% (very practical), and by students is 98% (very practical), and the percentage of effectiveness of PBL-based math comic worksheets from the student response questionnaire is 98% (effective). The learning outcome test, in the form of a pre-test and post-test using N-Gain, was completed by 16 people in the medium category, 10 in the high category, and 2 in the low category.

This study aimed to improve the mathematics learning outcomes of eighth-grade students in class VIII-B at SMPN 11 Mataram through the implementation of the Problem-Based Learning (PBL) model supported by Quizizz Paper Mode media. The research employed a Classroom Action Research (CAR) design conducted in two cycles, each consisting of the stages of planning, action implementation, observation, and reflection. The research subjects were 28 students of class VIII-B in the odd semester of the 2024/2025 academic year. Data were collected using mathematics learning outcome tests, while data analysis was carried out descriptively. The indicator of research success was determined based on the achievement of the Minimum Mastery Criteria (KKM) of 67 with a minimum classical mastery of 85%. The results showed that in the pre-cycle stage, the students’ average score was 54 with a mastery level of 29%. In Cycle I, the average score increased to 64 with a mastery level of 64%. Furthermore, in Cycle II, the students’ average score increased significantly to 89, with classical mastery reaching 89%. These findings indicate that the implementation of the Problem-Based Learning model supported by Quizizz Paper Mode media was effective in improving students’ mathematics learning outcomes on the topic of the Pythagorean Theorem.

The purpose of this study was to analyze students' mathematical abilities in solving HOTS problems on the Pythagorean theorem material for eighth grade junior high school students. The type of research used in this study is descriptive qualitative. The subjects of this study were 20 students. The data collection tool was a HOTS question sheet consisting of 30 multiple-choice questions, each consisting of 10 questions at each level starting from C4, C5, and C6. The data analysis technique used descriptive analysis of student test results. The results of this study concluded that students' mathematical abilities in working on HOTS problems on the Pythagorean theorem material were still very low. This can be seen from the average student score of 44.67. From these results, it is known that 13 students have very low abilities, 4 students have low abilities, 2 students have medium abilities, 1 student has high abilities, and 0 students have very high abilities. Of the three levels of HOTS questions, 50% of students answered correctly at the C4 level, 44% of students answered correctly at the C5 level, and 40% of students answered correctly at the C6 level.

Varus deformity can present a significant challenge for limb alignment correction and balancing in total knee arthroplasty (TKA). One technique to address these challenges is a medial reduction osteotomy. This article describes utilization of a robotic platform to perform a safe and accurate medial subtraction osteotomy prior to balancing and bony resections. Deformity correction can be predicted by the Pythagorean Theorem. Computed tomography-based robotic systems can be used to perform medial reduction osteotomy of the tibia in the setting of significant varus deformity in patients undergoing TKA. Prior to balancing and bony cuts, the tibial component is downsized "virtually" from the planned size. Through lateralization of the component, the excess medial bone can be mapped via tracking of the registration probe and removed. The amount of medial tibial bone resected determines the amount of laxity that will be created when the tibia is reduced under the femur when implants are placed. Following this, soft tissue tensioning, planning, bony resections, and trialing can progress as normal for a robotic total knee. Through the described technique, the authors have been able to predict the amount of coronal plane correction based on the size of the osteotomized fragment using the Pythagorean Theorem. Robotic guidance of a medial subtraction osteotomy provides a safe and predictable means of varus correction. This is beneficial in that it can be performed with great accuracy and prior to any further balancing maneuvers or bony cuts.

The Pythagorean Theorem and its generalizations have been widely studied. In this article we focus on one particular generalization that asserts that the area of a polygon with one side the hypotenuse of a right triangle is the sum of the areas of similar polygons whose corresponding sides are the other two sides [4]. Using this theorem together with interactive software or simply pencil and paper, we create some visually enticing artwork. This activity, we believe, can contribute positively to geometry teaching at all academic levels.

The article examines the integration of the Pythagorean theorem and its generalizations into STEM-education through the use of modern programming technologies. Current trends in the implementation of innovative methods of teaching mathematics are analyzed, with particular attention paid to the application of gamification, information and communication technologies, and mobile applications. A methodological framework is developed for designing a series of mini-projects that combine historical and mathematical context with programming tools in R and Python. The article presents a detailed analytical algorithm for creating a calendar of Pythagorean theorem days, based on calculating Pythagorean triples for calendar dates in the d.m.yy format, where d² + m² = y². The algorithm efficiently identifies valid dates without exhaustive search by computing the difference of squares. As a result, twelve Pythagorean dates were identified for the period from 2001 to 2026. The proposed mini-projects include generating Pythagorean triples, simulating the Pythagorean fractal tree, analyzing the Spiral of Theodorus, modeling the harmony of spheres, creating interactive proof animations, and exploring generalizations of the theorem in higher dimensions. The study demonstrates the effectiveness of an interdisciplinary approach for fostering students’ critical thinking, digital literacy, and mathematical competencies within the framework of modern STEM-education.

This study aims to develop and psychometrically validate an assessment instrument designed to measure students’ metaphorical thinking ability in mathematics learning, with a specific focus on the Pythagorean Theorem. The instrument was developed using a Research and Development (R&D) framework based on the Oriondo and Antonio model, encompassing test design, empirical tryout, and measurement stages. Six polytomous essay items were constructed according to six indicators of metaphorical thinking: connect, relate, explore, analyze, transform, and experience. Content validity was established through expert judgment using Aiken’s V coefficient, with all items exceeding the minimum validity threshold, indicating strong agreement among experts. Empirical validation was conducted using Item Response Theory (IRT) with the Graded Response Model (GRM), selected for its suitability in analyzing ordered polytomous response data. The results demonstrate that the instrument satisfies the unidimensionality assumption, exhibits strong item discrimination parameters, and shows good model fit across all items. Analysis of the Test Information Function indicates high measurement precision within the ability range of θ −1 to +1, confirming strong local reliability. These findings indicate that the developed instrument is valid, reliable, and capable of providing accurate diagnostic information regarding students’ ability to construct mathematical meaning through metaphors. The study contributes methodologically by demonstrating the applicability of GRM-based IRT analysis for essay-type instruments and substantively by supporting the assessment of higher-order cognitive processes in mathematics learning.

This paper focuses on determining the thickness of silicon carbide epitaxial layers using infrared interferometry. A mathematical model was established based on the principle of double-beam interference, incorporating Snell’s Law and the Pythagorean theorem to calculate the optical path difference between two reflected beams (from the epitaxial layer surface and the substrate interface). The phase difference was linked to the interference order, and conditions for constructive and destructive interference were analyzed to derive the fundamental thickness model. The Sellmeier dispersion model was introduced to account for the wavelength-dependent refractive index influenced by doping concentration. For experimental verification, the reflectance spectra were preprocessed to remove noise and convert wavenumbers into wavelengths. Extreme interference points were identified, and thickness was optimized globally using genetic algorithms in MATLAB, minimizing the least-squares error between modeled and measured reflectance. Reliability was confirmed through residual analysis and consistency across incidence angles. The impact of multi-beam interference was analyzed by comparing its characteristics with double-beam interference and identifying necessary conditions (e.g., high reflectivity and phase stability). The coefficient of variation in wavenumber spacing was used to multi-beam effects. For silicon wafers, the transfer matrix model and genetic algorithms were employed to calculate optimized thickness, accounting for multi-beam interference. This work provides a robust framework for high-accuracy, nondestructive thickness measurement of epitaxial layers.

Mathematical justification is a crucial skill for mathematics teachers because it emphasizes proof and reasoning. This study examines how using the media “Digital Pythagorean Tangrams,” a manipulable, multimedia tool on Polypad, can improve prospective teachers' justification skills when proving the Pythagorean theorem. This research used a one-group pretest-posttest design where students were given a pre-test, then received an intervention using the media, and subsequently completed a post-test. The participants included 44 mathematics education students at a university in Bandung. Instruments comprised a mathematical justification test (pretest and posttest) and Digital Pythagorean Tangram media. Pretest and posttest data were analysed to obtain the N-gain score to estimate the effect size. The N-gain data were analysed using a one-sample t-test to check whether the improvement was statistically significant. The results revealed an average N-gain score of 0.42, which falls into the medium category, demonstrating a notable improvement in students' mathematical justification skills before and after using the digital tangram media. The One-Sample t-test also indicated a significance level below 0.05, confirming that the increase in students' mathematical justification ability is significant. Integrating digital manipulatives into proof-oriented instruction can meaningfully enhance prospective teachers’ justification skills and may serve as a practical pedagogical approach in mathematics teacher education.

Mathematical literacy skills of Indonesian students are still below the OECD average (PISA 2022) , where students face difficulties relating their mathematical knowledge to real-life contexts. This indicates that learning is still dominated by a mechanistic approach. This research aims to design worked examples in mathematical literacy skills for solving contextual problems in the Pythagorean Theorem material. The research method used is design and development research with the ADDIE model (Analyze, Design, Development). The problem design was developed based on the three main processes of OECD mathematical literacy: formulating, employing, and interpreting. The results yielded a problem design that provides systematic and contextual solution steps , while minimizing cognitive load by avoiding split-attention and redundancy effects. This worked example strategy is recommended for novice learners because it serves as scaffolding that guides students to understand the reasoning process and application of concepts systematically in real-world situations