Pythagorean Theorem Research Articles

Pengaruh Pendekatan Problem Solving Terhadap Kemampuan Menyelesaikan Soal Cerita Materi Teorema Phytagoras

This study aims to find out that the use of problems solving approach has a positive effect on students' abilities in solving story problems on the Pythagorean theorem class VIII.5 UPTD SMP Negeri 18 Barru. This type of research is quasi-experimental without a control class conducted in one group. The variables studied in this study were students' ability to solve word problems on the Pythagorean theorem class VIII.5 UPTD SMP Negeri 18 Barru as the dependent variable and learning using the problems solving method as the independent variable. The population of this study were all students of class VIII UPTD SMP Negeri 18 Barru for the academic year 2022/2023 which consisted of 5 (five) classes. Sampling was carried out by cluster random sampling, from 5 classes one class was taken randomly namely class VIII.5 as the experimental class. The instruments used in this study were tests of the ability to solve the Pythagorean theorem word problems and validated student activity observation sheets. Based on the descriptive statistical analysis, the average score of students' ability to solve the Pythagorean theorem story problems before learning using the problem solving learning method was 63,37 and the average score of students' ability to solve the Pythagorean theorem story problems after learning using the problem solving method was 80,40. Based on inferential statistical analysis using the one sample t-test, a significant value was obtained, p = 0,004 < a = 0, this means that H0 is rejected H1 is accepted. So it can be concluded that there is a positive effect of the problem solving learning method on the mathematics learning outcomes of class VIII.5 UPTD SMP Negeri 18 Barru.

Capturing the subject-specific quality of mathematics instruction: How do expert judgments relate to students’ assessments of the quality of their own learning and understanding?

Based on an opportunity-use model of instructional quality, this study investigates the extent to which subject-specific instructional quality rated by experts is reflected in students’ assessments of their own learning and understanding, and how students’ perceptions predict their achievement. The analyses used data from a German-Swiss sample of 36 classes with around 900 lower secondary students, obtained as part of the so-called “Pythagoras study” in the school year 2002/2003. The teachers were instructed to introduce the Pythagorean theorem in three lessons, which were videotaped. Using the videos, the experts assessed the instruction quality with respect to the goal of promoting a deep understanding of the theorem. The students completed the questionnaires assessing their understanding of the content, their learning process, and the general comprehension orientation of the teacher. The results showed significant and moderate correlations on the class level between expert-rated subject-specific teaching quality and students’ perceptions of their own learning and understanding, as well as of the teacher’s general comprehension orientation. Multilevel models revealed that subject-specific expert ratings are reflected in individual students’ perceptions of their own learning and understanding. Student perceptions were also associated with achievement gains. The results suggest that the assessment of quality by students and experts is more closely linked if a distinction is made between the quality of the learning opportunities offered and their use and if subject-specific criteria are used instead of generic criteria. This study contributes to a more nuanced understanding of the validity of student perspective in assessing instructional quality.

On Heron’s Formula for the Area of a Plane Triangle

Summary This paper provides a review of proofs of Heron’s formula for the area of a plane triangle. We groups proofs together into analogy/experimental evidence, geometry (incircle radius multiplied by semi-perimeter), algebra (equivalence with the Pythagorean Theorem), trigonometric proofs (often via the cosine rule), and specialization from a more general theorem. Lastly, we discuss a neglected paper by Baker of 1885 that lists 110 distinct formulae for the area of a plane triangle and relate some of these formulae to those used in proving Heron’s formula.

Comparative Analysis of Contents on the Pythagorean Theorem in Mathematics Textbooks from America and China

Comparative Analysis of Contents on the Pythagorean Theorem in Mathematics Textbooks from America and China

Pythagorean Calc Aplication: Is It Effective in Mathematical Understanding Ability?

Understanding mathematics requires deep thinking to learn it meaningfully. However, empirically, many students face complex difficulties in understanding mathematical concepts. This study aimed to determine the quality of improvement before and after using the Pythagorean Calculator application and to find out the effectiveness of using the Calc application on the student's ability to understand mathematics. This research uses a quantitative approach with a quasi-experimental method. This study was conducted at one of the private MTs in Ciamis district, West Java, by taking samples using purposive sampling, namely Class VIII B in the 2023/2024 academic year with a total of 21 students. Data was collected through a high-level mathematical understanding ability test based on the Polya indicator. The results of the research show that the quality of improvement in mathematical understanding abilities after learning assisted by the Pythagorean Calc application on the Pythagorean theorem material with the level of difference is in the medium category. Using the Pythagorean Calc application is effective in improving mathematical understanding abilities and is classified as a strong effect category.

IMPROVING MATHEMATICS LEARNING OUTCOMES OF PYTHAGOREAN THEOREM THROUGH JIGSAW TYPE COOPERATIVE METHOD

This research aimed to determine the extent to which the learning outcomes of class VIII G students of SMP Negeri 14 Denpasar had improved through the Jigsaw Type Cooperative learning method in learning mathematics regarding the Pythagorean Theorem. This research was classroom action research with the research subject being class VIII G of SMP Negeri 14 Denpasar Semester 1 in the academic year 2023/2024. Data on student learning outcomes was collected using written tests in the form of multiple choices. The result of this research showed that the first increase occurred in cycle I, 22 of the 42 students who took the test completed the test with a percentage of 52.4%. In cycle II, there was an increase in students who completed the test by 34 students out of 42 students who took the test with a percentage of 81%. It could be concluded that there was an increase in student learning outcomes in learning mathematics regarding the Pythagorean Theorem by applying the Jigsaw Type Cooperative learning method.

Another Simple Proof of Heron’s Formula

Summary We give a proof of Heron’s formula using the Pythagorean theorem, via the medial triangle.

Four errors students make with inverse-square law vectors

In this paper, we discuss four errors introductory physics students make when attempting to add two inverse-square law vectors. We observe multiple instances in which students (1) add vectors as if they were scalars, (2) project the r (or r 2) in the denominator, instead of the entire vector, when attempting to find the vector’s components, (3) incorrectly apply the Pythagorean theorem when attempting to calculate the magnitude of the resultant vector, and (4) incorrectly relate the signs of the components of an electric field (or force) to the signs of the electric charges. While these are not the only errors students make, they are the most frequently occurring based on our analysis of 678 exams taken by students in either introductory mechanics or electricity and magnetism (E&M). We then show how these errors can be encoded into a new type of activity or assessment question which we call a ‘student error task.’ Introductory physics instructors can use the student error task in this paper as a way to engage or assess their students’ understandings of how to add two inverse-square law vectors.

Improving Mathematics Learning Outcomes through Learning Cycle Development

This study aims to describe about: (1) development of valid learning cycle teaching materials for improving student mathematics learning outcomes in grade VIII SMP North Tapanuli Regency, (2) development of practical learning cycle teaching materials for improving student mathematics learning outcomes in grade VIII SMP North Tapanuli Regency, (3) development of effective learning cycle teaching materials for improving student mathematics learning outcomes in grade VIII SMP North Tapanuli Regency, and (4) improving students' mathematics learning outcomes using teaching materials based on the learning cycle method developed in grade VIII of SMP North Tapanuli Regency. The subjects in this study are grade VIII students of SMP Negeri 2 Parmonangan for the 2021/2022 academic year. The subjects in this study are grade VIII students of SMP Negeri 2 Parmonangan for the 2021/2022 academic year. The object of this research is mathematics teaching materials developed based on learning cycle learning methods to improve mathematics learning outcomes of Pythagorean Theorem material. Research using the Plomp development model consists of investigation, design, realization/construction, test, evaluation and revision, and implementation phases. The product test consists of Teaching Material Validity Data Analysis consisting of three linguists, three material experts and three design experts, as well as data analysis of learning outcomes by calculating N-gain. The results of the study stated that (1) teaching materials based on the learning cycle method developed are valid, used to improve students' mathematics learning outcomes in grade VIII SMP North Tapanuli Regency, (2) teaching materials based on the learning cycle method developed practically, used to improve student mathematics learning outcomes in grade VIII SMP North Tapanuli Regency, (3) teaching materials based on the learning cycle method developed effectively, used to improve student mathematics learning outcomes in grade VIII SMP North Tapanuli Regency, (4) teaching materials based on the learning cycle method developed can improve student learning outcomes in grade VIII SMP North Tapanuli Regency. The acquisition of calculated results or N-Gain value of 0.64 with the category of sufficient or good learning improvement results.

Virtual Roller Coaster Design

This lesson offers a technology-enriched educational experience for 10th grade students that focuses on applying geometry, trigonometry, and physics. Students design a virtual roller coaster by applying geometric shapes (e.g., circles, triangles, and polygons), trigonometric ratios (e.g., sine, cosine, tangent), and physics principles (e.g., forces, energy conservation, motion dynamics). Students are challenged to make simulations that reach a height of 100 feet, a track length of 1000 feet, and incorporate one or more loops. They explore concepts like the Pythagorean Theorem, circular motion, and centripetal forces. Assessments focus on precision and creativity, presentation effectiveness, and depth of reflective discussions.

The matter of math: Guiding the blind to touch the Pythagorean theorem

Science teaching relies heavily on visual illustrations and visually organized assignments as methods of teaching and proving concepts like the Pythagorean theorem. As Visually Impaired Persons (VIPs) cannot see, they derive no benefit from the use of such illustrations alone. Often, material objects are used to help VIPs understand mathematical concepts by means of tactile and haptic perception. In this paper, I will show the practices employed in the pursuit of transforming a visually available material object into something of specific scientific relevance for tactile and haptic exploration. This article thus emphasizes the matter – in the original sense of materiality – of math. Based on video-recorded data, this paper provides an in-depth interactional analysis of a single case in which a VIP engages in a mathematics learning situation with an assistant. This research is based on and contributes to ethnomethodology and conversation analysis by critically examining how objects are used to teach VIPs in math classes. The research has implications for a broadened understanding of how learning mathematics is not just a matter of logics, verbal descriptions or the visual analysis of charts and diagrams, but can also inherently involve the tangible materiality of mathematical representations.

Pre-Schmidt Mode Parameters in Quantum Entanglement

We wrote this paper to study in detail the mathematical methods used in deriving the parameters 𝑝𝑝′1, 𝑝𝑝′2, 𝑥𝑥′1, 𝑥𝑥′2, 𝑀𝑀, 𝐾𝐾 and 𝑒𝑒2𝜂𝜂 in the research papers by Han et al. (1999), Makarov (2018a), Han et al. (1993), and Han et al. (1995) to find the Schmidt modes 𝛬𝛬𝑘𝑘 in quantum entanglement. Here, we have analysed and developed a thorough calculation in exploring the rationales behind the existence of these parameters in two and three-coupled harmonic oscillators. Various mathematical approaches were applied in the study, ranging from polynomials and linear algebra to trigonometry and the Pythagorean theorem. We found the parameters 𝐾𝐾 and 𝑒𝑒2𝜂𝜂 using the matrices’ determinants and eigenvalues. With these rationales in deriving the parameters 𝐾𝐾 and 𝑒𝑒2𝜂𝜂 in the research papers for the two-coupled harmonic oscillators, we have formulated similar parameters for three-coupled harmonic oscillators as the conclusion of our study.

Analysis of Students’ Mathematical Reasoning in Solving Pythagorean Theorem Problems

The purpose of this research is to describe students' understanding of the concept of the Pythagorean theorem. This research was carried out at Muhammadiyah 7 Bayat Middle School. This research uses descriptive qualitative research methods. The data collection technique is written test questions. The subjects of this research consisted of 20 class VIII students. The data analysis techniques used are data reduction, data presentation and conclusions. The results of this research were that 45% of students were able to solve questions on applying questions (C3), in the good category 25%, and 30% in the fair category. In the analysis questions (C4), 50% were in the very good category, 30% in the good category, and 20% in the fair category. In the evaluation questions (C5), 25% were in the very good category, 45% in the good category, and 30% in the fair category.

A novel method to evaluate the transverse pedicle angles of the lower lumbar vertebrae using digital radiography.

To investigate a novel approach for establishing the transverse pedicle angle (TPA) of the lower lumbar spine using preoperative digital radiography (DR). Computed Tomography (CT) datasets of the lower lumbar were reconstructed using MIMICS 17.0 software and then imported into 3-matic software for surgical simulation and anatomical parameter measurement. A mathematical algorithm of TPA based on the Pythagorean theorem was established, and all obtained data were analyzed by SPSS software. The CT dataset from 66 samples was reconstructed as a digital model of the lower lumbar vertebrae (L3-L5), and the AP length/estimated lateral length for L3 between the right and left sides was statistically significant (P = 0.015, P = 0.005). The AP length of the right for L4 was smaller than that of the left after a paired t test was executed (P = 0.006). Both the width of the pedicle and the length of the pedicle (P2C1) were consistent with TPA (L3<L4<L5). There were no significant differences in TAN-TPA and DR-TPA compared with real TPA. The ICCs for the real TPA and DR-TPA within L3 showed good reliability, and the ICCs for the real TPA and DR-TPA within both L4 and L5 showed moderate reliability. Our novel approach can be considered a reliable way to determine the transverse pedicle angle from routine DR, and the width and length of the pedicle within lumbar DR should be considered to determine the length and trajectory of the screw during preoperative planning.

Training in problem solving in kinematics at a modern children’s technology park

The article describes the issues of development and programming of robot manipulators based on robotics kits in a children’s technology park. It covers the fundamental knowledge elements from mathematics and mechanics for solving direct and inverse kinematics problems. A manipulator consists of several links and an end effector, which form kinematic pairs performing translational or rotational movements relative to each other. The direct kinematic problem involves calculating the coordinates of the end effector given the known values of the joint angles, while the inverse kinematics problem is the opposite, calculating the joint angles of the manipulator given the known coordinates of the end effector. Solving these problems requires knowledge of basic coordinate systems, the Pythagorean theorem, trigonometric functions and the cosine theorem, all of which are described in the article. The ability to solve kinematics problems is crucial for successful manipulator programming as a whole. The article also provides an example of converting step motor steps into end effector coordinates and discusses the dependence of manipulator accuracy on the characteristics of the step motor.

Comparative Analysis of the Content on the “Pythagorean Theorem” in Middle School Mathematics Textbooks from China, France, and South Korea

Comparative Analysis of the Content on the “Pythagorean Theorem” in Middle School Mathematics Textbooks from China, France, and South Korea

DIFFERENCES IN STUDENTS' MATHEMATICS LEARNING OUTCOMES BETWEEN OFFLINE AND ONLINE LEARNING AT SMP NEGERI 2 PALU

This research aims to determine whether there are differences in student mathematics learning outcomes between offline and online learning at SMP Negeri 2 Palu. The research used a comparative method with a population of all students in class VIII, totalling 332 students. The sample was class VIII students who did offline and online learning, with different samples but the same abilities. The sampling technique is Purposive Sampling. The researchers took 60 samples consisting of 30 students each from classes A and B. The data collection technique was carried out by taking the even semester midterm test scores for the 2021/2022 academic year obtained by students during online learning and the even semester midterm test scores for the 2022/2023 academic year obtained by students during offline learning, with the same subject matter, namely the Pythagorean theorem. Provided that the material taught is equivalent during offline and online learning and has the same academic abilities: (1). Mathematics learning outcomes obtained an average score of 64.13 and 68.27 (2). Sig value. (2-tailed) 0.187 0.05. This means that H0 is accepted, and Ha is rejected. So, the research results show no difference in student mathematics learning outcomes between offline and online learning.

ANALYSIS OF CONCEPT UNDERSTANDING IN THE PYTHAGORAS THEOREM AT STUDENTS SMPN 1 KOTABARU

The aims of this study were 1) to analyze the conceptual understanding of students in class VIII F in understanding the Pythagorean theorem and 2) to find out the factors that cause misconceptions in class VIII F students in solving problems regarding the Pythagorean theorem. The approach used in this research is qualitative research. The method used in this research is the descriptive qualitative method. There are 3 subjects in this research. Data was collected using research instruments in the form of learning achievement tests and interviews. The results of the study stated that students' conceptual understanding subject S1 had a high conceptual understanding even though S1 was less thorough in checking answers. Whereas the S3 subject was able to explain the description of the Pythagorean theorem, it's just that the S3 could not continue the solving procedure on the Pythagorean triple problem due to lack of thoroughness, being in a hurry when working on the problem, and not understanding the Pythagorean theorem material. S2 was not able to understand some of the questions presented. It could not continue the completion procedure according to the questions due to not being thorough, easily forgetting the material that had been taught, being in a hurry when working on the questions, and not understanding the questions that had been given.

The fourth dimension of space and the entanglement principle

This paper focuses on the reason behind the entanglement phenomena, which is still an unknown fundamental question. It argues that while two entangled particles are separated in the 3D space the flow of information between them is instantaneous through the 4D space. The fourth dimension of space has the unit of measurement of length like the other three space dimensions. We shall call it Q, from the word quatre (four) in French. Its magnitude is equal to the invariant proper time τ times the speed of light c. Q = cτ. Each point in the 4D space corresponds to an event. Distances between events in the four-dimensional space can be computed using the Pythagoras theorem. It is argued that time t as measured by clock in our 3D space times the speed of light gives the length of the position vector of an event in the 4D space. At time t, when an observation is made at one of the entangled particles, the length of the position vectors of both entangled particles, sharing the same wave function, in 4D space are equal to t times the speed of light, i.e., ct. The time t is also the same for all the points on the line connecting the two events in the 3D space. Consequently, the length of the position vectors of all the points on the line connecting the two events in the 3D space is equal to ct in the 4D space. The corresponding points in the 4D space make an arc of equidistance to the origin. It is conjectured that the flow of information between the two entangled particles through this arc in the 4D space is not carried by energy or mass particles. It is, therefore, not bound by the time nor the speed of light constraint in 4D space and is instantaneous.

Students' Creative Thinking Ability in Solving Open-Ended Problems Based on Personality Type

This study aims to determine the creative thinking ability of junior high school students in solving open ended problems in terms of students' different personality types. This type of research is descriptive qualitative with instruments in the form of Hippocrates - Galenus personality type questionnaire and written test of creative thinking ability on pythagorean theorem material. The subjects of this study consisted of 31 students of class VIII A of Walisongo 1 Semarang Junior High School. The results of this study showed that (1) the results of filling out the personality type questionnaire from 31 respondents obtained 29% of students with sanguinis personality type, 19% of students with choleric personality type, 23% of students with melancholic personality type and 29% of students with phlegmatic personality type. (2) subjects with melancholic personality type have high creative thinking ability, subjects with sanguinis and choleric personality type have moderate creative thinking ability, while subjects with phlegmatic personality type have low creative thinking ability. This study also shows that each student with a different personality type can affect the creative thinking ability of each student.