Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.

A guide to the first steps into the world of Geometry, Trigonometry and their lines-of-reasoning widely used through the high school and first years of college, in the exact-sciences context.

Elective mathematics has been an extra mathematics subject for pilot students of Eduardo Barretto Sr. National High School for quite some time now. Through this, many alumni testified how this helped them understand senior high school and college math. However, the teachers have also been struggling with the resources for specific areas of mathematics, such as Business Math, Statistics, Analytic Geometry, Trigonometry, and Calculus. When the pandemic hit the Philippines, contextualized learning material aligned with the Most Essential Learning Competencies (...) (MELC) helped elective mathematics teachers. The contextualized learning material was developed from April to August 2020 and was first validated in February 2021 by the students who participated in the pilot testing. However, following the due process, it was suggested that I revalidate the IDEA-formatted Analytic Geometry and Trigonometry instructional material. It examines Eduardo Barretto Sr. National High School mathematics teachers and SDO Calamba City experts' evaluation of the content's topics, objectives, concepts, directions, and activities and the format's prints, illustrations, design, and layout. Participants' perceptions of learning material were assessed using a 4-point Likert Scale survey. Hence, descriptive quantitative design. Participants were surveyed using a 4-point Likert Scale to assess their learning content impression. The findings reveal that the participants found the format and content of the developed learning material to be very satisfactory. The indicators "localized" and "can be done independently" were viewed as Fair and Satisfactory, respectively. The learning material was enhanced by localizing the activities and giving directions that could facilitate self-learning. (shrink)

Trigonometry serves as an important precursor to calculus as well as college level courses. However, a possible factor that affects students from learning is teachers’ strategies in teaching. Several studies reveal that cooperative learning such as study group can increase achievement of students in mathematics. This study focused on the effects of study groups on the grade 9 students’ achievement in trigonometric problems. It sought to identify the achievement of the students in problem solving before and after the interventions (...) and to compare the interventions, the group taught with study groups and taught with the existing method. The study used true experimental design particularly pretest-posttest control group design. The study was analyzed through frequency count and percentage of scores, Wilcoxon rank test, and t-test for independents samples. The subjects of the study were the Grade 9 students in a public high school in Nueva Vizcaya, Philippines. The results revealed that majority of the students’ achievements increased. Moreover, the intervention with a study group and the existing method can improve the students’ performance in solving trigonometric problems. Furthermore, the result suggests that study group method was more effective than the existing method. (shrink)

Triangles and trigonometry are always difficult topics for both mathematics students and teachers. Hence, students' performance in solving mathematical word problems in these topics is not only a reflection of their learning outcomes but also an indication of teaching effectiveness. This case study drew from two examples of solving word problems involving triangles by pre-service mathematics teachers in a foundation mathematics course delivered by the author. The focus of this case study was on reasoning implications of students' performances on (...) the effective training of pre-service mathematics teachers, from which a three-step interactive explicit teaching-learning approach, comprising teacher-led precise and inspiring teaching (or explicit teaching), student-driven engaged learning (or imitative learning), and student-led and teacher-guided problem-solving for real-world problems or projects (or active application), was summarized. Explicit teaching establishes a solid foundation for students to further their understanding of new mathematical concepts and to conceptualize the technical processes associated with these new concepts. Imitative learning helps students build technical abilities and enhance technical efficacy by engaging in learning activities. Once these first two steps have been completed, students should have a decent understanding of new mathematical concepts and technical efficacy to analyze, formulate, and finally solve real-world applications with assistance from teachers whenever required. Specially crafted professional development should also be considered for some in-service mathematics teachers to adopt this three-step interactive teaching-learning process. (shrink)

This is the first book in a two-volume series. The present volume introduces the basics of the conceptual foundations of quantum physics. It appeared first as a series of video lectures on the online learning platform Udemy.]There is probably no science that is as confusing as quantum theory. There's so much misleading information on the subject that for most people it is very difficult to separate science facts from pseudoscience. The goal of this book is to make you able to (...) separate facts from fiction with a comprehensive introduction to the scientific principles of a complex topic in which meaning and interpretation never cease to puzzle and surprise. An A-Z guide which is neither too advanced nor oversimplified to the weirdness and paradoxes of quantum physics explained from the first principles to modern state-of-the-art experiments and which is complete with figures and graphs that illustrate the deeper meaning of the concepts you are unlikely to find elsewhere. A guide for the autodidact or philosopher of science who is looking for general knowledge about quantum physics at intermediate level furnishing the most rigorous account that an exposition can provide and which only occasionally, in few special chapters, resorts to a mathematical level that goes no further than that of high school. It will save you a ton of time that you would have spent searching elsewhere, trying to piece together a variety of information. The author tried to span an 'arch of knowledge' without giving in to the temptation of taking an excessively one-sided account of the subject. What is this strange thing called quantum physics? What is its impact on our understanding of the world? What is ‘reality’ according to quantum physics? This book addresses these and many other questions through a step-by-step journey. The central mystery of the double-slit experiment and the wave-particle duality, the fuzzy world of Heisenberg's uncertainty principle, the weird Schrödinger's cat paradox, the 'spooky action at a distance' of quantum entanglement, the EPR paradox and much more are explained, without neglecting such main contributors as Planck, Einstein, Bohr, Feynman and others who struggled themselves to come up with the mysterious quantum realm. We also take a look at the experiments conducted in recent decades, such as the surprising "which-way" and "quantum-erasure" experiments. Some considerations on why and how quantum physics suggests a worldview based on philosophical idealism conclude this first volume. This treatise goes, at times, into technical details that demand some effort and therefore requires some basics of high school math (calculus, algebra, trigonometry, elementary statistics). However, the final payoff will be invaluable: Your knowledge of, and grasp on, the subject of the conceptual foundations of quantum physics will be deep, wide, and outstanding. Additionally, because schools, colleges, and universities teach quantum physics using a dry, mostly technical approach which furnishes only superficial insight into its foundations, this manual is recommended for all those students, physicists or philosophers of science who would like to look beyond the mere formal aspect and delve deeper into the meaning and essence of quantum mechanics. The manual is a primer that the public deserves. (shrink)

● Much more relevant than the simplicity or complexity of a method of measuring the houses is whether such a division remains true to the physics of the sky (i.e. whether it makes any sense at all). ● Because astrology has no central institution to decide what is valid and what is not, we believe that the least astrologists should do is to respect the truths confirmed by science. Physics teaches that we cannot separate time from space or space from (...) time, for they stand in an unbreakable or indissoluble relationship which we call continuum , a plane made up of four dimensions (i.e. the three of space plus the one of time) regarded as spacetime, and that time is not absolute but relative (dependent on frames of reference). ● Astrology — which is not horoscopism but one of the consequences of physics — appeals to different methods of measuring segments of our horizon known as houses, sectors, or regions and has classified them into space-based measurements and time-based measurements (according to which of the two concepts decides the subdivision of the diurnal arc) , but said classification is impertinent according to the reality that governs us. ● One that can be confirmed to be compatible with our reality is to be employed at all times for all types of charts, that is to say, one that is truly or essentially spacetime-based without, at the same time, presenting any of the errors characteristic of other house systems . We will see that only one will be compatible with relativity, for while the others would have us believe that time is absolute, the Placidus method of house measurement will show us that it is relative. ● The difference lies in the ability of a method to recognise the different ascensional times of not only the twelve segments of our ecliptic at different latitudes but also of each degree of said circumference simultaneously and uninterruptedly, that is, temporally simultaneously and spatially uninterruptedly during their trajectories from the horizon (ASC) to the midheaven (MC). We concluded, based on the house cusp analysis, that only the Placidean formula can carry out this promise, that is, make it possible to measure the lengths of time under said conditions. ● While the Alcabitius and Koch measurements of time assign fictional speeds — i.e. times of oblique ascension — to all degrees contained between the horizon (ASC) and the midheaven (MC), Placidus takes care to measure the speeds or true times for all. The measurement most Newtonian of all the methods of house division, Regiomontanus, in turn, clings to space at the expense of time, the reason for which it cannot reflect the true positions of the planets in spacetime (ignores the relationship between space and time). ● During the demonstrational process, we take the opportunity of pointing out that velocity (special relativity) and gravity (general relativity) do not constitute the only — two — primary forms of time dilation. The tilt of a reference frame relative to another (that of Earth relative to the plane of our ecliptic, although velocity remains a variable, whereas gravity, in turn, a possibility we will explore in the appendices) constitutes the third form. ● We encourage any physicist and/or astronomer to corroborate our observations with their own tools. For our part, we have employed the astrological/astronomical software Solar Fire Gold, although the astrographs come from the astrological/astronomical software Janus. (shrink)