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Appreciating University Entrance Exam Problems (Mathematics): Index

2022年1月22日 13:27

Preface

University entrance exam math problems. Representing problem statements in diagrams is an effective means for thinking about solutions. However, for problems one has done before, it is sometimes possible to find the result through calculation alone without drawing a diagram. In some cases, one might not understand the situation but can still perform the calculations to get the answer, and students who are only made to "solve" entrance exam problems are prone to falling into this situation. Even if they can do the calculations, if you tell them to "try drawing a diagram," they may be unable to do so.
Drawing diagrams is not only effective for thinking about solutions, but for some problems, the problem setting itself is interesting even without doing any calculations. Concepts like the focus of light reflection on a paraboloid (or parabola) and spatial figures with loci can be better understood by drawing many diagrams or creating animations. Even without doing calculations, they are interesting just to appreciate. For example, the header image for this page is a diagram of a 2002 University of Tokyo problem and it has resulted in a colorful and beautiful figure. However, the field of the problem is integration, specifically the method of exhaustion to find the sum of an infinite series using integration. In other words, you can express the conditions of the problem statement as an equation and get the answer through integration, but that alone might not reveal what it represents or what it looks like geometrically. When you try appreciating it by turning it into an interactive diagram like this, you can understand, "I see, that's what it means." Among university entrance exam problems, there are some that deal with deep subjects like the Koch snowflake curve or reflections related to circles, and introducing these in class would likely increase interest in mathematics.

Therefore, "Appreciating University Entrance Exam Problems" is a collection of such subjects picked out from university entrance exam problems and made to be interactively movable on the Web. The diagrams are created using the geometry software Cinderella (CindyJS).
Cinderella is a geometry construction software that began development in 1993. It was started by a university professor in Germany. In 2005, the programming language CindyScript was added, making it possible to create constructions through programming. The development history around this is described in the Cinderella manual. Also, I have posted a Japanese translation on the Web. Initially, there were parts that required payment, but now it has become freeware. It can be used on Windows, macOS, and Linux (there are versions corresponding to each).
Cinderella was written in Java, and the things created could be run on the Web. However, due to security issues and the like, this stopped working. The developer, Mr. Kortenkamp, described it as "Java is dead." Therefore, development began anew in JavaScript. That is CindyJS.Cinderella When you "export to HTML" from , a file written in CindyJS is created, which can be run on the Web. It is not perfectly compatible with CindyScript, but development is much easier compared to languages like Python. As similar construction software, there is Geogebra, which has many people involved in the project and many users, but because Cinderella allows for programming, it is possible to create works quite flexibly. It is also used to create inserted figures for TeX (KetCindy), and it can create inserted figures much more easily than using Tikz. This is being developed primarily by university professors in Japan. I also use it to create high school mathematics textbooks.

I had been using Cinderella to turn university entrance exam problems into diagrams and make them move even before the appearance of CindyJS, but the ones on that Web page stopped working because "Java died." This magazine is where I have rewritten them using CindyJS, added new ones, and am unfolding them on note.