# Mathematics

Department Chair: Dr. Dusan Nesic

The Mathematics Department's major goal is to prepare our students for their future study at college, especially at Keio University where most of our students wish to enter. To achieve this goal, the department integrates the best practices of the American, Japanese, and other international curricula. In particular, American textbooks contain various types of basic problems, which help students build a solid foundation for each mathematical concept. Japanese textbooks and problem-solving books, on the other hand, offer problems that require multiple steps for their solution. These examples help students understand complex mathematical concepts. The American and the Japanese textbooks complement each other, creating a more comprehensive approach to learning.

The curriculum takes into consideration the cognitive development of each student. In general, more abstract concepts are introduced at higher grade levels. In each grade, students are placed in the class that best suits their individual math abilities. The goal for each class is to ensure that each student can fulfill the requirements, provided that the student puts enough effort into class work and homework.

Algebra & Geometry is a required 9th grade course. The course is designed to develop students' fluency in algebraic manipulations with polynomials and irrational numbers. Each student will develop the ability to construct geometric proofs. Students are placed in either an intermediate or honors level class based on the results of a placement test. The course includes the following topics: equations; inequalities; exponents; polynomials; systems of equations; radical expressions and equations; relations and functions; quadratic equations; the introduction to probability and statistics; congruent triangles; quadrilaterals; similarity; and circles and polygons and their surface areas and volumes.

Algebra & Trigonometry is a required 10th grade course. The goal of this course is to continue developing the students' competence in dealing with mathematical expressions. The course develops fluency in algebraic manipulations with rational and radical expressions and in solving quadratic equations. Students are placed in either an elementary, intermediate, or honors class based on the results of a placement test. The course includes the following topics: solving linear equations and inequalities; solving systems of linear equations; polynomials; polynomial equations and polynomial functions; equations of second degree; rational expressions and equations; powers, roots, and complex numbers; quadratic equations and methods for their solutions; relations, functions, and graphs; quadratic functions and transformations; right triangle trigonometry; and trigonometric identities.

Precalculus is a required 11th grade course. The goal of this course is to develop the students' competence in logical and abstract thinking. Building upon the skills developed in the previous years, 11th grade students will fully utilize their ability to understand mathematically expressed abstract concepts and be able to express their own ideas mathematically. Students are placed in an elementary, intermediate, or honors class based on their Algebra & Trigonometry course grades. This course includes the following topics: solving trigonometric functions, an introduction to three dimensional geometry, an introduction to vector algebra, the basics of combinatorial algebra, permutations, combinations, binomial theorem, the basics of probability, the analysis of straight lines, conic sections, sequences and series, mathematical induction, and an introduction to limits, derivatives, and integration.

Advanced Problem Solving is an elective course for 11th grade students. It focuses on developing the students' problem solving ability to the extent that they are comfortable participating in the American Mathematics Competitions. In order for students to qualify for the course, they must (1) be in an honors level Algebra & Trigonometry class at the end of the 10th grade and (2) participate in the American Mathematics Competition (AMC-10) during the 10th grade. The content of the course is driven by the teacher's preferences and in the past included combinatorics, complex numbers, inversions in plane, proofs by contradiction, sequences, series, probability, the laws of sines and cosines, basic mod computation, basic functions and graphs, basic geometry, checking the validity of the answer, working backward, working with algebra and geometry on the same problem, analogy (finding a similar problem), and the use of symmetry.

Statistics is an elective 11th grade course. This course will prepare students to take statistics courses at Keio University by focusing on some of the fundamental concepts and techniques used in statistics. This course covers descriptive statistics, elementary probability, random variables, probability distributions, sampling, and hypothesis tests. As part of the course, spreadsheet work (with Microsoft Excel) is demonstrated in class and students are also encouraged to do part of their weekly homework assignments with spreadsheets. There is no prerequisite for this course.

Advanced Calculus & Linear Algebra for Science Majors is a 12th grade course which is a requirement for students applying to Keio University's Faculty Of Science and Technology, School of Medicine, or Faculty of Pharmacy. The calculus portion of this course deals with single variable calculus and includes the following topics: the limits of functions, continuity, differentiation, the sketching of graphs, integration, surface area, volume, polar coordinates, differential equations, and infinite sequences and series. The linear algebra portion of this course includes the following topics: basic notions of vector spaces, systems of linear equations, determinants, eigenvalues and eigenvectors, inner product, orthogonal projection, and Gram-Schmidt orthogonalization. The calculus portion will be taught in the first three quarters of the school year and the linear algebra part will be taught in the fourth quarter. This is a very demanding and fast-paced course. Students should expect to devote a lot of time and effort to solving problems outside of class.

Calculus is a 12th grade course required for all students who **do not** intend to apply to Keio University's Faculty Of Science and Technology, School of Medicine, or Faculty of Pharmacy. Students are placed in honors or elementary level classes based on their grades in the 11th grade precalculus course. This course covers the limits of functions, derivatives, applications of differentiation, indefinite integrals, and definite integrals and their applications (areas and volumes).

Linear Algebra for Non-Science Majors is a 12th grade elective course intended to prepare students who are enrolling in the Faculty of Economics and Commerce at Keio University. The course covers an introduction to vectors, systems of linear equations, matrices, determinants, eigenvalues and eigenvectors, orthogonality, and vector spaces.