Mathematics

To achieve the goal, the department develops and improves its curriculum integrating strengths of American, Japanese and other countries’ curricula. In particular, American textbooks contain various types of basic practice problems, which help students build a solid foundation of each mathematical concept, while Japanese textbooks and problem solving books contain many problems requiring multiple steps for their solution, which help students develop ability to apply their knowledge to solve complex problems. Both American and Japanese textbooks are used. Indian textbooks are also used to take advantage of their clear and easy-to-understand presentation of proofs of theorems and formulas.

The curriculum is constructed taking the cognitive development of each student into account. In general, more abstract concepts are introduced at higher grade levels. At each grade level, students are placed in one of two to three levels that best fits their cognitive and mathematical development. The goal for each level is set so that students will feel that they can reach the goal if, but only if, they put effort into studying mathematics. Each grade level has the following goal:

9th Grade

In the 9^th grade, all students are required to take the course “Algebra and Geometry.” The goal of this course is to develop student competence to deal with mathematical expressions, the basic language of mathematics. In particular, the course is designed to develop students’ fluency in algebraic manipulations with polynomials and irrational numbers and to develop the ability to construct geometric proofs. Students are placed in either an intermediate or an honors level class, based on the results of a placement test. The course includes the following topics: Equations, Inequalities, Exponents and Polynomials, Polynomials and Factoring, Systems of Equations, Radical Expressions and Equations, Relations and Functions, Quadratic Equations, Introduction to Probability and Statistics; Congruent Triangle, Applying Congruent Triangles, Quadrilaterals, Similarity, Circles, Polygons and Areas, Surface Area and Volume.

10th Grade

In the 10^th grade, all students are required to take the course “Algebra and Trigonometry.” The goal of this course is to continue developing student competence to deal with mathematical expressions. The course develops fluency in algebraic manipulations, especially with rational and radical expressions, and in solving quadratic equations. Students are placed in either an elementary, intermediate or honors level class based on the results of a placement test. The course includes the following topics: Equations and Inequalities, Systems of Equations and Problem Solving, Polynomials and Polynomial Equations, Equations of Second Degree, Rational Expressions and Equations, Polynomial Functions, Powers, Roots, and Complex Numbers, Quadratic Equations, Relations, Functions and Graphs, Quadratic Functions and Transformations, Exponential and Logarithmic Functions, Trigonometric Functions, Trigonometric Identities and Equations, Counting and Probability.

11th Grade

In the 11^th grade, all students are required to take the course “Pre-calculus.” The goal of this course is to develop student competence in logical and abstract thinking. Building on the competence developed in previous courses, 11^th grade students fully utilize their ability to understand mathematically-expressed abstract concepts and to express their own ideas mathematically. This training logical and abstract thinking will be extremely valuable throughout their lives. Students are placed either in an elementary, intermediate, or honors level class, based on their Algebra and Trigonometry course grades. The course includes the following topics: Trigonometric Functions, Introduction to Three dimensional Geometry, Vector Algebra, Permutations and Combinations, Binomial Theorem, Probability, Straight Lines, Conic Sections, Matrices, Determinants, Sequence and Series, Mathematical Induction, Three Dimensional Geometry.

The mathematics core curriculum is structured on mathematical content areas. To develop students’ problem solving ability of utilizing knowledge of various mathematical content areas, 11^th graders can elect “Advanced Problem Solving” course.  Although the course mainly focuses on mathematical problems, it may include real world social, economical, and environmental problems as well. For students to qualify for the course, they must (1) be in an honor level Algebra and Trigonometry class at the end of the 10^th grade and (2) participate in the American Mathematics Competition (AMC-10) during the 10^th grade. The course includes the topics as follows: Combinatorics, Complex Numbers, Inversion in Plane, Mathematical Induction, Proofs by Contradiction, Sequence/Series, Basic Probability, Law of Sine/Cosine, Basic Mod Computation, Basic Functions and Graphs, Basic Geometry, Checking the Validity of the Answer, Working Backward, Work with Algebra and Geometry on a Same Problem, Analogy (Find Similar Problem), and Use of Symmetry.

12th Grade

In the 12^th grade, all students are required to study calculus, and students who wish to major in science, mathematics, medicine, pharmacy, or engineering at college are required to study linear algebra as well. Students who wish to major in economics and commerce in college are strongly encouraged to take linear algebra. Other students may take linear algebra as an elective course. The goal of these courses is to introduce 12^th graders directly to their study of mathematics at Keio University and other colleges in Japan and the United States.

 “Calculus for Non-Science Majors” course includes the following topics and students will be placed either in an elementary or intermediate level class by grades of Pre-Calculus: Limits of Functions, Derivatives (The Chain Rule, Implicit Differentiation, Parametric Representation, Differentiation of Exponential and Logarithmic Functions, Higher Derivatives), Applications of Differentiation (Maxima and Minima, Inflexion Points, Graph Sketching), Indefinite Integrals (Method of Substitution, Integration by Parts, Partial Fractions), Definite Integrals and Applications (Areas and Volumes).

Elective “Linear Algebra for Non-Science Majors” course includes the following topics: Vectors, Systems of Linear Equations, Matrices, Eigenvalues and Eigenvectors, Orthogolity, Vector Space.

“Advanced Calculus & Linear Algebra for Science Majors” course is a requirement for students applying to faculty of science and technology, faculty of medicine and faculty of Pharmacy. The calculus part in this course deals with single variable calculus and includes the following topics: Limit of Functions, Continuity, Differentiation, Sketching a Graph, Integration including Integration by Parts and Integration by Substitution, Surface area, Volume, Polar Coordinates, Differential Equations and Infinite Sequences and Series. Linear algebra part in this course includes the following topics: Basic Notions of Vector Spaces, Systems of Linear Equations, Determinants, Eigenvalues and Eigenvectors, Inner Product Spaces up to Orthogonal Projection and Gram-Schmidt Orthogonalization. The calculus part will be taught in the first three quarters of the school year and the linear algebra part will be taught in the forth quarter. This is a very demanding fast paced course and all the students enrolled in the course are expected to devote a lot of time and effort to study outside of class by reading textbooks and solving problems in the textbooks.

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