﻿ Math | The Hudson School

# Math

Upper school mathematics courses at The Hudson School give equal emphasis to the development of technical skills, real world applications, conceptual understanding, and precise use of notation and vocabulary.

Students learn how to solve equations, represent data using graphs and charts, analyze and utilize functions, as well as communicate their findings and ideas clearly and concisely. Our goal is to encourage students to view mathematics as a tool to be used rather than a chore to be completed. To promote understanding, teachers encourage in-class discussion as well as project based learning to provide a more robust learning experience.

Requirement for Graduation: Three years of mathematics including statistics are required. Electives: AP Calculus (AB), AP Calculus (BC), Software for Engineering I and II, 3D Modeling and CAD.

This course provides a comprehensive introduction to algebraic functions. Throughout, emphasis is put upon good methodology, clarity of presentation, and fluency with mathematical language. Understanding of algebraic axioms and properties are developed through rigorous practice of techniques and precision in communication.

Topics covered include first degree equations and inequalities, systems of first degree equations, and solving quadratic equations. Students perform simplifications of various types of expressions, including extensive work with fractions, exponents, and radicals. Students will be able to write equations from graphs or from stated conditions. Accurate use of mathematical vocabulary is emphasized when students are called upon to explain methods and techniques

Formal geometry presents the following issues in the context of studying the geometry of Euclidean space: (1) postulate-proof model for mathematical endeavor; (2) development of logical method through (a) identification of hypothesis and conclusion of a mathematical statement, (b) proof of mathematical statements by formal two-column, indirect, analytic, algebraic and informal or paragraph proof, (c) converse, inverse, negative, and contrapositive of mathematical statements, and (d) interpreting and creating definitions within a mathematical system; (3) development of visual skills, interpretation of geometric drawings, etc.; (4) after formal proof of various formulas such as area and volume, Pythagorean Theorem, extensive work on specific problem solving; and (5) development of verbal skills: creation of definitions, interpretation, restatement of theorems, interpreting problems stated verbally by translating into drawings and/or mathematical symbolism. Prerequisite: successful completion of Algebra 1.

The systematic study of functions and relations and their applications begins in Algebra 2.

Topics include linear functions and linear systems of equations, quadratic, exponential and logarithmic functions, polynomial functions - including the rational roots theorem; rational functions (algebraic fractions and the functions that involve them); quadratic relations (the conic sections: circle, parabola, ellipse, and hyperbola), radical functions, and sequences and series. Prerequisite: successful completion of Algebra 1 and Geometry. (In special cases, with departmental approval, this course may be taken concurrently with Geometry.)

The purpose of this course is to prepare students of AP Calculus 1. The curriculum of this course is designed to reinforce students’ understanding of functions through algebraic modeling and graphical representation. Basic functions that are covered include linear, quadratic, exponential, logarithmic, polynomial, rational, power, and absolute functions. Precalculus also covers the basics of trigonometry and its applications.

Students learn how to use, graph, and analyze trigonometric functions with both triangles and circles. The course finishes with an introduction to limits in preparation for Calculus. Prerequisite: successful completion of Algebra 2.

The goal of this class is to teach the basic tools and procedures of statistics in the context of applications important in our lives. Statistics is interesting and useful because it provides strategies for using data to gain insight into real problems.

Each topic builds on preceding ones to expand students’ statistical knowledge and solidify their understanding of how the concepts fit together. Topics and skills to be covered include: displaying distributions with graphs; describing distributions with numbers: measures of center, spread, and standard deviation; density curves and normal distributions; correlation, causation, and least square regression; sampling and designing experiments; probability models and random variables; and confidence intervals and tests of significance. This course may be taken as an elective in conjunction with any math course after Algebra 1 or as an alternative to the Calculus track after Algebra 2

This course follows the AP Calculus curriculum. Trigonometry and other precalculus topics will be reviewed. Emphasis will be placed on interpretation of mathematical models.

Topics include calculating the derivative of a function as a rate of change, emphasis on curve sketching, determining a derivative at a point, determining the derivative of power, exponential, logarithmic, and trigonometric functions applications of the derivative to curve sketching, maximum and minimum problems, related rate problems, estimation of roots (Newton’s method), and linear approximations, calculating an integral as the limit of a Reimann sum, determining the antiderivative of a function, the Fundamental Theorem of Calculus, and applications of the definite integral. Prerequisite: successful completion of Precalculus.

Calculus (BC) is an extension of the AP Calculus (AB) curriculum. Topics include: advanced methods of integration; parametric and polar coordinate systems; vector calculus; first and second order differential equations; and infinite series. The course emphasizes real-world applications of all topics. Prerequisite: successful completion of Calculus 1.

The purpose of the AP course in statistics is to introduce students to the major concepts and tools for collecting, analyzing and drawing conclusions from data. Students are exposed to four broad conceptual themes:

• Exploring data - describing patterns and departures from patterns.
• Sampling and experimentation - planning and conducting a study.
• Anticipating patterns - exploring random phenomena using probability and simulation.
• Statistical inference - estimating population parameters and testing hypotheses.

Students who successfully complete the course and exam may receive credit, advanced placement or both for a one-semester introductory college statistics course. This course may be taken as an elective in conjunction with any math course after Algebra 1 or as an alternative to the Precalculus/Calculus track after Algebra 2.

This course will introduce students to various programming based utilities used extensively in college and academia, especially in the field of engineering. Students will learn LATEX, OpenSCAD and Sagemath.

• LATEX is a typesetting language used for creating professional documents and presentations and is the standard format required for most academic journals.
• OpenSCAD is a programming based 3-D Modeling program useful for creating complex mechanical objects.
• SageMath is a free open-source mathematics software system licensed under the GPL. It builds on top of many existing open-source packages: NumPy, SciPy, matplotlib, Sympy, Maxima, GAP, FLINT, R and many more. Students can access their combined power through a common, Python-based language or directly via interfaces or wrappers.

*Also counts towards Computer Science department requirements

This course covers advanced topics in numerical analysis and modeling of real world systems.

GNU/Linux and free software is central to the study and simulation of modern physics. Programming languages such as C++ and Python will be used to program simulations of complex physical phenomena. LATEX will be used throughout the course for documentation and presentation of results. Projects will be chosen based on topics of student interest.

*Also counts towards Computer Science department requirements