# Upper School Mathematics

The Philosophy

The primary concern of the Mathematics Department is the development of a student’s ability to think logically and rationally. Regardless of interest, this reasoning is invaluable in nearly any chosen field, as well as in everyday living. We also believe students benefit from a deep understanding of mathematical proofs, facts and concepts, and that logical reasoning provides the path to that understanding. While students need to master some mechanical processes and algorithms to succeed in mathematics, they are encouraged and guided to explore and discover patterns, abstractions, connections, and problem solving strategies that can also be applied to real-world problems. Our program provides students with flexibility to find courses that provide appropriate challenge for their individual capacity.

Prerequisite: Pre-Algebra equivalent and/or Middle School geometry, Mathematics Department Approval

This course begins with a brief refresh of pre-algebra skills, including the order of operations, operations with fractions, negative numbers, and solving simple equations. Advanced Algebra is designed to provide students with a foundation for their progression through the remainder of the Mathematics Curriculum.

Topics of study will include:

• Introduction to variables
• Real number axioms
• Linear equations and inequalities in one and two variables
• Graphing in the plane
• Operations with polynomials
• Factoring
• Laws of exponents including negative exponents
• Rational expressions
• Irrational numbers

## Geometry

The Geometry course is designed to support the students’ transition from concrete to abstract reasoning. Students review the building blocks of geometry and apply geometric and algebraic properties through hands-on, inquiry-based, and proof-based activities. Students master properties of two and three-dimensional figures through study of advanced measurement, construction, and comparison. Students practice their reasoning skills as they solve problems involving spatial relationships, area, volume, and coordinate geometry. They extend their study of similarity and proportion as they deepen their understanding of the Pythagorean Theorem, the special triangles, and right triangle trigonometry.

Topics of study will include:

• Angles in the plane
• Parallel and perpendicular relationships
• Congruent triangles
• Similar triangles
• Special triangles
• Circles and angles in circles
• Constructions and loci
• Coordinate geometry
• Areas of polygons and circles
• Deductive proof and applications to three dimensions emphasized throughout
• Trigonometry of right triangles
• Laws of sines and cosines

## Algebra II

Prerequisite: Advanced Algebra and Geometry or equivalent

The philosophy behind Algebra II lies in tying functions to their graphs to widen students’ perspectives from pinpointing specific values of functions to the wider relationships and trends represented by graphing including such concepts as asymptotes, end behavior, and continuity.

Topics of study will include:

• Linear equation and inequalities in 1, 2, and 3 variables
• Linear systems
• Irrational numbers and radical expressions and equations
• Complex numbers
• Techniques for solving polynomial equations
• Exponential and Logarithmic functions, equations, and properties
• Rational expressions and equations including negative exponents

## Algebra II Honors

Prerequisite: Geometry

This course begins with a quick review of Algebra I, moves to linear functions, and rapidly progresses through the Algebra II curriculum. It is designed for motivated students who are ready to work through material at a quick pace in preparation for advanced mathematics offerings. Group work is a major component of the course and students are expected to take significant personal responsibility for their own learning.

Topics of study will include:

• Foundations from Algebra I
• Linear Functions
• The Graphs of The Basic Functions
• Systems of Linear Equations and Inequalities
• Polynomial Algebra
• Factoring
• Rational Functions
• Exponential Functions
• Inverse Functions
• Logarithmic Functions
• Conic Sections

## Precalculus

Prerequisite: Algebra II

In this course, we will explore a variety of topics that reinforce prior mathematical skills and understanding, and prepare students for more advanced mathematics courses, Calculus in particular. For an appetizer, we begin with some review of algebra and the function concept, including equation solving and function composition. The main course of the meal is trigonometry, as we advance from triangular trigonometry to circular trigonometry. We enjoy dessert towards the end of the year with some discrete mathematics, probability, and an exploration of conic sections.

The emphasis throughout is using the exercise and reinforcement of mechanical skills to achieve deeper understanding of the connections to other parts of mathematics and other fields of study. Rote learning and memorization are not the goal, but success in this course requires practice and preparation.

Topics of study will include:

• Functional analysis including composition and inversion
• Polynomials with factor theorem and rational roots theorem
• Review of exponential and logarithmic functions
• Trigonometric functions and their inverses
• Solving equations involving trigonometric functions
• Trigonometric identities and proofs
• Solving triangles using the Laws of Sine and Cosine
• Sequences and series
• Probability and combinatorics
• Conic sections

## Precalculus Honors

Prerequisite: Honors Algebra II

Honors Precalculus will cover the same topics as the Precalculus course but it will cover the material in greater depth, at a faster pace, and at a greater level of abstraction. Additional topics like polar coordinates and polynomial theorems will be covered to enhance the depth and rigor of study. Honors Precalculus is a fast-paced course intended for highly-motivated students who have demonstrated an enthusiasm for math. The expectations and workload placed on the students are much higher than in Precalculus. This higher expectation of work, quality, and depth of ideas will directly challenge students’ conceptual understanding of higher-level mathematics.

Topics of study will include:

• Transformations of functions
• Composition of functions
• Inverse functions
• Polynomial and rational functions
• Exponential and logarithmic functions
• Trigonometric functions
• Analytic trigonometry
• Combinatorics and probability
• Sequences and series
• Conic sections
• Matrices

## Calculus

Prerequisite: Precalculus

This course provides students with an intuitive approach to the fundamentals of differential and integral calculus. Focusing on functions, students explore limits, leading to the definition of derivative. The concepts of average and instantaneous rates of change are investigated. We develop the rules of differentiation, including the chain rule and implicit differentiation, and apply them to problems in optimization, related rates, and curve sketching. We introduce the concepts of finding area under a curve, the integral regarded as the antiderivative, and the Fundamental Theorem of Calculus. Applications of integration are included.

Topics of study will include:

• Limits and continuity
• Derivatives including the chain rule and implicit differentiation
• Applications in curve tracing, related rates, and optimization problems
• Integration including area approximation and the substitution method
• The fundamental theorems of calculus
• Analytic and graphical solutions of simple differential equations (if time allows)

It is important to note that Calculus does not create a pathway to AT Calculus II; after completing this course, students may advance to AT Calculus I, Statistics, Software Design with Java, or AT Investment Math.

## Advanced Topics Mathematics: Calculus I

Prerequisite: Precalculus

AT Calculus I offers a rigorous study of differential and integral calculus at a college level. Success in Calculus is highly dependent on strong mechanical skills with algebra and trigonometry. That said, our goal in mastering these skills is to go beyond and recognize deeper patterns and abstractions relating to rates of change. This course is a prerequisite to Advanced Topics Calculus II.

Topics of study will include:

• Limits and continuity
• Limit definition of the derivative
• Derivatives of algebraic and trigonometric functions
• Chain rule, implicit differentiation
• Applications in curve sketching, related rates, and max-min problems
• Continuity and the mean value theorem
• Approximating areas with rectangles or trapezoids
• Integration and Riemann sums
• The fundamental theorems of calculus
• Calculus of circular functions, exponential and logarithmic functions

## Advanced Topics Mathematics: Calculus II

Prerequisite: AT Calculus I (or Honors Calculus Before the 2023-2024 School Year)

What skills, habits of mind, and experiences are needed to be an effective mathematician in the 21st century? How can theory, application and modern technology help us answer this question? In this Advanced Topics Calculus course, we will develop a framework for advanced theoretical understanding and application of calculus, and how to apply calculus in fields of study such as engineering, physics, biology, and economics.

Topics of study will include:

• Volumes of revolution and of a known base
• Improper integrals
• Conic sections and the general second degree equation
• Calculus of parametric, polar, and vector functions
• L’Hôpital’s rule and its application to convergence of improper integrals and sequences
• Integration by parts and partial fractions
• Application of integrals to area, volume, length of curve, and surface area
• Analytic solution of variable separable and logistic differential equations
• Solution of differential equations graphically by slope fields and numerically by Euler’s method
• Infinite series of numbers; tests of convergence
• Power series, Maclaurin and Taylor series with Lagrange remainder

## Statistics

Embedded Honors Option Available

Prerequisite: Precalculus

This course includes three major areas of emphasis: data collection, data description, and data analysis as described below. Students electing to take the embedded honors option will undertake longer homework assignments and need to be able to work independently.

Topics of study will include:

• One-variable statistics: measures of central tendency and variability
• Graphs—histogram, box plot, dotplot, normal quantile plot
• Two-variable statistics—measures of linearity and transformation to linear graphs
• Scatterplot, residual plot
• Surveys
• Comparative experiments
• Probability and probability distributions, including binomial and geometric distributions
• Normal density curves
• Sampling distributions and the Central Limit Theorem
• Hypothesis tests and confidence intervals for means and proportions
• Chi-squared analysis of categorical data
• Inference on slope of a regression line
• Power of a test, Type I and Type II errors

## Advanced Topics Mathematics: Investment Math

Prerequisite: Precalculus

This seminar-style course will begin with an exploration of the broader capital markets and an examination of the fundamental principles of investing (time value of money, efficient market hypothesis, risk vs. return, supply/demand dynamics, market cycles, etc.). The focus will then shift to the technical analysis of single security price data as an ideal application of precalculus and other mathematics. Students will be responsible for analyzing a specific stock over the course of the term using the tools developed in the class. Throughout the course, there will be an emphasis on relating current events to the financial markets. The class will explore the power of TradingView’s software. We will learn about some of the many different functionalities that TradingView offers, and we will apply these functionalities to different price series. We will begin to develop our first strategy by optimizing parameters of basic analysis techniques learned in the fall. We will apply an advanced statistical analysis to review our results. Then, we will learn how to code using PineScript. We will then develop hypotheses about what drives the markets and use our ability to code to write algorithmic trading programs that try to capture gains from these observations. We will backtest our programs and evaluate their performance. We will then learn about how to manage a portfolio through the application of many different non-correlated algorithms. This course is cross-listed with the STEAM X Department.

## Software Design with Java

This course offers an extensive introduction to computer programming and software design using the Java programming language. There are no prerequisites for this course, but some understanding of basic programming structures, such as the coding topics learned in Computer Science Practice and Principles, or the Computer Science elective courses, would be helpful. This course begins with the basic syntax of Java, including variables and types, simple commands, program flow and decision statements, and iterative looping structures. We then proceed to arrays and array lists, interfaces and polymorphism, inheritance hierarchies, recursion, analysis of algorithms, sorting and searching. While we learn the particulars of Java, we focus on more broad-based language and design concepts that apply to different higher languages. Required: a laptop running Windows, Linux, or MacOS. This course is cross-listed with the STEAM X Department.

## Advanced Topics Computer Science: Software Design with Java

Prerequisite: Permission from the teacher

Embedded in Introduction to Software Design with Java, this course allows students with a stronger background in computer science to learn the Java programming language. The AT students in the class will also work independently on an exploration of Theoretical Computer Science, using the textbook Introduction to the Theory of Computation by Michael Sipser. This is a high-level and mathematically challenging exploration of automata, regular expressions, context-free grammars, Turing machines, the halting problem, and the P=NP problem. No specific prior knowledge is required and the required mathematical techniques will be introduced, but some programming experience and mathematical maturity are highly desired. Students who have previously taken the non-AT version of this class will focus more on the Theoretical Computer Science topic. This course will require meeting times in addition to the regularly scheduled blocks. Required: a laptop running Windows, Linux, or MacOS. This course is cross-listed with the STEAM X Department.

## STEAM X Electives (For Math Credit)

In some cases, when a student has completed the three credits of math required in grades 9-12, the math department may recommend that a student take a STEAM X course as their math credit. The math department believes that, in these cases, the problem solving, design thinking, and critical analysis involved in the STEAM courses is a valuable extension of their math study.

• STEAM X: Airfoils and Wind Turbines (Fall)
• STEAM X: Gliders (Winter)
• STEAM X: Laminate Process and Application in Wind Energy (Spring)
• STEAM X: Legacy and Independent Projects (Year Long)