Preparing Boys for Life

Curriculum

Haverford students are required to complete at least three years of mathematics; most students take four years of math and many add a math elective. Below you will find descriptions of the advanced (denoted by an *) and standard course offerings in each level of a traditional mathematics journey from algebra to calculus, as well as math electives for those students who wish to pursue additional topics in mathematics.

Core Courses

Algebra I

About Algebra I and Geometry:
Algebra is important as a modeling and problem solving tool, and bridges the gap from computational mathematics to abstract understanding. Geometry introduces the spatial relationships that exist in two and three dimensions. The concepts learned in these introductory courses are used by each of us every day and form the foundation upon which subsequent math courses build.

Algebra I is an introductory course designed for incoming Third Formers who have had little or no algebra or who need a thorough review of basic algebra. After a review of arithmetic operations, the first semester focuses on the basic concepts of algebra: using variables to represent numbers, evaluating formulas, solving algebraic equations, and the graphing of linear equations and basic transformations. The second semester looks at systems of linear equations, functional notation, quadratic equations and rational expressions. Use of the graphing calculator will be developed as an aid to solving systems of equations and quadratic equations.

Geometry

About Algebra I and Geometry:
Algebra is important as a modeling and problem solving tool, and bridges the gap from computational mathematics to abstract understanding. Geometry introduces the spatial relationships that exist in two and three dimensions. The concepts learned in these introductory courses are used by each of us every day and form the foundation upon which subsequent math courses build.

This standard course provides a comprehensive introduction to Euclidean geometry. Topics covered include foundations of geometry, deductive reasoning and proof, transformations, coordinate geometry, congruence and similarity, polygons, circles, area, and volume. The advanced geometry course includes a rigorous treatment of mathematical proof, and students will be expected to justify the major theorems of the course. The students will also be expected to connect concepts and the most successful students will solve problems creatively.

Algebra II

About Algebra II:
The Haverford School offers two levels of Algebra II - honors and standard. The goal of each is to expand and deepen your existing knowledge of Algebra I and Geometry; both courses emphasize the computational and theoretical components of the subject matter. Successful completion of these courses will satisfy the Common Core requirements for Algebra (as set by the Pennsylvania Department of Education) and will prepare students to tackle more advanced coursework in the future.

The standard level course is a comprehensive curriculum with particular emphasis on the practical/computational components of the subject and on the use of functions as mathematical models for solving real-world problems. The advanced course covers all of the topics in the standard class, but in a much more rigorous fashion. This course delves much deeper into the theory underpinning the basics and considers a wider range of topics. The curriculum reaches well beyond the Common Core requirements and prepares the students to tackle PreCalculus at the advanced level the following year.

Functions, Statistics and Trigonometry

This course provides preparation for the study of Precalculus, geared toward those students needing further review of advanced algebra concepts. First semester topics include statistics, mathematical modeling, transformations, and the Unit Circle. During the second semester, elementary functions are emphasized along with probability, and trigonometry. Real-world models are developed throughout.

Precalculus

About Precalculus:
Precalculus builds on the concepts from Algebra and Geometry to create the foundation for the study of Calculus and is offered in standard and honors levels. This challenging course includes an examination of many types of functions including trigonometric, exponential, logarithmic, rational, quadratic, and higher degree polynomials. Students will be challenged to examine mathematics graphically, analytically, verbally and numerically. The use of the graphing calculator will be required in this course, and students will be expected to know the five basic graphical functions: minimum, maximum, value, zero, and intersection.

Standard-level Precalculus provides a comprehensive preparation for the study of Calculus at Haverford or an Introductory Calculus course in college. Polynomial, exponential, and logarithmic functions are emphasized in the first semester and trigonometry, sequences and series, and probability are the focus of the second semester. Real-world models are developed throughout. The advanced course covers all of the topics in standard course, as well as conic sections, parametric equations, polar coordinates, vectors, and the complex plane. Connections with the sciences, economics and other real world applications are developed throughout. This course will also develop the student’s skills in the use of the graphing calculator, in all of its modes.

Calculus

About Calculus:
Inspired by problems in astronomy, Newton and Leibniz developed the ideas of calculus more than 300 years ago. Since then, each century has demonstrated the power of calculus to illuminate questions in mathematics, the physical sciences, engineering, and the social and biological sciences. Calculus is an extraordinarily powerful tool when reducing complicated problems to manageable procedures.

The standard course begins with a brief review of logarithmic, exponential and trigonometric functions. After exploring the ideas of limits and continuity, the course will focus on the two major concepts of Differential and Integral Calculus. Students will learn methods for taking derivatives and antiderivatives and use these methods in various applications. Although not as theoretical as the advanced class, a strong working knowledge of previous courses, the ability to work independently, and a desire to learn high-level mathematics are required. The advanced level course is a thorough and challenging analysis of limits, derivatives, and Riemann integration. In addition to numerous applications, this course includes a theoretical component and advanced methods of differentiation and integration that will not be covered in Standard Calculus. This course will prepare students to take Calculus II* at THS or move into a more theoretical Calculus course in college, such as required for Mathematics, Engineering or applied science majors.

Calculus II*

This extremely rigorous and challenging course is an extension and development of the topics studied in Calculus I*. Advanced topics covered will include techniques of integration, special methods for finding limits, the application of calculus to polar, vector and parametric functions, infinite series (including Maclaurin and Taylor series, and tests for convergence), and applications and solutions of differential equations in physics, engineering, and biology.

Elective Courses

Statistics

About Statistics:
In a society inundated with information, the ability to analyze and interpret data is an invaluable tool. Statistics provides the opportunity for students of the subject to learn how to make good decisions with data. Statistics permeates every branch of the natural and social sciences, and the ability to infer from statistical analysis is crucial in business, economics, political science and medicine. It is very likely that you will be required to take a Statistics course in college and then use it in your career.

The standard level of statistics is intended to provide students a framework to think about the world “statistically.” Real-world problems will be solved using 21st century methodologies, i.e. by incorporating useful technologies and working collaboratively; the process will be project-based, highly interactive, and engaging. The advanced class is a comprehensive survey of the foundations of probability theory and statistical methods for collecting, organizing, displaying, analyzing and drawing conclusions from data. Emphasis is placed on clear and accurate reporting of the results obtained from these activities. Students, having successfully completed Statistics*, may successfully sit for the AP Examination in the spring. Technology will be used extensively for solving problems contemplated in the course. Students may take this course concurrently with Calculus, Calculus I* or Calculus II*.

Finance

About Finance and Economics:
Making sound fiscal and monetary decisions is an essential life skill, yet most people acquire it only with age and through a process of trial and error. Studying Finance and Economics will equip students with powerful mathematical and decision-making skills to help them take control of and proactively map their lives in an uncertain world. Clear financial and economic thinking will yield profound benefits for students of the subjects, as well as for society-at-large.

Finance: Theory of Interest:
This course explores the theories and applications of both simple and compound interest. We will learn the basics of general annuities and perpetuities; amortization tables and sinking funds will be developed; bonds and equity instruments will be compared and contrasted; and capital budgeting will be discussed. A major goal of the course will be to teach students effective problem-solving techniques using real-world monetary transactions. Technological solutions to all of the problems contemplated will be emphasized.

Finance: Portfolio Analysis (spring semester)
The main objective is to provide students with a sound understanding of the concepts and practices associated with making sound investments. We will contemplate topics such as: financial instruments, the markets and related indices, risk and return vs. pricing theory, performance evaluation, and efficient diversification. A wide variety of securities will be discussed. Among them are: common stocks, bonds, mutual funds, real estate, options and tax-advantaged investments. The capstone project for the course will be the design, construction, and management of a hypothetical portfolio by the students.

Economics

About Finance and Economics:
Making sound fiscal and monetary decisions is an essential life skill, yet most people acquire it only with age and through a process of trial and error. Studying Finance and Economics will equip students with powerful mathematical and decision-making skills to help them take control of and proactively map their lives in an uncertain world. Clear financial and economic thinking will yield profound benefits for students of the subjects, as well as for society-at-large.

Economics: Macro* (fall semester):
This conceptually challenging VI Form elective covers the main ideas of macroeconomics, the study of the large-scale structure of the national and world economy. The mathematical level is comparable to that of an introductory college class in macroeconomics. Topics include national income accounting (GDP), economic growth, unemployment and inflation, the financial sector, money and banking, aggregate supply and demand, and fiscal and monetary policy. Prerequisite: Students must be enrolled in or have completed a Calculus course.

Economics: Micro* (spring semester):
This mathematically demanding VI Form elective covers the main ideas of microeconomics, the study of the decision-making processes of consumers and producers in a market economy. The mathematical level is comparable to that of an introductory college class in microeconomics. Topics include market equilibrium, elasticity, taxes and price controls, international trade, consumer and producer decisions, competition and monopoly, and externalities, such as pollution and global climate change. Prerequisite: Students must be enrolled in or have completed a Calculus course.

Software Programming

Software Programming I:
The course focuses on the software development life cycle through exploring and utilizing the Python programming language. Students will explore and determine the scope of a project, design a solution, program the application, test and debug. It is expected that students will master algorithm building and develop their understanding of the object oriented nature of Python.

Software Programming II (spring semester):
Here students will learn to identify interesting problems that can be addressed with software. They will explore Python's usage of classes and objects and use them in programs they build. Working as a collective is an essential skill for future programmers and students taking this course will learn to develop this skill and practice it regularly.

Advanced Topics in Mathematics

In this semester course we will explore the foundations of linear algebra and multivariable calculus, interweaving topics drawn from both fields. We will begin with a study of three-dimensional vectors, continue on to matrices, determinants and linear systems, and conclude with a treatment of vector spaces, linear transformations and functions of several variables. Applications will be drawn from physics, economics and computer science.