# Math

Too often, mathematics is perceived as a discipline that one is either good at or bad at; you are either a “math person” or “not a math person.” Through a self-designed integrated program, the Pingree Math Department emphasizes multiple perspectives, giving each student a voice and an avenue through which to participate. Our curriculum pushes students to go beyond the vocabularies of algebra, geometry, trigonometry, statistics, and calculus to appreciate overarching skills and develop flexible problem solving strategies. Teachers use their expertise to guide student-led inquiry and to foster a collaborative environment that stresses the importance of patience, flexibility, and selflessness.

Through the use of technology and web-based resources—Mimio Boards, TI graphing calculators, laptops/iPads, GeoGebra, Google, Schoology, Pencilcode, and in-house tutorial videos—learning happens effectively and efficiently, affording the opportunity to explore more sophisticated, multi-skill problems. To further inspire and challenge motivated students, the Pingree Math department offers electives in the art of mathematics, financial literacy, and probability, as well as honors-level classes and AP courses in Calculus AB, Calculus BC, and Statistics. Pingree also fields a successful and popular math team.

## Sample Courses

- Math I: Methods in Algebra, Geometry, and Probability
- Math II: Methods in Algebra, Trigonometry, and Numerical Analysis
- Math III: Methods in Advanced Algebra, Geometry, and Proof
- Pre-Calculus
- Honors Introduction to Calculus
- Calculus
- AP Calculus AB
- AP Calculus BC
- Introductory Statistics
- Inferential Statistics
- Risk and Reward
- AP Statistics
- The Art of Mathematics
- Financial Literacy

## Math I: Methods in Algebra, Geometry, and Probability

Math 1 aims to give students deep understanding of, and confidence in, fundamental mathematical practices that will carry them through the rest of their mathematical coursework, and lives beyond school. The practical aspects of math topics (such as order of operations, calculating, using graphing utilities and computer applications), as well as theoretical concepts fundamental to algebra (such as coefficients, exponents, operations and inverses), are all covered in the first half of the year. Much of this is done through units on linear and quadratic equations and their graphs. The second semester explores data analysis and the many ways in which statistics can be applied and represented (box and whisker, stem and leaf, normal curve, etc.) to inform our daily lives. In addition, probability and counting, and an introduction to the essential concepts of geometry and basic right triangle trigonometry are explored. Extensive use of the web, graphing calculators, GeoGebra, tutorial videos, and a Math 1 Course Reader provide students with multiple approaches to the topics.

## Math II: Methods in Algebra, Trigonometry, and Numerical Analysis

Math 2 introduces students to the world of functions; students learn how to recognize functions graphically, numerically, and algebraically, and they become fluent in the language of function notation. Students delve deeply into the study of four function families–linear, quadratic, exponential, and radical; and an introduction to how algebraic transformations affect the graphs, domain and range of each family is also presented. In addition, Math 2 students begin their study of linear systems, matrices, conceptual probability and statistics, and trigonometry, with particular focus on right triangles, similarity, and the geometric mean. Students are regularly asked to investigate concepts collaboratively, and developing confident problem solvers, who are excited about taking risks, is a goal of the course. To further develop problem-solving skills and to encourage students to approach challenges with a patient and flexible attitude, students will take part in a 2-week computer programming “mini-term” in January. The introduction to computer programming is also intended to introduce students to the field of programming in the hopes that they may elect to take a full-credit programming course during their time at Pingree. To master the Math 2 syllabus, students will make use of multiple resources, including a Math 2 Course Reader, tutorial videos, the web, graphing calculators, and GeoGebra.

## Math III: Methods in Advanced Algebra, Geometry, and Proof

Math 3 builds upon the topics covered in both Math 1 and Math 2 and also introduces students to new material not covered in the previous courses. The year begins with inductive and geometric proof, centering upon the study of parallel lines and polygons, and extends to coordinate proofs and proofs by contradiction. Students then move onto trigonometry, reviewing right-triangle trigonometry before delving into the Law of Sines, the Law of Cosines, and polar coordinates. Further exploration of topics in statistics are incorporated into the course, as well as an essential refresher unit on functions and function notation through the review of quadratic, rational, and exponential functions and an introduction to logarithmic functions. To adequately prepare students for Pre-calculus, the unit circle is presented with particular attention paid to measuring angles in both degrees and radians. The year also includes units on sequences and series and transformational geometry, including an introduction to graphing both the sine and cosine curves by hand. To master the Math 3 syllabus, students will make use of multiple resources, including a Math 3 Course Reader, tutorial videos, the web, graphing calculators, and GeoGebra. Also offered: Honors Math III.

Beyond the Math 3 topics, the Honors Math 3 syllabus presents students with a unit on linear programming and introductions to limits and trigonometric identities.

## Pre-Calculus

The first semester of Pre-Calculus is dedicated to a complete study of trigonometry, which requires students to synthesize and apply the algebraic, graphical, and numerical skills developed in Math 1 – Math 3. The second semester offers an in depth study of functions (polynomials, exponential, logarithmic, rational, and radical). Additional topics covered in the course include complex numbers, polar coordinates, vectors and limits. When time permits, the course extends the study of limits to incorporate an introduction to the derivative. Also offered: Honors Pre-Calculus.

## Honors Introduction to Calculus

*Honors Introduction to Calculus *is designed to lay the groundwork for the study of AP Calculus at the AB or BC level. Students will be introduced to the fundamental concept of the derivative as a limit, including its definition, graphical representation, and application to the physical sciences. Students will also learn the basic rules of differentiation, including the power, product, quotient, and chain rules. (Trimester 3 only)

## Calculus

This course is a study of the concepts and skills of differential calculus, which deals with rates of change, and integral calculus, which deals with accumulation. In both branches, concepts learned in algebra and geometry are extended using the idea of limits, and The Fundamental Theorem of Calculus is presented in detail to bring the two branches together. Applications drawn from the physical, biological, and social sciences are investigated, and significant use of the graphing calculator is incorporated into the course to enhance learning and efficiency.

## AP Calculus AB

Designed for the most motivated and proficient mathematics students, this course prepares students for the successful completion of the Advanced Placement AB Examination in the spring. The AP course must move at a significantly faster pace than the non-AP calculus class to ensure that students have adequate time to merge and apply the skills of differential and integral calculus to more complex problems. Take-home problem sets throughout the year will require students to independently synthesize and apply skills developed throughout their high school careers to challenging AP Calculus questions.

## AP Calculus BC

## Introductory Statistics

This course will provide students with a foundation that will be useful long after leaving Pingree, whether they go on to study math, biology, economics, psychology, history, business, law or medicine. The course introduces students to the major concepts and tools for collecting, analyzing and drawing conclusions from data. Topics include experimental design, sampling, graphing, summarizing data, and linear regression. First priority is given to seniors, but if space permits, juniors may enroll in this course with departmental recommendation. Furthermore, the course can be taken in addition to Pre-calculus or Calculus.

## Inferential Statistics

This course continues to build upon the strong foundation started in Introductory Statistics. The course focuses on statistical inference, confidence intervals, and tests of significance. Topics include probability, random variables, probability distributions, hypothesis testing, and treatment testing. First priority is given to seniors, but if space permits, juniors may enroll in the course with departmental recommendation. Furthermore, the course can be taken in addition to Pre-calculus or Calculus. Prerequisite: Introductory Statistics.

## Risk and Reward

A well-known bumper sticker reads: "The Lottery - a tax on people who are bad at math." This course will convince students to steer clear of lotteries and casinos! It will focuson the topic of probability with an emphasis on the risks involved in games of chance. Some or all of the following topics may be covered: Definition of Probability, Odds and Random Chance, Erroneous Beliefs, Random Events, Patterns, Electronic and Online Gambling, Lotteries, The Gambler, and Sports Betting. There will be opportunities for students to branch out and investigate further topics, with instructor permission. Though a graphing calculator is required and the material will center around the mathematical components of probability, there will also be essential reading and writing components to this course. (Trimester 3 Only)

## AP Statistics

This full-year course is in preparation for the AP Statistics exam. The course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. First semester topics include interpreting, summarizing, comparing, and exploring distributions of data and methods of data collection, such as surveys, experimental, and observational study. Second semester topics explore statistical inference – confidence intervals, and tests of significance. Course supplements include newspaper and magazine articles. The course is open to seniors and can be taken in addition to Pre-calculus or Calculus.

## The Art of Mathematics

Since the first time one of our human ancestors used a piece of string to scratch a circular decoration, the concepts, techniques, and applications of mathematics have helped inform, influence, and create art. The *Art of Mathematics*will explore the many ways in which mathematics -- the “language of pattern”-- has led artists to express their aesthetic vision. The course has no prerequisites, and will require only an open mind and the patience needed to learn a few new perspectives -- literally. The following topics may be covered: symmetry, harmony, fractals, line design, mandalas, one and two perspective, op art, origami, tessellations, Islamic art -- and other culturally specific applications of mathematics in art. Also, the works of various innovative artists will be discussed -- from Brunelleschi and DaVinci to M.C. Escher and Erik Demaine.

## Financial Literacy

This course will expose students to the fundamental principles of personal finance. Students will learn about credit and debt, saving and investing, money management, risk management, income and careers, and taxes. The goal is to equip students to make responsible, informed financial decisions now and in their future.