# Mathematics

The goal of the mathematics department at Trinity-Pawling is to educate students in the fundamental skills necessary for the study of mathematics, the physical and social sciences, and any subject that requires the use of logic, sequential reasoning, abstract thought, and problem solving. Every student must successfully complete three years of study in mathematics. This may be accomplished through one of the course sequences outlined below. In all courses, students are required to use graphing calculators and other computer technology in their study of mathematics.

## Math 7 • Middle School

This 7th grade math course spends the fall term reviewing basic math concepts including the order of operations, solving first order equations, fractions, decimals, percents, and proportions. The course then moves into an elementary Algebra 1 curriculum, covering topics including the rules of exponents, polynomials, factoring of polynomials, linear equations, and solving systems of linear equations. 7th grade students exit this course prepared for Math 8, an algebra-focused course.

## Math 8 • Middle School

Math 8 is an Algebra 1 course for 8th grade students. Students begin with a solid review of arithmetic concepts, including fractions, negative numbers, and the meaning of a variable in an expression or equation. Students learn how to reason and manipulate symbolically. They solve equations with one variable to the first power to obtain a single answer, and then equations with multiple variables are solved for one variable in terms of the others. The students also learn functional notation and the meaning of a function. The students study lines — they learn the meaning of slope and x and y-intercepts, and they develop solution sets by graphical and algebraic approaches. They learn to solve systems of linear equations by graphing, substitution, and elimination. The students also solve quadratic equations by graphing and by factoring.  Students exit this course prepared to take Geometry or Honors Geometry in the Upper School.

## Algebra 1

Algebra 1 starts with a solid review of arithmetic concepts, including fractions, negative numbers, and the meaning of a variable in an expression or an equation. Students learn how to reason and manipulate symbolically. They solve equations with one variable to the first power to obtain a single answer, and then equations with multiple variables are solved for one variable in terms of the others. The students also learn functional notation and the meaning of a function. The students study lines — they learn the meaning of slope and x and y-intercepts, and they develop solution sets by graphical and algebraic approaches. They learn to solve systems of linear equations by graphing, substitution, and elimination. The students also solve quadratic equations by graphing and by factoring. Students exit this course prepared to take Geometry or Honors Geometry.

## Geometry

In Geometry, emphasis is placed on the deductive nature of the subject, as well as on the use of algebra to solve various types of geometry problems. Concepts covered include: Reasoning and Proof, lines and angles, triangles and quadrilaterals, similarity, the Pythagorean Theorem, Right Triangle Trigonometry, circles, and surface area and volume. Traditional instruction along with project-based learning allow students to take geometry concepts and apply them in the real world.

## Geometry Honors

In Geometry Honors, as in Geometry, emphasis is placed on the deductive nature of the subject, as well as on the use of algebra to solve various types of geometry problems. The differences between Geometry Honors and Geometry are mainly ones of pace and the amount and difficulty of the algebra used to solve the geometry problems. Concepts covered include: Reasoning and Proof, lines and angles, triangles and quadrilaterals, similarity, the Pythagorean Theorem, Right Triangle Trigonometry, circles, and surface area and volume. Traditional instruction along with project-based learning allow students to take geometry concepts and apply them in the real world. Students leaving Geometry Honors generally take Algebra 2 next, or, with the approval of the instructor and the Mathematics Department, Algebra 2 Honors.

## Algebra 2

Algebra 2 is designed for students who have already taken one year of Algebra. Students explore the basics of problem-solving, functional notation, fractions, and most importantly, factoring. The course then covers solving functions that are quadratic, exponential, and rational. We also cover polynomials, inequalities, systems of equations, and the basics of graphing. Students exit the course prepared for Precalculus or Functions, Analysis, and Trigonometry.

## Algebra 2 Honors

This course is an accelerated study, designed for the students to take an in-depth look at functions and problem-solving. They deal with the full family of functions, starting with linear and working through quadratic, exponential, logarithmic, rational, radical, and polynomial functions — both algebraically and graphically. Students also solve systems of equations and inequalities before ending the year with sequences and series. Students exit the course prepared for Honors Advanced Precalculus.

## Functions, Analysis, and Trigonometry

This course is an intermediate course, between Algebra 2 and Precalculus. The class begins with prerequisites, which includes a review of real numbers, exponents, radicals, polynomials, factoring, and rational expressions. The following topics are covered during the year: graphs, linear equations, quadratic equations, complex numbers, functions, polynomial functions, synthetic division, rational functions, exponentials, exponential functions, and an introduction to trigonometry. Problem-solving and thinking in different ways to solve problems is one of the goals for each student. Another goal of the course is to make sure each student has mastered these topics so that he can be successful in the next math course.

## Precalculus

The purpose of this course is to introduce students to the full family of functions algebraically, graphically, and numerically. Students explore functions that are linear, quadratic, exponential, rational, logarithmic, and trigonometric algebraically before discussing their graphs and how they could be translated. They solve polynomial functions of higher degrees and discuss their end behavior. Students exit this course feeling comfortable with any function they may encounter in an introductory Calculus course.

This is an accelerated course designed to prepare students both to take AP Calculus AB and the Math SAT II course, if they so choose. Students explore the full family of functions algebraically, graphically, and numerically to prepare them for an introductory Calculus course the following year. Students explore functions that are linear, quadratic, polynomial, exponential, rational, logarithmic, and trigonometric algebraically before discussing their graphs and how they could be translated.  Students then learn how to utilize trigonometric identities to solve equations before closing the year with sequences, series, and probability.

## Statistics

This class prepares students for upper level statistic courses that they may encounter at the college level. The focus of this class is the study of collecting, displaying, and analyzing data. Much of the learning is hands-on and project-based. Students expand their statistical knowledge through the study of material that is interesting and meaningful to them personally. This course is open to students in grades 11 and 12 with the recommendation of a mathematics teacher and the Mathematics Department Chair.

## Honors Calculus

Honors Calculus begins with an extensive review of the Precalculus skills fundamental to success in Calculus. Students study limits and the definition of the derivative. They learn the essence of differential calculus by learning most of the Leibniz rules of derivatives, including the product, quotient, and chain rules. They study these essential topics in order to mathematically model changing systems, obtain derivatives of implicitly defined relations, and solve related rates problems. Finally, as the last step in differential calculus, students put all the derivative skills to use to determine how to optimize functions by finding maxima and minima.  These topics are applied in real-world applications, including business models.

In the second half of the course, students begin Integration by investigating infinite limits of Reimann Sums and the First Fundamental Theorem of Calculus. They study basic integration techniques, including U-substitution, and use the techniques to work out applications requiring integration. Upon completing Honors Calculus, students have the equivalent of one semester of college-level calculus. Underclass students completing Honors Calculus have the option to take Statistics or, with Mathematics Department permission,  AP Statistics or AP Calculus AB.

## AP Calculus AB

Advanced Placement Calculus AB begins with an extensive review of the Precalculus skills fundamental to success in Calculus. Students then study limits and the definition of the derivative. They learn the essence of differential calculus by learning all of the Leibniz rules of derivatives, including the product, quotient, and chain rules. They study these essential topics in order to mathematically model changing systems, optimize functions, obtain derivatives of implicitly defined relations, and solve related rates problems. In the second half of the course, students begin integration by investigating infinite limits of Reimann Sums and the First Fundamental Theorem of Calculus. They study basic integration techniques, including U-substitution, and apply these techniques to various applications, including areas between curves, volumes by revolution, volumes with a constant cross section, and the average value of a function. The students move into the final topics of the course — solving separable differential equations and an advanced limit technique called L’Hospital’s Rule. Students finish the course with a comprehensive review of differential and integral calculus, for the purpose of preparing for the AP Exam. Students exit AP Calculus AB and move into AP Calculus BC or AP Statistics.

## AP Calculus BC

This course is intended for students who have already taken AP Calculus AB. Students follow the AP Calculus BC curriculum carefully, although the year begins with a study of advanced integration techniques. Students then move on to cover parametric and polar calculus before discussing separable equations and convergent and divergent series. This approach gives the class more time to highlight some of the more difficult topics highlighted in this course and puts the students in a good position to fully comprehend the big ideas discussed within the AP test at the end of the year. After AP Calculus BC, students have the option to either take AP Statistics or Multivariable Calculus.

## Multivariable Calculus

This is an advanced course reserved for students who have completed all of our AP mathematical offerings. The goal of this course is to get students comfortable working with functions of multiple variables. Students begin with an introduction to vectors and graphing in three dimensions. Next, they will learn the calculus component of the course, specifically partial derivatives, directional derivatives, gradients, optimization, Lagrange multipliers, double and triple integrals, vector fields, line integrals, flux integrals, and Green’s, Gauss’s, and Stokes' Theorems. Students exit this course prepared for Linear Algebra.

## AP Statistics

AP Statistics is designed to be an interactive, thought-provoking course that allows students to construct their own understanding of concepts and techniques of statistics. The goal of the course is to teach our students to think carefully about collecting and analyzing data. As such, examples, assignments, and hands-on projects are always tied to the real world. This is a college-level course that prepares students to complete the AP Statistics Exam for college credit. The class is open to students in grades 11 and 12 with recommendations from a mathematics teacher and the Mathematics Department.

My first goal for all Trinity-Pawling students is for them to be comfortable with numbers. The days of being able to avoid numbers are over. I am not saying that every boy will be an engineer or scientist; however, they will all be required to evaluate, think critically about, and make decisions from numbers and/or numerical models.

Dr. Glenn Mandigo,
Math Department Chair

## Meet the Mathematics Department

Math Teacher

### Brian Foster ’79

Athletic Director, Math Teacher

### Nick LaFontaine

Assistant Director of College Counseling, Math Teacher

### Dr. Glenn Mandigo

Mathematics Department Chair

### Jeff Mandigo

Mathematics Teacher

### Chris Murphy '13

Service Program Director, Mathematics and Science Teacher

### Taylor Pirie

Mathematics and Economics Teacher

### Patrick Ruwe II

Mathematics Teacher