# Mathematics

The Mathematics Department seeks to provide an education that provides all our students with a proficiency in mathematics while simultaneously conveying the beauty and elegance of mathematics. In the seventh grade, this might mean reviewing computational skills one day and investigating geometric transformations using Geometer’s Sketchpad the next. For a 12th-grader, it might mean making the connection between the infinite Taylor Series studied in Advanced Calculus and the repeating decimal numbers they learned about in seventh grade. One of the strengths of the mathematics department at Riverdale is that it is possible to have the same teacher in seventh grade and 12th grade. These teachers are in a position to make connections between topics that might otherwise go unnoticed and unexplored.

Our curriculum is designed so that each student is provided with appropriately challenging course of study. Classes are grouped by ability in order to allow students to meet their potential in mathematics. The most talented and dedicated students have the opportunity to study mathematics in greater depth and at an accelerated pace. It is possible for students who excel in mathematics at the regular level to make the transition to honors level mathematics; this is usually accomplished by completing summer work and with the permission of the Mathematics Department. Our program also allows talented students entering the school with a weak background in mathematics to join the honors program after having mastered the basic skills at an appropriate pace and with permission of the Department.

We place an increased emphasis on problem-solving skills, on reasoning and communication, on establishing connections between the various mathematical disciplines, and on the use of both computer and calculator technology as mathematical tools. To help students master these skills, a structured sequence of courses is offered, from pre-Algebra through Calculus, and beyond. Typically Geometry is taken in ninth grade, followed by Algebra II, Trigonometry/Precalculus and Calculus and/or Statistics. Honors students take similar courses (with a greater emphasis on problem solving) in a more accelerated sequence. Students in honors level mathematics generally take an Advanced Calculus course during their senior year. All Riverdale students are required to complete mathematics through Trigonometry and most students take at least one semester of Calculus. Also offered are a full-year Calculus course for students in the regular level and a coordinated Calculus/Physics class for honors students. Accelerated students may take Advanced Calculus during their junior year and an Advanced Mathematics course (topics vary from year to year) during their senior year. In a typical year, over 50% of the graduating seniors will have completed an advanced course in mathematics by the time they graduate.

## Love of Learning, Love of Community

Mathematics Department co-chair Kevin Bailey describes the interconnection of learning and community at Riverdale.

#### Geometry with Algebra

Geometry with Algebra

This course will include a full treatment of the major content and skills of a regular Geometry class while also incorporating and reviewing key Algebra I skills that are critical for success in future math courses. A particular focus will be placed on understanding geometric shapes and relationships as well as improving a student’s abilities with logic and argumentation, including various methods of proof. The algebraic content to be covered will include: linear equations, factoring, polynomials, rational expressions, and radical functions.

#### â€‹Geometry

Geometry

**Prerequisite: **Algebra I

A course designed to develop an understanding of proof, and to acquaint students with Euclidean geometry and its algebraic representations. The topics include lines, planes, and angles, congruence of triangles, parallel lines and planes, polygonal regions, quadrilaterals similarity, circles and spheres, areas and volumes, and right-triangle trigonometry. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes are to be expected and a major assessment will be given during each unit. Cumulative examinations, given at the end of each semester, will count for a total of 25% of the yearlong grade in the course.

#### â€‹Geometry (Honors)

Geometry (Honors)

**Prerequisites:** A minimum grade of B- in Algebra I (Honors) and/or the recommendation of the department

An honors level course that covers the same topics as Geometry, but is designed to develop a more rigorous understanding of proof, encourage critical thinking and creativity and to connect geometric and algebraic concepts. Additional topics studied include geometric inequalities and coordinate geometry. Geometer's Sketchpad is used throughout the year to enhance the curriculum. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes will be given and a test will be given at the completion of each chapter. Cumulative examinations, given at the end of each semester, will count for a total of 25% of the yearlong grade in the course.

#### Algebra II

Algebra II

**Prerequisite: **Geometry

A course that develops and enhances the skills and concepts of Algebra I and prepares students for a course that develops and enhances the skills and concepts of Algebra I and prepares students for more advanced work in both science and mathematics. The topics include a brief review of certain topics studied in Algebra I followed by: functions and relations, linear functions, systems of linear equations and inequalities, matrices, quadratic functions, exponential and logarithmic functions, polynomial functions, rational algebraic functions, irrational numbers, radical equations and variation, and the conic sections. All functions are explored both algebraically and graphically; the TI-Nspire is used extensively and students are responsible for learning how to use this tool throughout the year. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end a semester, will count for 20% of the semester grade.

#### Algebra II (Honors)

Algebra II (Honors)

**Prerequisites:** A minimum grade of B- in Geometry (Honors) and/or the recommendation of the department

An honors level course that covers the same topics as Algebra II but at a faster pace and in more depth. Students taking this course may use investigative approaches when introduced to new concepts. Supplementary problems will be given throughout the year to encourage critical thinking and creativity and to encourage students to extend their understanding of the material beyond the typical textbook problem. Significant application of real-world usage of course-specific skills, such as linear modeling and maximizing financial and business solutions, will be woven throughout the curriculum. The TI-Nspire calculator will be used extensively. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end of a semester, will count for 20% of the semester grade.

#### Discrete Mathematics, Algebra, and Trigonometry

Discrete Mathematics, Algebra, and Trigonometry

**Prerequisite:** Algebra II

This course focuses on circle and triangle Trigonometry as well as new and previously introduced topics in Algebra, including sequences and series, exponential and logarithmic functions, counting principles and probability. Both the pacing of the course and class size are structured to meet the needs of students who typically need more time to thoroughly assimilate new material and who benefit from more individual attention. Trigonometric functions and solutions to trigonometric equations are examined both algebraically and graphically. Applications of trigonometric functions are explored through topics like latitude, navigation, and surveying, as well as through the graphical modeling of real-world phenomenon. In the second semester, a significant amount of time is devoted to modeling data sets and solving real-world problems using Excel. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end of a semester, will count for 20% of the semester grade.

#### Precalculus with Trigonometry

Precalculus with Trigonometry

**Prerequisites** Successful completion of Algebra II and the recommendation of the department

Precalculus is a course that covers new and previously introduced topics in Algebra at a more complex level, and also provides an introduction to both circle and triangle trigonometry. Trigonometric functions and solutions to trigonometric equations are examined in detail, both algebraically and graphically. Applications of trigonometry to navigation and surveying is studied, as is using sinusoidal functions to model real-world phenomenon. Second semester typically focuses on: series and sequences, limits, counting principles, probability and the binomial theorem, exponential and logarithmic functions, operations with and transformations of functions, and both graphical and algebraic analysis of functions. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end of a semester, will count for 20% of the semester grade.

#### Precalculus with Trigonometry (Honors)

Precalculus with Trigonometry (Honors)

**Prerequisite:** Successful completion of Algebra II (Honors) and/or the recommendation of the department

An honors level course that covers the same topics as Precalculus with Trigonometry but at a faster pace and in more depth. Students taking this course will also study proof by induction, polar coordinates and complex numbers, parametric equations, and, if time allows, a more thorough study of limits and an introduction to continuity. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end of a semester, will count for 20% of the semester grade.

#### Statistics and Probability (fall semester)

Statistics

**Prerequisites: **Algebra II and departmental approval

Statistics is a semester course that covers descriptive and inferential statistics with special attention given to real world applications. The course begins with graphical and numerical descriptions of a sample including histograms and measures of central tendency and spread (mean, standard deviation, etc.). The normal distribution is then studied, followed by two-variable statistics including scatterplots and regression analysis. Sampling and experimental design are delved into. Additional topics may include probability theory, sampling distributions, binomial distributions, confidence, and hypothesis testing. New topics are generally introduced manually and then explored further with graphing calculators, spreadsheets and statistical applications and websites. Homework is assigned at virtually every class meeting and is usually due at the next meeting when it can be discussed in detail. Assessments including quizzes, tests, projects and presentations will occur approximately every two weeks.

#### Statistics: Analysis and Regression (spring semester)

Statistics: Analysis and Regression

Data is everywhere. There is more data being gathered now than in all previous human history. Getting useful information from all this data is vital, so we must learn to aggregate, analyze, and interpret it. In this course, students will examine the different ways in which data is gathered in the world around you, including datafiles, tables, online resources and repositories, and through the use of hardware sensors and recorders. Students will learn how to visualize and analyze data using principles and techniques of computer science, spreadsheet software, and database applications. Course topics include statistical and regression analysis, data visualization techniques, text analysis, filtering, querying, and developing patterns.

#### Introduction to Calculus

Introduction to Calculus

**Prerequisites: **Successful completion of Precalculus with Trigonometry, or a grade of A- or higher in Advanced Algebra with Trigonometry, and the recommendation of the department

Introduction to Calculus is a semester long course designed to prepare students for an introductory college level Calculus course. It is anticipated that the course will cover at least 50% of a typical Calculus I course. The semester begins with a review of functions from both a graphical and algebraic perspective. The focus then shifts to a study of limits and continuity, and is followed by a full treatment of differential calculus. While the formal definition of the derivative is introduced, more time is devoted to calculations and applications of the derivative. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end of the semester, will count for 20% of the semester grade.

#### Exploring Mathematics: Puzzles and Problem Solving

Exploring Mathematics: Puzzles and Problem Solving

What is it that mathematicians really do? What is mathematical thinking? In this course, students will explore basic concepts in the areas of number theory and geometry, extensive areas of study for mathematicians around the globe. Students will look at interesting and unique problems, and begin to pose solutions—from the idea that some infinities may be larger than others, to the question of how many guards are needed to guard an entire art gallery of any shape. Students will also explore concepts such as modular arithmetic, Fibonacci sequence, and symmetry. Students are expected to critically think about and explore ideas that are not usually thought of as "math." This elective is open to any Upper School students who are interested in exploring unique concepts in pure math, and who enjoy puzzles and riddles.

#### Mathematical Approaches to Economics (fall and/or spring semester(s))

Mathematical Approaches to Economics

**Prerequisite:** Successful completion of Precalculus with Trigonometry or Precalculus with Trigonometry and the recommendation of the department

This course, offered in the fall and/or spring semester(s), introduces students to the fundamental definitions and theories of micro- and macroeconomics, including supply and demand, cost versus benefit, profit and loss, efficient markets, market regulation (e.g., taxes, subsidies, price ceilings, minimum wages), capitalist competition and monopoly. While the course requires reading and discussions of economic theories, the goal of the class is to employ graphical and computational skills from Algebra and Precalculus to accurately and objectively quantify the costs and benefits of various financial scenarios and policies.

#### Quantitative Reasoning: Math in the Real World (fall semester)

Mathematical Approaches to Economics

Students often associate math with "solving for x." This course is intended to expand the minds of math students and allow them explore how math is used in writing, journalism, medicine, and art. Students will be able to improve their quantitative and qualitative literacy and become aware of the big impacts math has in the world around us. The class will begin by exploring the math of the MetroCard, then continue onward to formal logic. Students will then discuss the power of units and percentages and finish the course exploring how math in art. Students will be able to improve upon their number sense and learn about how mathematics has been used to create the "Mona Lisa," the tile work in mosques, and how math is shown in nature.

#### Stats in Sports (fall semester)

Stats in Sports

**Prerequisite:** Algebra II

Since the start of the 21st century, unprecedented access to information and the accompanying statistical revolution in sports—a field often known as sabermetrics—has changed the way professional athletic teams are built, how games are played, and how players are evaluated. When watching a sports telecast or reading a game recap, fans are inundated with numbers. Some of these numbers may provide fans a deeper understanding of the sport, while others claim to illuminate while doing exactly the opposite. What information is actually useful, and how much should we trust statistics that don’t match our perception?

In this course, students will be introduced to statistical analysis through the lens of sports. They will learn how data is collected and parsed, and how to reconfigure it for the purposes of various statistical studies. They will study probability theory, and analyze differences between theoretical predictions and empirical results. In-class assessments will ask students to quantitatively compute relevant statistics, while at-home writing assignments will ask students to more qualitatively reflect and write on specific articles or general trends in sports. Computer statistical software will be incorporated where appropriate. As a final project, students will be required to do their own statistical research project, and write a paper that would be fit for publication online.

#### Calculus

Calculus

**Prerequisites:** Successful completion of Precalculus with Trigonometry or Precalculus with Trigonometry (Honors) and the recommendation of the department

This full year course is designed for those students who wish to study a full year of Calculus but choose not to do so at the honors level, or have excelled in both our regular Precalculus course and their other mathematics courses throughout high school. This course requires that students have a solid foundation in algebra and are able to think both creatively and analytically at a high level. Calculus begins with a short review of functions from both a graphical and algebraic perspective. Limits and continuity are studied in depth, which is then followed by a thorough treatment of differential calculus. The limit definition of the derivative is emphasized, as are applications of the derivative such as related rates and optimization. The second semester is devoted to the study of integral Calculus and the important connections between derivatives and integrals. Homework is assigned daily and completion of it is essential for success in this course. Announced and unannounced quizzes will be given and a major assessment will be given during each unit. An examination, if given at the end of a semester, will count for 20% of the semester grade.

#### Calculus A (Honors)

Calculus A (Honors)

**Prerequisite:** Recommendation of the department, with the following expectations: a record of consistent effort and application in previous mathematics courses and a grade of B- or better in Precalculus with Trigonometry (Honors)

Calculus A is a college level course in differential and integral calculus. This course aims to cover all the topics recommended by the College Board for Calculus AB and some additional topics that might be covered in the first semester of a college calculus course. Students should have excellent algebra skills and the discipline to keep up with the demanding workload. Homework is assigned daily and typically accounts for about 10% of the grade. The remaining 90% of the grade is determined by quiz and test grades, except in the fourth quarter when a research project completed after the AP Examination accounts for approximately 25% of the quarter grade. Quizzes and tests routinely cover material learned in previous courses. Students may choose to take the Calculus AB AP Examination, although some additional work may be necessary outside of class in order to fully prepare for the examination.

#### Physics/Calculus (Honors)

Physics/Calculus (Honors)

**Prerequisite:** Departmental approval, with the following expectations: a record of consistent effort and application in previous mathematics courses and a grade of B- or better in Precalculus (Honors)

Advanced Calculus A & Advanced Physics is a combined math/science course that is team-taught and meets twice a day. Calculus and Physics share a common origin and given that so much of what is covered in each class is mirrored in the other, teaching the courses in an integrated manner is an effective strategy to promote deep student understanding. The combination of teaching Calculus and Physics together allows the students to explore concepts like differential equations in a more tangible setting and at a higher level than is commonly covered. Coordinating the courses allows students to use the concepts of physics to reinforce and provide context for the calculus that they study concurrently. This course combines Advanced Calculus A and Advanced Physics course and prepares students for the AP exam in calculus, although not for physics. The course offers students a significant lab experience with more involved (and independent) lab work.

#### Calculus B (Honors)

Calculus B (Honors)

**Prerequisite: **Recommendation of the department, with the following expectations: a record of consistent effort and application in previous mathematics courses, and a grade of B+ or better in Precalculus (Honors)

A college level course in differential and integral calculus. This course covers all the topics recommended by the College Board for Calculus BC and some additional topics that might be covered in the first year of a college calculus course. This course covers more material at a faster pace than the Calculus AB course. Students should have excellent algebra skills and the discipline to keep up with the demanding workload. Homework is assigned daily and typically counts for about 10% of the grade. The remaining 90% of the grade is determined by quiz and test grades except in the fourth quarter when a research project completed after the AP Examination accounts for approximately 25% of the quarter grade. Quizzes and tests routinely cover material learned in previous courses. Students may choose to take the Calculus BC AP Examination, although some additional work may be necessary outside of class in order to fully prepare for the examination.

#### Advanced Mathematics (Honors) (fall and/or spring semester)

Advanced Mathematics (Honors)

This course, offered in the fall and/or spring semester(s), is designed for our students who complete either Calculus A or Calculus B by the end of junior year, or students currently enrolled in Calculus with a particular interest in higher level mathematics. Students are required to assimilate the concepts they have covered in prior math courses with new, college-level disciplines. While the specific content covered varies from year to year based on staffing and student interest, the course is taught at an undergraduate level and aims to introduce students to the coursework they will encounter as college math students. Specific disciplines have included: Multi-variable Calculus, Continuous Probability & Statistics (with Calculus), Mathematical Argumentation, Graph Theory, Set Theory, Abstract Algebra and more.