Mathematics

The Mathematics Department at Northern Highlands is dedicated to cultivating strong problem-solving, analytical, and critical thinking skills in our students. Through a comprehensive curriculum that spans from foundational algebra and geometry to advanced courses like calculus, statistics, and computer science, we prepare students for success in both academic and real-world applications. Our program emphasizes deep conceptual understanding, the practical use of mathematical tools, and the development of logical reasoning to solve complex problems. By fostering a love of mathematics and encouraging collaboration, creativity, and perseverance, we aim to empower students to confidently tackle challenges in STEM fields and beyond. Additionally, we focus on building resilience and adaptability, preparing students for a rapidly changing world where quantitative literacy is increasingly essential. Our mission is to create lifelong learners who appreciate the beauty and utility of mathematics in everyday life and future endeavors.

Typical Sequence for Mathematics

Algebra I   Grades 9-12 

In this course, students build deep conceptual understanding around the topics of equivalence, solving equations, systems, and inequalities, and exploring the connections and differences between linear, quadratic, and exponential functions. Students begin the year by generalizing patterns, studying both arithmetic and geometric sequences. This sets the foundation for exploring linear relationships, and later on, both linear and nonlinear functions. Throughout each unit, there is an emphasis on real-world applications of skills and topics infused throughout the course, which serves to bridge the gap between concrete ideas of arithmetic and abstract ideas for higher mathematics. Approved for NCAA DI and DII athletic eligibility 

Geometry     Grades 9-12 

This course expands on first year algebra skills and introduces students to further foundational skills needed for future coursework on Algebra II/Trigonometry and Pre-Calculus. This course includes an in-depth study of Euclidean Geometry with an emphasis on the following: coordinate geometry, parallel and perpendicular lines, angles, transformations, triangles, reasoning and proof, polygons and quadrilaterals, circles, area of plane figures, lateral and surface area of solids, and volume of solids. This course is designed to allow students to use mathematics as a tool for problem-solving and make further preparations for solving real world applications. 

Prerequisites: Successful completion of Algebra I. 

Prerequisite for incoming freshmen: Multiple criteria will be used as determined and reviewed by the student’s middle school principal. 

Prerequisites for sophomores who wish to double up in sophomore year, taking both Geometry and Algebra II/ Trigonometry: Minimum grade of “95” on assessments or better in Algebra I and teacher recommendation. Approved for NCAA DI and DII athletic eligibility 

Honors Geometry     Grades 9-10 

The subject of this course is the development of Euclidean Geometry with an emphasis on logical structure using inductive and deductive reasoning. Topics include parallel lines, congruent triangles, quadrilaterals, inequalities, similar polygons, right triangles, circles, areas of plane figures, geometric probability, and areas and volumes of solids. Units in coordinate geometry and transformations will be introduced and infused throughout the curriculum, as well. Although direct and indirect proofs will be written, logical reasoning and applications in real world situations will also be emphasized. 

Prerequisites: Minimum grade of “95” on assessments or better in Algebra I and teacher recommendation. 

Prerequisite for incoming freshmen: Multiple criteria will be used as determined and reviewed by the student’s middle school principal. 

Approved for NCAA DI and DII athletic eligibility 

Honors Math Analysis     Grade 9 

This accelerated course presents topics covered in Honors Geometry and Honors Algebra II/Trigonometry. Students will study logic, deductive reasoning, parallel lines, congruent triangles, quadrilaterals, inequalities, similar polygons, right triangles, circles, constructions, coordinate geometry, area and volume. In addition, students will have an in-depth study of functions: general, linear, quadratic, piecewise, polynomial, and rational. Proofs and derivatives of formulas will be incorporated when appropriate. 

Prerequisite for incoming freshmen: Multiple criteria will be used as determined and reviewed by the math supervisor. 

Approved for NCAA DI and DII athletic eligibility 

Algebra II                                                                           Grades 11-12

This course reviews first year algebra skills and introduces students to further foundational skills needed for future coursework. This course includes an in-depth algebraic and graphical approach to general functions and transformations, linear functions, quadratic functions, and polynomial functions. The course is designed to allow students to use mathematics as a tool for problem-solving and make further preparations for solving real world applications.

Prerequisites: Successful completion of Algebra I and Geometry.

Approved for NCAA DI and DII athletic eligibility 

Algebra II/Trigonometry                                              Grades 10-11

This course expands on first year algebra skills and introduces students to further foundational skills needed for future coursework in Pre-Calculus and beyond. This course includes an in-depth study of the following: complex numbers, as well as algebraic and graphical approach to linear functions, quadratic functions, and polynomial functions; trigonometry of right and non-right triangles is also explored. This course is designed to allow students to use mathematics as a tool for problem-solving and make further preparations for solving real world applications.

Prerequisite: Minimum grade of a “80” in both Algebra I and Geometry and teacher recommendation.

Prerequisites for sophomores who wish to double up in sophomore year, taking both Geometry and Algebra II/ Trigonometry: Minimum grade of “95” or better in Algebra I and teacher recommendation.

Approved for NCAA DI and DII athletic eligibility

Honors Algebra II/Trigonometry                            Grades 10-11

This course expands on first year algebra skills and introduces students to further foundational skills needed for future coursework in Precalculus and Calculus. This course includes an in-depth study of the following: statistics, probability, and complex numbers, as well as algebraic and graphical approach to linear functions, quadratic functions, polynomial functions, rational functions, exponential functions, and logarithmic functions. A comprehensive study of trigonometry and circular trigonometry is explored. The course is designed to allow students to use mathematics as a tool for problem-solving and make further preparations for solving real world applications.

Prerequisites for rising juniors who wish to accelerate into Honors Algebra II/Trigonometry: Minimum grade of “95” or better in Algebra I, “95” or better in Geometry and teacher recommendation.

Prerequisites for rising sophomores: Minimum grade of “80” in Honors Geometry and teacher recommendation. Minimum grade of “95” or better in Geometry and teacher recommendation.

Approved for NCAA DI and DII athletic eligibility

Advanced Algebra/Trigonometry                                    Grade 12

Designed for those students who completed Algebra II as juniors, this course continues the study of functions, which includes trigonometry, basic rational, exponential and logarithmic functions, and college algebra. A scientific calculator (required) is used extensively in this course. This course is not open to students who completed CP Algebra II/Trigonometry.

Prerequisite: Algebra II.

Approved for NCAA DI and DII athletic eligibility

Precalculus   Grades 11-12 

This course expands on first year and second year algebra skills and introduces students to further foundational skills needed for future coursework in Calculus. This course includes an in-depth study of: trigonometry of right triangles and the coordinate plane, trigonometric functions and identities, rational functions, and logarithmic functions. Students are introduced to sequences and series as well as elementary concepts of calculus, including limits. There will be an emphasis on efficient and effective problem-solving strategies that will be necessary in future calculus courses. This course is designed to allow students to use mathematics as a tool for problem-solving and make further preparations for solving real world applications. For juniors, the course is designed to prepare students for Honors Calculus. For seniors, the course is designed to prepare students for college coursework and essential mathematical skills. 

Prerequisites for rising juniors and rising seniors: Minimum grade of “80” or better in Algebra II/Trigonometry. Not open to students who took Algebra II. 

Approved for NCAA DI and DII athletic eligibility 

AP Precalculus   Grades 10-12 

This course includes an in depth study of trigonometric functions and identities, rational functions, logs and exponentials, their connection with previous Algebra 2 and Geometric properties and their applications to real world scenarios. Students will be introduced to conics, series, sequences, and polar and parametric equations as well as an analytical study of limits. This course provides an excellent foundation for calculus but also serves as an appropriate capstone mathematics course that will open pathways to success in STEM fields. It is expected that the students in this course will seek college credit, college placement, or both, as a result of successful performance on the advanced placement examination. Students enrolled in this course are expected to take the AP Precalculus exam in May. Please see page 5 for additional information regarding advanced placement courses and exams.

Prerequisite: Minimum grade of “80” or better in Honors Algebra II/Trigonometry 

Approved for NCAA DI and DII athletic eligibility

Honors Calculus   Grade 12 

This course is designed for students who opt not to take the AP Calculus course in their senior year. The course will prepare students with the fundamentals of calculus in preparation for college calculus. The course will introduce the concept of limits, techniques of differentiation and integration and its applications. Derivatives and antiderivatives of trigonometric functions, derivatives of exponential and logarithmic functions, and trigonometric functions will be explored. Techniques of integration using real world examples will be studied. Problem solving and applications are emphasized. 

Prerequisite: Successful completion of AP Pre-Calculus or minimum grade of “80” or better in Precalculus. Approved for NCAA DI and DII athletic eligibility 

Statistics and Probability   Grades 11-12 

This course introduces major concepts and tools for collecting and analyzing data, and drawing conclusions. The main themes are: exploring data, describing statistics, sampling and experimentation, statistical inference and hypothesis testing. Basic concepts of probability and normal distributions are studied. Case studies in confidence intervals, correlation, and regression are also examined. Verbal communication, problem solving, and the use of technology are emphasized throughout the year. This is a practical and helpful course for many careers, including the social sciences business and engineering. 

Prerequisite: Successful completion of Geometry and Algebra II. 

Approved for NCAA DI and DII athletic eligibility 

AP Statistics   Grades 10-12 

This course introduces students to the major concepts and tools for collecting, analyzing and drawing conclusions from data. Problem solving and effective verbal communication is strongly emphasized and is necessary for success in the course. There is an abundance of interpretive reading that requires students to use inference skills. Students are exposed to four broad conceptual themes: exploring data; sampling and experimentation; anticipating patterns; and statistical inference. Topics in probability include geometric and binomial theorems, and the normal curve. Those students interested in social sciences, engineering, science and math are encouraged to enroll. Students enrolled in this course are expected to take the AP Statistics exam in May. Please see page 5 for additional information regarding advanced placement courses and exams. 

Prerequisite for sophomores: *Honors Math Analysis or minimum grade of “90” or better (as a final grade) in Honors Geometry and teacher recommendation. 

Prerequisites for juniors and seniors: Honors Algebra II/Trigonometry or minimum grade of “93” or better in Algebra II/ Trigonometry and teacher recommendation. 

Note: A summer assignment may be required. 

Approved for NCAA DI and DII athletic eligibility 

AP Calculus (AB)   Grades 11-12 

Calculus AB covers differential and integral calculus topics that are typically covered in a Calculus I course in college. The course emphasizes theory as well as the applications of differentiation and integration. Concepts and problems are examined from a verbal, geometric, numeric, and analytical perspective. This is a rigorous, challenging, and demanding course that requires an intuitive knowledge of mathematics. It is expected that the students in this course will seek college credit, college placement, or both, as a result of successful performance on the advanced placement examination. Students enrolled in this course are expected to take the AP Calculus AB exam in May. Please see page 5 for additional information regarding advanced placement courses and exams. 

Prerequisites: Minimum grade of “80” or better in AP Pre-Calculus and teacher recommendation. 

Note: A summer assignment may be required. 

Approved for NCAA DI and DII athletic eligibility 

AP Calculus (BC)   Grades 11-12 

The Calculus BC course covers Calculus AB topics, as well as additional topics in differential and integral calculus and infinite series. This course is rigorous, challenging, and demanding, and is recommended only for those who appreciate and understand the theoretical aspects of mathematics. Additional topics are L’Hopital’s Rule, logistic growth, Euler’s Method, improper integrals, series convergence, and Maclaurin and Taylor Series. Students who perform well may receive up to two semesters of college credit. Students who take the Calculus BC examination will receive a Calculus AB subscore grade in addition to the Calculus BC grade. Students are expected to take the AP examination in May. Please see page 5 for additional information regarding advanced placement courses and exams. 

Prerequisites: Minimum grade of “90” or better in AP Pre-Calculus and teacher recommendation. 

Note: A summer assignment may be required. 

Approved for NCAA DI and DII athletic eligibility 

Multivariable Calculus (DUAL ENROLLMENT: Seton Hall University)   Grade 12 

This course is run through Seton Hall University’s Project Acceleration and is designed for students who have successfully completed Advanced Placement Calculus BC and is weighted accordingly. It is intended for advanced students who have demonstrated thorough knowledge of Calculus I and Calculus II. The course expands upon single variable calculus while covering topics in more than one variable including vectors and matrices, parametric curves, partial derivatives, double and triple integrals, and vector calculus in two and three dimensional space. All topics are presented using multiple representations with the use of a graphing calculator. Topics are represented graphically, numerically, algebraically and verbally. 

Prerequisites: Successful completion of AP Calculus BC, teacher recommendation and a 3 or higher on the AP Calculus BC exam. This will be confirmed over the summer. 

Note: There is a financial obligation for students who wish to earn college credit for this course.

Approved for NCAA DI and DII athletic eligibility 

Mathematics Full Year Course Electives 

AP Computer Science Principles   Grades 10-12 

AP Computer Science Principles introduces students to the foundations of computer science with a focus on how computing powers the world. Along with the fundamentals of computing, students will learn to analyze data, create technology that has a practical impact, and gain a broader understanding of how computer science impacts people and society. Students are expected to take the AP examination in May. Please see page 5 for additional information regarding advanced placement courses and exams. 

Prerequisite for rising sophomores only*: Minimum grade of “90” or better in Honors Geometry or “80” or better in Honors Math Analysis (final grade will be checked in June). 

Prerequisites for rising juniors and seniors: Minimum grade of “90” or better in Algebra II/Trigonometry or a minimum grade of “80” in Honors Algebra II/Trigonometry. 

Approved for NCAA DI and DII athletic eligibility 

AP Computer Science A   Grades 10-12 

This course is designed for students who seek a challenge beyond Introduction to Computer Science and Honors Computer Science. The course focuses on comprehensive program development and implementation, using the JAVA programming language. Case studies and activities are used to analyze the logic behind effective data structure development using object-oriented programming. Debugging and efficient coding techniques are emphasized throughout the course, further incorporating in-depth analysis of real world applications and complex data structures. Students explore problem-solving in mathematics, business, and other disciplines. They also have the opportunity to construct a functional portfolio of programs to use when applying for admission to college level computer-science programs. Students are expected to take the AP examination in May. Please see page 5 for additional information regarding advanced placement courses and exams. 

Prerequisite for rising sophomores only*: Minimum grade of “90” or better in Honors Geometry or “80” or better in Honors Math Analysis (final grade will be checked in June) and demonstrated proficiency in computer science by passing a proficiency examination. 

Prerequisites for rising juniors and seniors: Minimum grade of “85” or better in Honors Computer Science, including JAVA or AP Computer Science Principles, and a minimum of Algebra II/Trigonometry. If this is your first Computer Science course you must demonstrate proficiency in computer science by passing a proficiency examination. 

Note: A summer assignment may be required. 

Approved for NCAA DI and DII athletic eligibility 

Honors Data Structures and Advanced Coding Concepts (DUAL ENROLLMENT: Ramapo College)   Grades 11-12 

This course offers a study of the basic data structures and related algorithms and is meant to function as the capstone course for students who are committed to computer science as an academic pathway. Stacks, queues, deques, arrays, linked lists, trees, graphs, strings, sorting, searching, and file structures are among the topics presented in this course. In addition, the topics of computer graphics, web development, computer security, and software development will be discussed throughout the course. 

Prerequisite: Successful completion of AP Computer Science A with a 4 or 5 on the AP exam. 

Note: There is a financial obligation for students who wish to earn college credit for this course; please see page 6 for additional information. Approved for NCAA DI and DII athletic eligibility 

Honors Quantum Computing (The Coding School in partnership with Quibit and IBM)   Grades 10-12 

Qubit by Qubit’s Introduction to Quantum Computing with Google AI is a cutting-edge course designed to introduce high school students to the exciting world of quantum computing. Quantum is the next frontier of computing technology and will revolutionize fields such as healthcare, finance, and cybersecurity. If students are interested in math, computer science, electrical engineering, or physics, you will find that quantum computing lies right at the intersection of all these fields. It also has important applications in biology, chemistry, and economics, which we will also explore in this class. Students do not need an extensive background in quantum computing or computer science to take this course. Topics covered include basic linear algebra, probability, and Python coding basics. Students will also spend time exploring the “weird” properties of quantum mechanics that make quantum computers so powerful. By the end of this course, students will be able to code quantum gates and circuits, understand quantum algorithms and even run code on a real quantum computer. While students are scheduled for this class during an instructional period, this course is predominantly self-paced and virtual with a faculty member as a facilitator. 

Prerequisite: Successful completion of Geometry. 

Note: Students interested in taking Honors Quantum Computing should let their counselor know during scheduling and also complete the Advancement and/or Enrichment application on the Career & Academic Pathways website. 

Approved for NCAA DI and DII athletic eligibility 

Mathematics Semester Course Electives 

Introduction to Computer Science (Fall)   Grades 10-12 

This is the beginning course for students who would like to explore the study of computer science. The programming concepts are taught using various block-based programming languages, which helps students learn coding concepts without having to memorize syntax for a given language. Students will first learn about fundamental coding concepts using one of many available block-based programming languages. Students will then further develop their understanding of user input and computer output in programs by creating mobile applications using a drag and drop mobile application development environment. Students will then explore game development using a block-based game development environment. In addition, students will visit topics of ethics in computer science throughout the course.

Approved for NCAA DI and DII athletic eligibility 

Honors Computer Science (Spring)   Grades 10-12 

This is the next course in the sequence following Introduction to Computer Programming. Students will learn programming concepts using a variety of text-based programming languages. In addition, more advanced computer science topics (including Arrays, Methods/Functions, Classes, Modules, etc.) are studied throughout the course. Students will also learn about object-oriented programming concepts and how to implement them. Students will learn how to develop an interactive web page. Students will create a variety of dynamic applications using one of the many integrated development environments available. Throughout the course, students will learn about and discuss computer security concerns and issues. 

Prerequisites: Introduction to Computer Science. 

Approved for NCAA DI and DII athletic eligibility