Mathematics

Department Chair: Kate Mitchell

The Mathematics Department at Holton-Arms offers a sequence of rigorous courses that cover traditional college preparatory material. At the same time, Holton recognizes that all girls do not acquire mathematical skills at the same rate or develop the ability for abstract thinking at the same age. There is no grade 9 mathematics course, for instance; rather, our offerings provide a variety of placement options at any given grade level. The goal of placement decisions is to offer each girl a course that challenges her and allows her to take risks, but at the same time provides her with a reasonable opportunity to experience the satisfaction of success.

In Middle School, students review and extend their understanding of skills and concepts in arithmetic, geometry, number theory, and probability and statistics begun in Lower School in order to apply their skills and conceptual understanding to the study of algebra. The Middle School Mathematics curriculum is designed to build a solid foundation in algebraic thinking and problem solving to enable students to be successful in studying advanced topics in mathematics.

After the building a solid foundation in algebra, all students study Geometry followed by Algebra 2 and Trigonometry. Goals for students throughout the sequence include mastery of fundamental operations along with development of calculator and problem solving skills and an appreciation for the structure and applications of mathematics. Students are required to have a TI-84 Plus graphing calculator beginning in Algebra 1. In addition, students will use computer and web-based technology to explore concepts.

The Mathematics Department recognizes the importance of having students work with data throughout the curriculum. In many courses, including Algebra 2 & Trigonometry and Precalculus, students are given the opportunity to learn about the modeling of functions by using data taken from real world situations. The girls will learn how to apply “continuous mathematics” to data taken from disciplines including science, politics, and current events.

Although any student’s greatest resource is herself, she will learn that even more can be accomplished through working with her classmates and teachers. Individual conversations and one-to-one help sessions between a student and her mathematics teacher are an integral part of the Holton experience.

Nine credits are required for graduation: one of which is Algebra 2 & Trigonometry.

Middle School

Pre-Algebra Fundamentals, Grade level: 7 (required if not taking Pre-Algebra 7)

This course is an integrated study of arithmetic, algebra, and geometry. Topics include properties of and operations with real numbers, exponents, square roots, and elementary geometric concepts, and problems. The concept of a variable is introduced to provide a background for elementary algebra. Emphasis is placed on reading a mathematics textbook and understanding how to picture mathematical concepts. This course utilizes activities, labs, games, and projects to help students comprehend and master the topics of the course.

Pre-Algebra / Pre-Algebra Accelerated, Grade level: 7, Prerequisite: Permission of the Department

This course continues students’ preparation for Algebra 1 and Geometry, and is designed to address four main topics: number theory, statistics, geometry and basic algebra. In number theory students explore the number system through use of patterns, fractions, decimals, and percentages. The statistics unit examines probability, data collection, data presentation, and decision-making. The geometry unit investigates properties and patterns of geometric shapes and theories. In the study of algebra, students are explore the use of variables and the associate properties to symbolically model problems. An honors section is available for qualified students, if numbers allow. Problem solving and abstracting thinking are emphasized in the honors section.

Intro to Algebra, Grade level: 8 (required if not taking Algebra 1)

This course concludes the general mathematics program with a review of the fundamentals of arithmetic. Word problems and the language, vocabulary, and notation used in mathematics are stressed. Topics from geometry include conceptualization of perimeter, area, and volume, and relationships in a circle. Emphasis is also given to equations in one variable and other basic abstractions of beginning algebra.

Algebra 1 / Algebra 1 Honors, Grade level: 7 or 8 Prerequisite: Permission of department, Credit: 3

This course includes the study of signed numbers, operations with literal expressions, factoring, fractions, linear equations and systems of linear equations, fractional equations, radicals, quadratic and exponential equations, and their applications to problem solving. Students are expected to become facile with the basic techniques of factoring and simplifying expressions. The ideas and rules of algebra learned in this course form the foundation for much of high school math and science. An honors section is available for qualified students, if numbers allow.

Geometry 8 Honors, Grade level: 8, Prerequisite: Algebra 1; permission of Department, Credit: 3

This rigorous course in geometry integrates coordinates and transformations with a traditional approach to 2-dimensional and 3-dimensional Euclidean geometry. A variety of methods will be used to prove geometric theorems. We approach and solve problems from different perspectives to gain an appreciation of the interconnectivity among concepts in mathematics. Writing will be used to help students develop their understanding of geometry by applying concepts to their own experiences and constructing meaning for mathematical symbols, procedures, and concepts. Technology will be incorporated to strengthen and explore geometric properties. The ability to absorb and master large amounts of material quickly and with little review is essential.

Upper School

Algebra 1, Grade level: 9, Credit: 3

This course is a study of elementary algebra, based on the elements, operations, and properties of the set of real numbers. Emphasis is placed on developing students’ skills in factoring and multiplying polynomials; solving linear, quadratic, and systems of equations; and solving inequalities and word problems. Offered in ninth grade if numbers allow.

Geometry, Grade level: 9, 10 (required), Credit: 3

This introductory course in geometry integrates coordinates and transformations with a traditional approach to 2-dimensional and 3-dimensional Euclidean geometry. A variety of methods will be used to prove geometric theorems. We approach and solve problems from different perspectives to gain an appreciation of the interconnectivity among concepts in mathematics. Writing will be used to help students develop their understanding of geometry by applying concepts to their own experiences and constructing meaning for mathematical symbols, procedures and concepts. Technology will be incorporated to strengthen and explore geometric properties.

Geometry Honors, Grade level: 9, Prerequisite: Algebra 1; permission of Department, Credit: 3

This rigorous course in geometry integrates coordinates and transformations with a traditional approach to 2-dimensional and 3-dimensional Euclidean geometry. A variety of methods will be used to prove geometric theorems. We approach and solve problems from different perspectives to gain an appreciation of the interconnectivity among concepts in mathematics. Writing will be used to help students develop their understanding of geometry by applying concepts to their own experiences and constructing meaning for mathematical symbols, procedures and concepts. Technology will be incorporated to strengthen and explore geometric properties. The ability to absorb and master large amounts of material quickly and with little review is essential.

Algebra 2 & Trigonometry, Grade level: 10, 11 (required), Prerequisites: Algebra 1 & Geometry, Credit: 3

This course includes a study of linear, quadratic, and polynomial functions, systems of equations in two and three variables, inequalities, complex numbers, variation, conic sections, exponential and logarithmic functions, and trigonometric functions, including applications. This is a more structured Algebra 2 and Trigonometry course than Algebra 2 and Trigonometry with Data Analysis.

Algebra 2 and Trigonometry with Data Analysis, Grade level: 9, 10, 11, Prerequisites: Algebra 1 & Geometry; permission of department required, Credit: 3

This course includes a study of functions (linear, quadratic, polynomial, and rational), systems of equations in two and three variables, inequalities and complex numbers, exponential and logarithmic functions, and trigonometric functions. The focus is on the representation of problems algebraically, numerically, and graphically. Applications include an introduction to data analysis and modeling through regression analysis.

Algebra 2 and Trigonometry with Data Analysis Honors

Grade level: 9, 10, 11, Prerequisites: Algebra 1 & Geometry; permission of department required, Credit: 3

This second-year algebra course consists of a thorough and fast-paced study of algebra and an extensive study of trigonometry. Its emphasizes the structure of mathematical systems and their underlying concepts. Problems are studies both algebraically and graphically. Topics include a study of functions (linear, quadratic, polynomial, and rational), systems of equations in two and three variables, inequalities and complex numbers, exponential and logarithmic functions, and trigonometric functions. Applications include an introduction to data analysis and modeling through regression analysis. Strong algebraic skills are presumed. The ability to absorb and master large amounts of material quickly and with little review is essential for success in an honors course.

Functions of Precalculus, Grade level: 11, 12, Prerequisite: Algebra 2 and Trigonometry or Algebra 2 and Trigonometry with Data Analysis. Credit: 3

This course is for students who want to develop a fuller understanding of earlier courses and to explore applications of mathematics. Topics include properties of functions, matrices, sequences and series, logarithms, trigonometry, and probability, as well as a review of elementary algebra.

Precalculus, Grade level: 10, 11, 12, Prerequisite: Algebra 2 and Trigonometry or Algebra 2 and Trigonometry with Data Analysis; permission of department required, Credit: 3

This rigorous course prepares students for study of the Calculus. Functions are emphasized, including polynomial and rational, circular (trigonometric), exponential, and logarithmic functions. Other topics may include matrices and determinants, sequences and series, the binomial theorem, probability, and limits. A strong recall and working knowledge of Geometry and Algebra 2 and Trigonometry are essential for success in this class.

Precalculus Honors, Grade level: 10, 11, Prerequisite: Algebra 2 and Trigonometry with Data Analysis Honors; permission of department required, Credit: 3

This course includes an extensive study of trigonometric, exponential, and logarithmic functions. Particular attention is paid to the properties of their graphs. Other topics include polynomials, analytic geometry, matrices and determinants, sequences and series, the binomial theorem, polar coordinates, and probability. During the fourth quarter, students begin their formal study of calculus. The ability to absorb and master large amounts of material quickly and with little review of Geometry and Algebra 2 and Trigonometry is essential. Offered if numbers allow.

Introduction to Calculus, Grade level: 12, Prerequisite: Precalculus or permission of department, Credit: 3

This course provides students with an intuitive approach to the fundamentals of differential calculus and integral calculus. The language of calculus will play an important role in developing the definitions of derivatives and integrals. Focusing on algebraic functions, students explore limits, leading to the definition of derivative. The concepts of average and instantaneous rate of change are investigated. Students develop the rules of differentiation, including the chain rule and implicit differentiation, and apply them to problems in optimization, related rates, and curve sketching. Students are introduced to the concepts of finding area under a curve, the integral regarded as the anti-derivative, and the Fundamental Theorem of Calculus. Applications of integration are also included. Generally, the content of this course is not as broad or as deep as the content of Calculus I. Offered if numbers allow.

Statistics, Grade level: 11, 12, Prerequisite: Algebra 2 and Trigonometry with Data Analysis and permission of department. Co-Requisite (for 11’s): Precalculus or higher. Credit: 3

The course provides a comprehensive, college level introduction to statistics. An introductory statistics course, similar to this course, is typically required for majors such as social sciences, health sciences and business. Students study the tools for collecting, organizing, and displaying data. They learn about planning and conducting surveys and experiments and drawing conclusions from their results. The course includes probability, the properties of the normal distribution, and statistical inference.

Calculus I, Grade level: 11, 12, Credit: 3, Prerequisite: Precalculus and permission of department*

This rigorous course follows the syllabus typical for a college level first semester calculus course as well as exploring some topics presented in the second semester of such a course. Topics include functions, analytic geometry, limits, differentiation, and integration.

* This permission is seldom granted unless a student has earned at least a B in Precalculus.

Calculus I & II, Grade level: 11, 12, Prerequisite: Precalculus Honors and permission of the department, Credit: 3

This course covers a syllabus that is both more intensive and more extensive than that of Calculus I. It completes the content that typically would be covered in two semesters of college calculus. In addition to the topics covered in Calculus I, other topics covered include vectors, parametrically defined curves, and infinite series. Students taking this course should come with a thorough knowledge of limits, continuity, and the derivative and its applications. Offered if numbers allow.

Multivariable Calculus, Grade level: 11,12 , Prerequisite: Calculus I & II, Credit: 3

This advanced level math course, designed for students who have completed Calculus I & II course, deals with functions of more than one independent variable. Topics will include partial differentiation, multiple integrals, vector-valued functions and differential operators, Stokes’ and Green’s Theorems and methods of differential equations. The physical concepts of flux, circulation, divergence and work will receive special attention, as will the central notion of a conservative field. Class will meet at 7 a.m. three mornings per week location TBD.

MATHEMATICS ELECTIVES

History of Mathematics, Grade Level: 9, 10, 11, 12, Credit: 1

Everybody is aware of the usefulness of mathematics. Its practical applications extend from engineering to astronomy, from business to medicine to urban planning. But not everyone is aware of its fascinating nature. In the words of Bertrand Russell, “mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” In this course, we will explore this side of mathematics, as an unmatched creative pursuit that will lead us to rediscover and relish the magnificence of its theorems and the brilliance of its scholars.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

The Mathematics of Social Choices, Grade Level: 9, 10, 11, 12, Credit: 1

This course examines applications of discrete mathematics focusing on the mathematics of social choices. Possible topics include mathematical models of fairness, elections, global data, consumer structures, mapping and graph theory, and optimization. Emphasis will be on problem solving, reasoning abstractly and quantitatively, modelling with mathematics, recognizing structures and patterns, collaborative work, and communication.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

The Mathematics of Everyday Life, Grade Level: 9, 10, 11, 12, Credit: 1

This course examines applications of discrete mathematics specific to models of the physical world as well as business and industrial applications. Some examples of curricular topics include numeration systems, modular arithmetic, fractals, counting, probability, and logistics. Emphasis will be on problem solving, reasoning abstractly and quantitatively, modelling with mathematics, recognizing structures and patterns, collaborative work, and communication.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

Model Thinking, Grade Level: 9, 10, 11, 12, Credit: 1

The “Model Thinking” course focuses on quantitative literacy, mathematical thinking and Fermi problems, and therefore fosters number sense and creative problem solving. As it will make use of linear and quadratic regression models, it requires only completion of Algebra 1. Some examples of such problems might include:

• Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed.
• Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player.
• Estimating how many piano tuners there are in NYC, or how many golf balls would fit in a math classroom.

Diagrams of various kinds, spreadsheets and other technology, and algebra are powerful tools for understanding and solving problems drawn from different types of real-world situations. Emphasis will be on problem solving, reasoning abstractly and quantitatively, modelling with mathematics, recognizing structures and patterns, collaborative work, and communication.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

Mathematical Modelling, Grade Level: 10, 11, 12, Credit: 1

This course is a mathematical modelling course with the same goals as the Model Thinking course but requiring more advanced techniques. Students in this course will have to have taken Algebra 2 in order to apply various types of regressions to analyze data. By its very nature this course will be interdisciplinary and will discuss applications from all realms and use global data, thus fostering global competencies. Emphasis will be on problem solving, reasoning abstractly and quantitatively, modelling with mathematics, recognizing structures and patterns, collaborative work, and communication.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

Architectural Design with SketchUp, , Grade Level: 9, 10, 11, 12, Credit: 1

The Architectural Design course will give students an introduction to drafting, modeling, and designing with digital design software. While learning to use the SketchUp program, the girls will be given the opportunity to digitally create several structures. A capstone project at the end of the course may include digitally creating a “dream home” or a “dream school.” The skills learned in this course are transferable and can then be applied to design projects for many courses in high school and beyond. This course will encourage the girls to practice their spatial skills and visual thinking, all foundational abilities for any STEM field. SketchUp, formerly Google SketchUp, is a 3D modeling computer program for a wide range of drawing applications such as architectural, interior design, landscape architecture, civil and mechanical engineering, film and video game design. It is available as a web-based application, SketchUp Free, a freeware version, SketchUp Make, and a paid version with additional functionality, SketchUp Pro. Emphasis will be on problem solving, modelling with mathematics, recognizing structures and patterns, collaborative work, and communication.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

Geometry in Nature, Art and Architecture, Grade Level: 9, 10, 11, 12, Credit: 1

The foundation of our geometric understanding of the world is Euclidean Geometry. Concepts of symmetry, perspective and shape underlie our sense of beauty in nature and art. On this foundation, mathematicians have built a vast edifice of mathematical ideas, including non-Euclidean Geometries such as spherical geometry and hyperbolic geometry. The visual world presents us with a rich array of topics for exploration and wonder. Geometric thinking allows us to see, understand, and appreciate more of the world around us. Possible topics for this course could include: nature numbers, the golden mean, the beauty of the golden rectangle, symmetry and regularity in crystals and Platonic solids, the fourth dimension in art, the Mobius Strip, and the geometry of origami. Emphasis will be on problem solving, modelling with mathematics, recognizing structures and patterns, collaborative work, and communication.

This course will be one trimester in length and will meet as a minor. Offered only if numbers allow.

FAQ’s

What is the difference between the Honors classes and the other course offerings?

Honors classes are offered to challenge students who are capable of working independently and at a much faster pace. For example, while the main content of the Geometry and Geometry Honors courses are very similar, Geometry Honors students will explore more open-ended, abstract questions and will be expected to have a stronger grasp of basic principles of Algebra I. The Geometry students will review key algebra skills as they work through the content of the course.

Can students move between the different levels (Honors, Regular, etc.) of mathematics courses?

Students have the ability to move between the levels when course selection decisions are made in the spring. Careful consideration is given to the course load of the student and to her performance in her current level of mathematics.

What is the difference between Calculus I and Calculus I & II?

Calculus I & II is an extension of Calculus I. The Calculus I & II curriculum differs in scope. Calculus I includes techniques and applications of the derivative, the definite integral, and the Fundamental Theorem of Calculus. Calculus I & II includes all topics in Calculus I, plus others such as parametric, polar, and vector functions, and series. Because Calculus I & II covers approximately two semesters of college calculus compared to one semester in Calculus I, a big difference in the courses is the pace and the requirements of students outside of class.

What is the difference between Calculus I and Introduction to Calculus?

Introduction to Calculus is offered to students who intend to continue their study of calculus in college. It provides a conceptual introduction to both differentiation and integration with an emphasis on applications. The slower pace of the course as well as the review of algebraic skills make the calculus concepts accessible to a broad spectrum of mathematics students.

Can you take more than one math class in a given year?

A student may take Statistics at the same time as she takes Precalculus or one of the Calculus course offerings. Approximately one-third to one-half of all students taking Statistics are taking another mathematics course concurrently. In addition, students can take additional mathematics trimester electives.

Is it a good idea to try to skip ahead by taking a summer class?

In general, it is not a good idea to take a full-year course over the summer. A majority of summer school programs are designed to review the basics of a subject for those students who did not adequately grasp the topics studied in the previous school year’s math course. The math department definitely intends for the student to understand much more of a course's content than just the basics. Even summer programs that are designed to present a full curriculum often fall short of preparing a student to skip ahead, because a typical summer program will last one or two months, whereas a Holton class will spend about eight to nine months studying a subject. Clearly a summer program would have to cover material much more quickly, giving the students little time to synthesize and understand the topics being covered or to review skills and concepts from previous years. Consequently, while the student might successfully complete the summer program, the student's performance in subsequent courses often suffers. The following procedure is in place for students considering summer classes in mathematics.

The student must take an accredited course with an approved syllabus and must pass the equivalent of a mid-term and final exam in the class they desire to skip. These are Holton exams given at the end of the summer. The student must demonstrate that she has mastered the content of a year-long course through a score of 80 or above on each of these exams. While this procedure is in place to enable a student to progress more quickly through the mathematics curriculum, it may not be in her best interest to do so. As discussed above, the opportunities for enriched practice and deeper exploration that are provided in a year-long course help to ensure not only a student's mastery of current material but also successful progression through subsequent courses.