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.

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.

LOWER SCHOOL

Math 3: Building Mathematical Thinking: Number Sense, Fact Strategies, Problem Solving, Grade level: 3

This course focuses on concepts, procedures, and applications in four areas:

• understanding of multiplication and division and strategies for multiplication and division within 100;

• understanding of fractions, especially unit fractions;

• understanding of the structure of rectangular arrays and of area; and

• describing and analyzing two-dimensional shapes.

Practice standards emphasize problem-solving, the use of multiple representations, reasoning, mathematical modeling, tool use, communication, and other ways of making sense of mathematics. Students learn how productive struggle and growth mindset benefit them as mathematicians.

Math 4: Exploring the why behind mathematics: Number Theory, Place Value Algorithms, Problem Solving, Grade level: 4

In Grade 4, instructional time will focus on four critical areas: (1) developing understanding and fluency with multi-digit multiplication including familiarity with patterns, factors and multiples, and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction/decimal equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; (3) understanding that geometric figures can be analyzed and classified on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry; and (4) solving problems involving length, weight, liquid, mass, volume, time, area, and perimeter.

The curriculum focuses on developing students’ deep understandings of mathematical concepts, proficiency with key skills, and ability to solve complex and novel problems. Our core math resource, Everyday Mathematics, blends direct instruction, structured investigation, and open exploration. Teachers tap into the intelligence and strengths of all students by presenting material that is as linguistically, visually, and kinesthetically rich as it is mathematically powerful. This course encourages students to verbalize ideas, ask questions, take risks, and actively engage in their own learning.

Math 5: Applying Mathematical Reasoning: Computation and Operations, Fractions/Decimals and Problem Solving, Grade level: 5

In Grade 5, instructional time will focus on three critical areas: (1) Developing an understanding of volume; (2) Developing fluency with addition, subtraction, multiplication and division of fractions (3) Extending division to 2-digit divisors; integrating decimal fractions into the place value system and developing fluency with whole number and decimal operations.

Our core math resource, Everyday Mathematics, focuses on both conceptual and procedural understanding. We continue to strive to explain the “why” before the “how”. This program provides meaningful learning experiences where students solve problems, reason about their mathematical thinking and communicate their ideas in oral and written form. An engaging selection of visual models, manipulatives, games, apps and other unique learning materials help bring math alive for all students! Our math program also offers opportunities for interested students to pursue additional math challenges through Math Olympiad- a popular international math competition focusing on problem solving strategies.

Math 6: Investigating Pre-Algebra, Grade level: 6

6^th Grade Mathematics is an integrated study of arithmetic, algebra, and geometry. Topics include properties of real numbers, proportions, integers, equations and expressions, probability and statistics, and elementary geometric concepts, problems, and constructions. The concept of a variable is introduced to provide a background for elementary algebra.

In this class, we will cross the bridge from the numerical world to the algebraic world. We will accomplish this goal by developing abstract thinking and problem-solving skills and exploring real-world connections to mathematical concepts. This course is rigorous and fast-paced and requires students to take responsibility for learning by recognizing their strengths and viewing any difficulties as opportunities for learning and growth.

MIDDLE SCHOOL

All middle school students at Holton take Algebra over two years:

Algebra 1A (Grade 7) and Algebra 1B (Grade 8)

In order for Holton students to develop a deep conceptual foundation in algebra, all of our students take ALG 1A in 7th grade and ALG 1B in 8th grade. Pathways for acceleration begin in the upper school. Click here to learn more about The Transition from Middle School to Upper School and Course Readiness.

Algebra is a generalization of arithmetic in which letters representing numbers are combined according to the rules and properties of arithmetic. In other words, anything you can do to numbers, you can also do to variables - this is algebra. Algebra provides development and practice of logical problem-solving, allowing us to consider those quantities that vary and those that remain constant in a given situation.

UPPER SCHOOL

Honors classes are offered to challenge students who are capable of working independently, require less review, and have a desire to think more deeply about mathematics. For example, while the content of the on-level courses and the honors courses are very similar, honors students will explore more open-ended, abstract questions and apply what they have learned to novel problems. Honors courses require an advanced fluency in Algebra and flexible thinking. Click here to learn more about The Transition from Middle School to Upper School and Course Readiness.

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.

Calculus 1A, 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 rigorous advanced level math course, designed for students who have completed Calculus I & II, deals with functions of more than one independent variable. Topics will include lines and planes, partial differentiation, multiple integrals, vector-valued functions and differential operators, and Green’s Theorem. The physical concepts of flux, circulation, divergence, and work will receive special attention, as will the central notion of a conservative field. Students will have opportunities to work both independently and cooperatively with their Landon counterparts in order to demonstrate mastery of concepts.

MATHEMATICS ELECTIVES - offered every two years on a rotating basis

Offered during the 2021-2022 school year:

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.

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, modeling 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 Art and Architecture Around the World 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, modeling 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.

Offered during the 2022-2023 school year:

Applications of Discrete Mathematics, 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, modeling 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 modeling 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, modeling 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, modeling 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.