# Mathematics

## Interior Masthead

The Friends Select Mathematics Department strives to meet the needs of each student through a flexible program within a supportive learning environment. In hopes of creating lifelong learners who appreciate the beauty of mathematics, our program emphasizes critical thinking and the development of problem solving skills, through the discovery of ideas and patterns. Electives are accessible to a variety of abilities and interests. We strive to meet the needs of students who would like to explore topics outside of the traditional curriculum, and those who would like to accelerate through the sequence. Faculty work to help students make connections between what they are learning in the classroom and how it will be used in everyday life. We encourage students to have tenacity, to be flexible, and to be willing to try a different approach to solve a problem. The sharing of ideas through class discussions is a critical component of the classroom and learning experience. Technology is used to enhance the discovery and learning experience.

## Curriculum

## Lower School

## Learning to think like true mathematicians

From a very young age, our students learn to think like mathematicians. Our students develop the mindset that math should make sense. They engage in constructing an understanding of number, number relationships and patterns. They learn how to approach complex problems, and how to clearly communicate their mathematical thinking. We ensure students develop fluency with math facts and computation, but in order for them to approach higher level mathematics, we teach toward the goal of understanding of math concepts.

The University of Chicago’s Everyday Math program focuses on numbers and operations, patterns and chance, and geometry and spatial sense. While building skills over the years, students explore algorithms, functions, and sequences and apply this learning to real-life situations. The program connects in a spiral pattern, so students are introduced to a concept at various points, spend time practicing it, and then become proficient to the point of mastery. A strong sense of number, problem solving skills, and conceptual understanding and problem solving are emphasized, so students are prepared to master the higher level mathematics that come in middle school, upper school and beyond.

## Middle School

## Mathematics - Grades 5 & 6

### Foundations for abstraction and sophisticated problem-solving

Our fifth and sixth grade mathematics program bridges students from the University of Chicago's Every Math program into the algebraic thinking necessary for seventh and eighth grade mathematics. Solidifying basic operations with real numbers is emphasized alongside a continued development of problem-solving skills. Students utilize both online and physical textbooks and additional resources to support their learning.

## Mathematics: Grades 7 & 8

### Developing Critical Mathematics Skills

Seventh and Eighth grade mathematics comprise our algebra program. In seventh grade pre-algebra, students explore basic number theory, properties of real numbers, polynomials, exponents, graphing linear functions and solving basic linear equations. In eighth grade algebra, students concentrate on a deeper level of complexity with: simplifying expressions, solving equations, and graphing functions, including quadratics. Students who successfully complete the algebra program take geometry in ninth grade.

*For those students who are academically and developmentally prepared, it is possible to take advanced math classes starting in 7th grade. Some middle school students may be placed in upper school math classes.

## Upper School

## Two tracks, intriguing electives

The math program in upper school is designed to help students master algebraic and geometric skills required for work at the college level. Along with skills, the mathematics department stresses the historical development of ideas and their aesthetic aspects through class discussions and projects. The Math/Science Symposium, held annually in April, offers students an opportunity to share their research into mathematical topics or problems. Core courses stress the development of problem-solving skills and the use of logical patterns needed to write a valid proof. Advanced and elective courses allow students to realize their potential in mathematics. A deep conceptual understanding of theory is emphasized alongside exposure to applications.

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## Math Courses

### Algebra I

This course focuses on strengthening students' computational skills in solving equations, and graphing linear equations. Topics include: simplifying expressions (polynomial, exponents, roots, rational), solving linear and quadratic equations and inequalities, solving systems of linear equations, and graphing linear functions. Students apply these skills to a variety of projects and problems throughout the year. Since the course is a prerequisite to the three-year required math sequence, it does not count as one of the three math credits required for graduation.

### Geometry

Accelerated Geometry

Geometry exposes students to topics in Euclidean geometry and to logic with deductive proofs. Students study polygon properties and theorems (especially triangles), circles, congruence and similarity, trigonometric ratios, and areas and volumes. Students work with compass and straightedge constructions and Geometer’s Sketchpad as investigation tools and proof verification. A variety of algebra skills are also reviewed. Daily homework requires students to apply concepts addressed in class to problems in creative ways. Student projects on various topics are included as part of their assessment. Accelerated Geometry stresses depth of coverage of applications, problems, and proofs. Strong Algebra I skills and a desire to be challenged are required, therefore, as well as the ability to work independently, creatively, and with genuine interest on difficult problems.

### Algebra II

Accelerated Algebra II

Algebra II develops algebraic and geometric skills as a preparation for Precalculus/Trigonometry. Students study the major functions: linear, quadratic, polynomial, rational, radical, exponential and logarithmic. Topics include function operations, inverse functions and domain and range. Simplifying, solving and graphing techniques are developed throughout the year. Students become familiar with using the TI-84 graphing calculator to supplement their understanding. Accelerated Algebra II will move at a brisk pace and will assume a strong foundation of algebraic skills. Students should only take Accelerated Algebra II if they have achieved a high level of success in previous mathematics courses, earning a B+ or higher in Geometry and a B+ or higher in Algebra 1, and if they are seeking ways of being challenged in mathematics. There will be an emphasis on abstraction, derivation, proof and problem solving and additional topics will be examined if time permits.

### Precalculus and Trigonometry

This course is a preparation and prerequisite for Calculus. Students study analytic geometry, conic sections, trigonometry, polar coordinates and complex numbers, sequences and series, and an introduction to limits and derivatives. Vectors and matrices will be discussed if time permits. Graphing calculators are used throughout the course. Students must supply A TI-84 or similar graphing calculator. Prerequisite: B or higher in Algebra II.

### Discrete Mathematics

Discrete Mathematics is the study of mathematical properties of sets and systems that have only a finite number of elements. This course will explore famous, modern day mathematics problems (four color theorem, traveling salesman problem, etc) and will introduce students to a variety of topics, which may include: topology and graph theory, game theory, social choice theory, probability, logic, cryptography and matrices. There will be an emphasis on problem solving, project-based learning, collaboration and applications. Prerequisite: completion of Algebra II or departmental permission.

### Statistics

Statistics is the science of data -- how to collect, organize, analyze, and interpret them. Students are encouraged, through theory and application, to employ methods of working with data and statistical reasoning to help separate “sense” (valid and reliable research) from “nonsense” (invalid and unreliable research) in the flood of data we live in. The following topics are studied: variables, measurement, scales, frequency distributions, central tendency, variability, normal distribution and standard scores, probability, correlation and regression, distribution of sample means, confidence interval estimates, and hypothesis testing. To supplement these studies, projects and computer work will constitute a major part of the course. In addition, students will (1) present case studies involving questionable data in an attempt to analyze, critique, and/or validate content, (2) create and implement their own research study, and (3) complete computer activities following specific units. Students must supply a TI-84 calculator or equivalent. Prerequisite: successful completion of Pre-Calculus or Discrete Mathematics or departmental permission.

### Calculus

This course builds upon concepts mastered in the algebra and precalculus courses. It covers limits of functions, and differentiation and integration of polynomial, rational, root, exponential, logarithmic and trigonometric functions. Students will work through the theory of differentiation and integration, and they will be exposed to applications of both. A TI-84 or similar graphing calculator is required. If students wish to take the AP exam, this course will prepare students for the AB Calculus exam. Prerequisite: B or higher in Precalculus. (AP)

### Advanced Calculus/UPenn Blended

This course, equivalent to a second or third course of college calculus, develops the ideas acquired in calculus and extends to further techniques. Topics include: advanced integration techniques, l’Hopital’s Rule, improper integrals, infinite series, conics, parametric and polar equations, vectors and vector-valued functions. If time permits, students may explore functions of several variables and multiple integration. This course is a “blended” course, and the Coursera online course, Calculus of a Single Variable, developed by the University of Pennsylvania will be integrated throughout the year as a way to review and explore applications, especially those related to engineering, of calculus. A TI-84 or similar graphing calculator is required. If students wish to take the AP exam, this course will prepare students to take the BC Calculus exam. Prerequisite: B or higher in Calculus. (AP)

### Introduction to Complex Analysis

The goal of this course is to understand and appreciate theory and applications of complex numbers. Topics covered in this course include complex numbers, Euler’s formula, transformations in Euclidean geometry, complex functions as transformations, Mӧbius transformations , and inversions. This course is designed to challenge and engage students who are very interested in advanced mathematics theory. Prerequisite: B or higher in Accelerated Algebra 2.

### Multivariable Calculus

This course builds upon concepts mastered in the calculus and advanced calculus courses. The following are topics covered in the course: vector-valued functions, functions of several variables, multiple integration, and an introduction to vector analysis. Students acquire a rigorous mathematical foundation so that they may pursue more advanced work. A TI-84 or similar graphing calculator is required. Prerequisite: B or higher in Advanced Calculus/UPenn Blended and a love of mathematics.