The Berkeley Carroll School

Mathematics



The Upper School mathematics program at Berkeley Carroll strives to develop life-long problem-solvers who approach math problems, unfamiliar and familiar, with confidence and perseverance. Students ask why and are pushed to discover mathematical concepts, with help from their teachers, through deductive proofs and experiments. Critical and analytic thinking demands a solid foundation of logic and reasoning, which students practice by regularly explaining and justifying their conclusions.

From geometry through calculus, our math classes have on-level and advanced options, and students’ placement is evaluated each year. Our most advanced students complete calculus by the end of junior year, most complete calculus by the end of senior year, and it is possible for junior or seniors to take two math classes if they wish.

The members of the Upper School math team participate in wide variety of competitions each year, allowing our most interested and committed math students the opportunity for city, state, or national recognition.


Courses

Algebra

Algebra I

How do you solve for the unknown?

How do you represent a real-world situation with an equation and a graph?

This foundational course focuses on the skills that students will need to be successful in all future math course. Students will explore writing and solving linear equations, powers and exponents, quadratic equations, polynomials and factoring, graphing and solving linear inequalities, and functions. Throughout the course, the mathematical concepts will be taught with an emphasis on real-world applications and technology. By the end of the this course, students will be prepared for Geometry.

Algebra II

How can you solve, graph, and formulate mathematical functions? What are the advantages and limitations when using mathematical functions to model real-life situations?

This course reviews and builds on topics presented in Algebra I and goes on to the study of linear programming and imaginary numbers. Students explore the behavior and applications of polynomial, exponential, logarithmic, and trigonometric functions.

Advanced Algebra II

How can you solve, graph, and formulate mathematical functions?
What are the advantages and limitations when using mathematical functions to model real-life situations?

This course reviews and builds on topics presented in Algebra I and goes on to the study of linear programming and imaginary numbers. Students explore the behavior and applications of polynomial, exponential, logarithmic, and trigonometric functions. Advanced level mathematics classes spend less time reviewing foundational material and investigate many topics at a greater depth than standard classes. Math department approval is required.

Geometry

Geometry

How can you use the formulas for the areas of traditional plane figures to find the areas of nontraditional plan figures?
How does the concept of similarity lead to trigonometry and proportions in a circle?

Geometry is a hands-on course in which students will explore the concepts of Euclidean geometry. This class also integrates the study of important algebra topics to prepare students for Algebra II. Throughout the year, students will use geometric tools and various manipulatives to discover and demonstrate geometric properties. Students will use computer programs, iPad apps and rulers, compasses, and protractors to complete projects and explore concepts. These concepts include the properties of parallel lines, the Pythagorean Theorem, right triangle trigonometry, area, volume, and the many interesting properties of circles. By the end of the course, students will have explored the basics of Euclidean geometry and will be prepared for Algebra II.

Advanced Geometry

What is proof? How do you use deductive reasoning to demonstrate the validity of previously accepted ideas?
How can you use the formulas for the areas of traditional plane figures to find the areas of nontraditional plane figures?
How does the concept of similarity lead to trigonometry and proportions in a circle?

This course covers geometric theory, its methods and applications. A careful development of deductive reasoning is presented and students will use two-column proofs to prove many important theorems and properties. Students will also use computer programs, iPad apps and rulers, compasses, and protractors to complete projects and explore concepts. These concepts include the properties of parallel lines, the Pythagorean Theorem, right triangle trigonometry, area, volume, and the many interesting properties of circles. By the end of the course, students will have thoroughly explored Euclidean geometry and will be prepared for Algebra II.

Pre-Calculus

Discrete Mathematics & Trigonometry

How can one use trigonometric functions to model real world phenomena?
How do small changes transform exponential and polynomial functions?
How can one best use a graphing calculator and other technology to gain a deeper understanding of mathematics?

In addition to reviewing some basic algebra skills this course covers functions (e.g. linear, quadratic, exponential) in great detail, with a strong emphasis on trigonometric functions and their applications. Additional topics include conic sections, exponential growth and decay, probability, and statistics.

Pre-Calculus

Why do certain functions model data better than others?
How can basic functions be transformed to represent complicated phenomena?
How can graphing calculators lead us astray?
Continuous functions vs. discrete mathematics: what, when, why?

This course expands the study of applications of polynomial, exponential, logarithmic, and trigonometric functions. Additional topics include sequences, series, conic sections, polar graphs, probability, and statistics. Students who successfully complete this course are prepared for the study of calculus.

Advanced Pre-Calculus

Why do certain functions model data better than others?
How can basic functions be transformed to represent complicated phenomena?
How can graphing calculators lead us astray?
Continuous functions vs. discrete mathematics: what, when, why?

This course expands the study of applications of polynomial, exponential, logarithmic, and trigonometric functions. Additional topics include sequences, series, conic sections, polar graphs, probability, and statistics. Students who successfully complete this course are prepared for the study of calculus. Advanced level mathematics classes spend less time reviewing foundational material and investigate many topics at a greater depth than standard classes.

Calculus

Calculus

What roles do zero and infinity play in mathematics?
What can derivatives and integrals tell us about functions?
What previously unapproachable problems are now very solvable using calculus?

Through multiple representations and extensive use of technology, this course develops the concepts and methods of differential and integral calculus. These results are applied to problems involving related rates of change, optimization, area, volume, and distance.

Advanced Calculus

What roles do zero and infinity play in mathematics?
What can derivatives and integrals tell us about functions?
What previously unapproachable problems are now very solvable using calculus?

This is a college-level mathematics course. Through multiple representations and extensive use of technology, the class rigorously develops the concepts and methods of differential and integral calculus, including transcendental functions. These results are applied to problems involving related rates of change, optimization, area, volume, and distance.

Advanced Calculus II

How do we see infinity in our everyday lives?
How can we represent trigonometric functions with polynomials?
How do we know when a sequence or series converges?

This course is a continuation of Calculus I. Students will work with derivatives of parametrically defined functions and further develop their integration techniques. As students study infinite series and Taylor polynomials, special attention will be paid to how we think about infinity in mathematics

Physical Applications of Calculus

How does one apply the tools learned in calculus to physics?
What questions posed in physics must be answered with calculus, and how does one answer them?

This course combines physics applications and advanced topics in calculus. We will consider realistic solutions to physical problems and we will weave in some of the historical connections between physics and math. This is a college level course that will require approval from both the Science and Math departments. It is taken concurrently with Advanced Calculus.

Electives

Statistics

How can statistics be used to predict the future and how certain can we be about our predictions?
How do we use a sample to make inferences about a population and when can we trust those inferences?

In this year-long course, students are introduced to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. The topics are divided into four major themes: exploratory analysis, planning a study, probability, and statistical inference.
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