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Mathematics

The Mathematics Department at Cate will encourage you to be a confident and competent mathematician, and, as a result, a better thinker.

The coursework is designed to prepare you for tasks in your life in which you will be using mathematics or be required to think mathematically.

You will find yourself using technology in class, studying in groups, competing in contests, and working on the board solving engaging problems.

You will take one course a year through your junior year, the final course determined by the initial Cate entry level.

For the last 25 years, Cate has been a leader in local California Mathematics League competitions, winning many county titles and finishing among the top schools in California. Last year, over 150 students and faculty participated – more than half of the School. Students also compete in the Westmont College Mathematics Contest and the American High School Mathematics Contest each winter.

Interested in more? Check out the Math on the Mesa!

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Inquiry in Mathematics

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Constructing understanding through a problem based learning (PBL) curriculum is challenging, inspires curiosity, and is very rewarding. Each day, our students arrive to class with a handful of problems prepared for sharing and discussion. We sit in a circle, showcasing work on the board using iPads with Apple TV, and talk through specific problem solving strategies, highlight content connections, and seek feedback on alternate methods.

Students see the true interconnectedness of math every day and they are also exposed to important life skills like risk-taking, collaboration, and effective written communication. There are a lot of “aha” moments and instances of students taking responsibility for their own learning.

Frequently Asked Questions about Cate PBL


This course introduces the basic concepts of algebra, including types of numbers and their properties, variables, operations with expressions, exponents, radicals, polynomials, solving linear and quadratic equations, solving single variable inequalities and linear inequalities, working with rational expressions, and graphing linear functions. This course introduces functions both algebraically and graphically. Emphasis is placed on developing skills needed for future work in math, problem-solving techniques, logic, and applications to real-world situations. Technology is used in the classroom to help students make and exploit connections among various representations of functions. This course is intended for students who have not had a full year of Algebra 1 or need more review.


Full year – .33 credits/trimester
Problem Based Learning 1: Foundations of Algebra and Geometry is a problem-solving course in which students take on the responsibility of investigative learning and get to experience the excitement of authentic discovery. We review basic geometry (surface area, volume) and the fundamentals of algebra, pushing students to discover how and why certain well-known processes work the way they do. We also study essential algebraic functions (lines, absolute value, quadratics) in-depth, emphasizing the role these patterns play in mathematical modeling and as problem-solving tools. Students are expected to be at the very center of a cooperative process, discussing, writing about, and presenting well-reasoned explanations. The course uses a variety of materials, primarily problems from the Math 1 text written by the math department at Phillips Exeter Academy and Desmos graphing technology. The course is intended as a transition to Geometry or Algebra 2 and Trigonometry for students who would benefit from a stronger algebraic foundation. For students who excel and find the curriculum rewarding, the class can also serve as an entry point to our Honors Problem Based Learning strand. Prerequisite: Algebra 1 or equivalent.


Full year – .33 credits/trimester
Geometry introduces and stresses the basic topics and concepts of plane and solid geometry, coordinate geometry, including angles, triangles, lines, circles, polygons, area, similarity, congruence, right angle trigonometry. Emphasis is placed on developing problem-solving skills, logical understanding of theorems and proofs, the deductive reasoning process, and relating the material to realistic applications. Algebra skills are reviewed and descriptive statistics are reviewed throughout the year to provide depth and connections. Prerequisite: Algebra 1 or PBL 1: Foundations of Algebra and Geometry.


Full year – .33 credits/trimester
Problem Based Learning 2: Honors Geometry and Algebra 2 is a problem-solving course that provides students with a more rigorous, integrated, and in-depth exposure to geometry in two and three dimensions. Students investigate lines, polygons, vectors, circles, and parabolas while also analyzing right-triangle trigonometry. Through the exploration of linear motion via parametric equations, students are introduced to optimization and transformations. We use a variety of materials, including iPad technology, graphing software like Desmos, and problems from “Math 2” written by the math department at Phillips Exeter Academy, which focuses on pattern-building through the integration of algebra and geometry. In this course, students take on the responsibility of thinking critically, creatively, and collaboratively to solve meaningful problems on their own, learning content and making connections through the problem-solving context. This course may serve as a transition into Algebra 2 and Trigonometry or, for those who excel and find the curriculum rewarding, it can also serve as an entry point to our Honors Problem Based Learning strand. Prerequisite: PBL 1 or Geometry or the equivalent and consent of the department.


Full year – .33 credits/trimester
Algebra 2 and Trigonometry builds a strong foundation of algebraic skills and understanding by reviewing and extending the topics from previous courses. Students work extensively with the “toolkit functions”. Linear, absolute value, quadratic, cubic, roots, rational, exponential, logarithmic, and trig functions are spiraled throughout the course, ensuring that students are able to connect their graphs, tables, and equations. Students learn to create new functions from their toolkit functions through transformations and explore how these new functions model real life situations. In all cases, the relationship between multiple representations (graphical, symbolic, numeric, and applied) is heavily emphasized. The mechanics of manipulating symbolic notation is traditionally the most challenging part of an Algebra 2 course, and we continually practice these cumulative skills throughout the year, connecting the symbolic representation of these functions to their more accessible graphical representations at every opportunity. The result is stronger algebraic, graphical, and problem-solving skills, all of which are essential to future studies in mathematics. Prerequisite: Geometry or the equivalent.


Full year – .33 credits/trimester
Problem Based Learning 3: Honors Algebra 2, Trigonometry and Pre-Calculus is a problem-based learning course that expands on the algebra and geometry content in PBL 2H to include nonlinear motion and nonlinear functions. Students investigate circular motion by using trigonometric functions, model various scenarios using exponential functions, straighten nonlinear data using logarithms, and describe geometric transformations using matrices. In preparation for the study of calculus, students are introduced to instantaneous rates of change through the exploration of slopes on nonlinear graphs. We use a variety of materials, including iPad technology and problems that focus on pattern-building through the integration of precalculus and trigonometry from the “Math 3” text written by members of the Math Department at Phillips Exeter Academy. As is the case in all our PBL courses, in this class students take on the responsibility of thinking critically, creatively, and collaboratively to solve meaningful problems on their own, learning content and making connections through the problem-solving context. Prerequisite: PBL 2H: Honors Geometry and Algebra 2 and consent of department.


Full year – .33 credits/trimester
Pre-Calculus: Functions builds on the foundation laid in Algebra 2 and Trigonometry. Students delve more deeply into transformations, inverse functions, and composition of functions while continuing to strengthen their graphical reasoning and symbolic manipulation skills. Writing equations, solving or evaluating them, and interpreting results are emphasized as students work with problems in context. Students write equations to model physical situations (tides, population growth, projectile motion, etc) and to set up and solve optimization problems, and they continue to use their TI-84 as a tool in problem solving. This course provides a thorough study of functions as a preparation for calculus. Entering students should have a strong background in Algebra, usually meaning B or better in Algebra 2 and Trigonometry. Prerequisite: Algebra 2 and Trigonometry.


Full year – .33 credits/trimester
Problem Based Learning 4: Advanced Problem Based Calculus begins with a foray into complex numbers, polar coordinates, recursion, functional notation, slope, velocity, asymptotes, the fundamental constant e and applications of the preceding before officially delving into differential and integral calculus. Through a carefully crafted sequence of problems, students become fluent in the conceptual and notational language of differential equations. Students also discover, explore, and apply the Fundamental Theorem of Calculus, which connects differential (rate problems) and integral (accumulation problems) calculus. Throughout the curriculum, students encounter problems in context (physics, economics, environmental studies) to emphasize the application power of calculus. They also continue to work with multiple representations of functions (graphical, numerical, symbolic) and are empowered to make strategic decisions about what tools they want to employ when solving problems. Students in PBL 4H may choose to take the AB Advanced Placement examination in the spring.


Full year – .33 credits/trimester
Probability, Statistics, and Calculus is intended as a mathematics elective for seniors who do not choose to pursue one of the advanced options. In the fall and winter trimesters, the course provides an introduction to the discrete math topics of probability and statistics, including the analysis of data, the conducting of surveys, sampling, experiments, and inference. In the spring the major themes of calculus (the limit, derivative, and integral) are introduced in a conceptual approach using applications, with extensive use of the graphing calculator. Prerequisite: Algebra 2 and Trigonometry


Full year – .33 credits/trimester
Advanced Calculus 1 AB is a college-level mathematics course designed as an introduction to a variety of topics relating to integral and differential calculus including functions, graphs, limits, the conception and application of derivatives, the interpretation and application of integrals, and the fundamental theorem of calculus. The course outline focuses on the tools of calculus for problem solving. Students will be prepared to take the Advanced Placement Calculus AB examination in the spring. Prerequisite: Pre-Calculus: Functions and consent of department.


Full year – .33 credits/trimester
Advanced Calculus 2 BC seeks to challenge our most advanced student by providing a rigorous course in college calculus with relevant applications and elegant connections. Emphasis is on problem-solving skills, preparation for college math, and using proof to understand why methods work. The course prepares students for the Calculus BC Advanced Placement exam but exceeds a typical AP course by including a higher level of theory, proof of methods, practical applications, and connections between Algebra, Geometry, Functions, and Trigonometry. Topics include limits, derivatives, integrals, series, parametric functions, polar curves, and vectors. Students are encouraged to discover ideas and connections through challenging problems, labs, and inquiry activities. Pre-requisite: PBL 4H: Advanced PBL Calculus or Advanced Calculus 1 AB and consent of the department


Full year – .33 credits/trimester
Advanced Statistics is equivalent to a college level, one-semester, introductory course in statistics. The purpose of Advanced Statistics is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: exploring data (observing patterns and departures from patterns), planning a study (deciding what and how to measure), anticipating patterns in advance (producing models using probability and simulation), and statistical inference (confirming models). Students will be prepared to take the Advanced Placement Statistics examination in the spring. Prerequisite: PreCalculus: Functions and consent of department. * This course is not offered every year, depending on student enrollment.


Full year – .33 credits/trimester
Advanced Statistics, Multivariable Calculus, and Linear (Math 61H) is intended for students who have completed Calculus BC and have a strong interest in higher mathematics. The fall and winter terms of this course comprise an equivalent to a college level, one-semester, introductory course in statistics. The purpose of the Advanced Statistics section of this course is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: exploring data (observing patterns and departures from patterns), planning a study (deciding what and how to measure), anticipating patterns in advance (producing models using probability and simulation), and statistical inference (confirming models). Students will be prepared to take the Advanced Placement Statistics examination in the spring. The spring term is designed to expose our most advanced students to areas of higher collegiate mathematics beyond Calculus and Statistics. We specifically introduce Linear Algebra and Multivariable Calculus through a series of questions and problems, emphasizing graphs, visuals, and technology. In the Linear Algebra portion, we connect matrices and vectors through applications in economics and science. The Multivariable unit includes gradient, partial derivatives, double and triple integrals with applications to Physics. We use Mathematica software to provide 3-D visuals and expose students to powerful computing tools used in college mathematics. Each student is provided a license for the term. Emphasis will also be placed on proof, a required component of theoretical mathematics. Prerequisite: Advanced Calculus 2 BC and consent of the department.


This advanced course in computational thinking will provide students with the programming skills to ask and answer a broader class of questions than can be addressed by conventional means. Assuming no prior background in programming, students will learn the core concepts of variables, loops, functions, recursion, and conditional execution. Smaller projects draw from physics and mathematics, but the first trimester concludes in a weeks-long video game development project. In the second term, students build on the skills of the first trimester as they extend into string manipulation, lists, dictionaries, and object-oriented design. Cryptography figures prominently as a context in which these skills are applied. Students also develop an understanding of algorithmic design, focusing on classic algorithms of searching and sorting. This winter term culminates in an interdisciplinary project of the student’s own design. For the third trimester, we will delve more deeply into data structures, algorithms, and other more advanced topics. Object-oriented design and the concepts of encapsulation and inheritance are fleshed out more completely. Students are given considerable leeway to pursue specific interests, for example learning a new language or scientific programming. Open to juniors and seniors. Prerequisite: Pre-Calculus: Functions and permission of the department