Upper School Mathematics
Problem solving and solving problems.
Our goal as a mathematics department is to foster an ongoing interest in the study of mathematics and the skills it engenders, with as much consideration as possible for the different learning styles and needs of the individual student. We recognize that students in today’s world need to be able to work collaboratively, synthesize information, analyze, spot connections between topics, and problem solve. In particular, our goal is to create students who take great joy in applying their skills in a logical and methodical way to complex and unfamiliar problems.
The vast majority of incoming students will start in the regular (non-honors) ninth grade math program—whether that is in Analytic Geometry & Algebra or Integrated Geometry & Algebra with the summer Geometry component—and will be on track to take AP Calculus in the 12th grade. Our sequence goes from those freshmen courses to Algebra 2 in the 10th grade; Functions, Statistics & Trigonometry or Pre-Calculus in the 11th; and Pre-Calculus, Statistics or Calculus (AP or non-AP) in the 12th grade. Often, families seek to accelerate their child through our math program. However, in our experience, skipping classes or seeking placement in honors when this is not indicated by our placement process and teacher recommendation is rarely a good idea and ultimately may do nothing but damage to a child’s self-esteem and confidence. Going faster is not going smarter in most cases, because concepts are often ill formed or poorly understood. Students need to build a skill set in proper sequence at an appropriate pace in order for them to learn most effectively.
Analytic Geometry and Algebra
A primary goal of the freshmen math program at Menlo is to shape a student’s conception of what it means to study mathematics. We want students to shift from thinking of their teacher as a sole locus of knowledge, to thinking that mathematics is a subject in which each student can construct his or her own mathematical understandings. To that end, the AG&A class is, by choice, textbook free. Within each unit of study, students are given daily problem sets from their teachers. New definitions are explained in the context of new problems. Students spend little to no time “taking notes” in a traditional sense. Class time is devoted to students solving problems and engaging in meaningful discussions about these problems, either with a nearby peer, in a small group of peers, or, sometimes, as an entire class. Because any study materials the students have are in large part self-created (they must work through the written problems, rather than reading a textbook author’s solution), we find that the materials are both relevant and meaningful. Topics studied include but are not limited to: systems of equations, angles in a plane, properties of quadrilaterals and regular polygons, properties of parallel lines, problem solving with circular sectors, triangle congruence, polygon similarity, right triangle trigonometry, coordinate geometry, transformations, graphing lines, and finding volumes of solid figures.
Prerequisites: Placement into this class happens via departmental placement test, or via completion of IGA.
Analytic Geometry and Algebra (H)
Honors Analytic Geometry & Algebra covers the same course content as the non-honors course. Students move through basic principles and new concepts quickly, spending less time gaining basic practice, and more time engaging with larger multi-step problems. The Honors Analytic Geometry and Algebra course is as much a course in mathematical problem-solving as it is a course in traditional Euclidean geometry.
Prerequisites: Place into this class via departmental placement test.
Integrated Geometry and Algebra
Integrated Geometry and Algebra is designed for students who enter the ninth grade needing additional review and practice in foundational algebra skills. Although the focus of the first several units is on developing mechanical proficiency, we expect students to move beyond basic procedural competence to develop a strong conceptual understanding of the material. In addition, students will learn how to document their work and how to study effectively for assessments in mathematics. Beginning in the second quarter, the curriculum is integrated with geometry through examination of the following topics: finding slopes of parallel and perpendicular lines, determining points of intersection by solving linear systems, manipulation of radicals and application of the Pythagorean Theorem, solution of Pythagorean inequalities, absolute value as a measure of distance, transformations of graphs of equations (lines, parabolas, and absolute value graphs), and the use of proportions in solving problems involving triangle similarity and right triangle trigonometry. An emphasis is placed on the development of problem solving strategies through applications of algebra to physical science, geometry, and finance. Connections to the ninth grade Physics curriculum are made through units covering mechanics and wave phenomena.
Prerequisite: Place into the class via departmental placement test.
This course introduces students to several topics in secondary mathematics: Functions and their transformations, Inverse Functions, Inequalities, Quadratics, Polynomials, Exponentials, Logarithms, Probability and Combinatorics, and Sequences and Series. Emphasis is placed on process, depth of understanding, and the development of mathematical intuition, not on memorization of rote facts. Students are encouraged to use mathematical methods that are meaningful for them.
Prerequisites: Completion of AGA or AGA (H) or completion of IGA plus recommendation from IGA instructor to take Summer Geometry plus successful completion of Summer Geometry.
Algebra 2 (H)
This is an Honors course in Algebra 2. Topics studied include those listed for Algebra 2 plus a thorough treatment of rational functions, principles of end behavior as a precursor to studying limits, radian angle measure and trigonometric functions and their transformations. Problem Sets are designed to challenge students depth and flexibility of understanding, in addition to their mathematical creativity.
Prerequisites: Recommendation from freshmen math instructor in conjunction with the department chair.
Building on the algebraic skills acquired in previous classes, this course attempts to deepen and strengthen students’ conceptual understanding and computational fluency. We extend and reinforce key algebraic concepts in the definition, application and manipulation of polynomials and rational functions, refining students’ graphical skills and exploiting technology as an aid to visualization and as an invaluable tool in tackling more complex problems. The heart of the course is devoted to a thorough presentation of the elementary transcendental functions: exponential, logarithmic, trigonometric, and inverse trigonometric functions. During the second semester students also explore some topics from discrete mathematics including sequences, series, elementary counting techniques and probability.
Prerequisites: Successful completion of Algebra 2.
This course offers rigorous preparation for the traditional calculus sequence. Students refine their computational skills, extend their ability to exploit appropriate technology, and practice communicating their insights in written and oral form. After a brisk review of the unifying concept of function, students explore the algebraic complexities of polynomials and rational functions and discover new applications of the exponential and logarithmic functions. Emphasizing careful derivations, students then embark on a sophisticated study of the trigonometric functions and their applications. Students prepare to tackle calculus by exploring limits. During the second semester students will also spend time studying advanced topics selected from areas such as conic sections, linear programming, series, vectors, matrices, and probability and statistics.
Prerequisites: Recommendation of Algebra 2 teacher.
Pre-Calculus (H) is an honors level pre-calculus course. It is aimed at the independent learners who are comfortable with handling symbolic language and abstract thinking challenges. Students work together in small groups in an effort to discover new concepts and explain new ideas from multiple perspectives. The course is aimed at honing the individual student’s mathematical creativity and providing a broad base of skills prior to taking calculus courses and higher. Students begin the year by engaging with contest level math problems that address many of the topics from Honors Algebra 2. In addition to extending previously studied topics such as transformations of functions, quadratic maximization, graphing rational functions, and exponential and logarithmic functions, the course includes a thorough introduction to limits and the definition of the derivative, an extension of trigonometry including trigonometric identities, the Law of Sines and the Law of Cosines, parametric and polar functions and their graphs, an exploration of methods of proof, and a thorough treatment of vectors and matrices.
Prerequisites: Recommendation from the A2H instructor.
One of the most beautiful and powerful branches of mathematics, calculus has long been the preeminent tool of scientists and engineers. In recent decades it has emerged to play a key role in the study of biology, medicine, economics, and finance. This course introduces students to the elements of differential and integral calculus, placing particular emphasis on applications drawn from the management, social, and life sciences. Students will sharpen (and develop a new appreciation for) their pre-calculus skills as they master and learn to apply derivatives, integrals, and the fundamental theorem of calculus. The focus is kept on conceptual understanding as students develop and apply new algebraic, numerical, and geometric skills. During the second semester the course also provides brief introductions to more advanced topics in mathematics, including partial derivatives, differential equations, and infinite series.
Prerequisites: Completion of Analytic Precalculus, or completion of Principles of Precalculus with a B or higher, or permission from the department.
Statistics is an application of mathematics for understanding the connections in business, the world around us, and the factors that affect change and consideration of options. Students make substantial use of the TI-83 calculator and Fathom statistical software. The course is designed to equip students with many skills:
• Quantitative literacy for use throughout their adult lives.
• Participation advantages for effective and efficient public policy debates.
• Evaluation skills for personal productivity in areas of insurance, health matters, banking, mortgage, leasing, and various other economic matters.
• Analysis of economic trends, predictions, and estimations. Students are exposed to the newspaper and various forms of media and the critical skills required for accurate interpretation and full comprehension of articles that require statistical thinking.
• Designing experiments based upon statistical findings, conducting polls, evaluating scientific claims, and presenting data. Students also examine a large number of case studies, both to appreciate the breadth and power of statistical techniques and to understand the widespread misuse of statistical ideas.
Prerequisites: Completion of Algebra 2.
AP Calculus AB
AB Calculus is a rigorous mathematics course that prepares students for the AP Calculus (AB) exam. We encourage students who have been successful with the previous pre-calculus course to consider an AP math class the following year. AB Calculus can be thought of as a turning point in a student’s study of mathematics, as the course demands a highly developed ability to think abstractly and aptly draw on skill sets developed in previous courses to tackle the calculus tasks before them. Teachers are dedicated to encouraging the development of a self-reliant learning style with strong inductive, deductive, and abstract reasoning skills to serve students well in a collegiate environment.
Prerequisites: Recommendation from Analytic Precalculus instructor or completion of Honors Precalculus.
AP Calculus BC
Beyond becoming prepared for the Advanced Placement examination, students in this course will be expected to acquire a deep understanding of the mathematics of single variable calculus. Topics studied include but are not limited to: the historical development of calculus, and its philosophical implications upon key topics in the history of both science and mathematics; the topics in single variable calculus as defined by the college board’s AP BC Calculus test; advanced Math Projects in areas of student interest.
Prerequisites: Recommendation from Precalculus instructor.
This course prepares students for the AP examination in statistics. The course covers many topics including: an overview of the same material as the non-AP course; advanced topics in probability, more extensive treatment of regression, applications of simulation and statistical modeling, and several additional topics in statistical inference; extensive practice with AP-style open-ended case studies.
If you believe ignorance is bliss, you will find this course upsetting. Learning statistics may render you skeptical of claims that you read in the news or hear from friends. You may wonder if one can ever really be certain about anything. In other math classes, there are clear right and wrong answers. In AP Statistics, the validity of an answer hinges on the clarity and logic of one’s argument.
AP Statistics is the study of data in the face of variation. Questions we ponder include: Could a result be due to chance? Could results be biased? How can you guess what will happen in the future and how confident can you be of that guess?
Prerequisites: Recommendation from A2H or Analytic Pre-Calculus teacher , or completion of Honors Pre-Calculus.
Advanced Topics in Math (H)
Advanced Topics in Mathematics is designed to provide students who have completed the traditional calculus sequence with the opportunity to continue their mathematical studies, deepening and broadening their understanding and preparing them for the possible further study of mathematics. Topics covered may include multivariable calculus, linear algebra, differential equations, topics in discrete mathematics, and calculus-based probability theory.
Prerequisites: Completion of Honors Precalculus plus either concurrently enrolled in BC Calculus or have completed BC Calculus.
Adv. Economic Theory (H)
Economics is the study of choices under constraints. We will employ mathematical optimization methods and multivariate calculus to analyze how consumers, producers, and governments make decisions. We will consider empirical studies and theoretical models. Topics include price determination, resource allocation, market competition, public welfare, government interventions, market failures, game theory, fiscal policy, monetary policy, international trading, and foreign exchange.
Prerequisites: Completion of AP Calculus BC or concurrent enrollment in Adv. Economic Theory and AP Calculus BC
Math placement is not an exact science, and no placement test ever gives enough information about how kids will thrive in our environment. The truth is that the vast majority of incoming students arrive with an A average in math, and we find ourselves having to tell students that while an A grade might reward due diligence at a lower level, it is only an indicator rather than a guarantee of success at higher levels. In the Upper School, we offer several different math classes, with fewer different levels in the lower grades and more for juniors and seniors. As our program is both challenging and inevitably quite different from the varied programs we see at our numerous feeder schools, our approach is to work with all our students in our core program for their first two years of high school before offering many different levels of classes.
The Honors Program
Students who are quite insightful in math will be placed in our honors program. The honors track is extremely rigorous and fast paced and is designed for mathematical enthusiasts. Honors Analytic Geometry & Algebra meets the needs of students who typically are able to understand new concepts in their math courses quickly and are yearning to go into more depth with just about every math topic that they meet. These students will develop their abstract thinking skills and will find plenty of challenge among their like-minded peers. Students who stay in our honors track go from Honors Analytic Geometry & Algebra to Honors Algebra 2 sophomore year to Honor Pre-Calculus in junior year to AP Calculus BC in senior year. Students in AP Calculus BC are also allowed to concurrently enroll in our highest math class, Advanced Topics in Math.
FAQs and Freshman Information
How do you determine which math class my child is placed in at the beginning of his Freshman Year?
Placement in math courses is based on three pieces of information: (1) standardized test scores; (2) teacher recommendations; and (3) performance on our placement exam. At the Welcoming Event in late April, every student will take a Placement Test to determine which math course they are best suited for. We believe in appropriately challenging students and do our best to place students in the best class for them. However, because the process is not perfect, during the first few weeks of school, freshmen teachers are very vigilant about determining if their students have been appropriately placed. Thus, inevitably there are a few changes that happen early on in the school year.
What is Menlo’s honors program in mathematics like?
Our honors program is designed for those students who are mathematics enthusiasts and are developmentally ready to delve deep into math concepts. Not only is the pace faster in these courses, but also every day students are expected to apply their knowledge to unfamiliar situations. In addition, honors students must be able to think flexibly across different strands of math, often picking and choosing from among a variety of mathematical tools in pursuit of a solution. Teachers of these courses usually serve as coaches, guiding the students along and offering help as needed, while the students work collaboratively during class to tackle challenging problems. As in our other courses, our honors students also are expected to display their work in a logical, organized manner.
My child has already taken Algebra and Geometry at her middle school. Does she have to take your Analytic Geometry & Algebra course?
Virtually all freshmen will start off in our Analytic Geometry & Algebra course. We recognize that many middle school students have had courses involving algebra and geometry. However, our course, and our program as a whole, is different from and substantially more challenging than courses at area middle schools. We believe in the importance of strong foundational skills, but more importantly our program emphasizes applying those skills to problem solving, synthesizing material, analyzing situations, working collaboratively, and moving flexibly between topics in mathematics. For those students who are still developing a solid foundation of algebra skills, we recommend taking our Integrated Geometry & Algebra course. For those who are ready to make the leap to problem solving with their algebraic and geometry knowledge, we believe Analytic Geometry & Algebra is the perfect course for them.
If my child does not start off in the honors track, can he/she move into that track down the road?
Yes, every year we have a few students who move into the honors track from the regular one. In order for this to happen, the student needs both to receive a strong recommendation from his/her previous math teacher at Menlo as well as to perform well on an honors placement test. In addition, usually some extra summer work is required.
If my child is not in the honors track, can she still take Calculus?
Yes! In fact, our top students in our non-honors track take AP Calculus AB as seniors. There also is the opportunity to take our non-AP Calculus course for those students not yet ready for the rigors of a college-level calculus course.
What is Integrated Geometry & Algebra?
We believe that strong algebra skills are the foundation of success both in our academic program here at Menlo as well as in advanced study. Mathematical development in students, like physical development, is a fits and spurts process. Some students arrive to Menlo with holes and weaknesses in their algebra knowledge that will prove problematic in future math and science classes. Integrated Geometry & Algebra is a course designed to develop strong foundational algebra skills that students can build upon. This course includes not only algebra, but also arithmetic/mental math skills and problem solving experience. In addition, beginning in the second quarter, the curriculum is integrated with geometry. Unlike most middle school Algebra I and Geometry courses, in Integrated Geometry & Algebra there will be a strong focus on learning the material on a deeper conceptual level and applying it to the freshman Physics curriculum. We want students to move beyond basic procedural competence to develop a strong understanding of material. In conjunction with the algebra and geometry topics, students will learn how to document their work and how to study effectively for assessments in mathematics.
If my child starts out in Integrated Geometry & Algebra, does that mean he always will be a year behind his peers?
Because we as a department and school are very sensitive to the potential social and confidence ramifications of a student being a year behind his/her peers in a particular subject, we view Integrated Geometry & Algebra as a 10 ½ month course for those students who are ready to make the leap to Algebra 2 as sophomores. Thus, for those students who excel in Integrated Geometry & Algebra and wish to catch up with their peers after Freshmen Year, we offer a three-week summer intensive Geometry course. If a student successfully completes this course, he/she can start in Algebra 2 Sophomore Year with his/her peers. Students who do not take the summer component of the course would take Analytic Geometry & Algebra as a sophomore.
My child really wants to be able to take all of your math courses, including AP Statistics and Advanced Topics. Is that possible if they start out in your Honors Analytic Geometry & Algebra class?
For those students who remain in our Honors Program, they have the opportunity to take AP Statistics concurrently with either Honors Pre-Calculus or AP Calculus. In addition, students can, and many do, take both AP Calculus BC concurrently with Advanced Topics during their Senior Year.
AP Statistics also can be taken by those students who complete Analytic Precalculus and receive a recommendation from their instructor.