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# Mathematics

The goal of the mathematics curriculum is to develop a sound understanding of mathematical concepts and foster the ability to apply them creatively to new situations. A major objective is for all students to gain confidence in their mathematical abilities and develop them to the fullest.

Courses are designed to provide students with a solid foundation in mathematical principles. By applying these principles in a variety of problem-solving situations, students become comfortable with mathematical techniques for evaluating problems presented in both practical and theoretical contexts. The department recognizes that the ability to reason logically and analytically is a skill valued beyond the confines of the mathematics classroom. To support critical thinking skills, teachers emphasize the ability to assess new problems and to determine how to proceed through a logical and organized path to a solution.

The use of technology, including the graphing calculator, is integrated throughout the curriculum. All Upper School students are required to own a TI-83 Plus or TI-84 Plus graphing calculator.
• ## Accelerated Math III

2 semesters, 1 credit
Prerequisite: Math II with a minimum grade of B+ and department recommendation

Accelerated Math III is the third course of a three-year sequence of courses. Students continue to develop skills working with linear, exponential and quadratic functions while beginning to work with higher-order polynomial functions. They learn the Fundamental Theorem of Algebra, apply strategies of factoring and polynomial division for finding roots, and learn to graph polynomials. Inverse functions are also stressed in this course and, through this lens, students are introduced to logarithmic functions. Students visualize 3-dimensional solids of revolution and practice using areas of cross sections to find an object’s volume. Students extend their skills with trigonometry by deriving and applying the Law of Sines and the Law of Cosines. They apply the sine, cosine and tangent functions to any angle, graph the sine, cosine and tangent functions under transformations, and use sine and cosine to model periodic behavior. Additionally, they explore trigonometric identities important to calculus and are introduced to the concept of limits. Compared to Math III, this course proceeds at a faster pace and with more depth on some topics and a greater emphasis on algebraic manipulation and trigonometry.
• ## AP Calculus AB

2 semesters, 1 credit
Prerequisite: Precalculus AB with a minimum grade of B- and department recommendation by current math teacher

This rigorous course follows the College Board Advanced Placement syllabus. It is equivalent to the first semester of a college-level calculus course. It begins with a study of limit theory and quickly moves on to differential calculus. The concept of the derivative is applied to related rates, extrema, optimization and curve-sketching problems. The second half of the year is an in-depth study of integral calculus, which includes calculating the area under the curve, the volume of solids and using differential equations in mathematical modeling problems. All students will sit for the AP Calculus AB exam in May.
• ## AP Calculus BC

2 semesters, 1 credit
Prerequisite: Precalculus BC Honors with a minimum grade of B and department recommendation by current math teacher

This rigorous, fast-paced course follows the College Board Advanced Placement syllabus. It is the equivalent of two semesters of college calculus. In addition to the work covered in the AP Calculus AB course, the following topics are included: additional techniques and applications of integration, differential equations, the calculus of polar coordinates and vector-valued functions, the convergence and divergence of infinite series of constants, and Taylor polynomials. All students will sit for the AP Calculus BC exam in May.
• ## AP Statistics

2 semesters, 1 credit
Prerequisite: Algebra II with Trigonometry with a minimum grade of A- or Precalculus AB/BC with B- and department recommendation by current math teacher

This course follows the College Board Advanced Placement syllabus and is equivalent to a one-semester, introductory, non-calculus-based college course in statistics. It is an introduction to statistical methods for business, health science and social science statistics.  The course introduces students to the major concepts and tools for collecting, summarizing, analyzing and drawing conclusions from data. There are four broad themes in this course: collecting data by sampling and experimentation, summarizing quantitative and categorical data, an introduction to probability and statistical inference. Students use technology, investigations, problem solving and writing as they build conceptual understanding. All students who take AP Statistics will sit for the AP exam given in May.
• ## Calculus

1 semester, 1/2 credit; 1st semester only
Prerequisite: Precalculus with a minimum grade of C

In this semester elective, students are introduced to limits through graphical analysis. They focus on derivatives, both graphically and algorithmically, with an emphasis on the general power rule and the chain rule, including the derivative of exponential functions. Students develop skills to understand applications of derivatives and their interpretations in context. Students apply derivatives to maximization and minimization problems in a variety of fields, including business and science.
• ## Data Science and Statistics

Prerequisite: Algebra II or Math III with a minimum grade of C

This introductory course emphasizes a modeling approach with the General Linear Model. Students work with authentic datasets, investigate relevant and timely questions, and build coding skills. Data analyses and visualizations will be completed throughout the course using the statistical programming language R, used in college and industry for data science applications. Additionally, students learn to find and clean data sets, store their work in Jupyter notebooks, apply critical thinking to interpret their observations and provide statistical support for their conclusions. No prior experience with coding is necessary.
• ## Linear Algebra (Honors)

2 semesters, 1 credit, offered every other year
Prerequisite: AP Calculus BC with a minimum grade of B and department recommendation  by current math teacher or teacher approval

This rigorous, college-level course explores various matrix methods of solving systems of equations in addition to covering matrix algebra, determinants, vector geometry, vector spaces, eigenvalues and linear transformations. Students examine the proofs of theorems and apply the theorems in solving problems and creating their own proofs. Students are also introduced to computer programming in C++.
• ## Math I

2 semesters, 1 credit
Prerequisite: Math B or Math 8

Math I is the first of a three-year sequence of courses. Students solidify and extend their understanding of functions, solving equations, geometric relationships and data analysis. Students continue to establish their understanding of linear and exponential sequences and functions. They expand their algebraic skills with a focus on solving equations and inequalities, including systems of linear equations and inequalities. Students develop explanations of geometric relationships of similarity and congruence through transformations and formal constructions. Students formulate and present mathematical arguments. Students extend their work from eighth grade working with linear models and linear regression equations. Additionally, students employ histograms, dot plots and box plots to describe and compare the shape, center and spread of univariate data. Students will have the opportunity to earn an honors designation through criteria shared in the first weeks of the course.
• ## Math I (Honors)

2 semesters, 1 credit
Prerequisite: Department recommendation by current math teacher

In this course, the curriculum of Math I is regularly extended through enrichment topics in each unit. Additionally, students in this course will focus on topics related to matrices. Students are introduced to matrix addition, subtraction and multiplication. Students use matrices to solve systems of linear equations with elementary row operations and the multiplicative inverse. Matrices are also used to represent vector transformations.
• ## Math II

2 semesters, 1 credit
Prerequisite: Math I with a minimum grade of C-

Math II is the second of a three-year sequence of courses. Students expand their knowledge of functions to include in-depth work with quadratics. Students build proficiency in moving between different forms of quadratic equations and identifying characteristics of the function from its equation. Students develop strategies to solve quadratic equations using a variety of methods and are introduced to complex numbers as solutions. They learn to distinguish between linear, exponential, absolute value and quadratic functions and apply transformations to the graphs of each. Students extend their work with transformations to include dilations. They write formal proofs for similarity and congruence. Additional topics in geometry include angle-arc relationships, development of formulas related to regular polygons, volume, radian measure and parabolas as conic sections. Students broaden their understanding of probability through a study of conditional probability using tree diagrams, Venn diagrams and two-way tables.
• ## Math II (Honors)

2 semesters, 1 credit
Prerequisite: Math I Honors with a minimum grade of B and department recommendation

Math II (Honors) is the second of a three-year sequence of courses. Students expand their knowledge of functions to include in-depth work with quadratics and build proficiency in moving between different forms of quadratic equations and identifying characteristics of the function from its equation. Students develop strategies to solve quadratic equations using a variety of methods and are introduced to complex numbers as solutions. They learn to distinguish between linear, exponential, absolute value, and quadratic functions and apply transformations to the graphs of each. Students extend their work with transformations to include dilations. They write formal proofs for similarity and congruence. Additional topics in geometry include angle-arc relationships, the development of formulas related to regular polygons, volume, radian measure and parabolas as conic sections. Students broaden their understanding of probability through a study of conditional probability using tree diagrams, Venn diagrams and two-way tables.

In this course, the curriculum of Math II is regularly extended through enrichment topics in each unit. Students in this course will expand on previous vector applications to work with complex numbers, angles and navigation. The topics of complex numbers and conic sections will also be studied.
• ## Math III

2 semesters, 1 credit
Prerequisite: Math II with a minimum grade of C

Math III is the third course of a three-year sequence of courses. Students continue to develop skills working with linear, exponential and quadratic functions while beginning to work with higher-order polynomial functions. They learn the Fundamental Theorem of Algebra, apply strategies of factoring and polynomial division for finding roots, and learn to graph polynomials. Inverse functions are also stressed in this course and, through this lens, students are introduced to logarithmic functions. Students visualize 3-dimensional solids of revolution and practice using areas of cross sections to find an object’s volume. Students extend their skills with trigonometry by deriving and applying the Law of Sines and the Law of Cosines. Then they learn to apply the sine, cosine and tangent functions to any angle, use sine and cosine to model periodic behavior, and explore fundamental trigonometric identities. Students also explore statistics with the Central Limit Theorem, the normal distribution and basic inferences related to means and proportions.
• ## Mathematics of Finance

2 semesters, 1 credit
Prerequisite: Algebra II with a minimum grade of C-

This course is a senior elective, covering a wide variety of mathematical concepts with an emphasis on finance. Over the first quarter, the course will provide a review of algebraic and geometric concepts in preparation for standardized testing while providing a firm foundation for the finance segment. During the remaining three-quarters of the year, students will cover the basics of investment in stocks and bonds, business modeling, loans (especially car loans and mortgages), tax calculations and retirement planning, all with an emphasis on the underlying mathematics. The year concludes with students preparing a budget using an Excel spreadsheet, which incorporates many of the concepts covered in the previous segments of the course.
• ## Multivariable Calculus (Honors)

2 semesters, 1 credit, (not offered in 2022–2023)
Prerequisite: AP Calculus BC with a minimum grade of B and department recommendation  by current math teacher or teacher approval

This advanced course begins with an investigation of vectors in XYZ space, their dot and cross products, and the use of vectors in determining equations for lines and planes in space. Students then study vector-valued functions, their derivatives, velocity and acceleration vectors, tangent and normal vectors, and the use of vector-valued functions in calculating arc length and curvature. Students explore multivariable functions, including limits and continuity, partial derivatives, differentiability, total differentials, the generalized chain rule, directional derivatives, tangent, and normal lines, tangent planes, extrema and optimization. Students then turn their attention to multivariable integral calculus with a study of iterated integrals, double integrals and volume, double integrals with polar coordinates, centers of mass and surface area. They investigate triple integrals and the volume bounded by surface curves. They study cylindrical and spherical coordinates and delve into how cylindrical and/or spherical coordinates make the computation of some triple integrals much more manageable. Lastly, students turn their attention to vector analysis, line integrals, vector fields, flow and flux, Green's Theorem, the Divergence Theorem, parameterized surfaces and Stokes' Theorem.
• ## Precalculus AB

2 semesters, 1 credit
Prerequisite: Algebra II with Trigonometry with a minimum grade of B and department recommendation by current math teacher

This course is designed to prepare students for AP Calculus AB or a calculus course in college. Students enrolling in this course should have a solid background in algebra and geometry. Students review, expand upon and apply their knowledge of polynomial, rational, exponential and logarithmic functions and sequences and series. This course extends the study of trigonometric relationships introduced in Algebra II with Trigonometry by emphasizing graphing functions, solving equations and verifying identities. New topics students are introduced to include vectors, probability, matrix operations and limits.
• ## Precalculus BC (Honors)

2 semesters, 1 credit
Prerequisite: Algebra II Honors with a minimum grade of B and department recommendation by current math teacher

This course is designed to prepare students for AP Calculus BC. It will build upon the students’ understanding and facility working with polynomial, rational, logarithmic and exponential functions by exploring those topics at a deeper level. Integration of ideas from previous mathematics courses is stressed, and students are given the chance to expand their abstract reasoning and spatial visualization skills while working in two- and three-dimensions. Topics of study include trigonometry, analytic geometry, vectors, polar and parametric equations, limit theory, differentiation rules, sequences, series and probability.
• ## Statistics

1 semester, 1/2 credit; 2nd semester only
Prerequisite: Algebra II with a minimum grade of C

This application-based elective introduces students to methods of data collection, display and analysis. Students explore regression models and gain the skills to test hypotheses using statistical inference related to means and proportions. Students learn how to use Excel to organize, display and analyze data. Students become critical consumers of data as they apply their knowledge to analyze how data and statistics are portrayed in the media.

## La Jolla Country Day School

9490 Genesee Avenue
La Jolla, CA 92037
858-453-3440

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