# Mathematics

The mathematics curriculum emphasizes an appreciation of the subject through experiences that encourage students to explore ideas, take risks, and think for themselves. We assert the importance of establishing fundamental skills and laying a solid foundation at every level, thereby allowing for subsequent study of mathematics and related fields. When we establish skills, the emphasis is not on rote memory, but on developing and working through logical processes that students understand. As such, daily practice is expected and needed to reinforce and further develop these mathematical skills and abilities. We emphasize that each course is important in and of itself, for its ability to engage and for its ability to enhance mathematical fluency. Applications, connections, and the use of technology make mathematics more relevant and more accessible. The creative and critical thinking, use of logic, and problem-solving skills developed in these courses are not only necessary for successful course completion, but are also applicable to other fields and other situations.

## Mathematics Faculty

## Courses

- Geometry
- Algebra 2
- Precalculus
- AP Calculus AB
- AP Calculus BC
- AP Statistics
- Advanced Topics in Mathematics
- Statistics
- Mathematical Models for Decision Making
- Advanced Statistical Methods and Data Analysis

## Geometry

## Year – 6 credits

**Prerequisites: **Successful completion of Algebra 1

In this course, students will analyze properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. Inductive reasoning is frequently used to discover new conjectures, while deductive reasoning (proofs) is also used to find new geometric relationships. Students will formulate conjectures and solve problems relating to the standard topics of Euclidean geometry. This course also includes the study of right triangle trigonometry. In addition, coordinate geometry methods will be used whenever possible, in order to emphasize the connections between algebra and geometry and to maintain students’ algebra skills. While students will learn traditional geometric constructions with the use of a compass and straight edge, they will also use the dynamic mathematics modeling program GeoGebra to construct and manipulate geometric figures in order to explore their properties and relationships. This course provides students with a balance among activities emphasizing discovery, application, and proof. Students will work individually and frequently work together in small cooperative learning groups. The course assignments include: homework assignments, in class discovery activities, quizzes, tests, and occasional projects. A special unit on tessellations provides students with the opportunity to design, create, and display a set of tessellating ceramic tiles. Students should expect to spend up to an hour each night completing nightly homework exercises.

**Students will: **

●Learn methods to reason in geometry

●Use tools of geometry

●Work with triangle properties

●Work with polygon properties

●Work with circle properties

●Study similarity, congruence, and associated applications

●Study the Pythagorean theorem in a geometric setting

●Study two-and three-dimensional shapes and associated area/volume calculations

●Study introductory trigonometry

●Study geometric transformations and applications

## Algebra 2

## Year—6 credits

**Prerequisite: **Geometry

In this course, students will represent and analyze mathematical situations numerically, graphically, and symbolically, building on the concepts and skills developed in Algebra 1. Students will use a variety of symbolic representations to model problem situations, including both explicitly defined and recursively defined functions. In addition, students will begin to investigate and compare the properties of classes of functions and their graphs, including quadratic, higher degree polynomial, exponential, and logarithmic functions. Students will maintain and build on proficiencies developed in Algebra 1 as well as learn to solve non-linear equations involving polynomial or exponential expressions. The course emphasizes using mathematical models to represent and understand quantitative relationships and to solve problems. Graphing calculators will be used to look at the table and graphs of explicitly and recursively defined functions. Students will work individually and in small groups. The course assignments include homework assignments, investigations, partial chapter quizzes, and chapter tests. Students should anticipate spending 30-60 minutes per day on homework completion and reviewing content.

**Students will: **

●Study lines and functions, including ordered pairs, graphing lines, equations of lines and applications, graphing linear inequalities, graphs and transformations of functions.

●Study systems of linear equations and matrices, including solving linear equations, applications, solving systems of linear inequalities, matrices.

●Study exponents and polynomial expressions, including properties of exponents, polynomial expressions, and factoring.

●Study quadratic and polynomial functions, including quadratic equations, quadratic functions, quadratic inequalities, real and complex zeros of polynomial functions, and graphs of polynomial functions.

●Study radicals and advanced functions, including radical functions, simplifying expressions, rational exponents, radical equations and functions, composition of functions, inverse functions.

●Study exponential and logarithmic functions, including properties of logarithms, exponential and logarithmic equations, applications of exponential and logarithmic functions, graphing exponential and logarithmic functions

●Study rational expressions and functions, including simplifying expressions, rational equations and inequalities, graphing rational functions, and applications.

Study conic sections, including midpoint and distance, parabolas, and circles.

## Precalculus

## Year - 6 credits

**Prerequisites: **Successful completion of Algebra 2

The central theme of this course is comparing, contrasting, and transforming functions in order to model change. We emphasize that functions can be grouped into families – linear, quadratic, exponential, power, polynomial, rational, and trigonometric – and that these functions can be used as models for real-world behavior. Students will use technology and analytic methods to develop a deeper conceptual understanding of the mathematical underpinnings of functional analysis. The inclusion of non-routine problems is intended to establish the idea that such problems are not only part of mathematics, but in some sense are the point of mathematics. Students taking this course should possess a willingness to work hard and be prepared to spend at least 6 hours weekly on this course outside of class time. Students taking this course should have completed both Algebra 2 and Geometry.

Students will:

•judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities;

•use symbolic algebra to represent and explain mathematical relationships;

•identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;

•use trigonometric relationships to determine lengths and angle measures;

•use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;

•understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices

## AP Calculus AB

## Year - 6 credits

This is a challenging college-level course, preparing students for the Calculus AB Advanced Placement exam. Applications and mathematical modeling are used to motivate topics and the graphing calculator is used extensively as a tool for investigating and applying the important concepts. Topics from the Calculus AB syllabus of the College Board’s AP curriculum will be covered. These include a review of the properties and graphs of elementary functions; limits of functions; the concept of the derivative, derivatives of elementary functions, applications of derivatives; interpretation and properties of definite integrals; applications of definite integrals; and techniques and applications of anti-differentiation.

## AP Calculus BC

## Year - 6 credits

**Prerequisite**: successful completion of Calculus AB.

Prerequisite: successful completion of AP Calculus AB. This course, which prepares students for the Calculus BC Advanced Placement exam, continues the study of calculus from AP Calculus AB. Applications and mathematical modeling are used to motivate topics and the graphing calculator is used extensively as a tool for conducting investigations. Topics from the Calculus BC syllabus of the College Board’s AP curriculum will be covered, along with additional topics from outside the usual BC Calculus curriculum.

## AP Statistics

## Year—6 credits

**Prerequisite:** successful completion of Algebra 2

This is a challenging college-level course that prepares students for the Advanced Placement Statistics exam. This course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes: (1) Exploring Data: Describing patterns and departures from patterns; (2) Sampling and Experimentation: Planning and conducting a study; (3) Anticipating Patterns: Exploring random phenomena using probability and simulation; and (4) Statistical Inference: Estimating population parameters and testing hypotheses. This course emphasizes statistical thinking and decision-making. Technology (TI graphing calculators and statistical software) is used extensively to foster active learning, and to carry out statistical analyses, including making graphs, finding numerical summaries, computing estimates, and testing hypotheses.

## Advanced Topics in Mathematics

## The specific topic rotates across a 3-year cycle

**Discrete Mathematics (2021-2022)**

Differential Equations (2022-2023)

Differential Equations (2022-2023)

Multivariable Calculus (2020-2021)

Year – 6 credits

Prerequisite: AP Calculus BC

This course is intended for advanced mathematics students who wish to further their studies beyond calculus. In this year's version of the course, students will learn multivariable calculus, which is a deeper exploration into the calculus covered in previous years in the setting of higher dimensions. Students will cover all of the material covered in a standard college-level multivariable calculus course, and also have the opportunity to utilize this exciting mathematics to engage in applied project work. This course will prepare students for work in a variety of disciplines, including science, engineering, mathematics, and computer science.

Students will:

●Engage with homework and problem sets to practice the various skills discovered in class

●Demonstrate mastery of material through tests, oral presentations, and written research

●Learn to apply this exciting field of mathematics to real-world applications in fields including science, engineering, and computer science

●Master the calculus concepts acquired in previous mathematics courses

●Present at the year-end STEM Symposium

## Statistics

**Prerequisites**: Successful completion of Algebra 2

This course is designed to introduce students to the major concepts and tools for collecting, analyzing and drawing conclusions from data. The course will prepare students for life in a world filled with data, as well as for many college majors that require statistics. Students will study the fundamental ideas of statistics in a context that relates to personal experiences and needs of an informed citizen of the 21st century. Statistics helps us to understand our world: Health professionals need to understand statistics to interpret accounts of medical research; business use statistical methods (business analytics) to efficiently crunch numbers to support their bottom line; and citizens need to understand statistics to accurately interpret opinion polls and the Consumer Price Index. Because data are omnipresent, everyone will find statistics useful and perhaps even profitable. In this course, students will study exploratory data analysis, learn the fundamentals of designing a study, use probability models, and use inferential statistics. Students will apply these concepts to such fields as business, biology, engineering, industry, the social sciences and many others. Students will use technology to calculate and display real data. Graphing calculators that can manipulate lists of data and contain statistical applications such as confidence intervals, tests of significance, and probability distributions will be our main technological hardware. In addition, we will be using online apps/programs. Students will work individually and frequently work together in pairs. Course assignments include: homework assignments, in class practice, computer simulations, hands-on activities, quizzes and tests. Students will be invited to join us when we are visited by guest speakers. Students should expect to spend an hour or more each night completing nightly homework exercises. These assignments occasionally include watching videos or taking notes.

**Students will: **

●formulate questions, design studies, and learn how to collect data about a characteristic shared by two populations or different characteristics within one population;

●evaluate the design of a study, the appropriateness of data analysis, and the validity of conclusions;

●select, create, and use appropriate graphical representations, including histograms, box plots, and scatter plots, for different kinds of datasets;

●know the characteristics of well-designed studies, including the role of randomization in surveys and experiments;

●compute basic statistics and understand the distinction between a statistic and a parameter.

●be able to display a scatter plot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools;

●identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.

●use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken;

●use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions;

●use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations;

- compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

## Mathematical Models for Decision Making

**Prerequisites**: Successful completion of Algebra 2

In this course, students are introduced to mathematical topics necessary for decision-making in today’s world. Mathematical topics are developed in the context of solving problems from business, science, and daily life. Major units of study include election theory, fair division and game theory, modeling with matrices, graphs and their applications, counting and probability, and personal finance and managing resources. Students will work individually and in small groups. The course assignments include class activities, small-scale research, homework assignments, quizzes, and unit tests. Students should anticipate spending 30 minutes per day on homework completion and reviewing content. Students should have successfully completed the Algebra 2 course before taking this class.

**Students will: **

●understand matrices as a system that has some of the properties of the real-number system;

●develop an understanding of properties of, and representations for, the addition and multiplication of matrices;

●judge the reasonableness of numerical computations and their results;

●develop fluency in operations with real numbers and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases;

●generalize patterns using explicitly defined and recursively defined functions;

●understand the meaning of equivalent forms of expressions, equations, and relations;

●use symbolic algebra to represent and explain mathematical relationships;

●judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology;

●use a variety of symbolic representations, including recursive equations, for functions and relations;

●use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;

●draw reasonable conclusions about a situation being modeled.

## Advanced Statistical Methods and Data Analysis

**Prerequisites: **Statistics, AP Statistics, or permission of instructor (this course is offered when there is sufficient enrollment (6 students or more)

This course extends statistical methods learned in prior statistics classes and introduces new data analysis tools. Students will study more sophisticated sampling and experimental design data collection methods, analysis of variance (ANOVA), nonparametric methods, and multiple and applied regression. Additional topics could be studied if time allows. This is a project-based course that emphasizes working with authentic data, and data analyses will be carried out using the R programming language. Students will have the opportunity to work with data sets of their choice in areas of interest to them (including, but not limited to, sports analytics, climate change, business/marketing, medicine/epidemiology). Third trimester will be focused on a data analysis project of the student’s choosing which could culminate in a display at the STEM Symposium in May.

Students will:

●Engage with homework and problem sets to practice the various skills discovered in class

●Demonstrate mastery of material through oral presentations and written research

●Learn to apply statistics to real-world applications

●Learn to analyze data and make graphical displays using the R programming language

●Possibly present at the year-end STEM Symposium