# US Mathematics

The Math department believes that students should have an awareness of problem-solving at all levels and that math is a universal tool for exploring the 21st century and exploring the structure within it. Every student discovers connections between their lives and the math they learn at Barrie. The integrated curriculum in the upper school highlights natural connections between different branches of math and other disciplines. These connections highlight the utility of mathematics and basic problem-solving and analytical skills.

BARRIE PREP MATHEMATICS CONTINUUM

Barrie's Middle and Upper School math scope and sequence supports our thematic curriculum. Rather than offer traditional courses in Algebra I, Geometry, and Algebra II, we have fully integrated mathematics courses: Foundations of Algebra and Geometry, Intermediate Algebra and Geometry, and Advanced Algebra and Geometry. These courses deliver the same scope and sequence presented in the traditional model, but in a more natural and continuous progression. Concepts from geometry are integrated throughout the three courses, eliminating the typical 'gap year' in a student's study of algebra. Earlier exposure to geometry skills allows for meaningful applications and project opportunities as students move throughout the thematic math curriculum. Problem-solving skills, data analysis, probability, and statistics thread throughout the courses; focus is placed on both conceptual and procedural knowledge, including the use of various mathematics technologies and software.

COURSE SEQUENCE

Intermediate Algebra and Geometry

In this integrated course, students build on their previous mathematical experience to explore new families of functions and trigonometry, and their applications to modeling and geometry. Problem-solving cuts across themes and connects them, and work in groups allows the students to discuss problems with their peers, present their ideas and provide and evaluate evidence supporting mathematical statements. Recursion offers a different perspective on linearity as well as a natural way to introduce exponential sequences.The analysis of objects in free fall motivates a thorough study of quadratic functions and of the effect on the graph of the basic transformations of a rule, which in turn, are necessary tools in analytic geometry. Trigonometry is used to calculate areas and perimeters of regular polygons, and to represent vectors in polar form. Graphing software and applets are used extensively to enhance students’ understanding of the mathematics underlying real situations, and video recording and frame-by-frame processing facilitate the modeling of physical phenomena. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts in lieu of a textbook. Students who complete this course successfully will advance to Advanced Algebra and Geometry.

In Advanced Algebra and Geometry, students work in small groups on a variety of hands-on activities to explore new families of functions and their connections with those studied in Intermediate Algebra and Geometry. Split-domain and polynomial functions offer an opportunity to review linear and quadratic functions in a new context. Exponential, logarithmic, radical, trigonometric, and elementary rational functions are studied and applied to physical phenomena, economics, and social and environmental problems. Transformations in the plane are studied with a focus on the images of geometric objects. Combinatorics constitutes a rich source of problems that introduce students to the techniques of systematic counting. The development of the ability to use multiple representations (numerical, graphical, verbal, and symbolic) and to move fluently between them is emphasized, both to enhance the students' understanding and as a problem-solving tool. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts in lieu of a textbook. Students who successfully complete this course will advance to Pre-Calculus

Pre-Calculus

In Pre-calculus, students extend or explore in more depth the families of functions studied in Advanced Algebra and Geometry, their properties, and their applications. The analysis of the properties of trigonometric functions and their inverses complements the study initiated in that course . The introduction of the concepts of average and instantaneous velocity opens the door to an in-depth analysis of rational functions and their graphs, including sign charts, continuity, asymptotes, and limits at infinity. Exponential growth and decay are reviewed in the context of regression using the log-transformation. Linear, circular, and projectile motion are modeled using parametric functions, and velocity vectors at selected points are calculated both numerically and symbolically. Projects connect functions to authentic applications that focus on integrating the themes of Discovery and Invention, and Global Issues and Diplomacy. Technology is used throughout the course to support student learning and to provide opportunities for application, visualization, and presentation of results. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts. Students who successfully complete this course will advance to Calculus (AP Calculus with recommendation) or Statistics (AP Statistics with recommendation).

MATHEMATICS ELECTIVES

Elective classes, including Advanced Placement courses, are based upon student interest/enrollment and may vary from year to year. Here is a sample of recent Math electives.

Calculus

Prerequisites: Successful completion of Pre-Calculus

In Calculus, students are introduced to the underpinnings of differential and integral calculus with an emphasis on its applications to rate of change and net change in a quantity in real or realistic situations, and modeling using elementary differential equations. The use of the calculator allows students to focus on ideas and solve significant problems without the distraction of lengthy operations. Interactive software provides a tool for the exploration of the problems discussed and enhances students’ understanding. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts.

AP Calculus AB

Prerequisites: Successful completion of Pre-Calculus and teacher approval

In the first half of this course, students learn the concept of derivative and its applications to the calculation of instantaneous velocity, rate of change, and acceleration as well as optimization with and without constraints. The second half of the course is devoted to the concept of integral and its applications to finding net change of a quantity; a variety of geometric problems, including the calculation of areas, length of curve and volumes of different solids; and the solution of elementary differential equations. Both concepts are introduced through concrete hands-on applications and software that allows increasingly fine discrete approximations. Students move fluently between verbal, numerical, graphic, and symbolic representations of the problems. Upon successful completion of this course, students are prepared to take the AP Calculus AB exam. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts.

AP Calculus BC

Prerequisites: Successful completion of Pre-Calculus and teacher approval.

Students taking AP Calculus BC will learn all the topics of AP Calculus AB with the addition of improper integrals, Taylor series and approximations, and the application of differential and integral calculus to parametric and polar curves. Upon successful completion of this course, students are prepared to take the AP Calculus BC exam. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts.

AP Statistics

Prerequisites: Successful completion of Pre-Calculus and teacher recommendation

AP Statistics introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. AP Statistics encompasses the four broad conceptual themes that the College Board requires: planning a study, exploring data, anticipating patterns, and statistical inference. Students work on projects that have immediate connections to the world around them. The use of technology allows the students to explore concepts realistically and efficiently. Upon successful completion of this course, students are prepared to take the AP Statistics exam. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts.

Multivariable Calculus

Prerequisites: Successful completion of AP Calculus

This course begins with a survey of geometry in three dimensions followed by calculus of two variables. The geometry behind calculus is emphasized, the proof or plausibility of the results discussed, and work on a variety of applications to physics, economics, and geometry is undertaken. This course includes a study of differential calculus for functions of two variables, including local, constrained, and global extrema; double and triple integrals; line integrals and Green's Theorem; and modeling using differential equations. The use of software enhances the learning experience allowing the student to visualize and explore the often complex geometry of the surfaces studied and investigate the effects of parameter changes in the situations modeled. A graphing calculator (TI-83 or 84) is required for this course. Students will use a variety of resources, prepared materials, and handouts.