# Mathematics

**AP Statistics Track****Year 1**- Quantitative Reasoning 1

**Year 2**- Quantitative Reasoning 2

**Year 3**- Functions, Finance, and Statistics

**Year 4**- Statistics (AP)

**AP Calculus****Year 1**- Geometry (H)

**Year 2**- Algebra II (H)

**Year 3**- Precalculus (H) or Precalculus Advanced (H)

**Year 4**- Calculus AB (AP) or Calculus BC (AP)

Electives: Statistics (AP) or Computer Science (AP)

## Quantitative Reasoning 1 and 2

During these courses, students will learn traditional material from Algebra 1 and 2 and Geometry, with the focus being on problem solving and critical thinking. However, our ultimate goals are for students to develop a growth mindset, to cultivate a curiosity/passion for mathematics, and to create an excellent foundation for further study of applied mathematics. In addition, we will prepare students to excel on the quantitative reasoning sections of ACT/SAT standardized tests.

## Functions, Finance, and Statistics (F2S)

The Functions, Finance, and Statistics (F2S) course focuses on mathematical applications. During the first semester, students delve into topics discussed in traditional a Precalculus course (Functions), and also add material from Finite Mathematics (specifically, the Finance part of the class, but also material in linear modeling). Unlike a traditional Precalculus course, however, the focus here is on applications, especially the use of technology (such as the TI-84+ graphing calculator and Microsoft Excel) to tackle challenging problems. Project-based learning is used extensively. The course shifts full-time to Statistics in January, which prepares them nicely to take the second year of this course—AP Statistics. Students will already have learned some of the material in the AP Statistics curriculum, and will spend the next year exploring more deeply the details of more advanced Statistics.

## AP Statistics

Having already begun the AP Statistics curriculum in F2S, we allow students to delve deeply into the subject matter during the senior year. Past AP problems are used extensively, as well as problem-based learning. The potential for the application of statistics is broad, and we see examples of this application in many different fields. Students develop analytical and critical thinking skills as they learn to describe data patterns and departures from patterns; plan and conduct studies; use probability and simulation to explore random phenomena; and estimate population parameters, test hypotheses, and make statistical inferences.

## AP Calculus Track (All Honors and AP courses)

The honors- and AP-level mathematics courses in the AP Calculus Track at Asheville School are both demanding and rewarding for students desiring rigorous coursework in the com

pany of like-minded students. Some of our students have shown excellent achievement and aptitude for mathematics at previous schools, and therefore begin their careers at Asheville School in honors-level courses. This process is by far the preferred means by which students will take honors-level courses. Successful students will demonstrate the following characteristics:

● Dedication to pursuing difficult subjects, consistent persistence, and the tenacity to understand and to solve problems

● Ability to connect ideas and topics, to demonstrate detailed and logical problem solving, and to work at a quick pace

● Willingness to take problem-solving risks, especially when uncertain regarding the correct approach

● Desire and ability to read and to interpret mathematical theorems

● Willingness to work with peers on a regular basis

● Commitment to presenting work accurately and neatly, demonstrating a step-by-step method

● Willingness to self-teach material, using a textbook and other resources to find similar problems and examples

● Independence and ownership of the material

● Dedication to mastery on a daily basis

● Enthusiasm for learning mathematics

## Geometry (Honors)

During the study of Honors Geometry, students develop the problem-solving skills required for future advanced mathematics courses at Asheville School. The course is based on the traditional topics of Euclidean geometry. Emphasis is placed on problem solving and the development of logical arguments through deductive and inductive reasoning. Students learn and use the concepts of point, line, and plane in several dimensions. Through deductive reasoning, students learn to write valid proofs and solve problems, using critical thinking supported by definitions, postulates, and theorems as they study parallel lines, congruent triangles, quadrilaterals, similar polygons, right triangles, circles, and coordinate geometry. Students also investigate area and volume. Algebra skills are reinforced throughout the course. Students use their own TI-83/84 calculator or a type of graphing or scientific calculator to solve trigonometric problems. Several projects related to the study of geometry are completed during the year.

## Algebra II (Honors)

The objective of this class is to build upon the skills developed in Algebra I and Geometry and to provide a solid foundation for higher levels of math and science. Topics covered will include: relations and functions, absolute value, quadratic functions, polynomials, rational expressions, logarithmic and exponential functions, conic sections, and other algebraic topics. This class will seek to make connections between our coursework and the real world while also applying uses of technology to our studies.

## Precalculus (Honors)

The Precalculus Honors course includes a review of second-year algebra; first- and second-degree equations with applications; inequalities including absolute values with applications; functions; graphs; polynomial and rational functions with applications; exponential and logarithmic functions with applications; and trigonometric functions with applications. In addition to several mathematical modeling projects, students also study sequences and series, permutations, combinations, and probability. Programmable graphing calculators are used extensively in the course. Most students completing this course take Calculus AB (AP) the following year.

## Precalculus Advanced (Honors)

The Precalculus Advanced (Honors) course includes a review of second-year algebra; first- and second-degree equations with applications; inequalities including absolute values with applications; functions; graphs; polynomial and rational functions with applications; exponential and logarithmic functions with applications; and trigonometric functions with applications. In addition to several mathematical modeling projects, students also study graphing in polar coordinates, vectors and their applications, mathematical induction, sequences and series, permutations, combinations, and probability. Programmable graphing calculators are used extensively in the course. Most students completing this course take Calculus BC (AP) the following year.

## Calculus AB (AP)

The underlying theme used throughout AP Calculus is to have students understand functions from the Rule of Four perspectives: graphic, numeric, algebraic and verbal. Students develop maximum understanding when they are provided opportunities to explore, discover, investigate, and discuss challenging new concepts. The material studied in this course includes limits; continuity; differentiation; applications of differentiation to increasing/decreasing functions, relative extrema, applications of extrema, concavity, inflection points, related rates, differentials, and local linearity; Rolle's theorem; mean value theorem for derivatives; indefinite and definite integration; the fundamental theorems of calculus; mean value of a function; mean value theorem for integrals; applications of integration to areas and volumes; definite integral as an accumulator; logarithmic and exponential functions including differentiation and integration; trigonometric functions including differentiation and integration; techniques of integration including algebraic and trigonometric substitutions, partial fractions (non-repeating linear factors) and integration by parts; solutions and applications of simple separable differential equations; numerical integration using Riemann sums and the trapezoidal rule; and L'Hopital's rule.

## Calculus BC (AP)

The underlying theme used throughout AP Calculus is to have students understand functions from the Rule of Four perspectives: graphic, numeric, algebraic and verbal. Students develop maximum understanding when they are provided opportunities to explore, discover, investigate, and discuss challenging new concepts. The material studied in this course includes limits; continuity; differentiation; applications of differentiation to increasing/decreasing functions, relative extrema, applications of extrema, concavity, inflection points, related rates, differentials, and local linearity; Rolle's theorem; mean value theorem for derivatives; indefinite and definite integration; the fundamental theorems of calculus; mean value of a function; mean value theorem for integrals; applications of integration to areas and volumes; definite integral as an accumulator; logarithmic and exponential functions including differentiation and integration; trigonometric functions including differentiation and integration; techniques of integration including algebraic and trigonometric substitutions, partial fractions (non-repeating linear factors) and integration by parts; solutions and applications of simple separable differential equations; numerical integration using Riemann sums and the trapezoidal rule; and L'Hopital's rule. In the BC Calculus course, students study the topics listed above and in addition they consider arc length and surfaces of revolution; applications of integration to work and to fluid pressure and fluid force; improper integrals; infinite series and the question of convergence; Taylor and Maclaurin series; power series, application of the ideas of calculus to functions given in parametric, polar and vector form; slope fields; numerical solution of differential equations using Euler's method; and solution of logistic differential equations and their use in modeling.

## Computer Science Principles (AP)

Computer Science Principles is a rigorous curriculum set designed to meet the standards of College Board’s AP Computer Science Principles framework. This curriculum aims to foster college and career readiness by teaching students to build real world apps using the same language and tools as professional iOS developers. The course places a heavy emphasis on pair programming, design thinking, and encouraging students to iterate on their ideas, with the goal of creating intrinsic motivation in students while also giving them a solid understanding of what it’s like to work as a professional developer in the real world. To help foster student motivation and build confidence, the course makes clear that in addition to submitting their Create Task to the College Board for review, students can and should also submit their app for review to Apple, with the ultimate goal of putting their app on the App Store and getting real world users!

## Computer Science A (AP)

In AP Computer Science, students focus on developing the critical thinking and other necessary problem-solving skills that allow us to view the world in a different manner. We will do this through the study of the Java programming language, designing programs to solve real world problems, and rigorous preparation for the AP Computer Science examination. Students will not only develop basic competency in elementary computer science topics, but they will also develop the ability to see how computer scientists have the capacity to change the world.

## Advanced Topics in Mathematics (Honors)

This post-AP Calculus course will function as an introduction to mathematical proof, linear algebra, and ordinary differential equations with a project-based emphasis on mathematical modeling. Students who complete this course will leave with an understanding of how to think like a mathematician, how to present mathematical ideas clearly and rigorously, and how to conduct mathematical research. Students will also be exposed to scientific computing software (MATLAB/Octave).