Math Department

The goals of the Math Department at Colorado Rocky Mountain School are to provide students with the tools to be logical thinkers and to help them gain a solid foundation in problem-solving. The department aims to help students become more confident in their mathematical abilities as well as appreciate ways in which the math they are studying relates to the world around them. Emphasis is placed on both mastery of mathematical operations and understanding of the underlying reasoning of the operations themselves. All topics are explored visually, symbolically, and verbally. Classwork, homework, quizzes, tests, and projects are designed to develop critical thinking. Graphing calculators and computer technology are an integral part of the curriculum and are used regularly as tools to help students further comprehend the topics being studied. Completing three years of math and passing Algebra 2 are the minimum requirements, but students are encouraged to continue math classes up to and including their senior year.

Algebra 1
This course teaches students the structure of the real-number system and the manipulative skills to simplify and solve numerical and variable expressions containing exponents, radicals, and fractions. Students also learn to manipulate linear, quadratic, exponential, and absolute-value expressions and equations. The definition of a function and graphing functions are introduced, as well as operations on polynomials, including factoring. Practical applications and word problems are explored and incorporated throughout the year.

Geometry
This course begins by introducing concepts of plane and coordinate geometry through such topics as angles, triangles, lines, circles, polygons, area similarity, congruence, and right-triangle trigonometry. The course concludes with a study of geometric solids. Setting up and writing proofs to justify logical arguments and conclusions helps to expand fundamental geometric concepts. Students use Geometer’s Sketchpad software as a vehicle to increase understanding. (Prerequisite: Algebra I)

Algebra 2
This course builds on the students' foundation in Algebra 1 to prepare them for future math courses. Students study various classes of functions, including linear, quadratic, exponential, and logarithmic. They apply these functions to real-world situations through mathematical modeling that provides them with an answer to the question, “What is all this algebra good for?” Integrated throughout the year are the fundamental algebraic skills of solving equations and simplifying expressions. Technology is used as an invaluable learning tool for discovering, rather than simply being told, algebraic concepts. By making and testing conjectures, students see the connection between the symbolic, numerical, and the geometric meanings of the concepts of algebra. (Prerequisite: Geometry)

Environmental Statistics
In this course, students use statistics and mathematical modeling to examine environmental data. Linear, exponential, logistic, and trigonometric functions are used to model change in different populations, environmental impacts, and natural phenomena. Students also study measures of center and spread, and probability theory to help understand and describe different sets of data, as well as to think critically about the statistics presented by the media. This course is entirely project-based. Students are asked to gather, examine, and explain relevant data about a population or ecosystem. In all these topics, technology is an invaluable tool in understanding the concepts as well as solving problems. Technology utilized includes the TI 84 calculator, Excel, PowerPoint, and the Internet. (Prerequisite: Algebra II)

Precalculus
This course studies, in depth, functions and their applications. Emphasis is on modeling polynomial, rational, exponential, logarithmic, and trigonometric functions. In addition, operations, transformations, and inverses of functions are covered. The graphing calculator is used as a tool to verify problem-solving techniques and to help interpret problems graphically and tabularly. Polar coordinates, conic sections, and sequences and series are introduced, time permitting. (Prerequisite Algebra II)

Calculus
This course is comparable to a first-semester college-level course in the calculus of functions of one independent variable. It is centered on the four central concepts to be mastered in the first course in calculus—limit, derivative, definite integral, and indefinite integral. For each concept, students are asked to know the precise definition, have an intuitive understanding of what the concept means, be able to “do” the concept, and be able to apply it in the mathematical or real world. There are four ways these concepts are presented to the student—numerically, graphically, algebraically, and verbally. By using these multiple representations, students with a wide range of abilities can be successful in Calculus. Students may earn college credit through successful performance on the Advanced Placement Calculus AB exam. (Prerequisite: Precalculus). Please check out the new Calculus Weblog

Honors Calculus and Advanced Topics*
This course is a continuation of the Calculus course and is intended to prepare students to sit for the Advanced Placement Calculus BC examination. The topics covered include the analysis of functions in parametric, polar, and vector forms; Euler’s method for solving differential equations; L’Hospital’s Rule; integration by parts; partial fractions; improper integrals; solving logistic differential equations; and the Taylor and Maclaurin series. Students are evaluated through tests, quizzes and projects. (Prerequisite: Calculus) *This course may be canceled if demand is insufficient or staffing is not possible for the upcoming year.

Advanced Placement Statistics
This course is intended to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. The course is structured along 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). In summary, it aims to prepare students for a passing grade on the AP exam, and to develop statistics as a tool that will help students become more critical thinkers. This course, equivalent to a one-semester college course in statistics, is ideal for students who plan to study the social sciences. Taking the Advanced Placement test is required. (Prerequisite: Precalculus) *This course may be canceled if demand is insufficient or staffing is not possible for the upcoming year.

Computer Programming
This course covers the basic concepts of modular programming. Students will study programming structure, use of variables and constants, and design of an intuitive user-interface. (Prerequisite: Algebra II)

Science Department

The Science Department at Colorado Rocky Mountain School emphasizes experiential investigation of the natural world. Students explore the nature of scientific inquiry and examine the function of models in scientific understanding. CRMS students use scientific methods to investigate and verify fundamental principles. They utilize computers, library resources, electronic probes, and mathematics to evaluate, quantify, and present their research. Finally, faculty encourage students to become critical thinkers and citizens who use the content, models, and skills of science throughout their lives.

Students start with Biology, followed by Chemistry. They are encouraged to continue with other science courses throughout their years of study here.

Biology
Biology is an introduction to life on earth. In CRMS’s first science course, students explore the rationale, logic, and assumptions of the scientific method, scientific inquiry, and scientific research and reporting. In the first semester, students study principles of populations, ecosystems, evolution, taxonomy and systematics, biochemistry, cells and cell division, photosynthesis, and metabolism. In the second semester, students investigate principles of genetics, infectious disease and the immune system, and human anatomy and physiology, including the endocrine system, reproduction, growth and development, neurobiology, and cardiopulmonary and skeletomuscular systems.

Chemistry
The composition and behavior of matter in all its phases is the focus of this course. Topics include the distinction between metals and nonmetals, bonding, the periodic table, basic types of chemical reactions, gas laws, the mole and stoichiometry, organic chemistry, nuclear chemistry, solution (acid/base and redox) chemistry, and energy changes. Studies emphasize qualitative and quantitative lab work, critical analysis of experimental results, and use of abstract models as a basis of explanation.

Geology
The geology of Carbondale and its surroundings is the focus of the fall semester. Field trips highlight more than two billion years of geologic history, including the origin of the Rocky Mountains, eruption of volcanoes, glaciations, development of the Colorado River system, and evidence for ancient rivers, lakes, deserts, coal swamps, beaches, and oceans. The second semester is global in scope, including topics such as Earth’s climate, meteorology, plate tectonics, volcanism, earthquakes, hydrology, and environmental and natural-resource issues. Biology and Chemistry are prerequisites.

Physics
Physics is an investigation of fundamental natural phenomena expressed through the language of mathematics. The development and experimental verification of physical theory is an integral part of the course. Topics include the description of motion and its causes as seen through the models of classical mechanics, the properties and behavior of light, and the quantum mechanical model of the atom. The notion of models as a description of the physical world leads to the discussion of the nature of knowledge and humanity’s understanding of the world. Algebra 2, Biology, and Chemistry are prerequisites.

Advanced Placement Environmental Science
The AP Environmental Science course is designed to be the equivalent of a college-level environmental science course. The class will explore the concepts of ecological interdependence, population dynamics, and geological processes. The course focuses on the skills and concepts needed to identify, assess, and resolve environmental problems at a local, regional, national, and global level. Field and laboratory work allow for firsthand observation and analysis of environmental phenomena. Biology and Chemistry are prerequisites.