MATHEMATICS AND COMPUTER STUDIES 2007 – 2008
Students with questions about course selections may see any member of the department for advice. Permission from a student’s current teacher or the department head is required for all Intensive courses as well as for Geometric Concepts and Precalculus Fundamentals. Permission from the Mathematics Department is required for independent study. Graphing calculators and computers are required for all courses. The TI-83 Plus or TI-84 are the departmental choices for calculators. Courses listed below are identified as either year-long or semester. Semester courses do not count toward fulfillment of the Lakeside mathematics requirement.
Appropriate Placement
The department’s criteria for appropriate placement in fundamental, regular, or intensive-level courses ask students to reflect on the following questions:
· Is the pace of the course challenging yet sufficiently comfortable for me to enjoy and thrive from the challenges?
· Is the level of abstraction demanded in the course appropriate for my level of understanding and notational facility?
· Is the content of the course appropriate to my interest and ability?
· Is there enough time available in my daily/weekly schedule to give sufficient time to the course’s homework requirements?
· Am I able to do most of the work required by the course independently (i.e., with no tutor support)?
Intensive Courses
Lakeside offers intensive courses in mathematics. These courses offer additional challenge to students who show special interest and aptitude for learning in mathematics.
The biggest difference between intensive and regular courses is that in an intensive section there is very little class time spent going over foundational material. This means that students must be able to master concepts quickly and learn from mistakes without much teacher support (and certainly with no tutor support). This allows most class time to be spent on advanced topics explored in greater depth and breadth. Students must demonstrate the ability to keep pace with the accelerated tempo of the class and to work independently.
A student interested in taking an intensive course should consult with his or her teacher. At a minimum, students should have compiled an excellent record in previous math courses. Generally this means a grade of an A in a regular course and a grade of at least a B- in an intensive course. Students moving from a regular course to an intensive course are usually asked to do additional catch-up work over the summer, and to pass a placement test in August. A grade of C+ or lower in an intensive course suggests that a student is not achieving at the expected level for an intensive course and should probably move to a regular section. However, regardless of previous grades, consultation with and approval from the student's current teacher is required.
For students new to Lakeside the process is different. For each incoming student the Mathematics Department, in consultation with the Admissions Office, considers the student’s test scores and academic background to decide an appropriate starting level (Algebra I, Algebra II, Geometry, Precalculus, Calculus), and whether the regular or intensive versions of these courses are the best fit for that student. Students or families who question this placement will be offered the opportunity to take a placement test in May, not for a grade or credit, but for diagnostic and placement purposes.
Note: All mathematics classes meet for two 45-minute periods and one 75-minute period per week unless otherwise noted.
Year-Long Courses
M110 Algebra 1: The course develops the algebra of real numbers including exponents, ratio and proportion; introduces the analysis of functions including linear, quadratic, and exponential functions; and begins a study of basic right trigonometry. Students explore functions and relations analytically and through applications, and analyze data graphically and numerically. The explorations make use a graphing calculator (TI-83 Plus or TI-84), and of Geometer’s Sketchpad© on the laptop. An emphasis is placed on building problem solving skills, writing mathematical reports, reading, studying, learning from a math text, and increasing the mathematical confidence of the individual students. (This course meets for three 45-minute periods and one 75-minute period per week.)
M210 Algebra 2: The course focuses on the analysis of functions and their applications while introducing students to a variety of topics in discrete mathematics. After exploring the algebraic, graphical, and numerical properties of general functions, specific types of functions will be examined from these perspectives. The course will examine each of the following families of functions: linear, quadratic, exponential, polynomial, logarithmic, rational, and trigonometric. Additional topics in discrete mathematics such a matrices, sequence and series, combinatorics, and probability will give students the tools to analyze interesting, highly relevant problems. Both computers (Excel©, Geometer’s Sketchpad©) and graphing calculators (TI-83 Plus or TI-84) will be used extensively and are required for the course. (Prerequisite: Algebra 1 or permission of department.) (This course meets for three 45-minute periods and one 75-minute period per week.)
M220 Intensive Algebra 2: This course covers the same topics as the regular course but with less review of algebra and more in-depth coverage of analysis of functions, discrete functions, data analysis, and parametric equations. Additional topics also include programming on the graphing calculator, inverse trigonometric functions, matrix algebra, vectors, classical fractals, and dynamical systems. Greater emphasis is placed on the solution of challenging problems: NCTM Calendar problems, AMC problems and Problems of the Week (POWs). Use of both computers (MS-Word©, Excel©, and TI-Interactive©) and graphing calculators (TI-83 Plus or TI-84) is required. The course requires about 1.5 times the amount of homework time as Algebra II. Three Portfolio requirements: classical fractals, finance, dynamical systems. (This course meets for three 45-minute periods and one 75-minute period per week.)
M300 Geometric Concepts: The course covers topics in plane geometry: parallel and perpendicular lines and planes, congruence and similarity in two and three dimensions, coordinate geometry, and some review of algebra/ trigonometry. The approach here is more visual and intuitive than in the Geometry course. The course emphasizes problem solving, pattern recognition, algebraic geometry and constructions. Geometer's Sketchpad© is used throughout the course. (Prerequisite: Algebra II or permission of the department)
M310 Geometry: The course covers new as well as traditional topics in geometry: logic, parallel and perpendicular lines and planes, congruence and similarity in two and three dimensions, coordinate geometry, transformations including linear transformation matrices, and networks. Previous work with trigonometry and vectors will be revisited in a geometric context. A major theme of this course is patterns in reasoning, including formal proof (direct and indirect), visual analysis, and problem solving. Computer explorations (primarily using Geometer’s Sketchpad©) are also included. (Prerequisite: Algebra II or permission of the department)
M320 Intensive Geometry: This course uses a variety of mathematical approaches to study plane and solid geometry. The course emphasizes different forms of mathematical proof and techniques of problem solving. In the process students are introduced to symbolic logic, linear algebra (through transformations), graph theory and networks, fractal geometry, conic sections, and the geometry of the complex plane. Students use Geometer’s Sketchpad extensively and complete two projects on the history and applications of geometry. (Prerequisite: an “A” in Algebra II or a “B” or higher in Intensive Algebra II, or permission from the department.)
M400 Precalculus Fundamentals: This course is designed specifically to meet the current levels of understanding of the students and to enhance their achievement and success as learners of mathematics. It is a bridge between Intermediate Algebra and Precalculus. The focus is on the concept of a function and the use of functions as mathematical models. There are also units on matrices and probability and statistics. Some SAT preparation is also included. Use of both computers (Excel©, Geometer’s SketchPad©) and graphing calculators (TI-83 Plus or TI-84) is required. Successful completion of this course enables a student to take either Precalculus or Statistics the following year. (Prerequisite: A course in the M300s or permission of the department)
M410 Precalculus: This problem based inquiry course focuses on advanced study of polynomial, exponential, logarithmic, power, and trigonometric functions, conic sections, parametric representation of functions, complex arithmetic, probability, sequences and series – finite and infinite, limits and an introduction to instantaneous rate of change. Additional topics such as matrices, vectors, and statistics, are introduced as time allows. The course materials are designed to promote and emphasize the following values: communication of mathematical ideas, persistence in solving challenging problems, self reliance, resourcefulness, and collaboration. Use of both computers (Excel, TI-Interactive©, Geometer’s SketchPad©) and graphing calculators (TI-83 Plus or TI-84) is required. (Prerequisite: Geometry, Intensive Geometry, or permission of the department.)
M420 Intensive Precalculus: The course rapidly, though thoroughly, reviews analysis of the elementary functions and their applications, and then extends the analysis to an introduction to the concepts and techniques of elementary differential calculus. The course also extends the study of dynamical systems, vectors, matrices and transformations, conic sections, and algebra and geometry in the complex plane. The course covers the content of both Precalculus and Differential Calculus, in addition to the enrichment topics listed above. Use of both computers (Excel©, Geometer’s SketchPad©) and graphing calculators (TI-83 Plus or TI-84) is required. The course requires about 1.5 times the amount of homework time as Precalculus. (Prerequisite: a year-grade of “A” in Geometry or a year-grade of “B” or higher in Intensive Geometry; and permission of the department)
M510 Calculus: This course covers an introduction to differential and integral calculus. Topics in elementary derivatives and integrals are studied from analytic, numerical, and graphical points of view. Importance is placed on intuitive understanding of calculus concepts as well as applications and problem solving. For example, students must identify the appropriate mathematical function to model a situation, and work with rates of change. Students who do particularly well in this course may take the AP Calculus AB exam in May. (Prerequisite: Precalculus and permission of the department.)
M520 Intensive Calculus: The course thoroughly covers the theory and applications contained in a first year college course in differential and integral calculus including: formal proofs for both derivative and definite integral; numerical methods including Simpson’s rule; the calculus of vectors and parametric equations; and infinite series, with particular emphasis on power series. Analytic, numerical, and graphical approaches to problem solving require students to work comfortably with graphing calculators and spreadsheet programs. The course requires about 1.5 times the amount of homework time as Calculus. This course often leads to advanced placement in college based on the Calculus BC Advanced Placement exam that is given each May. (Prerequisite: A year-grade of “A” in Precalculus or a year-grade of “B” or higher in Intensive Precalculus; and permission of the department)
M530 Statistics: The course covers descriptive statistics, data analysis, experimental design, and inferential statistics including hypothesis testing and confidence intervals. Topics on the AP syllabus are covered; students who do particularly well in this course and who are willing to do some extra work may take the AP Statistics exam given in May if they wish. Coursework is supported by use of graphing calculators (TI-83 Plus or TI-84), Excel, TI-Interactive!, and computer software as appropriate. Additional topics such as ANOVA may be included as time permits. (Prerequisite: Precalculus or Precalculus Fundamentals)
M540 Discrete Mathematics: The course introduces students to the mathematical thinking and applications often needed for the solution of contemporary problems. Six areas of mathematics are explored: management science (networks, planning scheduling); probability; the Digital Revolution (cryptography, electronic transmittal of information, the logic of Internet search engines); social choice and decision making (voting systems, game theory); symmetry and patterns; and mathematical models for resources (consumer finance, limitations to biological growth, dynamical systems). Students will maintain a portfolio of their explorations from different areas and will create a capstone interdisciplinary project to demonstrate their understanding of complex systems. (Prerequisites: Precalculus Fundamentals, or Precalculus, proficiency in the use of Excel and Internet research, or permission of department)
M620 Advanced Mathematics: The course is for students who have completed a calculus course. Course content depends upon the interests of the students taking the class. Material covered in past years has included probability, statistics, differential equations, cryptography, game theory, chaos and fractals, and topics from discrete mathematics such as graph theory, recursion, and algorithm development. (Prerequisite: A course in the M500s and/or permission of the department)
M730 Economics: The first semester of this course will focus on topics in Microeconomics, including Opportunity Cost, Supply and Demand, Consumer Behavior, Costs of Production, Market Structures, and Government Policy. While students will be better equipped to make decisions about spending and saving, this will not be a course in personal finance. Students should be comfortable working with algebra and graphs, and they will be challenged to use and develop their mathematical reasoning skills. The second semester will focus on topics in Macroeconomics, including Productivity, Inflation, Unemployment, Money and Banking, and International Trade. While this course will encourage students to think critically about current debates, it will not be a forum for political arguments. Rather this course will give students a solid framework to better analyze and comprehend global events, past and current. Students who are successful in this course will be well prepared to take the AP exams in Micro- and Macro-economics. (Prerequisite: Precalculus (may be taken concurrently) or permission of the department)
Fall Semester Elective Courses
M750 Computer Science I: A semester-long lab course offered in the fall. This course introduces students to computer programming through the Java language. While the course is intended for students with little or no programming experience, students work through a series of projects at their own pace so that a student with a programming background in another language could also take this class. The course begins with elementary data structures, algorithms, and console applications, and then moves on to object oriented programming and graphical applications.
M810 Independent Study I: A semester tutorial course offered in the fall. This course may be organized by an individual student or by a small group of students to study a chosen topic under the supervision of a teacher. Requirements for department approval of this arrangement must include: a contract with the supervising teacher, a list of required topics, homework assignments, projects and assessments (exams, portfolios, seminars). The requirements in this course must be equivalent to that in a standard Lakeside course. An independent study will or will not receive departmental approval within two weeks of the start of the semester. (Prerequisites: A course in the M600s or 700s (may be taken concurrently) and permission of the department)
Spring Semester Elective Courses
M755 Computer Science II: A semester-long lab course offered in the spring. This course is a continuation of Computer Science I covering more advanced data structures (classes, arrays, lists, trees), algorithms (searching, sorting), and object-oriented programming. Problem analysis, planning, coding, and debugging will be emphasized for each project. Students successfully completing this course should be well prepared for the Computer Science AB Advanced Placement exam. (Prerequisite: Programming I or permission of the instructor)
M815 Independent Study II: A semester tutorial course offered in the spring. This course may be organized by an individual student or by a small group of students to study a chosen topic under the supervision of a teacher. Requirements for department approval of this arrangement must include: a contract with the supervising teacher, a list of required topics, homework assignments, projects and assessments (exams, portfolios, seminars). The requirements in this course must be equivalent to that in a standard Lakeside course. An independent study will or will not receive departmental approval within two weeks of the start of the semester. (Prerequisites: A course in the M500s or 600s (may be taken concurrently) and permission of the department)