# Mathematics

## Algebra and Geometry • Intermediate Electives • Advanced Electives • Teaching Assistants

## Algebra and Geometry

**Algebra I**

This is a comprehensive course in which students master fundamental algebraic topics and techniques. These include evaluation and simplification of algebraic expressions, solving and graphing linear equations, linear systems, operations with polynomials, radical and rational expressions, and factoring. Throughout the course students encounter many opportunities to gain problem-solving skills and number sense. Students use manipulatives to gain an understanding of abstract concepts. Those who successfully complete this course enroll in Algebra II/Geometry, Year One the following year.

**Algebra IB**

This is a course in which students finish mastering fundamental algebraic topics and techniques. These include evaluation and simplification of algebraic expressions, solving and graphing linear equations, linear systems, operations with polynomials, radical and rational expressions, and factoring. Throughout the course students encounter many opportunities to gain problem-solving skills and number sense. Students use multiple methods to gain an understanding of abstract concepts. Those who successfully complete this course enroll in Algebra II/Geometry, Year One the following year.

**Algebra II/Geometry, Year One**

This course introduces and integrates concepts of geometry and intermediate algebra emphasizing an inductive approach. Geometer’s Sketchpad and Fathom computer programs, a graphing calculator, and manipulative tools such as patty paper, compass and straightedge are used to help students discover fundamental geometrical and algebraic relationships. Topics include properties of parallel and perpendicular lines, triangle, polygon, and circle properties, right triangle trigonometry, transformations, linear and quadratic functions, arithmetic sequences, variation, proportion, and similarity. Coordinate geometry is emphasized throughout. If time allows, students will also study some elementary statistics, including measures of central tendency and fitting data to a line. Prerequisite: Algebra I or the equivalent.

**Accelerated Algebra II/Geometry, Year One (Honors Level)
**

This course will cover all of the topics of Algebra II/Geometry, Year One, at an accelerated pace and a greater level of depth. We will place extra emphasis on deductive reasoning and the role of proof in mathematics. Additional topics may be included. Prerequisite: Algebra I or the equivalent; consent of the instructor and the department chair.

**Algebra II/Geometry, Year Two**

This course is the second in a two-year sequence and continues to integrate geometry and intermediate algebra concepts, but now emphasizing deductive reasoning. Students write various forms of formal proofs in order to establish many of the geometrical and algebraic conjectures they formed in the previous course, as well as additional principles. New topics include congruent triangles, inequalities in triangles, solid geometry, real number exponents, inverse functions, higher degree polynomial functions, exponential and logarithmic functions, complex numbers, Pythagorean proofs, rational functions, and coordinate geometry proofs. Prerequisite: Algebra II/Geometry, Year One, or the equivalent.

**Accelerated Algebra II/Geometry, Year Two (Honors Level)**

This course will cover all of the topics of Algebra II/Geometry, Year Two, at an accelerated pace and a greater level of depth. Additional topics may be included. Prerequisite: Algebra II/Geometry, Year One, or the equivalent; consent of the instructor and the department chair.

## Intermediate Electives

*One intermediate elective is offered each year. The prerequisite for all intermediate electives is completion of Algebra II/Geometry, Year Two, or the equivalent.*

**Functions, Statistics, and Trigonometry **

This year-long course provides instruction on functions, statistics, probability, and trigonometry for the general college preparatory student. Emphasis is placed on polynomial, exponential, logarithmic, and rational functions, and the development and use of the trigonometric functions on the unit circle (including the study of right and oblique triangle applications). It also includes a component on the gathering and use of data to address real-world issues, statistical influence, and probability.

**Game Theory (Global Online Academy; Spring 2015)**

Do you play games? Ever wonder if you’re using “the right” strategy? What makes one strategy better than another? In this course, we’ll explore a branch of mathematics known as game theory, which answers these questions and many more. Game theory is widely applicable in the real world as we face dilemmas and challenges every day, most of which we can mathematically treat as games! We will consider significant global events like the Cuban Missile Crisis, Mandela’s rise in South Africa, or the rise of Nobel Peace Prize winner Sirleaf in Liberia from a math perspective. Specific mathematical ideas we'll discuss include two person zero sum games, utility theory, two person non-zero sum games, multi-player games, game trees, matrix algebra, linear optimization, and applications of game theory techniques to a plethora of real world problems. (Prerequisite: Comfortable with Algebra) Note: This is an online course.

**Statistics 1 (Fall 2014)**

This course will cover gathering, describing, and displaying data, and topics in probability. Students will learn how to gather data by conducting censuses, surveys, and experiments around their school. We will also cover topics including, but not limited to, boxplots, the Normal model, and linear regression. We will always strive to connect the statistical material learned in class with real world applications to economics, elections, weather, and other themes. The semester will culminate with a unit on probability in which students will calculate the expected value of casino games. The two major goals of this course are for students to see the connection between mathematics and the real world, and for them to gain the tools to discern between reliable and questionable data that they are confronted with daily in the media and in everyday life.

**Statistics 2 (Spring 2015)**

This course will cover how to analyze data using statistical methods. Students will study confidence intervals and tests of inference including, but not limited to, hypothesis tests for proportions and means and the Chi-squared test. With the tools from this course, students will be able to form educated opinions from data on questions ranging from, “Is global temperature increasing?” to “Do SAT scores predict success later in life?” We will always strive to connect the statistical material learned in class with real world applications to economics, elections, weather, and other themes. The two major goals of this course are for students to see the connection between mathematics and the real world, and for them to gain the tools to discern between reliable and questionable data that they are confronted with daily in the media and in everyday life. Statistics 1 is a useful, but not mandatory, prerequisite.

## Advanced Electives

**Precalculus
**

A short review of the concepts of functions and their properties is followed by a thorough study of circular and triangular trigonometry. Students study conic sections, logarithmic and exponential functions, the graphs of rational functions, Binomial Theorem, arithmetic and geometric series and sequences, polar coordinates, 2-D vectors, polynomial graphs and functions, and parametric equations. Students use paper, pencil, and graphing calculators. Completion of this course prepares students to take Honors Statistics and/or Honors Calculus I. Prerequisite: Algebra II/Geometry, Year Two, or the equivalent.

**Accelerated Precalculus (Honors Level)**

Topics covered include all of those listed for Precalculus. In addition, Accelerated Precalculus includes three-dimensional vectors, DeMoivre’s Theorem, and mathematical induction. This course is for students who have a strong interest in mathematics and want to pursue advanced topics in great depth. Students are prepared to take Statistics and/or Calculus I upon successful completion of this course. Prerequisite: Algebra II/Geometry, Year Two, or the equivalent.

**Calculus**

This course will introduce students to the basics of differential and integral calculus. Concepts of the derivative as a slope and the integral as area will be explored using real-world examples as well as from a numerical, algebraic, visual, and verbal perspective. Activities using technology (Geometer’s Sketchpad, Mathematica, Desmos, etc.) will be utilized to help students understand concepts. Introductory rules for finding derivatives and integrals will be mastered and applied. This course is for students who want an introduction to calculus, but without the rigor required of preparing for an AP level exam.

**Honors Calculus I (Honors Level)**

Students enrolling in this course are assumed to have strong fundamental algebra and precalculus skills. Topics include limits, continuity, derivatives, integrals and their applications, slope fields, and separable differential equations. Concepts are approached through a three-step process: graphically, numerically, and analytically. Graphical analysis plays a major part in the development of many concepts. Students are prepared to take the Advanced Placement Calculus AB exam in May. Prerequisite: Precalculus or Accelerated Precalculus.

**Honors Calculus II (Honors Level)**

This course is a continuation of Calculus I and includes infinite sequences and series; parametric, polar, and vector function calculus; slope fields; Euler’s method; L’Hôpital’s rule; improper integrals; integration techniques; and an introduction to differential equations. If time permits, multivariable calculus is introduced. Students are prepared to take the Advanced Placement Calculus BC exam in May. Prerequisite: Calculus I or the equivalent.

**Multivariable Calculus (Global Online Academy)**

In this course, students will explore vector algebra and functions, matrices, curves in space, arc length and curvature, and velocity and acceleration. Further topics include partial differentiation, local extrema, exact differentials, the chain rule, directional derivatives, gradients, double and triple integration, line integrals, and volume. Students must have access to a computerized 3-D graphing utility, such as Grapher (a standard utility on Mac computers) or Autograph, and must be comfortable using learning to use new technology independently. Prerequisite: Calculus I. Note: This is an online course.

**Honors Statistics (Honors Level)**

This course begins with an in-depth study of descriptive statistics, variation, and probability, which leads into the study of inferential statistics. Topics include the concepts of statistical models and use of samples, variation, statistical measures, sampling distributions, probability theory, tests of significance, one-way and factorial analysis of variance and covariance and elementary experimental design, multiple linear regression and correlational design, and chi-square. If time permits, a few of the following topics will be presented based on student interests: Continuous random variables, Monte Carlo Methods, nonparametric statistical methods, multivariate analysis of variance and covariance, hierarchical linear modeling, and exploratory factor analysis. In addition, students will learn how to critically analyze quantitative research, evaluate the evidence on which generalizations are made, and write a quantitative methods paper.

**Advanced Statistics and Data Science (Global Online Academy; Fall 2014)**

This course is designed to help students discover the power, diversity, and broad applicability of statistics. Students will learn a variety of data analysis techniques, such as multiple regression, simulation studies, and survival analysis, with an emphasis on showing how these methods have revolutionized the use of statistics in fields such as engineering, environmental studies, economics, and medicine. Within each unit, guided activities assist students in working through the entire process of a real-world case study. The course uses an inquiry-based approach that teaches advanced statistical techniques through group work and hands-on exploration of current research questions. By researching the literature, planning and carrying out experiments, and presenting their results, students in this course will experience data analysis as it is actually practiced. Note: Students should have taken at least one semester or equivalent of an introductory statistics course. *Students taking this course are eligible to receive transferable Grinnell College credit.

## Teaching Assistants

**Math Teaching Assistants**

Teaching assistants are vital contributors to our algebra and geometry classes. TAs attend class each day, help students with practice problems and resolve homework difficulties, answer questions, and grade daily homework. In addition, they run review and extra-help sessions. As the year progresses, TAs plan and teach full lessons to the class. Upon completion of the year, students will receive one math credit. Prerequisite: Invitation of the department.