Mathematics: Scope and Sequence

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Algebra I

Recommended For: Freshmen
Prerequisites: None

The Algebra course is composed of four major units of study: Solving equations and inequalities, linear functions, quadratic functions, and rational expressions. In each of the major units of study, concepts will be explored using multiple representations so that students develop essential procedural and conceptual understandings in Algebra. The basic foundations of the algebra curriculum are developed in the first unit of study and quickly built upon in successive units. The central theme of this unit involves solving multistep equations and inequalities. Students will become adept at identifying and defining the algebraic properties and principles used to simplify and solve multistep equations and inequalities. These skills will then be applied to writing and solving multistep equations and inequalities for word problems. Each of the concepts in the first unit will be continuously revisited and reinforced throughout the remainder of the course and in every unit.

 


Geometry

Recommended For: Freshmen, Sophomores
Prerequisites: Algebra I

The course will allow students to strengthen their inductive and deductive reasoning as they examine and develop arguments, contradictions, and proofs. A significant amount of definitions, postulates, and theorems will need to be mastered by students as they perform basic proofs and then apply these proofs to real world problem solving situations. The course includes several major units of study beginning with the basic components of geometry and then proceeding to concepts involving two and three-dimensional geometric figures. The basic components unit includes a review of key notations and visual representations that will be used through out the course. Central to this unit are the angles relationships and properties that emanate from parallel lines cut by transversals.

Building on the basic components of geometry, the next unit relates to an extensive examination of triangles. Students will work extensively with two column proofs of triangle congruence and similarity. The triangle unit continues with a closer examination of right triangles. Students will know and apply the Pythagorean theorem, Distance Formula, special right triangle relationships, and trigonometric functions to find unknown lengths and angles in right triangles. The focus of the course then transitions to a more general investigation of the properties of two-dimensional figures including the relationships between angles and sides, area, and perimeter. Students then investigate the relationships and properties of three-dimensional figures involving computations and problem solving related to volume and surface area. Finally the course concludes with the circle unit. Students will develop theorems related to chords, secants, tangents, inscribed angles and polygons. These theorems will then be applied to problem solving situations that involve missing angle and arc measures, as well as finding the length of arcs, chords, tangents, and secants.

 


Honors Geometry

Recommended For: Freshmen, Sophomores
Prerequisites: Algebra I

Our Honors Geometry class will cover everything that Geometry covers but will be taught at a faster pace and with less support as our college prep section. The course will allow students to strengthen their inductive and deductive reasoning as they examine and develop arguments, contradictions, and proofs. A significant amount of definitions, postulates, and theorems will need to be mastered by students as they perform basic proofs and then apply these proofs to real world problem solving situations. The course includes several major units of study beginning with the basic components of geometry and then proceeding to concepts involving two and three-dimensional geometric figures. The basic components unit includes a review of key notations and visual representations that will be used through out the course. Central to this unit are the angles relationships and properties that emanate from parallel lines cut by transversals.

Building on the basic components of geometry, the next unit relates to an extensive examination of triangles. Students will work extensively with two column proofs of triangle congruence and similarity. The triangle unit continues with a closer examination of right triangles. Students will know and apply the Pythagorean theorem, Distance Formula, special right triangle relationships, and trigonometric functions to find unknown lengths and angles in right triangles. The focus of the course then transitions to a more general investigation of the properties of two-dimensional figures including the relationships between angles and sides, area, and perimeter. Students then investigate the relationships and properties of three-dimensional figures involving computations and problem solving related to volume and surface area. Finally the course concludes with the circle unit. Students will develop theorems related to chords, secants, tangents, inscribed angles and polygons. These theorems will then be applied to problem solving situations that involve missing angle and arc measures, as well as finding the length of arcs, chords, tangents, and secants.

 


Algebra II

Recommended For: Freshmen, Sophomores, Juniors
Prerequisites: Applicant must have finished Geometry & Algebra I

Algebra II provides a review and extension of the concepts taught in Algebra I and Geometry. Throughout this course, students will develop learning strategies, critical thinking skills, and problem solving techniques to prepare for future math courses in high school and college. The course begins with an extensive review of Algebra I concepts including equation and inequalities, linear equations and functions, systems of equations, radical expressions, quadratic equations and functions, polynomials, and rational expressions. A few new concepts such as complex and imaginary numbers and solving systems of equations in two and three variables, are introduced in order to build on students basic Algebra knowledge and skills.

The Algebra II course then explores the algebraic and geometric concept of conic sections. This includes the equations and graphing for several functions that define the conic section units including the circle, ellipse, parabola, and hyperbola functions. Students will develop an understanding of inverse functions and relations including an introduction to exponential and logarithmic functions, and in particular, natural logarithms. These functions will also be used in problem solving situations. The emphasis then shifts towards a study of matrices and determinants. Students will be required to master the addition, subtraction, and multiplication of matrices. In addition to using determinants and Cramer’s Rule, students will use inverse matrices to solve systems of two or three equations.

 


Trigonometry/Pre-Calculus

Recommended For: Sophomores, Juniors, Seniors
Prerequisites: Applicant must have completed Algebra I, II, and Geometry

The course is designed to strengthen student conceptual understanding and mathematical reasoning of techniques used in trigonometry, geometry, and algebra. Mathematical Analysis standards require students to know and apply to problem solving situations: polar coordinates and vectors; complex numbers; the fundamental theorem of algebra; conic sections; roots and poles of rational functions; functions and equations defined parametrically; and the limit of a sequences and functions. Trigonometry standards build on those concepts previously learner in the Geometry course. Students develop an understanding of angle measurements in degrees and radians and use this concept to graph in a variety of forms the sine, cosine, tangent, cotangent, secant, and cosecant functions.

Several more trigonometry identities are introduced. Students will prove these identities and use them to simplify other similar identities. The trigonometric functions will be revisited and used in problem solving situations and word problems in order to find the missing angle, side, or area of right triangles. Students must be familiar with polar coordinates and complex numbers and be able to multiply complex numbers in their polar form. Finally, students will apply these skills as they work with complex numbers in polar form using the DeMoivre’s theorem. In the Linear Algebra portion of the course the standards indicate an extensive examination and application of the algebraic and geometric interpretations of matrices and vectors. The goal of Linear Algebra is for students to learn the techniques of matrix manipulation so that they can solve systems of linear equations in any number of variables. Students must understand and know how to apply the Gauss-Jordan method and the Cramer’s rule of solving matrices.

 


Honors Pre-Calculus/Trigonometry

Recommended For: Sophomore, Juniors, Seniors
Prerequisites: Applicant must have a final grade of an A or B in Algebra II

Topics in Mathematical Analysis, Trigonometry, and Linear Algebra are often combined to create a pre-calculus course needed to prepare students for the study of Calculus. The course is designed to strengthen student conceptual understanding and mathematical reasoning of techniques used in trigonometry, geometry, and algebra. Mathematical Analysis standards require students to know and apply to problem solving situations: polar coordinates and vectors; complex numbers; the fundamental theorem of algebra; conic sections; roots and poles of rational functions; functions and equations defined parametrically; and the limit of a sequences and functions. Trigonometry standards build on those concepts previously learner in the Geometry course. Students develop an understanding of angle measurements in degrees and radians and use this concept to graph in a variety of forms the sine, cosine, tangent, cotangent, secant, and cosecant functions.

 


Honors Calculus AB

Recommended For: Juniors and Seniors
Prerequisites: Applicant must have a final grade of an A or B in Trigonometry/Pre-Calculus

The prerequisites to learning and using calculus are the algebra, trigonometry, and analytical geometry skills students have developed in the preceding Algebra II and Pre calculus classes. In addition to the rigor and depth that will permeate all aspects of this course students will hopefully also develop an appreciation for the versatility and usefulness that the study of Calculus provides to professional fields related to mathematics, science, design, technology, and engineering. The course begins with an examination of limits and continuity. Students will be required to calculate limits of function values and to test functions for continuity. Once students are able to calculate limits, they can then proceed to finding derivatives. The derivatives unit illustrates the role calculus plays in measuring the rates at which things change. Students will explore the circumstances in which derivates exist, the basic derivative techniques, rates of change, trigonometric derivatives, major rules and laws, common differentiation tasks, and an extensive application of derivatives in real world situations.

The focus of the course then shifts from derivates to finite sums and integrals. Students will examine the close connections between derivatives and integrals though the examination of the contributions of Leibniz and Newton to the study of Calculus. During the integral unit students will be required to work extensively with integration and derivatives as these concepts relate to the graphs of exponential, inverse, logarithmic, inverse trigonometric, and hyperbolic functions. Students will know and apply several major integration rules and theorems including the Fundamental Theorem of Calculus, L’Hopital’s rule, Mean Value theorem, and Rolle’s theorem. In addition, students will apply all the above techniques and theorems of integration to finding the volumes of rotational solids and arc lengths. Calculus students then transition to the study of differential equations, sequences, and series. The section pertaining to differential equations requires students to have knowledge of the separation of variables, the types of solutions, and exponential growth and decay. Students must also be able to visualize differential equations in terms of linear approximations, slope fields, and Euler’s method. The sequence and series section allows student the opportunity to examine basic examples of infinite series such as geometric series, P-series, and the telescoping series. Students will also be able to perform a variety of infinite series convergence test. Finally an exploration of special series such as the power series, the Maclaurin series, and the Taylor series will conclude the unit.

 


Honors Calculus BC

Recommended For: Juniors and Seniors
Prerequisites: Applicant must have a final grade of an A or B in Honors Calculus AB

Honors Calculus BC is a second course in a single-variable calculus that is equivalent to a second semester calculus course at most colleges and universities. This course will provide a deeper understanding of the concepts of limit, continuity, derivatives, and integrals which were covered in Honors Calculus AB. The major topics covered in Honors Calculus BC are Parametric, polar, and vector functions; slope fields; Euler’s method; L’Hopital’s Rule; Improper Integrals; Logistic differentiable equations; Polynomial approximations and Series; and Taylor Series.

 


Probability and Statistics

Recommended For: Juniors and Seniors
Prerequisites: Applicant must have completed Trig/Pre-Calculus

This course covers the study of probability, interpretation of data, and fundamental statistical problem solving. Students must know the definitions of the notions of independent events, conditional probability, mean, median, mode, variance of a discrete random variable, and the mean of a discrete random variable. Each of these definitions will then be used to solve for probabilities and events under a diversity of statistical circumstances. Throughout the course the distributions of data will be described using different methods including frequency tables, histograms, standard line and bar graphs, stem and leaf displays, scatter plots, and box and whisker plots. For the each distribution of data students must be able to identify the standard distribution and compute the variance and standard deviation. In an advanced placement probability and statistics class students must be able to determine P-value for a statistic and be familiar with and understand the uses of a chi-square distribution and the chi-square test.

 


Honors Probability and Statistics

Recommended For: Juniors and Seniors
Prerequisites: Applicant must have completed Trig/Pre-Calculus

Probability and Statistics is a unique mathematical course combining lessons and activities that incorporate elements from a wide range of subjects including Algebra, English, Science, Technology, and History. The course will include extensive topics in Statistics defined as the study of collecting, organizing, analyzing, and interpreting numerical information from data. The statistical elements will also be applied to the study of Probability as the likelihood that an event will occur. Together probability and statistics are tools that allow us to analyze data within a specific context in order to make informed decisions or predictions.Students will be expected to demonstrate mastery of the content by taking detailed and reflective notes, analyzing studies and experiments, gathering and organizing data, problem solving, writing detailed constructed responses/reflections; and creating and designing their own statistical studies.