# Mathematics: Scope and Sequence ### 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. 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.

During the second unit, students use algebra to generalize, interpret, and analyze key patterns observed when working with linear functions. Particular attention is paid to patterns that relate to the concept of slope and how this concept manifests in graphs, tables, and equations. Students will also explore multiple methods of graphing linear functions including: creating a table; finding the x- and y- intercepts; using the slope- intercept form; and point slope form. With a strong linear functions foundation, students will transition into applying procedural graphing knowledge and skills to more conceptual tasks as they solve systems of equations and inequalities both graphically and algebraically.

During the quadratic functions unit, students begin to master the basic factoring techniques used extensively in the remainder to the Algebra curriculum. The concept of factoring will then be applied to graphing, analyzing, and interpreting the relationship between quadratic equations and their graphs. Students will also need to master multiple factoring techniques including completing the square and using the quadratic formula. Students will then begin to apply their procedural knowledge to more conceptual tasks as they solve physical problems including motion, force, gravity, and acceleration.

The final unit of study emphasizes computational mastery in a more complex algebraic manner. Students apply basic techniques of adding, subtracting, multiplying, and dividing as they simplify rational expressions. Students also expand their skills and knowledge of operations with fractions as they apply these skills to solving rational equations.

### 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

Honors Geometry will provide both breadth and depth of exploration in the subject area, developing writing, research, and analytical skills. The honors level course content and student experience will be demonstrably more challenging than what is offered through the regular college preparatory courses in the same field. Assignments and evaluations will include more challenging problem solving and writing requirements. Specific to the Geometry Honors Course student will be responsible for mastery of college level logic proofs in all sections and units of the course. Students will be required to demonstrate mastery using multiple perspectives that include but are not limited to: written analysis of concepts and connections: visual representation and manipulation; symbolic notation and justification; and relevant connections to real world situations.

### 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 Algebra II course focuses on four critical areas: relate arithmetic of rational expressions to arithmetic of rational numbers; expand understandings of functions and graphing to include trigonometric functions; synthesize and generalize functions and extend understanding of exponential functions to logarithmic functions; and relate data display and summary statistics to probability and explore a variety of data collection methods.

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. Students explore the structural similarities between the system of polynomials and the system of integers. They draw on analogies between polynomial arithmetic and base- ten computation, focusing on properties of operations, particularly the distributive property. Connections are made between multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The Fundamental Theorem of Algebra is examined.

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. Students also expand their problem solving strategies through the 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.

By identify appropriate types of functions to model a situation, students adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this Algebra II course. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context.

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.

Students will then transition to the study of sequences, series, and mathematical induction unit. Students learn to find a particular term in an arithmetic or geometric sequence. They will also compute sums of finite arithmetic and geometric series as well as of infinite geometric series.

### Honors Algebra II

Recommended For: Freshmen, Sophomores, Juniors
Prerequisites:

Honors Algebra II will provide both breadth and depth of exploration in the subject area, developing writing, research, and analytical skills. The honors level course content and student experience will be demonstrably more challenging than what is offered through the regular college preparatory courses in the same field. Assignments and evaluations will include more challenging problem solving and writing requirements. Honors level Algebra II classes require students to discover and build the formulas we use instead of being directly given them. Honors classes often move at a faster pace and students work more independently during practice exercises instead of being led by the teacher. Honors Algebra II assessments require students to demonstrate a deep understanding of the content and apply their knowledge to unique cases of each problem type. The writing assignments at the Honors level are held to a more rigorous standard in terms of vocabulary, evidence and precise language to describe their reasoning.

### 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 Trigonometry/Pre-Calculus

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. Foundational concepts for the course relate to the study of collecting, organizing, analyzing, and interpreting numerical information from populations or samples. Building on these techniques for gathering data students will begin to explore ways of organizing and presenting data as part of a branch of statistics called descriptive statistics. Graphs provide an important way for student to show how data is distributed. Several graphic displays will be examined and student will be required to construct these displays by hand and using technology (graphing calculators and Excel or Google Sheets). Students master the commonly used measures of the center know as mean, median, and mode. In addition, commonly used measure of spread such as variance, standard deviation, and range are examined as they relate to the mean of a data set.

The box and whisker plot is introduced as an alternative method of examining spread and distribution about the median for data sets that are skewed or bimodal. With the introduction of linear regression models and inferences related to these models, students will analyze scatter diagrams in terms of the linear relationships between x and y data points. The correlation coefficient will further highlight the mathematical features of the linear relationship and allow students to further analyze the strength of the relationship. Students will connect the linear algebra concepts of slope and y-intercept while finding the least square line, the linear regression model, and the variation in the explanatory variable.

In the intermediary stages of the course the concept of chance and likelihood is examined using theoretical and empirical experiment and calculations related to probability. Students will explore how the law of large numbers relates to relative frequencies and distributions. Students will have opportunities throughout the unit to apply basic rules of probabilities in everyday life spanning across multiple fields of interest (competition, games, medical, engineering, insurance, finance etc.). Multiple connections between statistical concepts previously studied and probability concepts allow students to solve for probabilities and events under a diversity of statistical circumstances.

The final stages of the course shift the focus to critical analysis of statistical and probability processes and results. Using point estimates and confidence intervals student are able to make effective estimations about a population using data that is gathered from a sample of the population. Hypothesis testing is a major component of this type of inferential statistics. Students will be required to define and use all the terms associated with hypothesis testing and be able to explain the logic behind hypothesis testing. The concept of Null Hypothesis will be examined in depth as well as the relationship between the outcomes of the hypothesis testing and Type I and Type II errors. Students will be able to apply the critical value and P- value approach to hypothesis testing. Tests of significance will be performed in order to determine the probability of rejecting the Null Hypothesis, when it is in fact true. Paired data samples will be introduced and applied problems in social science, natural science, and business administration will be examined frequently in the study of matching pairs as students look at inferences involving paired differences, differences of mean, and differences of proportions. Statistical inferences will conclude with the applications of the chi-square probability distribution and inferences for correlations and linear regression. Students will perform multiple tasks and experiments to develop and understanding and apply 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

Honors Probability and Statistics will provide both breadth and depth of exploration in the subject area, developing writing, research, and analytical skills. The honors level course content and student experience will be demonstrably more challenging than what is offered through the regular college preparatory courses in the same field. Assignments and evaluations will include more challenging problem solving and writing requirements. Honors Probability and Statistics is a unique mathematical course combining lessons and activities that incorporate elements from a wide range of subjects including psychology, 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 required to demonstrate mastery using multiple perspectives that include but are not limited to: written analysis of concepts and connections: visual representation and manipulation; symbolic notation and justification; and relevant connections to real world situations. Students will be required to take detailed and reflective notes, analyze studies and experiments, gather and organize data, problem solve, write detailed constructed responses/reflections; and create and design their own year long statistical study.