Mathematics

Departmental Statement

Mathematics is an all-encompassing field in today’s dynamic society. Students need to acquire a foundation to discern, adapt and make rational decisions. Increasing the knowledge and understanding of a variety of data and technology enables the development of independent, responsible, problem solving and decision-making individuals. Mathematics helps students to successfully understand the society and the environment they live in and provides opportunities for creative thinking.

The Mathematics program involves understanding the pervasive impact of Mathematics on society and in daily life, higher-order thinking skills, as well as the ability to seek, evaluate, organize, interpret and present information. The curriculum is designed to enable students to become creative problem solvers. More specifically, the goals are to increase students’ understanding of mathematical concepts, develop their ability to discern, reason logically, address a variety of issues that occur in learning, and to apply mathematic techniques in the real world.

Course Selection

Students follow one sequence of courses in their mathematic education, consisting of MAG1C (Algebra I), MAG2C (Algebra II), MGA2C (Algebra/Geometry), MAG3C (Algebra III), MST4C (Statistics), MPC4C (Pre-calculus), MAB5C (AP Calculus AB) and MCA5C (AP Calculus AB/BC). Under normal circumstances, three credits in any seven of the above courses are required to receive a diploma from AIS.

The Mathematics Department requires the minimum of MAG1C (Algebra I), MAG2C (Algebra II) and MGA2C (Algebra/Geometry). However, students are encouraged to take a Mathematics course for each academic year of study. For some students, it is permissible for them to be allowed to take two courses concurrently with possible combinations of Algebra II and Algebra/Geometry or Algebra III and Pre-calculus. This “accelerated track” will permit the students to take Advanced Placement classes in senior year. Entrance to the “accelerated track”, is dependent upon the student achieving 85% and the approval of the Mathematics Department.

Students transferring from other schools to AIS are given an entrance placement test, and will be placed according to their ability as well as their previous academic record.

Mapping the Mathematics Program

Course Description

The objectives of this course are to prepare students for either Algebra II or Geometry. Expressions, equations and applications represent the content of this course. Students write expressions representing variable quantities in real-life situations, write mathematical expressions in terms of x, or find x when the value of the expression is known. Students develop skills in solving realistic word problems in which questions involve real numbers, expressions with two variables, factoring problems, radical equations and inequalities, probability, scattered data, and linear functions. They are engaged in a variety of learning tasks such as discussions, group work and presentations. Students learn to use graphing calculators to solve problems.

Learner Outcomes

Students should be able to:

• Identify patterns and use them as a basis of predictions
• Solve multi-step equations
• Simplify expressions by using the rules and properties of operations
• Understand the concepts and components of linear functions and their graphs
• Understand the concept of proportion and percent from an algebraic perspective and apply them to basic principles of experimental probability and to fundamental techniques to the study of statistics
• Solve and graph absolute value equations and inequalities (on a number line)
• Become familiar with different techniques for solving systems of linear equations in 2 variables
• Simplify exponential expressions and polynomials
• Understand how to factor special polynomials and be able to solve equations by factoring and quadratic formula

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

The objectives of this course are to enable students to further develop their literacy skills as they experiment with various types of reading and writing. The Honors track of this course encourages students to seek greater analytical depth in the coursework.  Students are provided with enrichment activities and opportunities to further enhance their skills.  The course moves at an accelerated pace, and students will study a greater breadth of higher-level literature compared to that in ENG 1C.  Major texts under study include Falling Leaves, Romeo and Juliet, and The Odyssey.  During the course, students develop skills in process writing, interpreting and comparing texts, writing for different audiences and purposes, presenting and supporting an argument, and synthesizing information from a variety of sources. Students engage in a variety of learning tasks such as discussions, role-plays, dramatizations, group work, presentations, independent research, and individual writing assignments.Various resources are used throughout the year, including Elements of Literature, 3rd Course, films, and non-fiction supporting materials.

Learner outcomes

Students will be able to:

• Comprehend, respond to, and analyze a wide variety of literary and informational texts through the employment of reading strategies and will apply knowledge of word use in written English to become fluent readers;
• Write in clear, concise, organized language that varies in content and form for different audiences and purposes;
• Listen actively to information from a variety of sources in a variety of situations;
• Speak in clear, concise, organized language that varies in content and form for different audiences and purposes;
• Create, access, view, evaluate, and respond to print, non-print, and electronic texts and resources; and
• Demonstrate a command of research skills.

Assessment

Students will be assessed on written and oral responses to literature, nonfiction texts, and media, formal essays and other types of writing, comprehensive tests, student-facilitated presentations, and a research project.

Course Description

The objectives of this course are to reinforce and expand topics from Algebra I and prepare students for Algebra III (MAG) or Pre-Calculus (MPC). Topics include elementary Algebra techniques, linear functions, quadratic functions, exponential functions, logarithm functions, polynomial functions, and complex numbers. Students develop skills in manipulating variables, sketching graphs with and without the use of calculators, deriving equations of functions from graphs or data, and transforming them into various forms. In addition, they investigate and evaluate different concept problems such as continuity and slopes of tangent lines of different graphs. Students engage in a variety of learning tasks such as discussions, and independent research. Various resources include graphing calculators, videos, and the Internet.

Grade 9 students may take this course and Geometry concurrently.  Successful completion enables them to enter the Advanced Placement (AP) Mathematics course offered in Grades 11 and 12.

Learner Outcomes

Students should be able to:

• Define, analyze, graph, and apply in mathematical and real-world situations (linear function; inequalities; absolute value equations and inequality; quadratic functions; exponential functions; piece-wise functions, step functions; parametric equations)
• Learn basic function concepts: functions as relations; equations with functions; composition of functions and inverses of functions; transformations

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

The objectives of this course are to introduce basic concepts in such a way that students relate Algebra and Geometry to their everyday world, and to help students develop critical thinking. Students learn the basic concepts and skills, develop reasoning, and apply what they learn to various subjects in Algebra and Geometry. Students will develop skills in writing logic “if-then” statements, their converses, inverses, and contra-positives as well as defining postulates to deduce proofs of theorems.  Applying theorems to areas such as triangles, quadrilaterals, and circles is also a focus.

Students engage in a variety of learning tasks such as discussions, group work, independent research, presentations, and project research. Various resources include graphing calculators, videos, and the Internet.

Learner Outcomes

Students should be able to:

• Use definitions, postulates, and theorems and construct diagrams to justify the validity of a statement and give counter examples to disprove a statement
• Know how to complete two column proofs
• Prove congruency and similarity of triangles and use theorems on corresponding parts of congruent concept triangles
• Find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems
• Prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles
• Use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles
• Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms and trapezoids
• Perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line
• Prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles
• Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles
• Prove relevant theorems by using coordinate Geometry, including the midpoint of a line segment, the distance formula and various forms of equations of lines and circles
• Know the definitions of the basic trigonometric functions defined by the angles of a right triangle and how to use elementary relationships between them
• Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side
• Understand and use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles, and 45°, 45° and 90° triangles
• Use the law of cosines, together with the law of sines, to solve triangles.
• Use vectors and vector addition to solve problems

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

The objectives of this course are to emphasize the role of Algebra and Trigonometry as the foundation for Calculus. Representing both Algebra and Trigonometry as the study of classes of functions, students learn the essential unity of the two subjects and understand the connections between trigonometric functions, complex numbers and series.  Students apply Trigonometry to situations involving triangles and explore real world phenomena using the sine and cosine functions to understand the connections between trigonometric and circular functions. Students use circular functions to model periodic real world situations and apply general techniques to trigonometric functions to solve trigonometric equations and verify trigonometric identities. In addition, students investigate properties of circles, ellipses, hyperbolas, sequences, series, binomial formula, counting principles, permutations, combinations and probability.

Students engage in a variety of learning tasks such as discussions, group work, independent research, presentations, use of graphing calculators and computer in solving problems. Various resources include graphing calculators, videos and the Internet.

Learner Outcomes

Students should be able to:

• Know the definition of all six trigonometric functions
• Deduce the graph of the function in the form f(t) = A sin(B+C)+D and interpret A, B, C, and D in term of amplitude, frequency, period, horizontal shift, and vertical shift
• Know the trigonometric formula and use them to prove identities
• Use different methods to solve trigonometric equations
• Be adept at using Trigonometry in a variety of applications and word problems

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

The objectives of this course are to expose students to graphical, numerical, analytical and verbal representations of different functions. Students are exposed to polynomials, logarithmic, exponential, trigonometric, inverse trigonometric, piecewise, and vector value functions. The concept of the function’s rate of change is introduced.  Students develop skills in manipulating Algebra, properties of functions, the Algebra of function, the language of function (domain and range, odd, even, periodic, symmetry, zeros, intercepts etc), reasoning about new functions derived from familiar ones via composition, inverse and arithmetic combinations, modeling a wide variety of functions and interpreting results and verify conclusions. They engage in a variety of learning tasks such as group work, individual and group presentations, and the preparation of a portfolio consisting of a combination of assignment projects. Various resources include websites, graphing software, and graphing calculators.

Learner Outcomes

Students should be able to:

• Know the characteristics of and be able to graph quadratic, polynomial, logarithmic, exponential, circular, and trigonometric functions
• Use trigonometric, exponential, polynomial, and rational functions to model real-life data
• Know how derivatives (rate of change) may be used in curve sketching, in the solution of maximum / minimum problems and in developing the idea of instantaneous rate of change
• Apply simple transformations, including a • ƒ(x), ƒ(x) + d, ƒ(x+c), ƒ(b • x), |ƒ(x)|, ƒ(|x|),ƒn(x),1/ƒ(x) to basic  functions
• Perform operations including composition and decomposition on functions, find inverses and describe these procedures and results verbally, numerically, algebraically, and graphically
• Investigate identities graphically and verify them algebraically, including logarithmic properties, trigonometric identities, and exponential properties
• Work with vectors algebraically and graphically and be able to graph vector valued functions
• Apply sequences and series to solve problems including sums, binomial expansion, binomial theorem, combinations and Pascal’s Triangle

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

This is an Algebra-based course that covers basic statistical concepts and techniques. It is designed to provide students with a foundation in core statistical topics such as descriptive statistics, probability, estimation, hypothesis testing, and linear regression. Students use statistical methods to interpret real life data from newspapers, magazines and other sources. Students are expected to know how to use computer software for graphing, worksheets and presentation (e.g. MS Office software). Projects require students to work collaboratively to discuss problems, analyze data, and propose solutions.

Learner Outcomes

• Identify the way technology is utilized to handle large data sets or complex, real life questions
• Organize and describe of data sets
• Utilize of data to predict the probability that an event will occur
• Create and utilize of probability distributions
• Recognize normal distributions and how to use their properties in real-life applications
• Make meaningful estimations of population parameters utilizing confidence intervals
• Test a claim about a parameter
• Test a hypothesis that compares two populations
• Describe and test the significance of relationships, between two variables when data are presented as ordered pairs

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

This course prepares students for the Advanced Placement (AP) Calculus AB examination. It familiarizes students with graphical, numerical, analytical and verbal representations of different functions and the connections among those representations. Students learn the meaning of the derivative in terms of a rate of change and local linear approximation and will be able to use derivatives to solve a variety of problems.  They also learn the meaning of the definite integral both as a limit of Riemman sums and as the net accumulation of rate of change and will be able to use integrals to solve a variety of problems.   The relationship between the derivative and the integral as expressed in both parts of the Fundamental Theorem of Calculus are also studied.

Students develop skills in using technology to help solve problems, and experiment, interpret results and verify conclusions.  They also develop an awareness of the applications of Calculus in the field of Engineering, Physics, Biology and areas such as Business and Economics.  Students engage in a variety of tasks such as group work, individual and group presentations, and preparation of a portfolio (combined assignment projects) and practice through homework. Various resources include websites, graphing software, and graphing calculators.

Learner Outcomes

Students should be able to:

• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal and understand the connections between these representations
• Compute limits of functions and prove the existence of limits
• Identify continuous functions
• Apply the definition of derivative and compute derivatives of algebraic, logarithmic, exponential and trigonometric functions using the product, quotient, and chain rules
• Sketch curves by applying the first and second derivative tests and characteristics derived from the first and second derivative
• Apply the derivatives to related problems, relative extrema problems and applications in rectilinear motion
• Find anti-derivatives of functions
• Use the fundamental theorem of Calculus to evaluate proper integrals
• Apply integration to problems following the law of natural growth and decay (logistic growth), to find the areas of region, and to compute the volumes of solids of revolutions
• Compute integrals using integration by substitution and by approximations
• Solve separable differential equation by integration and by the Euler’s method

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.

Course Description

This course prepares students for the Advanced Placement (AP) Calculus AB/BC examination. It familiarizes students with graphical, numerical, analytical and verbal representations of different functions and the connections among those representations. Students learn the meaning of the derivative in terms of a rate of change and local linear approximation and will be able to use derivatives to solve a variety of problems.  They also learn the meaning of the definite integral both as a limit of Riemman sums and as the net accumulation of rate of change and will be able to use integrals to solve a variety of problems.   The relationship between the derivative and the integral as expressed in both parts of the Fundamental Theorem of Calculus are also studied.

Students develop skills in using technology to help solve problems, and experiment, interpret results and verify conclusions.  They also develop an awareness of the applications of Calculus in the field of Engineering, Physics, Biology and areas such as Business and Economics.  Students engage in a variety of tasks such as group work, individual and group presentations, and preparation of a portfolio (combined assignment projects) and practice through homework. Various resources include websites, graphing software, and graphing calculators.

Learner Outcomes

Students should be able to:

• Work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal and understand the connections between these representations
• Compute limits of functions and prove the existence of limits
• Identify continuous functions
• Apply the definition of derivative and compute derivatives of algebraic, logarithmic, exponential and trigonometric functions using the product, quotient, and chain rules
• Sketch curves by applying the first and second derivative tests and characteristics derived from the first and second derivative
• Apply the derivatives to related problems, relative extrema problems and applications in rectilinear and plane motion
• Find anti-derivatives of functions
• Use the fundamental theorem of Calculus to evaluate proper and improper integrals
• Apply integration to problems following the law of natural growth and decay (logistic growth), to find the areas of region, and to compute the volumes of solids of revolutions and find the length of  a smooth curve specified parametrically
• Compute integrals using integration by substitution and  by estimating finite sums
• Solve separable differential equation by integration and by estimating particular solutions using the Euler’s method
• Manipulate the Taylor Series ad shortcuts to compute the Taylor Series, including differentiation, anti-differentiation and the formation of a new series from a known series.
• Explore the convergence or divergence of a series of non negative terms using different tests
• Explore the alternating series with the error bound
• Sketch the graph and analyze an equation in polar co-ordinates and calculate the area enclosed by polar graphs

Assessment

Students will be assessed according to their performance in chapter tests, quizzes, homework, projects, participation as well as semester and final exams.