A. Course Information Course Title: Mathematics IB HL1 Department: Mathematics Grade Level: 11 Length of Course: Two Semesters Type(Academies/CTE): IB Academic Type: Honors Level Pre-requisites: 1. The completion of Math 3 with a grade of B+ or better and Teacher Recommendation 2. The completion of Math 3 Honors with a C or better and Teacher Recommendation Modeled after another institution, program or after a UC approved course: IBO approved curriculum for Mathematics Higher Level B. Textbook: Title: Mathematics Higher Level Core Edition: 4th Edition Publication Date: 2012 Publisher: IBID Press Author: Fabio Cirrito, Nigel Buckle, Iain Dunbar Title: Mathematics Higher Level Topic 9 Option Calculus for the IB Diploma Publication Date: 2013 Publisher: Cambridge University Press Author: Paul Fannon, Vesna Kadelburg, Ben Woolley and Stephen Ward C. Supplementary Instructional Materials: Items listed below are commonly used as supplementary materials and are coordinated with the adopted course objectives: 1. Released exams from IBO and publisher s tests 2. Resources found on the IBO Forum 3. Formula Booklet from IBO D. Content Standards: Subject Area Content Standards Students will demonstrate an acceptable level of understanding of the following key concepts: 1. basic algebraic concepts and applications including arithmetic and geometric sequences and series, sigma notation, logarithms, counting principles, permutations and combinations, proof of mathematical induction, complex numbers, complex plane and systems of equations. 2. functions as a unifying theme in mathematics and to apply functional methods to a variety of mathematical situations. 3. circular functions to introduce trigonometric identities and to solve triangles using trigonometry. 4. basic concepts and techniques of differential and integral calculus and their applications. E. Description of the Course: IB Math HL Year 1 is the first year of a two-year course designed to compete the rigorous curriculum for the IB Diploma Program. The course focuses on developing important mathematical concepts in a comprehensible, coherent and rigorous way. Students are encouraged to apply their
mathematical knowledge to solve problems set in a variety of meaningful contexts. Development of each topic will feature justification and proof of results. Students in this course are expected to develop insight into mathematical form and structure, and will be intellectually equipped to appreciate the links between concepts in different topic areas. Students will be encouraged to develop the skills needed to continue their mathematical growth in other learning environments. This course is a demanding one, requiring students to study a broad range of mathematical topics through a number of different approaches and to varying degrees of depth. F. Description of technology and digital media implementation: 1. Graphic Display Calculator: A graphing calculator will be used regularly in class as well as on homework. 2. Equation Editor: A program in Microsoft word that allows students to write mathematical equations in the correct format. This program is particularly useful for their Internal Assessment. 3. GeoGebra: An interactive geometry, algebra, statistics and calculus program G. Describe how this course supports students academic literacy. Problem-solving is central to learning mathematics and involves the acquisition of mathematical skills and concepts in a wide range of situations, including non-routine, open-ended and real-world problems. Having followed a DP mathematics HL course, students will be expected to demonstrate the following: 1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts. 2. Problem-solving: recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems. 3. Communication and interpretation: transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation. 4. Reasoning: construct mathematical arguments through use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions. 5. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analyzing information, making conjectures, drawing conclusions and testing their validity. H. Complete Outline of Course Content: Topic 1 Core: Algebra The aim of this topic is to introduce students to some basic algebraic concepts and applications: 1. Arithmetic Sequences and Series: sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. a) Sigma notation b) Applications 2. Exponents and Logarithms a) Laws of Exponents and Laws of Logarithms b) Change of Base 3. Counting Principles, including permutations and combinations a) The Binomial Theorem b) Circular arrangements c) Proof of Binomial Theorem 4. Proof of Mathematical Induction
5. Complex numbers: the number I, the terms real part, imaginary part, conjugate, modulus and argument a) Cartesian form z = a + ib b) Sums, products and quotients of complex numbers 6. Modulus argument (polar) form z = r(cos θ + i sin θ) = rcisθ = re iθ a) The Complex Plane 7. Powers of Complex numbers: DeMoivre s Theorem a) n th roots of a complex number 8. Conjugate roots of polynomial equations with real coefficients 9. Solutions of systems of linear equations (a maximum of three equations and three unknowns), including cases where there is a unique solution, an infinity of solutions or no solution. Topic 2 Core: Functions and Equations The aims of this topic are to explore the notion of function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of technology in both the development and the application of this topic. 1) Concept of function f: x f(x): domain, range, image (value) a) Odd and even functions b) Composite of functions c) Identity function d) One-to-one and many-to-one functions 2) The graph of a function; its equation y = f(x). Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range. a) The graphs of the functions y = f(x) and y = f( x ) b) The graph of y = 1 given the graph of y = f(x) f(x) 3) Transformations of graphs; translations; stretches; reflections in the axes a) The graph of the inverse function as a reflection in y = x 4) The rational function x ax+b, and its graph cx+d a) The function x a x, a > 0 and its graph b) The function x log a x, x > 0, and its graph 5) Polynomial functions and their graphs a) The factor and remainder theorem b) The fundamental theorem of algebra 6) Solving quadratic equations using the quadratic formula a) Use of the discriminant to determine the nature of the roots. b) Solving polynomial equations both graphically and algebraically c) Sum and product of the roots of polynomial equations d) Solution of a x = b using logarithms e) Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach 7) Solutions of g(x) f(x) a) Graphical or algebraic methods, for simple polynomials up to degree 3 b) Use of technology for these and other functions Topic 3 Core: Circular Functions and Trigonometry The aims of this topic are to explore the circular functions, to introduce some important trigonometric identities and to solve triangles using trigonometry. 1) The Circle: radian measure of angles
a) Length of arc; area of a sector 2) Definition of cos θ, sin θ and tan θ in terms of the unit circle a) Exact values of sin, cos, and tan of 0, π, π, π, π and their multiples. 6 4 3 2 b) Definition of the reciprocal trigonometric ratios sec θ, csc θ, and cot θ c) Pythagorean Identities: cos 2 θ + sin 2 θ = 1; 1 + tan 2 θ = sec 2 θ; 1 + cot 2 θ = csc 2 θ 3) Compound angle identities a) Double angle identities 4) Composite functions of the form f(x) = asin(b(x + c)) + d a) Applications 5) The inverse functions x arccos x, x arctan x; their domain and ranges; their graphs 6) Algebraic and graphical methods of solving trigonometric equations in a finite interval, including the use of trigonometric identities and factorization. 7) The cosine rule a) The sine rule including the ambiguous case b) Area of a triangle as ½ ab sin C c) Applications Topic 6 Core: Calculus The aim of this topic is to introduce students to the basic concepts and techniques of differential and integral calculus and their application. 1) Informal ideas of limit, continuity and convergence a. Definition of derivative from first principles f (x) = lim h b. The derivative interpreted as a gradient function and as a rate of change c. Finding equations of tangents and normals. d. Identifying increasing and decreasing functions e. The second derivative f. Higher derivatives 2) Derivatives of x n, sin x, cos x, tan x, e x, and ln x a. Differentiation of sums and multiples of functions b. The product and quotient rules c. The chain rule for composite functions d. Related rates of change e. Implicit differentiation f. Derivatives of sec x, csc x, cot x, a x, log a x, arcsin x, arccos x, and arctan x 3) Local maximum and minimum values a. Optimization problems b. Points of inflexion with zero and non-zero gradients c. Graphical behavior of functions, including the relationship between the graphs of f, f, and f" 4) Indefinite integration as anti-differentiation a. Indefinite integral of x n, sin x, cos x, and e x b. Other indefinite integrals using the results from 6.2 c. The composites of any of these with a linear function 5) Anti-differentiation with a boundary condition to determine the constant of integration a. Definite integrals b. Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves c. Volumes of revolution about the x-axis or y-axis 6) Kinematic problems involving displacement s, velocity v and acceleration a h 0 f(x+h) f(x)
a. Total distance travelled 7) Integration by substitution a. Integration by parts *Content subject to change as determined by IBO curriculum reviews. I. Key Assignments: 1. Homework 2. Examinations designed by the instructor or by the textbook authors or a mixture of both 3. Internal Assessment J. Assessments and assessment procedures: 1. Homework: Designed to give students practice on concepts that they have recently learned. Used as a formative assessment. 2. Examinations: Designed as a summative assessment used at the end of a chapter or unit. 3. Internal Assessment: This is a piece of written work that involves investigating an area of mathematics. This area of investigation is chosen by the student giving students the opportunity for developing independence in their mathematical learning. Students are encouraged to take a considered approach to various mathematical activities and to explore different mathematical ideas. The internal assessment also allows students to work without the time constraints of a written examination and to develop the skills they need for communicating mathematical ideas. K. List college and career skills that students will be acquiring: 1. Problem-solving skills, decision-making skills, and the ability to plan and complete a project, 2. An understanding of the principles and nature of mathematics 3. The ability to communicate clearly and confidently in a variety of contexts 4. Logical, critical and creative thinking skills, and patience and persistence in problem-solving 5. The ability to employ and refine their powers of abstraction and generalization 6. The ability to apply and transfer skills to alternative situations, to other areas of knowledge