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Home , Fluent (mathematics)

P.O. BOX AA  70 CHURCH STREET  MILLBROOK, NEW YORK 12545

IB HL Analysis and Approaches

2019-2020

Mr. McPherson The IB Mathematics: Analysis and Approaches HL course content has been developed to meet the requirements of the new IB Diploma Subject Guide (first exams May 2021). There is a strong emphasis on and on algebraic, graphical and numerical approaches. In this course, students will develop strong skills in mathematical thinking and become fluent in the construction of mathematical arguments.

The Mathematics: Analysis and Approaches course is designed for students who wish to study mathematics as a subject in its own right or to pursue their interests in areas related to mathematics. It will appeal to students who are interested in exploring real and abstract applications of mathematical concepts. They will enjoy problem solving and generalisation. This course is suitable for students who may go on to further study in subjects that have a significant level mathematics content, for example mathematics itself, engineering, physical sciences or economics.

Students who wish to take the course at HL should have strong algebraic skills and really enjoy solving challenging problems. Topics covered through the course The five topics below are covered during the SL and HL courses. Each of these topics has sub-topics with HL students covering some additional sub- topics or the same sub-topics at greater depth. •Number and Algebra •Functions •Geometry and Trigonometry •Probability and Statistics •Calculus There is some content that is common with the Mathematics: Applications and Interpretations course but the Mathematics: Analysis and Approaches has a greater emphasis on calculus, numerical and algebraic approaches.

Activities and assessment Teacher and students have regular opportunities to assess progress and attainment. Some activities assess student achievement against formal criteria, such as worksheets, problem sets or quizzes.

Advice from our teachers In order to succeed, it is recommended students commit seven to ten hours a week.

Benefits to students Taking Mathematics: Analysis and Approaches HL will enable students to: •develop independence in learning and time-management •develop their ability to work in a connected world •become experienced in a wide range of digital learning tools •gain skills for success at university, work and life

Course Syllabus – Year 1

Ch 1 – Quadratics Functions (9 days) Homework – (A B – Quadratic Functions and Their Graphs) Homework – (C – Discriminant) Homework – (D – Finding a Quadratic from Its Graph) Homework – (E – Simultaneous Equations) Homework – (F G – Problem Solving and Optimization) Classwork – Quadratic Functions and Equations (4Qs) (1 day) Classwork – Quadratic Functions and Equations (8Qs) (1 day) Review for Test (1 day) TEST (1 day) [9]

Ch2 – Functions 1 (6 days) Homework – (A B C – Relations & Functions, Notation, Domain & Range) Homework – (D – Composition) Homework – (E – Sign Diagrams) Homework – (F – Rational Functions) Homework – (G – Inverse Functions) Homework – (H – Graphs of Functions) [15]

Ch3 – Transformations (2 days) Homework – (A B C – Translations, Stretches, Reflections) Homework – (D E – Miscellaneous Transformations, Reciprocals of Functions) [17]

Ch4 – Functions 2 (9 days) Homework – (A B – Even & Odd Functions, Graphing the Square of a ) Homework – (C - Absolute Value) Homework – (D – Rational Functions) Homework – (E – Partial Fractions) Homework – (F – Inequalities) Classwork – Functions Basics Composition and Inverse (1 day) Classwork – Rational Functions (1 day) Review for Test (1 day) TEST (1 day) [26]

Ch5 – Right Triangle Trigonometry (3 days) Homework – (A B C – Trig Ratios, Inverse Trig Ratios, Right Angles in Geometry) Homework – (D – Problem Solving) Homework – (E F – True Bearing, Angle Between a Line and a Plane) [29]

Ch6 – Unit Circle and (3 days) Homework – (A B – Radians, , Sector Area) Homework – (C D – Unit Circle and Special Angles) Homework – (E F – Pythagorean Identities, Finding Angles) [32] Ch7 – Non-Right Triangles (9 days) Homework – (A – Area) Homework – (B – Cosine Rule) Homework – (C – Sine Rule) Homework – (D – Problem Solving) Classwork – Angles, Circles, Arcs, and Sectors (1 day) Classwork – Measure, Arc Length and Sectors (1 day) Classwork – Triangle Trigonometry (1 day) Review for Test (1 day) TEST (1 day) [41]

Ch8 – Further Trigonometry (9 days) Homework – (A – Reciprocal Trigonometric Functions) Homework – (B – Inverse Trigonometric Functions) Homework – (C – Algebra with Trigonometric Functions) Homework – (D - Double Angle Identities) Homework – (E – Compound Angle Identities) Classwork – Triangle Trigonometry 2 (1 day) Classwork – Trigonometry (1 day) Review for Test (1 day) TEST (1 day) [50]

Ch9 – Sequences and (8 days) Homework – (A B C – Sequences, Arithmetic and Geometric) Homework – (D E – Growth and Decay, Financial Mathematics) Homework – (F G – Series and Arithmetic Series) Homework – (H I – Finite and Infinite ) Classwork – Sequences and Series 1 (1 day) Classwork – Sequences and Series 2 (1 day) Review for Test (1 day) TEST (1 day) [58]

Ch10 – Exponentials (9 days) Homework – (A – Rational Exponents) Homework – (B – Algebraic Expansion and Factorization) Homework – (C – Exponential Equations) Homework – (D E – Exponential Functions, Growth and Decay) Homework – (F – Natural Base) Classwork – Exponential Graph Review (1 day) Classwork – Exponential Functions and Modeling (1 day) Review for Test (1 day) TEST (1 day) [67] Ch11 – Logarithms (10 days) Homework – (A B – Base 10 and base a logarithms) Homework – (C – Laws of Logarithms) Homework – (D – Natural Logarithms) Homework – (E F – Logarithmic Equations and Change of Base) Homework – (G - Solving Exponential Equations with Logs) Homework – (H – Logarithmic Functions) Classwork – Exp and Log Expressions and Equations (1 day) Classwork – Exp and Log Functions and Equations (1 day) Review for Test (1 day) TEST (1 day) [77]

Ch12 – Intro to Complex Numbers (1 day) Homework – (A B C D E – Operations, Factorization, Complex Conjugates) [78]

Ch13 – Real Polynomials (12 days) Homework – (A B E – Polynomials, Operations, Division) Homework – (C – Zeros, Roots, and Factors) Homework – (D – Polynomial Equality) Homework – (F G – Remainder and Factor Theorems) Homework – (H – The Fundamental Theorem of Algebra) Homework – (I – Sum and Product of Roots) Homework – (J K – Graphing Cubics and Quartics) Homework – (L – Polynomial Equations) Classwork – Sum and Product of Roots of Polynomial Equations (1 day) Classwork – Factor and Remainder Theorems (1 day) Review for Test (1 day) TEST (1 day) [90]

Ch14 – The Counting Principle (3 days) Homework – (A B – Product and Sum Principles) Homework – (C D – and Permutations) Homework – (E – Combinations) [93]

Ch15 – The (7 days) Homework – (A – Binomial Expressions) Homework – (B – Binomial Theorem with Integer Exponents) Homework – (C – Binomial Theorem with Rational Exponents) Classwork – Counting Principle and the Binomial Theorem (1 day) Classwork – Sequences, Series, and the Binomial Theorem (1 day) Review for Test (1 day) TEST (1 day) [100] Ch16 – Reasoning and Proof (4 days) Homework – (A B – Logical Connectives and Deduction) Homework – (C D – Equivalence and Definitions) Homework – (E F – Exhaustion and Counter Example) Homework – (G H – Contrapositive and Contradiction)

Ch17 – Principle of Mathematical Induction (6 days) Homework – (A – The Process) Homework – (B – The Principle of Mathematical Induction) Classwork – Mathematical Induction 1 (1 day) Classwork – Mathematical Induction 2 & 3 (1 day) Review for Test (1 day) TEST (1 day) [110]

Ch18 – Linear Algebra (2 days) Homework – (A B – Systems of Linear Equations and Row Operations) Homework – (C D – 2x2 and 3x3 Systems Using Row Operations) [112]

Ch19 – Vectors (13 days) Homework – (A – Vectors & Scalars) Homework – (B – Geometric Operations) Homework – (C D – Vectors in a Plane, Magnitude) Homework – (E – Operations with Plane Vectors) Homework – (F G – Vectors in Space and Operations with Vectors in Space) Homework – (H – Vector Algebra) Homework – (I J – Vector Between Points and Parallelism) Homework – (K L – Scalar Product, Angle Between Vectors) Homework – (M N – Proof Using Vectors, Vector Product) Classwork – Scalar Product and Applications (1 day) Classwork – Vector Basics (1 day) Review for Test (1 day) TEST (1 day) [125]

Ch20 – Vector Applications (13 days) Homework – (A – Lines in 2D and 3D) Homework – (B – Angles Between Lines) Homework – (C – Constant Velocity) Homework – (D – Shortest Distance from a Point to a Line) Homework – (E – Intersecting Lines) Homework – (F – Relationship Between Lines) Homework – (G – Planes) Homework – (H – Angles in Space) Homework – (I – Intersecting Planes) Classwork – Vector Product and Applications (1 day) Classwork – 3D Vectors Lines and Planes (1 day) Review for Test (1 day) TEST (1 day) [138] Ch21 – Complex Numbers (11 days) Homework – (A – The Complex Plane) Homework – (B – Modulus and Argument) Homework – (C – Geometry in the Complex Plane) Homework – (D – Polar Form) Homework – (E – Euler's Form) Homework – (F – DeMoivre's Theorem) Homework – (G – Roots of Complex Numbers) Classwork – Complex Numbers 1 (1 day) Classwork – DeMoivre's Theorem (1 day) Review for Test (1 day) TEST (1 day) [149]

Ch22 – Probability (10 days) Homework – (A B – Experimental Probability and Two-Way Tables) Homework – (C D – Sample Spaces and Events, Theoretical Probability) Homework – (E F – Making Predictions and the Addition Law) Homework – (G H – Independent and Dependent Events) Homework – (I J – Conditional Probability and the Formal Definition of Independence) Homework – (K – Baye's Theorem) Classwork – Probability and Counting Principle (1 day) Classwork – Probability of Events 2 (1 day) Review for Test (1 day) TEST (1 day) [159]

END OF YEAR 1 Course Syllabus – Year 2

Ch 23 – Limits (7 days) Homework – (A B – Limits and the Existence of Limits) Homework – (C D – Limits at Infinity and Trigonometric Limits) Homework – (E – Continuity) Classwork – Limits from Graphs and Continuity (1 day) Classwork – Methods of Evaluating Limits (1 day) Review for Test (1 day) TEST (1 day) [7]

Ch24 – Introduction to (10 days) Homework – (A – Rates of Change) Homework – (B – Instantaneous Rates of Change) Homework – (C – The of a ) Homework – (D – The Function) Homework – (E – Differentiation from First Principles) Homework – (F – Differentiability and Continuity) Classwork – Derivative from First Principles (1 day) Classwork – Intro to Differential Calculus (RevC)) (1 day) Review for Test (1 day) TEST (1 day) [17]

Ch25 – Rules of Differentiation (14 days) Homework – (A – Simple Rules of Differentiation) Homework – (B – The ) Homework – (C – The ) Homework – (D – The ) Homework – (E – of Exponential Functions) Homework – (F – Derivatives of Logarithmic Functions) Homework – (G – Derivatives of Trigonometric Functions) Homework – (H – Derivatives of Inverse Trigonometric Functions) Homework – (I – Second and Higher Derivatives) Homework – (J – Implicit Differentiation) Classwork – Differentiation Basics 1 (1 day) Classwork – Differentiation Practice 1 (1 day) Review for Test (1 day) TEST (1 day) [31] Ch26 – Properties of Curves (11 days) Homework – (A B – and Normals) Homework – (C D – Increasing and Decreasing, Stationary Points) Homework – (E F – Shape, Inflection Points) Homework – (G – Understanding Functions and Their Derivatives) Homework – (H – L'Hopital's Rule) Classwork – Tangents and Normals (1 day) Classwork – (1 day) Classwork – Further Differentiation Methods 1 (1 day) Classwork – Implicit Differentiation (1 day) Review for Test (1 day) TEST (1 day) [42]

Ch27 – Applications of Differentiation (7 days) Homework – (A – Rates of Change) Homework – (B – Optimization) Homework – (C – ) Classwork – Optimization 1 (1 day) Classwork – Implicit Differentiation and Related Rates (1 day) Review for Test (1 day) TEST (1 day) [49]

Ch28 – Introduction to Integration (5 days) Homework – (A – Approximating the Area Under a Curve) Homework – (B – The Riemann ) Homework – (C – Antidifferentiation) Homework – (D – The Fundamental Theorem of Calculus) Classwork – Riemann Sums 2 (1 day) [54]

Ch29 – Techniques of Integration (15 days) Homework – (A – Discovering ) Homework – (B – Rules for Integration) Homework – (C – Particular Values) Homework – (D – Integrating f(ax + b)) Homework – (E – Partial Fractions) Homework – (F – Integration by Substitution) Homework – (G – ) Classwork – Integration Basics 1 (1 day) Classwork – Integration by Inspection (1 day) Classwork – Integration by Substitution 1 (1 day) Classwork – Integration by Substitution 2 (1 day) Classwork – Integration by Substitution 3 (1 day) Review for Test (1 day) TEST (1 day) [69] Ch30 – Definite Integrals (14 days) Homework – (A B – Definite Integrals with and without Substitution) Homework – (C D – Area Below and Above a Curve) Homework – (E – Area Between Two Functions) Homework – (F – Area Between a Curve and the y-axis) Homework – (G – Solids of Revolution) Homework – (H – Problem Solving using Integration) Homework – (I – Improper Integrals) Classwork – Areas 1 (1 day) Classwork – Areas Volumes (1 day) Classwork – Volumes Solids of Revolution (1 day) Classwork – Improper Integrals (1 day) Classwork – Integration Practice 2 (1 day) Review for Test (1 day) TEST (1 day) [83]

Ch31 – Kinematics (8 days) Homework – (A – Displacement) Homework – (B – Velocity) Homework – (C – Acceleration) Homework – (D – Speed) Classwork – Kinematics 1 (1 day) Classwork – Kinematics 2 (1 day) Review for Test (1 day) TEST (1 day) [91 days]

Ch32 – Maclaurin Series (11 days) Homework – (A – Maclaurin Series) Homework – (B – Convergence) Homework – (C – Composite Functions) Homework – (D – Addition and Subtraction) Homework – (E – Differentiation and Integration) Homework – (F – Multiplication) Homework – (G – Division) Classwork – Maclaurin Series 1 (1 day) Classwork – Maclaurin Series 2 (1 day) Review for Test (1 day) TEST (1 day) [102] Ch33 – Differential Equations (14 days) Homework – (A – Differential Equations) Homework – (B – Euler's Method for Numerical Integration) Homework – (C – Differential Equations of the form dy/dx = f(x)) Homework – (D – Separable Differential Equations) Homework – (E – Logistic Growth) Homework – (F – Homogeneous Differential Equations of the form dy/dx = f(y/x)) Homework – (G – The Integrating Factor Method) Homework – (H – Maclaurin Series Developed from a ) Classwork - Differential Equations 1 (1 day) Classwork – Differential Equations 2 (1 day) Classwork – Differential Equations 3 (1 day) Classwork – Differential Equations 4 (1 day) Review for Test (1 day) TEST (1 day) [116]

Ch34 – Bivariate Statistics (4 days) Homework – (A B – Association Between Numerical Values, Pearson's Coefficient) Homework – (C – Line of Best Fit by Eye) Homework – (D – The Least Squares Regression Line) Homework – (E – The Regression Line of x against y) [120]

Ch35 – Discrete Random Variables (10 days) Homework – (A B – Random Variables, Discrete Probability Distributions) Homework – (C – Expectation) Homework – (D – Variance and Standard Deviation) Homework – (E – Properties of aX+b) Homework – (F G – The Binomial Distribution, Technology to Find Binomial Probs) Homework – (H – The Mean and Standard Deviation of a Binomial Distribution) Classwork – Linear Regression (1 day) Classwork – Binomial Distributions (1 day) Review for Test (1 day) TEST (1 day) [130]

Ch36 – Continuous Random Variables (10 days) Homework – (A - Probability Density Functions) Homework – (B – Measures of Center and Spread) Homework – (C – The Normal Distribution) Homework – (D – Calculating Normal Probabilities) Homework – (E – The Standard Normal Distribution) Homework – (F – Normal Quantiles) Classwork – Normal Distributions 2 (1 day) Classwork – Probability Distributions 2 (1 day) Review for Test (1 day) TEST (1 day) [140]

End of Year 2