MATHEMATICS HIGHER LEVEL

Transcription

1 IB Diploma Scots College COURSE HANDBOOK MATHEMATICS HIGHER LEVEL

2 Course description Mathematics, while it is a subject in its own right, is also a subject which facilitates the development and understanding of other subjects such as the sciences and some humanities such as economics and geography. Thus a major aim of this course is to provide students with the necessary knowledge and skills to be able to develop these other subjects to their fullest. The students will learn the appropriate techniques for solving problems, they will learn to be critical thinkers in their selection of these appropriate techniques, they will improve their logical thought processes to allow them to develop logical chains of reasoning which will allow them to communicate their ideas in a meaningful way. Over the two year period, the students will study six core topics and one optional topic some of these will be studied as a single unit while others will be split into different units to be studied at different times during the two years. (The optional topic will be statistics and probability because that is closest to the NZ NCEA course, and if they get to university in NZ they will want to have solid statistics behind them, rather than a solid background in discrete maths or set/group theory which will be picked up at university if needed.) The internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics of the student s own choice. The difference between mathematics at Standard Level and Higher Level Mathematics at Higher Level caters for students with a very strong background in mathematics who are highly competent in a range of analytical and technical skills. The majority of these students will be expecting to include mathematics as a major component of their university studies, either as a subject in its own right or within courses such as physics, engineering and technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems. The IB says: 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. Students wishing to study mathematics in a less rigorous environment should therefore opt for one of the standard level courses, mathematics SL or mathematical studies SL. Internationalism Mathematics is also a subject that can be readily transported into the international arena - most of the notation and the techniques are recognizable worldwide, which means that it is a subject readily accessible to students studying in another country and in a different language from their own. It is also important for students to realize that contributions to the development of mathematics have come from so many different civilizations. The differing approximations to the value of π over the centuries (for example the Rhind Papyrus from Egypt in 150BC, the Greek Archimedes in 0BC, the Indian Madhava in the 15 th century) bear testimony to that - similarly the contemporaneous discovery of differential calculus by Sir Isaac Newton in England and Gottfried Leibniz in Germany Theory of Knowledge TOK is central to the diploma and also plays a part in the teaching and learning in mathematics. The development of mathematical thinking comes largely in two ways (a) the starting with a premise and by a series of logical deductions working through to a conclusion which can then be applied in a variety of situations, and (b) the development of a model to mirror a situation and then the application of the properties of that model, with the accuracy of the model being paramount. Although logic is no longer taught per se, it would be expected to be a major plank in the students thinking things out for themselves. Textbooks Maths for the International Student Mathematics HL (Core) by Urban et al (Haese and Harris) Maths for the International Student Mathematics HL (Option): Statistics & Probability by Quinn et al (Haese)

3 Course Outline Year 1 Term 1 Algebra (1a), Trigonometry (1) Algebra (1b) Complex Numbers Functions & Equations (1) Calculus (1) Statistics and Probability (1) Algebra () Internal Start Mathematical Exploration Vectors Vectors Algebra () Revision, EOY Exam, Mathematical Exploration Year 1 Statistics and Probability () Statistics and Probability (Option) Internal Complete Mathematical Exploration Draft Statistics and Probability (Option) Functions & Equations () Trigonometry () Calculus (), Revision, Mock Exam Revision + Exams Assessment Outline The assessment will follow the IB pattern of three externally assessed papers lasting a total of five hours (and worth 80% of the final mark) and the internally assessed exploration (representing 0% of the final mark). Paper 1: hrs 0% Based on the compulsory core of the syllabus. No calculator allowed. Section A (15%): Compulsory short-response questions. Section B (15%): Compulsory extended-response questions. Paper : hrs 0% Based on the compulsory core of the syllabus. Graphic Display Calculator (GDC) required. Section A (15%): Compulsory short-response questions. Section B (15%): Compulsory extended-response questions. Paper : 1 hr 0% Mainly based on the syllabus option. Graphic Display Calculator (GDC) required. Extended-response questions. Internal Assessment: Maths Exploration, 0% The internally assessed component is a single mathematical exploration. This is a piece of written work that involves investigating an area of mathematics of the student s own choice which takes the form of a mini extended essay. This task offers students an opportunity to develop independence in their mathematical learning through engagement in a mathematical topic. An important component of the work is for students to develop skills in communicating mathematical ideas. About 10 hours of class time will be used for this task and students will need to spend additional time writing their exploration at home.

4 ALGEBRAIC TECHNIQUES (1a) Year Sequences Arithmetic sequences: n th term, sum of n terms 5 (review for Geometric sequences: n th term, sum of n terms, S Ch Week some students) Sigma notation P5-71 Week 1.,. Exponents y=x b b = log xy Ch 5 Logarithms Rules of logs and exponents P78-88 Change of base log ba = log ca log cb Ch Solving exponential & log equations P Week Presumed Absolute value Elementary treatment Worksheet knowledge Surds Manipulation ( ab = a. b, ab = b a) Alg fractions Addition Presumed Rearranging Rearranging formulae Worksheet knowledge Quadratics Factorisation (inc x + x + 1) expansion (review) Quadratic Equations (inc use of formula) Completing the square FUNCTIONS AND EQUATIONS (1) Year 1.1 Basic concepts Definitions of function, domain, range, mapping Odd & even functions Ch 1 One to one, one to many etc P17-1,-9 Composite functions (f g)(x) = f(g(x)) Week 5 Inverse functions f -1, including domain restriction, and self-inverse functions Week. Graphing Use of graphics calculator functions Key features of straight line, basic parabola, cubic Ch 1 Absolute value P5-7,1- Rectangular Hyperbola (asymptotes) Week.,. Transformations Shift in x or y directions 11 7&8 of graphs Reflections in both axes Ch 5 One way stretches P1-19 Inverse and Applications of above to parabola 1/f(x) graphs Inverse functions as reflection in the line y = x 1/f(x) has asymptotes when f(x) = 0 Week. Quadratics Use of quadratic formula (previously done) Ch 8 Discriminant Δ = b ac P1-17 Nature of roots TRIGONOMETRY (1) Year 1.1,.7 Radian measure concept of π radians = 180 o (review) arc length Ch 10,11 Week 9 Trig of non-right area of sector P7-79 angled triangles sine Rule (including ambiguous case) (review) cosine Rule area of triangle = ½absinC Week 11.,. Trig graphs definition of sin, cos, tan in terms of unit circle definition of tan as sin/cos Ch 1 exact values P85-07 graphs of sin, cos, tan including the transformations period, amplitude applications, including angles of elevation & depression Week 1. Trig equations solutions in a finite interval (e.g. sin x = 1)

5 ALGEBRAIC TECHNIQUES (1b) Year Complex Definition of i (using quad formula as introduction) 1 Term numbers Terminology (real, imaginary, complex) Ch7 Modulus, argument and conjugates P180-7 Week Sums, products and quotients of complex numbers Ch 15 1,, 1. Cartesian, Euler and polar forms. Argand diagram P De Moivre s Theorem Powers and roots of a complex number 1.8 Conjugate root theorem CALCULUS (1) Year 1.1,. Derivative limits, continuity and convergence 8 Term differentiation by first principles (only polynomials) Ch 0 Week drawing graphs of the derived functions (f, f, f ) P590-0,5 differentiation of x n for n ε Q P11-15 derivatives as rates of change P- second derivatives and higher derivatives equations of straight like y-y 1 = m(x x 1) equations of tangents and normals identifying increasing and decreasing functions Week. Local max/min local maximum and minimums using f ' (x) = 0 identifying max/min using change of sign of f ' (x) Ch 1 identifying max/min using sign of f "(x) P- optimization problems with applications to area, profit, etc Week 7.,.5 Integration integration as the reverse of differentiation constant of integration and its evaluation Ch application to area under a curve P Ch 1. Kinematics v = ds/dt, a = dv/dt, interpreting integration constants P7- total distance travelled Ch 5 P77-78 STATISTICS & PROBABILITY (1) Year 1 Presumed Data review pie charts, box & whisker, line graphs Term knowledge presentation review bar graphs and simple review mean, median, mode, quartiles, IQR, cumulative statistics frequency diagrams, histograms, percentiles Week Populations, definitions of population, sample, random sample samples frequency distribution of discrete and continuous data Ch 17 Data grouped data P85-99 presentation Mean, standard deviation & variance calculation of standard deviation understanding the meaning of standard deviation and dispersion variance calculating statistics using calculator Week 9 5. Probability trials, outcomes, sample space, equally likely, event definition of P(A) as n(a)/n(u) complementary events 5. Probability tree and Venn diagrams, contingency tables diagrams use of the above to solve probability problems use of counting principles 5 P P Ch 18 P58-5 Ch 18 P57-55

6 Week 10 Year 1 Term Week Random variables 5. Binomial Poisson discrete and continuous random variables probability density functions Expected value (mean), mode, median, variance and standard deviation binomial distribution condition Poisson distribution conditions mean, variance of binomial and Poisson Ch 8 P Ch 8 P Week 5.7 Normal distribution z represents the number of SD away from the mean standardization of the normal distribution use of the normal distribution to find probabilities inverse normal distribution use of graphics calculators (tables no longer given) Ch 9 P81-89 ALGEBRAIC TECHNIQUES () Year Solving solutions using row reduction & GDC (max x) Worksheet Term Week systems of equations unique solutions, no solutions, infinite solutions & Ch 1 P5- Year 1 Mathematical Exploration Choose exploration, start planning & research Term (additional time in term after exams) VECTORS Year 1 Presumed Coordinate gradients of parallel and perpendicular lines Worksheet Term knowledge geometry midpoint of a line segment distance between two points.1 Vector displacement in the plane and in dimensions concepts column vectors and in terms of unit vectors Ch 1 adding and subtracting vectors algebraically P71-98 Week adding and subtracting vectors geometrically zero vector, negative vector multiplication by a scalar magnitude of a vector position vectors Week 5. Scalar product scalar product (dot product) of two vectors algebraic properties of scalar products Ch 1 perpendicular and parallel vectors P0-0 angle between two vectors application proof of geometric properties Week.5 Vector calculating the vector (cross) product, including the products component (& determinant) representation Ch 1 properties of vector product P07-15 geometric interpretation of vxw areas of triangles and parallelograms Week 7. Vector lines in the plane and D space equation of a parametric and Cartesian forms of lines Ch 1 5 line r = a + b angle between two lines P51-7 Year 1. Lines coincident, parallel, intersecting and skew lines Term distinguishing between the above Week 1 points of intersection. simple applications to kinematics

7 Week. Vector equation of the plane.7 Plane intersections vector equation of line r = a + b + c use of normal vector to obtain the form r.n =- a.n Cartesian equation of the plane intersections of a line with a plane intersections of two planes intersections of three planes (row reduction method) angle between a line and a plane angle between two planes Ch 1 P51-7 ALGEBRAIC TECHNIQUES () Year 1 1. Counting permutations, including formula for n P r Term principles combinations, including formula for n C r Ch 8 Week binomial theorem expansion of (a + b) n, n ε N P1-0 Week 1. Mathematical proving various formulae eg 1+++ = ½ n(n+1) Induction proving sums of series, divisibility, complex numbers Ch 9 P- YEAR STATISTICS & PROBABILITY () Year 5. Probability for P(AB) = P(A) + P(B) P(AB) combined events P(AB) = 0 for mutually exclusive events Ch 18 Week P55-5 1, 5. Conditional definition P( AB) = P(AB) / P(B) probability independence P( AB) = P(A) = P( AB ) Use of P(AB) = P(A)P(B) to show independence Bayes Theorem (now for up to three events) Year Mathematical Exploration Complete exploration draft (ongoing but due before the end of term) STATISTICS & PROBABILITY (OPTION) Year 7. Expectation Random variables discrete & continuous algebra Expectation and variance of random variables Stats & Prob Week Linear transformations E(aX+b) = ae(x)+b, and Var(aX+b) = a Var(X) Pg 9-5 Expectation of product E(XY) = E(X)E(Y) Linear combinations of random variables Continuity correction for continuous random variables Year 7.1 Probability Cumulative distribution functions of discrete random generating variables for various probability mass functions: Stats & Prob Week functions Discrete Uniform, Bernoulli & Binomial (Core Recap) Pg -5 Geometric & Negative Binomial Poisson (Core Recap) Week 7.1 Probability Mean and variance of discrete random variables Stats & Prob functions generating Poisson approximation to the binomial distribution Pg -1 7

8 STATISTICS & PROBABILITY (OPTION) Year 7.1 Probability Cumulative distribution functions of continuous random generating variables for various probability density functions: Stats & Prob Week functions Continuous Uniform Pg -51 Exponential Normal Normal approximation to the binomial distribution (including continuity correction) Week Probability Probability generating functions Stats & Prob generating G(t) = E(t x ) = P(X=x)t x Pg 5-5 functions E(X) = G (1), Var(X) = G (1) + G (1) [G (1)] Week 7. Central Limit Sampling distribution of sample means Stats & Prob Theorem Central Limit Theorem Pg -81 Application to proportions Week 7 7. Unbiased Unbiased estimators and estimates estimators is an unbiased estimator for μ and S is an unbiased Stats & Prob estimator of Pg Confidence intervals Week 8 7. Hypothesis testing Week 9 7. Errors in hypothesis testing Week Bivariate statistics T is an unbiased estimator for parameter θ if E(T) = θ T 1 more efficient estimator than T if Var(T 1) < Var(T ) Confidence intervals for the mean of a normal population Stats & Prob Confidence intervals for the mean when variance is Pg unknown of population is unknown (t-distribution) Matched pairs Null H 0 and alternative H 1 hypotheses Type I and type II errors Stats & Prob 5 Significance level Pg Critical regions and p-values Confidence intervals Hypothesis testing with unknown (t-distribution) Matched pairs Calculating probability of type I and type II errors Stats & Prob Pg 11-1 Scatter diagrams and relationships Correlation Product moment correlation coefficient R Stats & Prob Pg 1-1 Interpreting values of r Covariance COV(X,Y) = E(XY) E(X)E(Y) If X and Y are independent then COV(X,Y) = 0 (or =0) 8

9 STATISTICS & PROBABILITY (OPTION) Year 7.7 Regression Equation of regression line of Y on X is given by Term Stats & Prob Week 1 where Pg Bivariate normal distribution Equation of of regression line of X on Y Use of GDC for regression calculations Prediction of values using regression lines Bivariate normal distribution Hypothesis testing for dependence of X and Y using Stats & Prob Pg which has a t-distribution with n- degrees of freedom FUNCTIONS AND EQUATIONS () Year. Reciprocal f(x) = 1/x for x 0 Term function graph; self-inverse (symmetry in line y = x) Ch 1 Week f(x) = a x (a 0) the function and its inverse f(x) = log ax) P1- & rational graphs of these two functions functions rational functions & asymptotes solving equations of form a x = b using logarithms Week. Exponential graphs of exponential and log functions f(x) = e x and its a x = e xlna Ch inverse f(x) = ln applications (compound interest, growth, decay) P88-98 x.7 Inequalities in one variable Ch 1 use of the absolute value sign P-5 solution of g(x) f(x) (f,g linear or quadratic) P9-1 Week.5,. Polynomials remainder theorem.7 factor theorem Ch 7 factorizing and solving polynomials, including P fundamental theorem of algebra & sum & products of roots of polynomials Worksheets solving polynomial inequalities (up to degree ) 9

10 TRIGONOMETRY () Year. Trig functions definition of tan as sin/cos Term definition of sec, cosec, cot Ch 1 Pythagorean identities involving the above P07-18 Week 5.,. Compound compound angle formulae Week angle double angle formulae trig identities, including proofs given sin, finding possible values of other ratios (eg sin without finding ).5 Inverse trig f(x) = arcsin x, etc ratios their domains and ranges their graphs Week 7. Trig equations solving equations using double angle formulae solving equations using trig identities to transform solving equations involving factorizing for above use both analytical and graphical methods CALCULUS () Year. Derivative derivatives of trig functions, e x, ln x 5 Term derivatives of reciprocal trig functions, a x log ax Ch19 Week 8 derivatives of arcsin x, arcos x, arctan x P Ch P71-8 P90-9 Week 9. Chain rule, use of chain rule product, use of product and quotient rules Ch 0 quotient rules second derivatives P0-11 relationships between graphs of f, f and f P1-18 awareness of higher derivatives Year Term. Points of inflexion use of terms concave up and concave down points of inflexion at f (x) = 0 special case for y = x n Ch 1 P-1 Week 1. Implicit diffn implicit differentiation P1- Week.,.5 Integration integration of trig functions, e x, 1/x integration of functions involving linear functions Ch ax + b P70-7 application to volumes of revolution (x and y axes) Ch P Week.7 Further integration by substitution Ch integration integration by parts P7-7 repeated integration by parts Ch 7 P