Mathematics HL is a two year course for juniors and seniors with a good background in mathematics who are competent in a range of analytical and technical skill. 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 or technology. Others may take this subject because they have a strong interest in mathematics and enjoy meeting its challenges and engaging with its problems It will be paired with a Mathematics SL class. (Provo High School has a split schedule with A and B days, Each day covers four of the eight classes taught.) The mathematics HL class would be on days opposite of the Mathematics SL class. The SL class will complement the HL class as they have no overlapping of material

Students will experience internationalism through mathematics by having teacher directed discussions of a) the differences in notation, b) the lives of mathematicians set in a historical and/or social context, c)the cultural context of mathematical discoveries, d) the ways in which specific mathematical discoveries were made and the techniques used to make them, e) how the attitudes of different societies towards specific areas of mathematics are demonstrated, f) the universality of mathematics as a means of communication. Students will experience fully integrated mathematics e.g. when they learn matrices and then learn statistics they will then see statistical problems using matrices, everything they learn can be crossed with anything else they have learned in the past. This will result in continual review of past material and an attitude of learning full mastery not just passing this week’s test. Each type of problem will be analyzed from an algebraic approach, from a numerical approach and from a graphical approach to enhance full mastery. We have graphing calculators in classroom sets for each class as well as calculators to rent for the year for anyone who cannot afford to buy a calculator but wishes to have one at home. We have a few math reference books in our math department and more in our media center. We have funding set aside to purchase whatever books are deemed helpful. We are just across the street from Brigham Young University, a large international school with a multimillion volume library that they allow us to use. We also have support from their Mathematics and Math Education departments on a limited bases. In the past we have been able to have Mathematics professors with national and international reputations spend time working with our students in extracurricular projects. We would look to extend this involvement into the IB program.

Prior to entering the Mathematics HL class the students will have the opportunity to take Algebra 2 and Precalculus classes with strong vector development and integrated topics. Provo High has a strong background in the use of technology especially graphing calculators in teaching math and solving problems. We will expand this knowledge and use into the IB Program.

Topics:

1. Algebra

a. Arithmetic and geometric sequences and series; sigma notation

b. Exponents and logarithms; laws of logarithms; change of base

c. Binomial Theorem; counting principles, including permutations and combinations

d. Proof by mathematical induction; forming conjectures to be proved by mathematical induction

e. Complex numbers; conjugate, modules and argument; Cartesian form

f. Sums, products and quotients of complex numbers

g. De Moivre’s theorem; powers and roots of a complex number

h. Conjugate roots of polynomial equations with real coefficients

2. Functions and Equations

a. Concept of function, domain and range, image, composite functions, inverse function

b. Graph of a function, use of GDC to investigate properties of graphs and their solutions

c. Transformations of graphs: translations, stretches; reflections in the axes inverse functions—reflection across y = x; graph of the reciprocal of a function; absolute value graphs

d. Reciprocal function, its graph; its self-inverse nature

e. Quadratic function, graph, axis of symmetry, vertex form, factored form—x-intercepts

f. Solutions to quadratic functions, quadratic formula, use of its discriminant

g. Exponential and logarithmic functions

h. The exponential function of e raised to the x power and its inverse lnx, x > 0

i. Inequalities in one variable, using their graphical representation

j. Polynomial functions; the factor and remainder theorems, with application to the solution of polynomial equations and inequalities

3. Circular Functions and Trigonometry

a. Circles, radian measure, arc length, sector area

b. Define sin & cos terms of unit circle

c. Compound angle identities and double angle identities

d. Periodic nature of trig functions, their domains, ranges and graphs

e. Solutions to trig equations over finite interval; Use of trig identities and factorisation to transform equations

f. Solution of triangles, cosine rule, sine rule, area of triangle

4. Matrices

a. Definition – element, row, column, and order

b. Matrix algebra -- =, +, -, X scalar, X by another matrix, identity & zero matrices

c. 2x2 & 3x3 determinants, inverse of 2x2 matrix, conditions for inverse existence

d. Solution to systems of linear equations to a maximum of 3 equations in 3 unknowns; Conditions for the existence of a unique solution, no solution and an infinity of solutions

5. Vectors

a. Vectors as displacements in 2 and 3 dimensions

b. Scalar product of two vectors – Algebraic properties of the scalar product; perpendicular vectors, parallel vectors, and the angle between vectors

c. Line as r = a + l b, the angle between two lines

d. Distinguishing between coincident, parallel intersecting and skew lines, points of intersection

e. Vector product of two vectors, v x w; determinant representation; geometric interpretation of êv x w ê

f. Vector equation of a plane r = a + lb + mc; use of normal vector to obtain the form r ° n = a ° n; Cartesian equation of a plane ax + by + cz = d

g. Intersections of: a line with a plane; two planes; three planes. Angle between: a line and a plane; two planes

6. Statistics and Probability

a. Concept of population, sample, random sample and frequency distribution of discrete and continuous data

b. Presentation of data: frequency tables and diagrams, box and whisker plots

c. Measures of central tendency: mean, median, ode; quartiles, percentiles; range; interquartile range; variance; standard deviation

d. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.

e. Concepts of trial, outcome, equally likely outcomes, sample space (U) and event; probability of an event, complementary events

f. Combined events

g. Conditional probability; independent events; use of Bayes’ theorem for two events

h. Venn diagrams, tree diagrams and tables of outcomes to solve problems

i. Concept of discrete and continuous random variables and their probability distributions; definition and use of probability density functions; expected value (mean), mode, median variance and standard deviation

j. Binomial distribution; its mean and variance. Poisson distribution; its mean and variance

k. Normal distribution; its properties; standardization of normal variables

7. Calculus

a. Limits and convergence

b. Differentiation of a sum and a real multiple of the functions in VII. A, chain rule, application of the chain rule to related rates of change; product rule, quotient rule, second derivative; awareness of higher derivatives

c. Local maxima & minima points, first & second derivatives in optimization problems

d. Indefinite integration as anti-differentiation; indefinite integrals of exponential, and trig functions, the composite of any of these with the linear function ax + b

e. Anti-differentiation with a boundary condition to determine the constant term; definite integrals; areas under curves; areas between curses; volumes of revolution

f. Kinematic problems involving displacement, s, velocity, v, and acceleration, a

g. Graphical behavior of functions; tangents and normals, behavior for large êx ê, asymptotes; significance of the second derivative; distinction between maximum and minimum points; points of in flexion with zero and non-zero gradients.

h. Implicit differentiation

i. Further integration: integration by substitution; integration by parts

j. Solution of first order differential equations by separation of variables

Optional Topic -

8. Statistics and Probability

a. Expectation algebra; linear transformation of a single random variable. Mean and variance of linear combinations of two independent random variables. Extension to linear combinations of n independent random variables.

b. Cumulative distribution functions. Discrete distributions: uniform, Bernoulli, binomial, negative, binomial, Poisson, geometric, hypergeometric.

c. Distribution of the sample mean. The distribution of linear combinations of independent normal random variables. The central limit theorem. The approximate normality of the proportion of successes in a large sample.

d. Finding confidence intervals for the mean of a population. Finding confidence intervals for the proportion of successes in a population.

e. Significance testing for a mean, Significance testing for a proportion. Null and alternative hypotheses. Type I and Type II errors. Significance levels; critical region, critical values, p-values; one-tailed and two-tailed tests.

f. The chi-squared distribution: degrees of freedom, v.

Assessment:

Basic Description--

Students will be assessed by internal and external measures. The internal assessment will be based on a student portfolio containing two pieces of exemplary work assigned by the teacher. One will be in mathematical investigation and the other will be in mathematical modeling. The two papers will be based on different areas of the syllabus. Students will be given at least two assignments in each category. The portfolio is internally assessed by the teacher and externally moderated by the IBO.

The external assessment will consist of three written papers to be given at the end of the school year. Students will be allotted a total of five hours to complete the three papers. Paper one– twenty compulsory short-response questions based on the compulsory core of the syllabus, two hours to complete. Paper two– five compulsory extended-response questions based on the compulsory core of the syllabus, two hours to complete. Paper three– Extended-response questions based mainly on the syllabus option Statistics and Probability.

In addition to the internal and external assessments, students will take frequent tests and quizzes throughout the year for the purpose of providing feedback to both student and teacher regarding progress toward the aims and objectives of the course.

Detailed Description—

External assessment details 5 hrs 80%

General

Paper 1, paper 2 and paper 3

These papers are externally set and externally marked. Together they contribute 80% of the final mark for the course. These papers are designed to allow students to demonstrate what they know and what they can do.

Calculators

For all examination papers, students must have access to a GDC at all times. Regulations covering the types of calculator allowed are provided in the Vade Mecum.

Mathematics HL information booklet

Each student must have access to a clean copy of the information booklet during the examination. One copy of this booklet is provided by the IBO as part of the examination papers mailing.

Awarding of marks

Marks may be awarded for method, accuracy, answers and reasoning, including interpretation.

In paper 1, paper 2 and paper 3, full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations (in the form of, for example, diagrams, graphs or calculations). Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. All students should therefore be advised to show their working.

Paper 1-- 2 hrs 30%

This paper consists of 20 compulsory short-response questions based on the core of the syllabus.

Syllabus coverage

Knowledge of all topics in the core is required for this paper. However, not all topics are necessarily assessed in every examination session.
The intention of this paper is to test students’ knowledge across the breadth of the core. However, it should not be assumed that the separate topics from the core are given equal emphasis.

Question type

· A small number of steps is needed to solve each question.

· Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

Mark allocation

· This paper is worth 120 marks, representing 30% of the final mark.

· Questions of varying levels of difficulty are set. Each question is worth 6 marks.

Paper 2 2 hrs 30%

The paper consists of 5 compulsory extended-response questions based on the core of the syllabus.

Syllabus coverage

· Knowledge of all topics from the core is required for this paper. However, not all topics are necessarily assessed in every examination session.

· Individual questions may require knowledge of more than one topic from the core.

· The intention of this paper is to test students’ knowledge of the core in depth. The range of syllabus topics tested in this paper may be narrower than that tested in paper 1.

· To provide appropriate syllabus coverage of each topic, some questions in this section are likely to contain two or more unconnected parts. Where this occurs, the unconnected parts will be clearly labelled as such.

Question type

· Questions require extended responses involving sustained reasoning.

· Individual questions may develop a single theme or be divided into unconnected parts.

· Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

· Normally, each question reflects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problem solving.

Mark allocation

· This paper is worth 120 marks, representing 30% of the final mark.

· Questions in this section may be unequal in terms of length and level of difficulty. Therefore, individual questions may not necessarily be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of each question.

Paper 3--1 hr 20%

This paper consists of four sections, one on each of the options in the syllabus. Each section has a small number of extended-response questions based mainly on the option topic. Where possible, the first part of each question will be on core material leading to the option topic. When this is not readily achievable, as for example with the discrete mathematics option, the level of difficulty of the earlier part of a question will be comparable to that of the core questions.

Students must answer questions on one option topic only. Students must answer all the questions in the section chosen.

Syllabus coverage

· Students must answer all the questions based on the option they have studied.

· Knowledge of the entire content of the option studied is required for this paper, as well as the core material.

Question type

· Questions require extended responses involving sustained reasoning.

· Individual questions may develop a single theme or be divided into unconnected parts. Where this occurs, the unconnected parts will be clearly labelled as such.

· Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.

Mark allocation

· This paper is worth 60 marks, representing 20% of the final mark. Approximately 15 marks are allocated to core material (or work of a similar level).

Questions in this section may be unequal in terms of length and level of difficulty. Therefore, individual questions may not necessarily be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of each question. Each section is worth 60 marks.

Internal assessment 20%

Portfolio

A collection of two pieces of work assigned by the teacher and completed by the student during the course. The pieces of work must be based on different areas of the syllabus and represent the two types of tasks:

· mathematical investigation

· mathematical modelling.

The portfolio is internally assessed by the teacher and externally moderated by the IBO. Procedures are provided in the Vade Mecum.

Resources:

Calculus: Graphical, Numerical, Algebraic by Finney, Demana, Waits, and Kennedy

Precalculus with Limits, A Graphing Approach by Larson, Hostetler, Edwards

Pure Mathematics Volumes 1 and 2 by Bostock and Chandler

A Concise Course in A Level Statistics by Crawshaw and Chambers

graphing calculators

Brigham Young University Library and Math Department

Standards:

http://www.ibo.org

Helpful Websites:

Provo High School

An “IB World School”

1125 N. University Ave.

Provo, UT 84604

Phone: 801-373-6550

Fax: 801-374-4880

IB Coordinator:: Lori Rich

LoriR@provo.edu