CIC Mathematics Curriculum Documentation

IB Mathematical Methods 2

COURSE OVERVIEW

This senior year course is the second of a two-year program that leads to external examinations in IB mathematical methods (standard level). The program is intended to provide a sound mathematical basis for those students planning to pursue further study in such fields as chemistry, economics, geography and business administration. The workload is extensive and demanding. Topics include geometry and trigonometry, probability and statistics, sequences and series and further differentiation and integration.

SKILLS OVERVIEW

Students will be able to:

1. Differentiate sums, products and quotients of
   1. polynomial functions
   2. rational functions
   3. sine, cosine and tangent
   4. Exponential function base e and natural logarithm function
2. Graph
   1. the functions above including asymptotes, intercepts and stationary points
   2. the conic sections
3. Integrate
   1. the functions above
   2. using methods of substitution and parts
4. Calculate
   1. vector addition, multiplication by a scalar, dot product, normal and unit vectors and the angle between two vectors
   2. matrix addition, multiplication and multiplication by a scalar, determinants and scale factor for areas under transformations
   3. slopes, tangents and normals to functions at a given point
   4. area under a curve using definite integration
   5. probabilities including conditional probability
   6. mean, standard deviation and area under the normal distribution curve
   7. terms and sums for arithmetic and geometric sequences and series
   8. arc length and sector area
   9. approximations using trapezium rule, Newton-Raphson process and linear interpolation
5. Solve
   1. simple trigonometric equation on a given interval
   2. triangle problems using right and non-right triangle methods
   3. min/max problems
   4. conic section problems.
6. Justify conclusions using proper language, symbols and methods

COURSE CONTENT

The order in which the topics are listed is roughly the order in which they have been taught. Some topics are given different placement as the result of time constraints. Some of the topics listed are introduced in the IB Math 1 course and the two documents should be read together. More detail may be found in the IB Mathematical Methods syllabus (for testing in May 94 - May 99)

1. Geometry and Trigonometry
   1. similarity: relation between lengths and areas of similar plane figures and volumes and surface areas of similar solids
   2. arcs and sectors: circular measure, arc length and sector area (formulae given)
   3. trigonometry: sine, cosine and tangent as circular functions, graphs of the functions including periodic and symmetry properties Solution of simple equations on a given interval Right triangle ratios, Cosine and sine rules including ambiguous case and their applications and formula for finding the area of a non-right triangle (formulae given)
2. Vectors and Matrices
   1. vectors: vectors in dimensions up to 4 Addition subtraction, multiplication by a scalar, length, zero, negative and unit vectors; calculation of a resultant scalar (dot) product and angle between two vectors
   2. matrices: as an (m x n) array of real numbers; addition, subtraction, multiplication by a scalar and of matrices; matrices as data storage zero matrix
   3. square (2 x 2) matrices: unit, inverse and singular matrices; determinants and solutions of simultaneous linear equations; transformations including scale factors
3. Functions and Calculus (1)
   1. functions: composites and inverses; domain and range
   2. differentiation: differentiation of sums, products and quotients (includes a review of logarithms)
   3. applications of differentiation: equations of tangents and normals to a point on a given function; stationary values and tests for minima, maxima and point of inflexion including non-zero gradients and second derivative test (part of optional topic); min/max problems; curve sketching including asymptotes, intercepts and stationary points; approximation of roots by graphical method or linear interpolation and further approximation using Newton-Raphson process (theoretical treatment of convergence not required)
4. Probability and Statistics
   1. probability: Mutually exclusive and independent events, conditional probability Solutions using tree diagrams, Venn diagrams, and formulae (not given).
   2. statistics: Frequency diagrams including histograms Mean, variance and standard deviation of grouped data Cumulative frequency curves, median and interquartile range; the normal distribution
5. Analytic Geometry ( Section 2 Review and Optional Topic Introduction)
   1. review: linear equations, distance, midpoint and slope; graphing quadratic functions in standard form
   2. conic sections: definitions, equations and graphs of circles, ellipses, parabolas and hyperbolas; simple problems on all conic sections
6. Number and Algebra
   1. sequences and series: arithmetic and finite geometric sequences and simple and compound interest; arithmetic and finite geometry series (formulae given)
   2. binomial theorem: up to the eighth power; Pascal's triangle may be used to obtain coefficients
7. Calculus (Section 4 Review) and Further Calculus (Optional Topic Review and Introduction)
   1. integration as inverse of differentiation
   2. using substitution and by parts
   3. definite integrals and area under a curve
   4. estimating area using the trapezium rule
   5. calculating volumes of revolution

ASSESSMENT TECHNIQUES AND TEACHING METHODOLOGIES

1. Assessment Techniques
   1. Tests (90%): Tests in this class are created using questions from past IB papers. They are cumulative and cover material that has been reviewed since the beginning of the school year. The marks are usually composed 50% from paper one questions and 50% from paper two questions. All tests are timed using IB timing and grading reflects the IB scale. (The square root of the raw percentage is used to convert to the CIC grading scale.) During third and fourth quarters, students should sit at least three full mock exams.
   2. Problem Sets (10%): May be graded homework assignments, practices exams, or test corrections.
2. Teaching Methodologies: This course includes the essential elements of university level classes that combine periods of direct instruction and periods of discussion section. During discuss sections, students work on and ask questions about problem sets. Direct instruction may include only lecture or lecture in addition to investigation and/or discussion.

KEY RESOURCES

1. Primary Students Texts:
   1. Pure Mathematics 1., Bostock and Chandler (ST(P) 1994)
   2. Pure Mathematics 2., Bostock and Chandler (ST(P) 1994)
2. Teacher Resource Texts: (packets may be copied for students)
   1. Pure Mathematics 1., Hanrahan, Porkess and Secker (MEI Structured Mathematics 1995)
   2. Pure Mathematics 2., Hanrahan, Porkess and Secker (MEI Structured Mathematics 1995)
   3. Finite Mathematics, A Search for Meaning., Egsgard, Flewelling, Newell, Warburton (Gage Educational Publishing Company 1988)
   4. Algebra and Geometry, A Search for Meaning., Egsgard and Ginestier (Gage Educational Publishing Company 1994)
   5. Any standard US based Calculus textbook
3. Graphics Calculator

ESL STRATEGIES AND CONSIDERATIONS

At this level, CIC students are required to be proficient in English at grade level.

FUTURE NEEDS AND CHANGES

1. For the 1999 - 2000 school year:
   1. New primary texts have been ordered
   2. Course content summary and skills overview should be modified so as to be appropriate for the level of the new IB mathematical methods syllabus
   3. Assessment techniques and teaching methodologies must be revised to reflect the required IB coursework component.
2. For the 2000 - 2001 school year, the content summary and skills overview should be modified to complete the two-year sequence begun by IB Mathematical Methods 1, a new course for the 1999 — 2000 school year.