Discrete Mathematics Course overview

Discrete Mathematics – IB Course Overview

The nature of mathematics can be summarized in a number of ways: for example, it can be seen as a well-defined body of knowledge, as an abstract system of ideas, or as a useful tool. For many people it is probably a combination of these, but there is no doubt that mathematical knowledge provides an important key to understanding the world in which we live.

Discrete Mathematics is a course that has an emphasis on applications of mathematics, including a large section that focuses on statistical techniques and is designed for students with varied mathematical backgrounds and abilities. It prepares students to be able to solve problems in a variety of settings, to develop more sophisticated mathematical reasoning, and to enhance critical thinking. The course includes a project each student completes based upon his or her own research. The project provides an opportunity for students to carry out a mathematical study of their own choice using their own experience, knowledge and skills acquired during the course. This process allows students to take sole responsibility for a part of their studies in mathematics. Frequently lessons will be inquiry-based, to promote an understanding of mathematics by providing a meaningful context as well as a basis for students to understand more fully how to structure their work for the project.

The Aims of the Discrete Mathematics course are to enable students to:

  • enjoy mathematics, and develop an appreciation of the elegance and power of mathematics
  • develop an understanding of the principals and nature of mathematics
  • communicate clearly and confidently in a variety of contexts
  • develop logical, critical and creative thinking, and patience and persistence in problem-solving
  • employ and refine their powers of abstraction and generalizations
  • apply and transfer skills to alternative situations, to other areas of knowledge and to future developments
  • appreciate how developments in technology and mathematics have influenced each other
  • appreciate the moral, social and ethical implications arising from the work of mathematicians and the applications of mathematics
  • appreciate the international dimension of mathematics and its contribution to other disciplines

Assessment Objectives

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. Students will be expected to demonstrate the following:

  • Knowledge and understanding:  Students shall recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.
  • Problem solving:  Students shall recall, select and use their knowledge of mathematical skills, results and models in both real and abstract contexts to solve problems.
  • Communication and interpretation:  Students shall transform common realistic contexts into mathematics; comment on the context; sketch or draw mathematical diagrams, graphs of constructions both on paper and using technology; record methods, solutions and conclusions using standardized notation.
  • Technology:  Students shall use technology accurately, appropriately and efficiently both to explore new ideas and to solve problems.
  • Reasoning:  Students shall construct mathematical arguments through the use of precise statements, logical deduction and inference, and by the manipulation of mathematical expressions
  • Investigative approaches:  Students shall investigate unfamiliar situations involving organizing and analyzing information or measurements, drawing conclusions, testing their validity, and considering their scope and limitations.[1]

Syllabus Outline - Topics

Logic, Sets and Probability

Descriptive Statistics

Statistical Applications

Project

Geometry and Trigonometry

Mathematical Models

Differential Calculus

Voting and Apportionment

Graph Theory


[1] Contents reference IB Mathematical Studies guide March 2012.

 

Syllabus Outline – Topics Defined

Logic, Sets and Probability

       1) Set Theory – Venn diagrams, subsets, set operations

       2) Logic – inductive, deductive reasoning, quantified/compound statements

       3) Truth Tables – equivalence, tautology, contradiction

       4) Probability

 Descriptive Statistics

       1) Data displays

       2) Measures of central tendency

       3) Measures of dispersion

Statistical Applications

       1) The Normal distribution/standard deviation

       2) Correlation/line of regression

       3) Chi-squared hypothesis testing

Project

       1) Project overview/scoring rubric

       2) Mini project activities

       3) Sample projects/scoring practice

       4) Project assignment/timeline and due dates

Geometry and Trigonometry

       1) 2-dimensional figures

       2) Right triangle trigonometry

       3) Law of sines, cosines

       4) 3-dimensional solids – volume, surface area

Mathematical Models

       1) Arithmetic/geometric sequences

       2) Linear/quadratic models

       3) Exponential Models                            Voting and Apportionment

       4) Polynomial models                                   1) Voting methods/flaws

       5) Financial models                                          2) Apportionment methods

Differential Calculus                                                 /flaws

       1) Derivatives                                   Graph Theory

       2) Equations of tangent lines                            1) Euler paths/circuits

       3) Maxima, minima, optimization                            2) Hamilton paths/circuits