# 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

** **