# Problems and Proofs

**Semester Elective Unit**
## Prerequisites

Year 9 Mathematics.

## Course Description

This course provides a rich experience of where the key discoveries in mathematics have come from historically and their applications to science, engineering and computing. Students become familiar with the process of generalisation, conjecture and proof, and develop an understanding of the key problems that mathematicians are still trying to solve in the 21st century.

## Essential Questions

- How does mathematics relate to science, engineering and computing?
- How did some of the big ideas in mathematics develop over time?
- What are the key problems mathematics is still trying to solve?
- How can we systematically generalise solutions to a problem to develop conjectures and form irrefutable proofs?

## Areas of Study

**Generalisations, Conjectures and Proofs**

- Investigate the process of generalisation and hypothesis or conjecture when solving problems.
- Develop an understanding of the notation, form and uses of a variety of proofs, including geometric proofs, proof by induction, proof by contradiction, proof using contrapositive and direct proof.
- Become familiar with key proofs such as divisibility proofs, the sum of the first
*n* square numbers, the proof of irrationality for some real numbers and proofs of circle theorems and Pythagoras’ Theorem.

**Applications of Mathematics in Science, Engineering and Computing**

- An introduction to complex numbers and their connections to engineering.
- Infinitude of primes and their connections to computing.
- Applications of statistics and probability in science.

**History of Mathematics**

- Investigate some historical techniques for arithmetic algorithms.
- Investigate the life, work and influence of a notable mathematician.
- Develop an understanding of how mathematical ideas have developed over time.
- Develop an appreciation for how social, cultural and historical factors have influenced the development of mathematics.
- Understand how mathematics contributed to society and human culture.

## Assessment

Task |
Description |

History of Mathematics Essay |
A short essay outlining how one particular area of mathematics developed historically and its relevance to real world applications today. |

Assignments |
A series of assignments involving formal proofs and problem-solving tasks. |

Semester Examination |
Students undertake an examination at the end of the semester. |