MATH107 - Introduction to Logic and Algebra

Year

Credit points

Campus offering

Prerequisites

Nil

Teaching organisation

Unit rationale, description and aim

Commencing students of Mathematics require a solid foundation in several aspects of Mathematics, including logic and algebra, both to support later units in the sequence and to start their study of abstract algebra which is a requirement of all ITE Mathematics courses. This unit also extends students’ knowledge of number systems to include complex numbers. Numerous important real-world applications of mathematics will be considered in the Graph Theory topic. This unit will also link these topics to problems faced by various cultures including Australia’s first people.

This unit forms the first unit in the standard Mathematics sequence and introduces several ideas and objects that form a base for further mathematics. These skills will be developed with the express purpose of being able to solve relevant problems. Topics covered will include matrices, graphs (networks), vectors, sets, functions and complex numbers together with examples motivated by the development of Mathematics across ancient cultures and indigenous knowledges. 

The aim of this unit is to provide the introductory logic and algebra knowledge required for further study in Mathematics, in particular for ITE students of Mathematics. 

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Use matrices to solve systems of linear equations (GA5, GA9) 

LO2 - Use their knowledge of matrices, graphs (networks), vectors and complex numbers to solve simple problems and prove simple results (GA5) 

LO3 - Apply vector ideas to solve simple problems in geometry (GA5, GA9) 

LO4 - Apply graphs (networks) to the solution of simple problems, particularly real-world problems (GA5, GA9) 

LO5 - Apply their knowledge of complex numbers both to solve simple equations and to successfully manipulate complex numbers (GA5) 

LO6 - Use their knowledge of logic and proof, including mathematical induction, to prove simple results (GA5, GA8) 

Graduate attributes

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA9 - demonstrate effective communication in oral and written English language and visual media 

Content

Topics will include: 

1. Matrices – definitions and arithmetic 
2. Systems of linear equations and the use of matrices in their solution 
3. Other matrix operations (determinant and inverse) and transformations of the plane 
4. Logic and the basic techniques of proof
5. Proof, including mathematical induction
6. Introduction to vectors, arithmetic and basic properties 
7. Scalar product, equation of lines and planes, vector geometry, cross product 
8. Introduction to graphs (networks) and their applications 
9. Solving polynomial equations – introduction to complex numbers and their arithmetic 
10. Complex numbers – Argand plane, polar representation 

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology. 

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours of attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study. 

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students. The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure. 

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy, or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit. 

Overview of assessments

Representative texts and references

Recommended references 

Anton, H. & Kaul, A (2019). Elementary Linear Algebra, 12th Edition New York:John Wiley & Sons 

Croft, A., & Davison, R. (2016). Foundation Maths. 6th Edition Pearson Education 

Eccles, P. (1997). An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge:Cambridge University Press 

Lay, D.C. & Lay, S.R. (2014). Linear Algebra and Its Applications. 5th Edition New York:Pearson Education 

Liebeck, M. (2010). A Concise Introduction to Pure Mathematics. Boca Raton:CRC Press 

Lisle, J. (2018). Introduction to Logic. Master Books 

Smullyen, R.M. (2011) What Is the Name of This Book? Dover Publications 

Strang, G. (2016) Introduction to Linear Algebra. 5th Edition Wellesley:Wellesley-Cambridge 

Wilson, R. (2015) Introduction to Graph Theory. 5th Edition Prentice Hall.