MATH205 - Geometry

Year

Credit points

Campus offering

Prerequisites

MATH104 Differential and Integral Calculus OR MATH107 Introduction to Logic and Algebra

Teaching organisation

Unit rationale, description and aim

Geometry has arisen in all cultures as a means of solving problems and/or through decoration. It has been a cornerstone of mathematics for millennia and represents a fundamentally important way of viewing and investigating the world. Combining the pictorial, numeric and algebraic aspects also demonstrates the benefits of viewing problems from a variety of standpoints. By considering Euclidean and non-Euclidean geometry students will gain insight into the importance of axioms in mathematics.

This unit applies techniques acquired in earlier units to investigate and extend student’s knowledge of geometry. Initially studying Euclidean geometry; lines, polygons and circles are considered as well as Euclidean constructions, giving a rationale for each one. Major theorems on concurrence and collinearity follow. Suitable geometry software packages will be introduced to support the study of this unit. The second part of this unit will extend the study of geometry to 3 dimensions, conic sections and similar topics. The unit will also briefly look at non-Euclidean geometries as examples of how the world works and as examples of the importance of axioms.

The aim of this unit is to strengthen students’ knowledge of Euclidean geometry and to introduce them to axiomatic considerations and the benefits of approaching problems from several points of view. The unit will also consider how first peoples’ knowledge can form the foundation of a subject ultimately leading to the most abstract ideas.

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 their knowledge of Euclidean theorems and the nature of formal proofs to solve simple problems and prove simple results (GA4, GA5, GA8)

LO2 - Perform constructions using classical criteria (GA5, GA10)

LO3 - Solve problems involving conics (GA5)

LO4 - Explain the historical development of geometry, the significance of Euclid’s work and the importance of non-Euclidean geometry (GA4, GA5, GA8)

LO5 - Explain clearly their own solutions to problems (GA5, GA7, GA9)

LO6 - Use a geometrical software package or app competently (GA9, GA10)

Graduate attributes

GA4 - think critically and reflectively 

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

GA7 - work both autonomously and collaboratively 

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

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

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics may include:

1. Straight lines, congruence, similarity
2. Pythagoras’ Theorem, parallels
3. Circles: basic theorems
4. Euclidean constructions
5. Theorems on concurrency and collinearity
6. Nine-point circle and other recent theorems
7. 3 dimensional geometry, regular polyhedra
8. Conic sections
9. Astronomy, spherical geometry and other non-Euclidean geometries
10. Projective geometry and perspective

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 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 geometrical 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

Besant, W. (2009). Conic sections, treated geometrically. BiblioLife.

Brannan, D., Esplen, M., & Gray, J. (2011). Geometry (2nd ed.). Cambridge: Cambridge University Press.

Coxeter, H. (1989). Introduction to geometry. New York: John Wiley & Sons.

Coxeter, H., & Greitzer, S. (2014). Geometry revisited. Washington DC: Mathematical Association of America.

Cromwell, P. (2004). Polyhedra. Cambridge, UK: Cambridge University Press.

Euclid. (2010). The thirteen books of Euclid’s elements (T. Heath Trans.), Lawrence, MA: Digireads.com

Gibson, C. (2004). Elementary Euclidean geometry: An introduction. Cambridge: Cambridge University Press.

Loomis, E. (2008). Elements of geometry and conic sections. BiblioBazaar.

Ostermann, A. and Wanner, G (2012). Geometry by its history. Springer Verlag