68105 Algebra
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Credit points: 6 cp
Result type: Grade and marks
Requisite(s): 68102 Mathematics for Secondary Education Foundations
These requisites may not apply to students in certain courses. See access conditions.
Anti-requisite(s): 37233 Linear Algebra
Description
In this subject, students develop an understanding of the theory of linear algebra, applications of linear algebra, and some of the main computational techniques used in these applications. Topics include systems of linear equations (LU factorisation and iterative methods); vector spaces; inner product spaces; Gram-Schmidt orthogonalisation, QR decomposition; approximation theory: least squares and orthogonal polynomials; the eigenvalue problem; singular value decomposition and applications.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | apply fundamental concepts of mathematics to solve problems involving linear algebra |
|---|---|
| 2. | apply knowledge of the conceptual development of linear algebra to explain procedures and calculations |
| 3. | use mathematical terminology and symbols to define concepts |
| 4. | apply mathematical knowledge and skills in a variety of situations, in both familiar and new contexts |
| 5. | communicate mathematical knowledge clearly, logically and critically |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Analyse: Demonstrate critical engagement with mathematical knowledge in the secondary classroom context. (1.1)
- Synthesise: Develop a professional identity as a leader in mathematics and mathematics education. (1.2)
- Evaluate: Evaluate the role of mathematics and mathematics education demonstrating awareness of the links to society and classroom practice. (1.3)
- Analyse: Critically evaluate information in the investigation of mathematical and pedagogical problems. (2.1)
- Synthesise: Investigate real-world problems by analysing and critically evaluating different solutions to complex challenges. (2.2)
- Analyse: Derive innovative solutions to complex mathematical and educational problems. (4.1)
- Synthesise: Through reflection, own and understand their learning journey. (4.2)
- Evaluate: Critique approaches to meeting educational needs. (4.3)
- Analyse: critique approaches for communicating with students, parents, peers, mathematicians, educationalists, and the public. (5.1)
- Synthesise: Develop and Communicate complex ideas. (5.2)
- Evaluate: Judge the use of interpersonal communication skills with students, parents, peers, mathematicians, educationalists, and the public. (5.3)
Contribution to the development of graduate attributes
1.0 Disciplinary knowledge
The online contents and tutorial activities will allow students to develop practical and theoretical skills in Algebra.
2.0 Research, inquiry and critical thinking
The collaborative approach to problem formulations and solutions used in the tutorials helps students develop skills in identifying and evaluating alternative approaches to solving problems.
4.0 Reflection, innovation, creativity
The micro-teaching assessment task requires students to think about how to present material creatively and engagingly, and reflect afterwards on its effectiveness in communicating and teaching the concepts.
5.0 Communication
Presentation of written and oral solutions to problems using appropriate professional language is emphasised in the tutorials and assessment. All assessment tasks require the appropriate presentation of information, reasoning and conclusions and require students to gain meaning from instructions (written or verbal) and problem statements.
Teaching and learning strategies
This subject requires about 1-2 hours on Canvas per week and a 2 hour tutorial once a week. There is also about 2-3 hours of written homework each week. Students are required to read the Canvas pages online and engage with asynchronous discussions via discussion boards. As students complete the interactive elements on Canvas they are provided with formative feedback on the comprehension of concepts, and on skill development.
Through the weekly homework tasks, students will build their problem solving and modelling skills. Students will also develop a high standard of written communication to explain their solutions and the steps taken to arrive that those solutions.
In the interactive and collaborative weekly tutorials students will learn the professional skills of presenting short mathematical explanations to their peers, as though to a high school class. These skills will be assessed in the Micro-teaching task, where students will demonstrate their discipline knowledge in the context of professional practice. Oral and written communication to a target audience is key. Students will prepare and present to their colleagues solutions of relevant mathematical problems. Each student will do this at least twice.
Feedback on microteaching will be provided via discussions in the tutorials with peers and the tutor.
Content (topics)
- Operations with vectors and matrices;
- Systems of linear equations;
- Linear spaces and subspaces; linear dependence/independence;
- Basis, dimensions, coordinate systems;
- Linear transformations, eigenvectors and eigenvalues;
- Orthogonality, projections, orthogonalisation and orthogonal decomposition;
- Least-squares solutions;
- Quadratic forms;
- LU factorisation and iterative methods.
Assessment
Assessment task 1: Understanding and skills assessments
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking 4. Reflection, Innovation, Creativity |
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| Objective(s): | This assessment task addresses subject learning objective(s): 1, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1, 2.2, 4.1 and 4.2 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 35% |
| Criteria: |
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Assessment task 2: Micro-teaching
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary Knowledge 2. Research, inquiry and critical thinking 4. Reflection, Innovation, Creativity 5. Communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 1.2, 1.3, 2.1, 2.2, 4.1, 4.3, 5.1, 5.2 and 5.3 |
| Type: | Demonstration |
| Groupwork: | Group, individually assessed |
| Weight: | 35% |
| Length: | The micro-teaching will take about 15 minutes at the start of each two hour tutorial. Students will know in advance which questions they are to present. |
| Criteria: |
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Assessment task 3: Final exam
| Intent: | This assessment task contributes to the development of the following graduate attributes: 2. Research, inquiry and critical thinking 4. Reflection, Innovation, Creativity
|
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 2.2, 4.1 and 4.2 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 30% |
| Criteria: |
|
Minimum requirements
In order to pass this subject, a final result (the sum of all the marks with all the assessment tasks) of 50% or more must be achieved.
Recommended texts
Any textbook with title "Linear Algebra" should be useful for the first half of this subject.
For the second half we will make use of the textbook:
Title: Linear Algebra Done Right
Author: Axler, Sheldon
ISBN: 3319110802
Edition: 3rd ed. 2015
This is available as free PDF via the UTS Library (see Canvas for a link).