1. VCE Mathematical Methods School-assessed Coursework

2. Copyright
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3. VCE assessment principles
Assessment is an integral part of teaching and learning at the senior secondary level that:
identifies opportunities for further learning
describes student achievement
articulates and maintains standards
provides the basis for the award of a certificate.

4. VCE assessment principles
As part of VCE studies, assessment tasks enable:
the demonstration of the achievement of an outcome or set of outcomes
judgment and reporting of a level of achievement for school-based assessments at Units 3 and 4.

5. VCE assessment principles
VCE assessment will be:
valid
equitable
balanced
efficient.

6. Validity
School-assessed Coursework and examinations make distinct and complementary contributions to student assessment. School-assessed Coursework enhances validity of student assessment by providing the opportunity for a context, scenario or situation to be explored mathematically in greater breadth and/or depth than is possible in an examination, with non-routine and open-ended elements explored more fully.

7. VCE Mathematics assessment
Two types of formal assessment:
School-based: prescribed task types, School-assessed coursework, locally devised and assessed
Examinations: externally set and marked.
Both types of assessment are based on the outcomes in relation to the areas of study.

8. Timeframe for tasks
The tasks for School-assessed Coursework for Mathematics are to be implemented over a longer continuous period, where investigative, modelling or problem-solving techniques or approaches are employed and the related use of technology as a tool for working mathematically is suitably incorporated.

9. Context for a task
A practical or theoretical context, scenario or situation is central to the development of School assessed coursework tasks in Mathematical Methods. This context, scenario or situation is to be explored in some breadth and depth, with natural connection to stimulus items and questions of interest.

10. Content and task design
A task for School-assessed Coursework will not cover all of the content from an area of study or topic, or all of the key knowledge and key skills for an outcome. The context, scenario or situation on which the task is based will afford connection to some of these more than others.

11. Mathematical Methods Unit 3

12. Application task
A function and calculus-based mathematical investigation of a practical or theoretical context involving content from two or more areas of study. The task has three components of increasing complexity:
introduction of the context through specific cases or examples
consideration of general features of the context
variation or further specification of assumption or conditions involved in the context to focus on a particular feature or aspect related to the context.
The application task is to be of 4-6 hours duration over a period of 1-2 weeks.

13. Mathematical investigation
Is the process of using mathematical constructs, structures, concepts, processes and skills to represent and explore aspects of a situation or context in a way that enables one to investigate characteristics, features and behaviour of systems and objects related to the context. This includes the use of data, sets, dependent and independent variables, constants and parameters, relations and functions, graphs, tables, charts and diagrams, equations and inequalities, variation, generalization, specialisation, experiment, computation and proof.

14. Mathematical investigation schema
A simple representation for the mathematical investigation process is shown below:

15. Mapping the schema to outcomes
The following tables map key knowledge and key skills for Outcome 2 and Outcome 3 to the stages of the mathematical investigation process. Key knowledge and key skills for Outcome 1 will be relevant to the stages of the mathematical investigation process depending on the context chosen, and the applicable content from the areas of study. Each key knowledge and key skill statement is mapped to a particular stage of the process, or in some cases to two stages of the process as applicable. The numbers in the table correspond to the order of occurrence of that key knowledge or key skill statement in the list for each outcome. This mapping should be used to inform the design and development of application tasks that incorporate the outcomes and ensure that related mark weightings are suitably addressed in the development of the corresponding approach to assessment.

16. Identify context for investigation

17. Represent mathematically

18. Systematically explore and analyse
Outcome
Key knowledge dot point
Key skill dot point 2 3
3, 4
4, 5
2, 3, 4, 6, 8, 9

19. Interpret and communicate results

20. Application task – indicative content coverage: topics and bullet points
For each component of the task note the relevant topics and content:

21. Application task – indicative content coverage: topics and bullet points (continued)

22. Application task – indicative content coverage: topics and bullet points (continued)

23. Application task outcome – component mark allocation (weighting)
Use the table to indicate the mark allocation for the outcomes with respect to each component of the task:

24. Task design …1
For the application task suitable constants, parameters, variables, functions and relations and operations for a topic of interest, situation or context should be used and related questions investigated. The task should not be too directive or mirror an extended set of examination-style questions. Multiple-choice items are not suitable for either an application task or a modelling or problem-solving task and are not to be used.

25. Task design … 2 some sample contexts
Bezier curves
splining pathways, such as a pathway along a creek
modelling with combined functions, for example, a pendulum or swing
approximating one function with another function
phenomena with casual relationships between variables
optimisation with different container designs
mean value theorems

26. - investigate the behaviour of a family of functions and
Task design … 3
- investigate the behaviour of a family of functions and
key features (systematic variation of parameters)
make comparisons
formulate, explore and discuss conjectures
establish results, find counter-examples
generalise and consider special cases

27. Task design … 4 Elements of formulation
amount of direction (tell or ask)
choice (… using suitable set of information about a function and its derivative…..)
openness (don’t specify approach … using a suitable measure for …)
options (… consider different transformations or combinations of functions …)

28. Aligning the task with assessment
The Mathematical Methods Advice for teachers provides
performance criteria for the application task and a sample record sheet based on these criteria.
These performance criteria may be used in several ways:
directly in conjunction with the sample record sheets and teacher annotations for pointers with respect to key aspects of the task related to each criterion for the outcomes
directly with the descriptive text for each criterion modified to incorporate task specific elements as applicable
as a template for teachers to develop their own criteria and descriptive text for each criterion, including their own allocation of marks for the criteria with the total mark allocation for each outcome as specified in the study design.

29. Outcome 1 – pointers for assessment
Identify aspects of likely response related to the criteria for the outcome:

30. Outcome 2 – pointers for assessment

31. Outcome 3 – pointers for assessment

32. VCAA sample application tasks
Five sample application tasks can be accessed from the Mathematical Methods Advice for teachers section of the VCAA website.
These sample tasks are downloadable and editable word documents that can be used as templates for designing new tasks.
Each sample task includes:
introduction to the context
three components
mapping with respect to areas of study and outcomes

33. School-assessed Coursework Resource (2000)
This resource contains previous VCE Mathematics Investigative Projects from VCE Change and approximation and Mathematical Methods studies These are downloadable and editable Word documents and can be used as a basis for developing Application tasks.

34. Mathematical Methods Unit 4

35. Modelling
Mathematical modelling is the process of using mathematical constructs, structures and techniques to represent and describe a real-world context or system, in a simple and concise way that enables one to investigate features and characteristics of its behaviour, analyse particular aspects or solve problems of interest, and to make predictions related to the context or system. For Mathematical Methods, this includes the use of real numbers, sets, variables, constants and parameters, relations and functions, graphs, tables and diagrams, equations and inequalities, derivatives, anti-derivatives and integrals, random variables, probability distributions and elementary statistical inference.

36. Modelling schema

37. Mapping the schema to outcomes
The following tables map key knowledge and key skills for Outcome 2 and Outcome 3 to the stages of the mathematical modelling process.
Key knowledge and key skills for Outcome 1 will be relevant to the stages of the modelling process depending on the context chosen and the applicable content from the areas of study.
Each key knowledge and key skill statement is mapped to a particular stage of the process, or in some cases to two stages of the process as applicable. The numbers in the table correspond to the order of occurrence of that key knowledge or key skill statement in the list for each outcome.
This mapping should be used to inform the design and development of modelling tasks that incorporate the outcomes and assist teachers to ensure that related mark weightings are suitably addressed in the development of the corresponding approach to assessment.

38. Formulate the model

39. Apply the model

40. Interpret and refine the model

41. Modelling or problem-solving task outcome – component mark allocation (weighting)
Use the table to indicate the mark allocation for the outcomes with respect to each part/section of the task:

42. Problem-solving
Problem solving is a process that occurs in a context where a question, task or issue needs to be solved or resolved, and there is a motivation, but not yet the means, to do so. Questions or tasks for which there are already recognised methods or approaches for solution or resolution, do not require problem-solving in this sense. In mathematics problems are generated from questions, conjectures and hypotheses within and across areas of study. New problems may arise in their own right, or as a variation, re-formulation, extension or generalisation of a known problem or class of problems.

43. Problem-solving schema

44. Mapping the schema to outcomes
The following tables map key knowledge and key skills for Outcome 2 and Outcome 3 to the stages of the problem-solving process. Key knowledge and key skills for Outcome 1 will be relevant to the stages of the problem-solving process depending on the context chosen, and the applicable content from the areas of study. Each key knowledge and key skill statement is mapped to a particular stage of the process, or in some cases to two stages of the process as applicable. The numbers in the table correspond to the order of occurrence of that key knowledge or key skill statement in the list for each outcome. This mapping should be used to inform the design and development of problem-solving tasks that incorporate the outcomes and assist teachers to ensure that related mark weightings are suitably addressed in the development of the corresponding approach to assessment.

45. Define the problem

46. Devise a plan

47. Implement the plan

48. Review/extend the plan

49. Modelling or problem-solving task – indicative content coverage: topics and bullet points
For each component of the task note the relevant topics and content:

50. Modelling or problem-solving task – indicative content coverage: topics and bullet points (continued)

51. Modelling or problem-solving task outcome – component mark allocation (weighting)
Use the table to indicate the mark allocation for the outcomes with respect to each part/section of the task:

52. Modelling and problem-solving are complementary processes
Mathematical modelling and problem-solving are complementary processes. Developing a model may be a strategy that is employed to solve a problem, and problem-solving may be required in developing and applying aspects of a model. Some sample contexts for mathematical modelling and problem-solving can be found here.

53. The International Mathematical Modelling Challenge (IMMC)
The IMMC website provides a range of resources and ideas to support modelling and problem solving, including a selection of sample problems.

54. The International Mathematical Modelling Challenge (IMMC) – modelling framework
This elaborates the modelling process:
Specify the mathematical problem. Frame the real-world scenario as an appropriate, related mathematical question(s).
Formulate the mathematical model. Make simplifying assumptions, choose variables, estimate magnitudes of inputs, justify decisions made.
Describe the real-world problem. Identify and understand the practical aspects of the situation.
Solve the mathematics.
Interpret the solution. Consider mathematical results in terms of their real-world meanings.
Evaluate the model. Make a judgment as to the adequacy of the solution to the original question(s). Modify the model as necessary and repeat the cycle until an adequate solution has been found.
Report the solution. Communicate clearly and fully your suggestions to address the real-world problem.

55. Aligning the task with assessment
The Mathematical Methods Advice for teachers provides
performance criteria for a Modelling or problem-solving task and a sample record sheet based on these criteria.
These performance criteria may be used in several ways:
directly in conjunction with the sample record sheets and teacher annotations for pointers with respect to key aspects of the task related to each criterion for the outcomes
directly with the descriptive text for each criterion modified to incorporate task specific elements as applicable
as a template for teachers to develop their own criteria and descriptive text for each criterion, including their own allocation of marks for the criteria with the total mark allocation for each outcome as specified in the study design.

56. Outcome 1 – pointers for assessment
Identify aspects of likely response related to the criteria for the outcome:

57. Outcome 2 – pointers for assessment

58. VCAA sample modelling or problem-solving tasks
Four sample Modelling or problem-solving tasks can be accessed from the Mathematical Methods Advice for teachers section of the VCAA website. These sample tasks can be used as templates for designing new tasks.

59. School Assessed Coursework Resource (2000)
This resource contains previous VCE Mathematics Challenging problems for VCE Change and approximation and Reasoning and data studies 1989 – These are downloadable and editable Word documents, and can be used as a basis for developing modelling or problem-solving tasks.