# Mathematical Methods

I. INTRODUCTION

1. Subject description

Mathematical Methods is a 20-credit subject at Stage 2.

Mathematical Methods develops an increasingly complex and sophisticated understanding of calculus and statistics. By using functions and their derivatives and integrals, and by mathematically modelling physical processes, students develop a deep understanding of the physical world through a sound knowledge of relationships involving rates of change. Students use statistics to describe and analyse phenomena that involve uncertainty and variation.

Mathematical Methods provides the foundation for further study in mathematics, economics, computer sciences, and the sciences. It prepares students for courses and careers that may involve the use of statistics, such as health or social sciences. When studied together with Specialist Mathematics, this subject can be a pathway to engineering, physical science, and laser physics.

2. Mathematical options

The diagram below represents the possible mathematical options that students might study at Stage 1 and Stage 2. Notes:

Although it is advantageous for students to study Australian Curriculum 10 and 10A in Year 10, the 10A curriculum per se is not a prerequisite for the study of Specialist Mathematics and Mathematical Methods. The essential aspects of 10A are included in the curriculum for Specialist Mathematics and Mathematical Methods.

Mathematical Methods can be studied as a single subject; however, Specialist Mathematics is designed to be studied together with Mathematical Methods.

3. Capabilities

The capabilities connect student learning within and across subjects in a range of contexts. They include essential knowledge and skills that enable people to act in effective and successful ways.

The SACE identifies seven capabilities. They are:

• literacy
• numeracy
• information and communication technology (ICT) capability
• critical and creative thinking
• personal and social capability
• ethical understanding
• intercultural understanding.

3.1. Literacy

In this subject students develop their literacy capability by, for example:

• communicating mathematical reasoning and ideas for different purposes, using appropriate language and representations, such as symbols, equations, tables, and graphs
• interpreting and responding to appropriate mathematical language and representations
• analysing information and explaining mathematical results.

Mathematics provides a specialised language to describe and analyse phenomena. It provides a rich context for students to extend their ability to read, write, visualise, and talk about situations that involve investigating and solving problems.

Students apply and extend their literacy skills and strategies by using verbal, graphical, numerical, and symbolic forms of representing problems and displaying statistical information. Students learn to communicate their findings in different ways, using different systems of representation.

3.1. Numeracy

Being numerate is essential for participating in contemporary society. Students need to reason, calculate, and communicate to solve problems. Through the study of mathematics, they understand and use mathematical skills, concepts, and technologies in a range of contexts that can be applied to:

• using measurement in the physical world
• gathering, representing, interpreting, and analysing data
• using spatial sense and geometric reasoning
• investigating chance processes
• using number, number patterns, and relationships between numbers
• working with graphical, statistical and algebraic representations, and other mathematical models.

3.3. Information and communication technology (ICT) capability

In this subject students develop their information and communication technology capability by, for example:

• understanding the role of electronic technology in the study of mathematics
• making informed decisions about the use of electronic technology
• understanding the mathematics involved in computations carried out using technologies, so that reasonable interpretations can be made of the results.

Students extend their skills in using technology effectively and processing large amounts of quantitative information.

Students use ICT to extend their theoretical mathematical understanding and apply mathematical knowledge to a range of problems. They use software relevant for study and/or workplace contexts. This may include tools for statistical analysis, algorithm generation, data representation and manipulation, and complex calculation. They use digital tools to make connections between mathematical theory, practice, and application; for example, to use data, address problems, and operate systems in particular situations.

3.4. Critical and creative thinking

In this subject students develop critical and creative thinking by, for example:

• building confidence in applying knowledge and problem-solving skills in a range of mathematical contexts
• developing mathematical reasoning skills to think logically and make sense of the world
• understanding how to make and test projections from mathematical models
• interpreting results and drawing appropriate conclusions
• reflecting on the effectiveness of mathematical models, including the recognition of assumptions, strengths, and limitations
• using mathematics to solve practical problems and as a tool for learning
• making connections between concrete, pictorial, symbolic, verbal, written, and mental representations of mathematical ideas
• thinking abstractly, making and testing conjectures, and explaining processes.

Problem-solving in mathematics builds students’ depth of conceptual understanding and supports development of critical and creative thinking. Learning through problem-solving helps students when they encounter new situations. They develop their creative and critical thinking capability by listening, discussing, conjecturing, and testing different strategies. They learn the importance of self-correction in building their conceptual understanding and mathematical skills. 3.5. Personal and social capability

In this subject students develop their personal and social capability by, for example:

• arriving at a sense of self as a capable and confident user of mathematics through expressing and presenting ideas in a variety of ways
• appreciating the usefulness of mathematical skills for life and career opportunities and achievements
• understanding the contribution of mathematics and mathematicians to society.

The elements of personal and social competence relevant to mathematics include the application of mathematical skills for informed decision-making, active citizenship, and effective self-management. Students build their personal and social competence in mathematics through setting and monitoring personal and academic goals, taking initiative, and building adaptability, communication, and teamwork.

Students use mathematics as a tool to solve problems they encounter in their personal and working lives. They acquire a repertoire of strategies and build the confidence needed to:

• meet the challenges and innovations of a rapidly changing world
• be the designers and innovators of the future, and leaders in their fields.

3.6. Ethical understanding

In this subject students develop their ethical understanding by, for example:

• gaining knowledge and understanding of ways in which mathematics can be used to support an argument or point of view
• examining critically ways in which the media present particular perspectives
• sharing their learning and valuing the skills of others
• considering the social consequences of making decisions based on mathematical results
• acknowledging and learning from errors rather than denying findings and/or evidence.

Areas of ethical understanding relevant to mathematics include issues associated with ethical decision-making and working collaboratively as part of students’ mathematically related explorations. They develop ethical understanding in mathematics through considering social responsibility in ethical dilemmas that may arise when solving problems in personal, social, community, and/or workplace contexts.

3.7. Intercultural understanding

In this subject students develop their intercultural understanding by, for example:

• understanding mathematics as a body of knowledge that uses universal symbols which have their origins in many cultures
• understanding how mathematics assists individuals, groups, and societies to operate successfully across cultures in the global, knowledge-based economy.

Mathematics is a shared language that crosses borders and cultures, and is understood and used globally.

Students read about, represent, view, listen to, and discuss mathematical ideas. They become aware of the historical threads from different cultures that have led to the current bodies of mathematical knowledge. These opportunities allow students to create links between their own language and ideas and the formal language and symbols of mathematics.

4. SACE numeracy requirement

Completion of 10 or 20 credits of Stage 1 Mathematics with a C grade or better, or 20 credits of Stage 2 Mathematical Methods with a C grade or better, will meet the numeracy requirement of the SACE.

II. LEARNING SCOPE AND REQUIREMENTS

1. Learning requirements

The learning requirements summarise the key skills, knowledge and understanding that students are expected to develop and demonstrate through learning in Stage 2 Mathematical Methods.

In this subject, students are expected to:

• understand mathematical concepts, demonstrate mathematical skills, and apply mathematical techniques
• investigate and analyse mathematical information in a variety of contexts
• think mathematically by posing questions, solving problems, applying models, and making, testing, and proving conjectures
• interpret results, draw conclusions, and determine the reasonableness of solutions in context
• make discerning use of electronic technology to solve problems and to refine and extend mathematical knowledge
• communicate mathematically and present mathematical information in a variety of ways.

2. Content

Stage 2 Mathematical Methods is a 20-credit subject.

Stage 2 Mathematical Methods focuses on the development of mathematical skills and techniques that enable students to explore, describe, and explain aspects of the world around them in a mathematical way. It places mathematics in relevant contexts and deals with relevant phenomena from the students’ common experiences, as well as from scientific, professional, and social contexts.

The coherence of the subject comes from its focus on the use of mathematics to model practical situations, and on its usefulness in such situations. Modelling, which links the two mathematical areas to be studied, calculus and statistics, is made more practicable by the use of electronic technology.

The ability to solve problems based on a range of applications is a vital part of mathematics in this subject. As both calculus and statistics are widely applicable as models of the world around us, there is ample opportunity for problem-solving throughout this subject.

Stage 2 Mathematical Methods consists of the following six topics:

• Topic 1: Further differentiation and applications
• Topic 2: Discrete random variables
• Topic 3: Integral calculus
• Topic 4: Logarithmic functions
• Topic 5: Continuous random variables and the normal distribution
• Topic 6: Sampling and confidence intervals.

The suggested order of the topics is a guide only; however, students study all six topics. If Mathematical Methods is to be studied in conjunction with Specialist Mathematics, consideration should be given to appropriate sequencing of the topics across the two subjects.

Each topic consists of a number of subtopics. These are presented in the subject outline in two columns as a series of key questions and key concepts, side by side with considerations for developing teaching and learning strategies.

The key questions and key concepts cover the prescribed areas for teaching, learning, and assessment in this subject. The considerations for developing teaching and learning strategies are provided as a guide only.

A problem-based approach is integral to the development of the mathematical models and associated key concepts in each topic. Through key questions teachers can develop the key concepts and processes that relate to the mathematical models required to address the problems posed.

The considerations for developing teaching and learning strategies present problems for consideration and guidelines for sequencing the development of the concepts. They also give an indication of the depth of treatment and emphases required.

Although the material for the external examination will be based on the key questions and key concepts outlined in the six topics, the considerations for developing teaching and learning strategies may provide useful contexts for examination questions.

Students should have access to technology, where appropriate, to support the computational aspects of these topics.

2.1. Calculus

The following three topics relate to the study of calculus:

• Topic 1: Further differentiation and applications
• Topic 3: Integral calculus
• Topic 4: Logarithmic functions

Calculus is essential for developing an understanding of the physical world, as many of the laws of science are relationships involving rates of change. The study of calculus provides a basis for understanding rates of change in the physical world, and includes the use of functions, and their derivatives and integrals, in modelling physical processes.

In this area of study, students gain a conceptual grasp of calculus, and the ability to use its techniques in applications. This is achieved by working with various kinds of mathematical models in different situations, which provide a context for investigating and analysing the mathematical function behind the mathematical model.

The study of calculus continues from Stage 1 Mathematics with the derivatives of exponential, logarithmic, and trigonometric functions and their applications, together with differentiation techniques and applications to optimisation problems and graph sketching. It concludes with integration, both as a process that reverses differentiation and as a way of calculating areas. The fundamental theorem of calculus as a link between differentiation and integration is emphasised.

2.2. Statistics

The following three topics relate to the study of statistics:

• Topic 2: Discrete random variables
• Topic 5: Continuous random variables and the normal distribution
• Topic 6: Sampling and confidence intervals

The study of statistics enables students to describe and analyse phenomena that involve uncertainty and variation. In this area of study, students move from asking statistically sound questions towards a basic understanding of how and why statistical decisions are made. The area of study provides students with opportunities and techniques to examine argument and conjecture from a statistical point of view. This involves working with discrete and continuous variables, and the normal distribution in a variety of contexts; learning about sampling in certain situations; and understanding the importance of sampling in statistical decision-making. III. ASSESSMENT SCOPE AND REQUIREMENTS

All Stage 2 subjects have a school assessment component and an external assessment component.

1. Evidence of learning

The following assessment types enable students to demonstrate their learning in Stage 2 Mathematical Methods.

School assessment (70%)

• Assessment Type 1: Skills and Applications Tasks (50%)
• Assessment Type 2: Mathematical Investigation (20%)

External assessment (30%)

• Assessment Type 3: Examination (30%)

Students provide evidence of their learning through eight assessments, including the external assessment component. Students undertake:

• six skills and applications tasks
• one mathematical investigation
• one examination.

2. Assessment design criteria

The assessment design criteria are based on the learning requirements and are used by:

• teachers to clarify for students what they need to learn
• teachers and assessors to design opportunities for students to provide evidence of their learning at the highest possible level of achievement.

The assessment design criteria consist of specific features that:

• students should demonstrate in their learning
• teachers and assessors look for as evidence that students have met the learning requirements.

For this subject, the assessment design criteria are:

• concepts and techniques
• reasoning and communication.

The specific features of these criteria are described below.

The set of assessments, as a whole, must give students opportunities to demonstrate each of the specific features by the completion of study of the subject.

2.1. Concepts and Techniques

The specific features are as follows:

• CT1: Knowledge and understanding of concepts and relationships.
• CT2: Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts.
• CT3: Application of mathematical models.
• CT4: Use of electronic technology to find solutions to mathematical problems.

2.2. Reasoning and Communication

The specific features are as follows:

• RC1: Interpretation of mathematical results.
• RC2: Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations.
• RC3: Use of appropriate mathematical notation, representations, and terminology.
• RC4: Communication of mathematical ideas and reasoning to develop logical arguments.
• RC5: Development, testing, and proof of valid conjectures.*

* In this subject students must be given the opportunity to develop, test, and prove conjectures in at least one assessment task in the school assessment component.

3. School assessment

Assessment Type 1: Skills and Applications Tasks (50%)

Students complete six skills and applications tasks.

Skills and applications tasks are completed under the direct supervision of the teacher.

The equivalent of one skills and applications task must be undertaken without the use of either a calculator or notes.

In the remaining skills and applications tasks, electronic technology and up to one A4 sheet of paper of handwritten notes (on one side only) may be used at the discretion of the teacher.

Students find solutions to mathematical problems that may:

• be routine, analytical, and/or interpretative
• be posed in a variety of familiar and new contexts
• require discerning use of electronic technology.

In setting skills and applications tasks, teachers may provide students with information in written form or in the form of numerical data, diagrams, tables, or graphs. A task should require students to demonstrate an understanding of relevant mathematical concepts and relationships.

Students select appropriate techniques or algorithms and relevant mathematical information to find solutions to routine, analytical, and/or interpretative problems. Some of these problems should be set in context, for example, social, scientific, economic, or historical.

Students provide explanations and arguments, and use correct mathematical notation, terminology, and representations throughout the task.

Skills and applications tasks may provide opportunities to develop, test, and prove conjectures.

For this assessment type, students provide evidence of their learning in relation to the following assessment design criteria:

• concepts and techniques
• reasoning and communication.

Assessment Type 2: Mathematical Investigation (20%)

Students complete one mathematical investigation.

Students investigate mathematical relationships, concepts, or problems, which may be set in an applied context. The subject of a mathematical investigation may be derived from one or more subtopics, although it can also relate to a whole topic or across topics.

A mathematical investigation may be initiated by a student, a group of students, or the teacher. Teachers should give some direction about the appropriateness of each student’s choice, and guide and support students’ progress in an investigation. For this investigation there must be minimal teacher direction and teachers must allow the opportunity for students to extend the investigation in an open-ended context.

Students demonstrate their problem-solving strategies as well as their knowledge, skills, and understanding in the investigation. They are encouraged to use a variety of mathematical and other software (e.g. computer algebra systems (CAS), spreadsheets, statistical packages) to enhance their investigation. The generation of data and the exploration of patterns and structures, or changing parameters, may provide an important focus. From these, students may recognise different patterns or structures. Notation, terminology, forms of representation of information gathered or produced, calculations, evidence of technological skills, and results are important considerations.

Students complete a report for the mathematical investigation.

In the report, students interpret and justify results, and draw conclusions. They give appropriate explanations and arguments. The mathematical investigation may provide an opportunity to develop, test, and prove conjectures.

The report may take a variety of forms, but would usually include the following:

• an outline of the problem and context
• the method required to find a solution, in terms of the mathematical model or strategy used
• the application of the mathematical model or strategy, including
• relevant data and/or information
• mathematical calculations and results, using appropriate representations
• the analysis and interpretation of results, including consideration of the reasonableness and limitations of the results
• the results and conclusions in the context of the problem.

A bibliography and appendices, as appropriate, may be used.

The format of the investigation report may be written or multimodal.

The investigation report, excluding bibliography and appendices if used, must be a maximum of 15 A4 pages if written, or the equivalent in multimodal form. The maximum page limit is for single-sided A4 pages with minimum font size 10. Page reduction, such as two A4 pages reduced to fit on one A4 page, is not acceptable. Conclusions, interpretations and/or arguments that are required for the assessment must be presented in the report, and not in an appendix. Appendices are used only to support the report, and do not form part of the assessment decision.

For this assessment type, students provide evidence of their learning in relation to the following assessment design criteria:

• concepts and techniques
• reasoning and communication.

4. External assessment

Assessment Type 3: Examination (30%)

Students undertake a 130-minute external examination.

The examination is based on the key questions and key concepts in the six topics. The considerations for developing teaching and learning strategies are provided as a guide only, although applications described under this heading may provide contexts for examination questions.

The examination consists of a range of problems, some focusing on knowledge and routine skills and applications, and others focusing on analysis and interpretation. Some problems may require students to interrelate their knowledge, skills, and understanding from more than one topic. Students provide explanations and arguments, and use correct mathematical notation, terminology, and representations throughout the examination.

A formula sheet is provided with the examination booklet.

Students may take two unfolded A4 sheets (four sides) of handwritten notes into the examination room.

Students may have access to approved electronic technology during the external examination. However, students need to be discerning in their use of electronic technology to find solutions to questions/problems in examinations.

All specific features of the assessment design criteria for this subject may be assessed in the external examination.

5. Performance standards.

The performance standards describe five levels of achievement, A to E.

Each level of achievement describes the knowledge, skills, and understanding that teachers and assessors refer to in deciding how well students have demonstrated their learning, on the basis of the evidence provided.

During the teaching and learning program the teacher gives students feedback on their learning, with reference to the performance standards.

At the student’s completion of study of each school assessment type, the teacher makes a decision about the quality of the student’s learning by:

• referring to the performance standards
• assigning a grade between A+ and E- for the assessment type.

The student’s school assessment and external assessment are combined for a final result, which is reported as a grade between A+ and E-.

 Concepts and Techniques Reasoning and Communication A Comprehensive knowledge and understanding of concepts and relationships. Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts. Successful development and application of mathematical models to find concise and accurate solutions. Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems. Comprehensive interpretation of mathematical results in the context of the problem. Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations. Proficient and accurate use of appropriate mathematical notation, representations, and terminology. Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments. Effective development and testing of valid conjectures, with proof. B Some depth of knowledge and understanding of concepts and relationships. Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts. Some development and successful application of mathematical models to find mostly accurate solutions. Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems. Mostly appropriate interpretation of mathematical results in the context of the problem. Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations. Mostly accurate use of appropriate mathematical notation, representations, and terminology. Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly effective development and testing of valid conjectures, with substantial attempt at proof. C Generally competent knowledge and understanding of concepts and relationships. Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts. Successful application of mathematical models to find generally accurate solutions. Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems. Generally appropriate interpretation of mathematical results in the context of the problem. Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations. Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy. Generally effective communication of mathematical ideas and reasoning to develop some logical arguments. Development and testing of generally valid conjectures, with some attempt at proof. D Basic knowledge and some understanding of concepts and relationships. Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts. Some application of mathematical models to find some accurate or partially accurate solutions. Some appropriate use of electronic technology to find some accurate solutions to routine problems. Some interpretation of mathematical results. Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations. Some appropriate use of mathematical notation, representations, and terminology, with some accuracy. Some communication of mathematical ideas, with attempted reasoning and/or arguments. Attempted development or testing of a reasonable conjecture. E Limited knowledge or understanding of concepts and relationships. Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems. Attempted application of mathematical models, with limited accuracy. Attempted use of electronic technology, with limited accuracy in solving routine problems. Limited interpretation of mathematical results. Limited understanding of the meaning of mathematical results, their reasonableness or limitations. Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy. Attempted communication of mathematical ideas, with limited reasoning. Limited attempt to develop or test a conjecture.

6. Assessment integrity

The SACE Assuring Assessment Integrity Policy outlines the principles and processes that teachers and assessors follow to assure the integrity of student assessments. This policy is available on the SACE website (www.sace.sa.edu.au) as part of the SACE Policy Framework.

The SACE Board uses a range of quality assurance processes so that the grades awarded for student achievement, in both the school assessment and the external assessment, are applied consistently and fairly against the performance standards for a subject, and are comparable across all schools.

Information and guidelines on quality assurance in assessment at Stage 2 are available on the SACE website (www.sace.sa.gov.au).

IV. SUPPORT MATERIALS