# Units 3 and 4

## Performance criteria

The following performance criteria have been developed to assist teachers in the assessment of School Assessed Coursework for Further Mathematics Units 3 and 4. The performance criteria for the Application task and for the three modelling or problem-solving tasks relate the mark weightings for the outcomes to these assessment tasks and provide a basis for decision about level of achievement.

### Unit 3

#### Outcome 1

On completion of this unit the student should be able to define and explain key concepts and apply related mathematical techniques and models as specified in Area of Study 1 in routine contexts.
Mark range Criterion
0 - 2 Appropriate use of mathematical conventions, symbols and terminology

Use of correct terminology and conventions in diagrams, tables and graphs, such as axes labels, scales, identification of types of data, representations and summary statistics. Description of key features of data sets including symmetry, skew and outliers. Accurately defining statistical and other mathematical terms or expressions. Use of correct expressions in representations, summaries, computations and symbolic manipulations.

0 - 4 Definition and explanation of key concepts

Definition and explanation of statistical and other mathematical concepts and use of appropriate phrases and symbolic expressions. Provision of examples which illustrate key concepts and explain their role in the development of related data analysis. Statement of conditions or restrictions which apply to the definition of a concept. Identification of key concepts in relation to data analysis and explanation of the use of these concepts in applying mathematics in different contexts.

0 - 4 Accurate use of mathematical skills and techniques

Use of numerical and symbolic terms to carry out computations, evaluate expressions and formulas, construct tables, produce graphs and solve equations. Use of mathematical routines and procedures for statistical analysis and to solve practical problems involving data analysis.

### Unit 3

#### Outcome 2

On completion of this unit the student should be able to select and apply the mathematical concepts, models and techniques as specified in Area of Study 1 in a range of contexts of increasing complexity.
Mark range Criterion
0 - 4 Identification of important information, variables and constraints

Identification of key characteristics of a problem, task or issue and statement of any assumptions underlying the use of relevant statistics and mathematics in the given context. Choice of suitable variables, constants, parameters and statistics for the development of mathematics related to various aspects of data analysis in a given context. Specification of any conditions for use of statistics related to aspects of the given context.

0 - 8 Application of mathematical ideas and content from the specified areas of study

Demonstration of understanding of key statistical and mathematical content from one or more areas of study in relation to a given context. Use of specific and general formulations of concepts and content drawn from Data analysis to derive results for analysis in this context. Appropriate use of examples to illustrate the application of a mathematical process, or use of a counter-example to disprove a proposition or conjecture.

0 - 8 Analysis and interpretation of results

Analysis and interpretation of results obtained from considering a data set or subsets of the data set. Generation of possible inferences from analysis to draw conclusions related to the context for investigation, and to verify, refute or modify hypotheses. Discussion of the validity and limitations of any models.

### Unit 3

#### Outcome 3

On completion of this unit the student should be able to select and appropriately use numerical, graphical, symbolic, and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.
Mark range Criterion
0 - 4 Appropriate selection and effective use of technology

Relevant and appropriate selection and use of technology, or a function of the selected technology for the statistical context being considered. Use of appropriate specifications which illustrate key features of the mathematics under consideration.

0 - 6 Application of technology

Use and interpretation of the relationship between numerical, graphical and tabular forms of information produced by technology. Analysis of the relationship of the results from an application of technology to the nature of a particular mathematical question, problem or task. Use of tables of values, families of graphs or other results produced using technology to support data analysis in investigative contexts. Production of results efficiently and systematically which identify examples or counter-examples which are clearly relevant to the task.

A sample record sheet for the Data analysis application task (doc - 72.72kb) can be used to record student level of achievement with respect to the available marks for the performance criteria relating to each outcome, and to indicate pointers corresponding to relevant aspects of the task.

### Modelling or problem-solving task performance criteria for the core recursion and financial modelling

These criteria and mark ranges can be used for the modelling or problem solving task for the core recursion and financial modelling in Unit 3.

### Unit 3

#### Outcome 1

On completion of this unit the student should be able to define and explain key concepts and apply related mathematical techniques and models as specified in Area of Study 1 in routine contexts.
Mark range Criterion
0 - 1 Appropriate use of mathematical conventions, symbols and terminology

Application of mathematical conventions in diagrams, tables and graphs, such as headings, axes labels and points. Use of symbolic notation appropriately and accurately in defining mathematical terms or expressions, and describing representations of objects such as first order linear recurrence relations for financial modelling. Use of correct expressions in symbolic manipulation or computation in mathematical work.

0 - 2 Definition and explanation of key concepts

Definition of mathematical concepts using appropriate phrases and mathematical expressions. Provision of examples which illustrate key concepts and explain their role in the development of related mathematics. Statement of conditions or restrictions which apply to the definition of a concept. Identification of key concepts in relation to each area of study and explanation of the use of these concepts in applying mathematics in different contexts. Construction of mathematical expressions involving concepts such as interest, growth and decay, annuities and perpetuities and first order linear recurrence relations.

0 - 2 Accurate use of mathematical skills and techniques

Use of algebraic expressions and numerical values to carry out computations, evaluate expressions and formulas, construct tables, produce graphs and solve equations. Use of mathematical routines and procedures such as generating sequences, and time value of money calculation.

### Unit 3

#### Outcome 2

On completion of this unit the student should be able to select and apply the mathematical concepts, models and techniques as specified in Area of Study 1 in a range of contexts of increasing complexity.
Mark range Criterion
0 - 2 Identification of important information, variables and constraints

Identification of key characteristics of a problem, task or issue and statement of any assumptions underlying the use of relevant financial mathematics in the given context. Choice of suitable variables and constants for the development of mathematics related to various aspects of a given context. Specifications of constraints, such as initial values, domain and range constraints, and variations related to aspects of a context.

0 - 4 Application of mathematical ideas and content from the specified areas of study

Demonstration of understanding of key mathematical content from one or more areas of study in relation to a given context. Use of specific and general formulations of concepts and mathematical content drawn from the areas of study to derive results for analysis in this context. Appropriate use of examples to illustrate the application of a mathematical process, or use of a counter-example to disprove a proposition or conjecture. Use of a variety of approaches to develop  relations and functions as models for financial computations presented in tabular or graphical form and to set up and solve practical problems.

0 - 4 Analysis and interpretation of results

Analysis and interpretation of results obtained from examples or counter-examples to establish or refute general case propositions or conjectures related to a context for investigation. Generation of inferences from analysis to draw conclusions related to the context for investigation, and to verify or modify conjectures or explore variations.  Discussion of the validity and limitations of any models.

### Unit 3

#### Outcome 3

On completion of this unit the student should be able to select and appropriately use numerical, graphical, symbolic, and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.
Mark range Criterion
0 - 2 Appropriate selection and effective use of technology

Relevant and appropriate selection and use of technology, or a function of the selected technology for the mathematical context being considered, in particular first order linear recurrence relations and time value of money calculations. Distinction between exact and approximate results produced by technology, and interpretation of these results to a required accuracy. Use of appropriate range and domain and other specifications which illustrate key features of the mathematics under consideration.

0 - 3 Application of technology

Use and interpretation of the relation between numerical, graphical and symbolic forms of information produced by technology about relations, functions and equations and the corresponding features of those relations, functions or equations. Analysis of the relationship of the results from an application of technology to the nature of a particular mathematical question, problem or task. Use of tables of values, families of graphs, solutions of equations or collections of other results produced using technology to support analysis in problem solving, investigative or modelling contexts. Production of results efficiently and systematically which identify examples or counter-examples which are clearly relevant to the task.

A sample record sheet for a modelling or problem solving task for the core recursion and financial modelling (doc - 69.69kb) can be used to record student level of achievement with respect to the available marks for the performance criteria relating to each outcome, and to indicate pointers corresponding to relevant aspects of the task.

### Modelling or problem-solving task performance criteria

These criteria and mark ranges can be used for the modelling or problem solving tasks for the two selected modules in Unit 4.

### Unit 4

#### Outcome 1

On completion of this unit the student should be able to define and explain key concepts as specified in the content from the two selected modules, and apply related mathematical techniques and models in routine contexts.
Mark range Criterion
0 - 1
Appropriate use of mathematical conventions, symbols and terminology

Application of mathematical conventions in diagrams, tables and graphs, such as axes labels, angles, arcs and points. Use of symbolic notation appropriately and accurately in defining mathematical terms or expressions, and describing representations of objects such as matrices, regions or networks. Use of correct expressions in symbolic manipulation or computation in mathematical work.

0 - 1
0 - 2
Definition and explanation of key concepts

Definition of mathematical concepts using appropriate phrases and mathematical expressions. Provision of examples which illustrate key concepts and explain their role in the development of related mathematics. Statement of conditions or restrictions which apply to the definition of a concept. Identification of key concepts in relation to each area of study and explanation of the use of these concepts in applying mathematics in different contexts. Construction of mathematical expressions involving concepts such as matrices and recurrence relations, geometric representations of shapes and objects or directed graphs and networks.

0 - 2
0 - 2
Accurate use of mathematical skills and techniques

Use of algebraic expressions and numerical values to carry out computations, evaluate expressions and formulas, construct tables, produce graphs and solve equations. Use of mathematical routines and procedures such as solving triangles, ratio and proportion, sequences, interest calculations and constructions of networks to obtain results and solve problems.

0 - 2

### Unit 4

#### Outcome 2

On completion of this unit the student should be able to select and apply the mathematical concepts, models and techniques from the two selected modules in a range of contexts of increasing complexity.
Mark range Criterion
0 - 2
Identification of important information, variables and constraints

Identification of key characteristics of a problem, task or issue and statement of any assumptions underlying the use of relevant mathematics in the given context. Choice of suitable variables and constants for the development of mathematics related to various aspects of a given context. Specifications of constraints, such as domain and range constraints, related to aspects of a context.

0 - 2
0 - 4
Application of mathematical ideas and content from the specified areas of study

Demonstration of understanding of key mathematical content from one or more areas of study in relation to a given context. Use of specific and general formulations of concepts and mathematical content drawn from the areas of study to derive results for analysis in this context. Appropriate use of examples to illustrate the application of a mathematical process, or use of a counter-example to disprove a proposition or conjecture. Use of a variety of approaches to develop functions as models for data presented in tabular or graphical form. Use of algebra, coordinate geometry, matrices, and relations to set up and solve problems.

0 - 4
0 - 4
Analysis and interpretation of results

Analysis and interpretation of results obtained from examples or counter-examples to establish or refute general case propositions or conjectures related to a context for investigation. Generation of inferences from analysis to draw conclusions related to the context for investigation, and to verify or modify conjectures. Discussion of the validity and limitations of any models.

0 - 4

### Unit 4

#### Outcome 3

On completion of this unit the student should be able to select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.
Mark range Criterion
0 - 2
Appropriate selection and effective use of technology

Relevant and appropriate selection and use of technology, or a function of the selected technology for the mathematical context being considered. Distinction between exact and approximate results produced by technology, and interpretation of these results to a required accuracy. Use of appropriate range and domain and other specifications which illustrate key features of the mathematics under consideration.

0 - 2
0 - 3
Application of technology

Use and interpretation of the relation between numerical, graphical and symbolic forms of information produced by technology about relations, functions and equations and the corresponding features of those relations, functions or equations. Analysis of the relationship of the results from an application of technology to the nature of a particular mathematical question, problem or task. Use of tables of values, families of graphs, matrices, geometric shapes or objects or collections of other results produced using technology to support analysis in problem solving, investigative or modelling contexts. Production of results efficiently and systematically which identify examples or counter-examples which are clearly relevant to the task.