1. 1 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Work with a range of patterns and functions to solve problems.

2. 2 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 US:9007Work with a range of patterns and functions to solve problems TF209.57Defining functions Notional Hours TF209.58Evaluating functions TF209.59Representing functions TF209.60Working with variations in functions TF209.61Working with the transformation of functions Total Notional Hours P&P2FM24 Understanding and using functions in our daily lives P&P2FM25 Using functions to solve more complex problems Modules TF209.62Working with the extreme values of functions 5 8 10 9 10 50 Topics

3. 3 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Critical Cross-field Outcomes Identify and solve problems Work effectively with others Communicate effectively Organise and manage myself and my activities Collect, analyse, organise and critically evaluate information Use science and technology effectively and critically Demonstrate an understanding of the world as a set of related systems Use mathematics Learner and societal development US:9007Work with a range of patterns and functions to solve problems Be able to generate examples of functions. Understand the concept of mutual dependency. TF209.57 Defining functions Understand why functions are important in science. Be able to discover the rules that relates to specific sets of numbers. Be able to clearly define functions and demonstrate what role they play in daily work and personal life.

4. 4 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Topic Specification Learning Outcomes On completion of this topic the learner will be able to: Demonstrate understanding of the concept of mutual dependency; Generate simple examples of functions; Demonstrate understanding of the practical use of functions in science; Discover the “rules” that relates to specific sets of numbers; Clearly define what a function is and how they can be used in daily work and personal life. Assessment Criteria : (The assessor will observe and confirm) The learners understanding of the concept of functions and their ability to describe how this concept can be used in daily work and personal life. Notional Hours 3 Assignment Description WHATWHAT WHYWHY HOW A group exercise of attending a lecture on the subject of functions and then jointly working through examples and case studies. To ensure that the learners develop a very practical understanding of this key concept. Learners attend the joint work session; Learners do the practice exercises and then go out and generate practical examples of functions and their application at home and in the workplace; Learners present their work during a structured work session for review and feedback. TF209.57 Defining functions

5. 5 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Critical Cross-field Outcomes Identify and solve problems Work effectively with others Communicate effectively Organise and manage myself and my activities Collect, analyse, organise and critically evaluate information Use science and technology effectively and critically Demonstrate an understanding of the world as a set of related systems Use mathematics Learner and societal development US:9007Work with a range of patterns and functions to solve problems Be able to explain a function in terms of inputs and outputs. Understand what an independent and dependent variable is. TF209.58 Evaluating functions Be able to use arrow diagrams to picture functions. Be able to explain what is meant by the domain of a function. Be able to explain what is meant by the range of a function. Be able to do basic evaluations of algebraic functions.

6. 6 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Topic Specification Learning Outcomes On completion of this topic the learner will be able to: Demonstrate understanding of what independent and dependent variables are; Explain a function in terms of inputs and outputs; Use arrow diagrams to picture functions; Explain what is meant by the domain of a function. Explain what is meant by the range of a function. Do basic evaluations of algebraic functions. Assessment Criteria : (The assessor will observe and confirm) The learners ability to correctly describe and evaluate functions and relate this to daily work and private life. Notional Hours 4 TF209.58 Evaluating functions Assignment Description WHATWHAT WHYWHY HOW A group exercise of attending a lecture on the subject of functions and then jointly working through examples and case studies. To ensure that the learners develop a very practical understanding of this key concept. Learners attend the joint work session; Learners do the practice exercises and then go out and generate practical examples of functions and their application at home and in the workplace; Learners present their work during a structured work session for review and feedback.

7. 7 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Critical Cross-field Outcomes Identify and solve problems Work effectively with others Communicate effectively Organise and manage myself and my activities Collect, analyse, organise and critically evaluate information Use science and technology effectively and critically Demonstrate an understanding of the world as a set of related systems Use mathematics Learner and societal development US:9007Work with a range of patterns and functions to solve problems Understand what is meant by representing a function. TF209.59 Representing functions Be able to explain the four ways of expressing functions. Be able to define what the graph of a function is. Be able to describe the difference between a linear and constant function. Be able to find the domain and range of a function from a graph. Be able to determine if a graph is a graph of a function using the vertical line test.

8. 8 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Topic Specification Learning Outcomes On completion of this topic the learner will be able to: Demonstrate understanding of what is meant by representing a function; Explain the four ways of expressing functions; Define what the graph of a function is; Describe the difference between a linear and constant function; Find the domain and range of a function from a graph; and Determine if a graph is a graph of a function using the vertical line test. Assessment Criteria : (The assessor will observe and confirm) The learners ability to properly represent functions graphically, verbally and numerically using practical data from the workplace and personal life. Notional Hours 5 TF209.59 Representing functions Assignment Description WHATWHAT WHYWHY HOW An individual exercise of self study and working through examples and doing exercises that are then presented in group context for correction, review and feedback. To ensure that the learners can independently apply the representation of functions in a practical way that makes sense to them in their private and work environment. Learners do self study and work through the examples; Learners develop representations of functions using data from their personal and work life and prepare these examples for presentation; Learners present the results of their work in a group discussion and receive feedback.

9. 9 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Critical Cross-field Outcomes Identify and solve problems Work effectively with others Communicate effectively Organise and manage myself and my activities Collect, analyse, organise and critically evaluate information Use science and technology effectively and critically Demonstrate an understanding of the world as a set of related systems Use mathematics Learner and societal development US:9007Work with a range of patterns and functions to solve problems Understand the concept of increasing and decreasing functions. TF209.60 Working with variations in functions Be able to explain what a step function is and give practical examples. Be able to sketch and explain what is meant by a stretched step function. Understand the concept of direct and inverse variation. Be able to apply the given formulas of direct and inverse variations to practical problem solving.

10. 10 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Topic Specification Learning Outcomes On completion of this topic the learner will be able to: Demonstrate understanding of the concept of increasing and decreasing functions; Explain what a step function is and give practical examples; Sketch and explain what is meant by a stretched step function; Demonstrate understanding of the concept of direct and inverse variation; Apply the given formulas of direct and inverse variations to practical problem solving. Assessment Criteria : (The assessor will observe and confirm) The learners ability to use the concepts of functions to solve a variety of practical day to day problems. Notional Hours 6 TF209.60 Working with variations in functions. Assignment Description WHATWHAT WHYWHY HOW An individual exercise of self study and working through examples and doing exercises that are then presented in group context for correction, review and feedback. To ensure that the learners can independently apply the concept of the variation of functions in a practical way that makes sense to them in their private and work environment. Learners do self study and work through the examples; Learners solve relevant problems using data from their personal and work life and prepare these examples for presentation; Learners present the results of their work in a group discussion and receive feedback.

11. 11 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Critical Cross-field Outcomes Identify and solve problems Work effectively with others Communicate effectively Organise and manage myself and my activities Collect, analyse, organise and critically evaluate information Use science and technology effectively and critically Demonstrate an understanding of the world as a set of related systems Use mathematics Learner and societal development US:9007Work with a range of patterns and functions to solve problems Understand what is meant by the shift in functions. Be able to apply the transformation of functions for the solving of practical day to day problems. Understand the implications of the vertical shift in function graphs. Understand the implications of the horizontal shift in function graphs. Understand the concept of shrinking and stretching of graphs. Understand the concept of even and odd functions. TF209.61 Working with the transformation of functions.

12. 12 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Topic Specification Learning Outcomes On completion of this topic the learner will be able to: Demonstrate understanding of the concept of increasing and decreasing functions; Explain what a step function is and give practical examples; Sketch and explain what is meant by a stretched step function; Demonstrate understanding of the concept of direct and inverse variation; Apply the given formulas of direct and inverse variations to practical problem solving. Assessment Criteria : (The assessor will observe and confirm) The learners ability to use the concepts of functions to solve a variety of practical day to day problems. Notional Hours 8 TF209.61 Working with the transformation of functions. Assignment Description WHATWHAT WHYWHY HOW An individual exercise of self study and working through examples and doing exercises that are then presented in group context for correction, review and feedback. To ensure that the learners can independently apply the concepts of increasing and decreasing values of functions in a practical way that makes sense to them in their private and work environment. Learners do self study and work through the examples; Learners solve practical problems using data from their personal and work life and prepare these examples for presentation; Learners present the results of their work in a group discussion and receive feedback.

13. 13 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Critical Cross-field Outcomes Identify and solve problems Work effectively with others Communicate effectively Organise and manage myself and my activities Collect, analyse, organise and critically evaluate information Use science and technology effectively and critically Demonstrate an understanding of the world as a set of related systems Use mathematics Learner and societal development US:9007Work with a range of patterns and functions to solve problems Understand the concept of extreme values of functions. TF209.62 Working with the extreme values of functions Be able to express a quadratic function in a standard form. Understand what a quadratic function is. Understand the concept of extreme values of quadratic functions. Understand the concept of maximum and minimum values of quadratic functions. Be able to use graphic devices to find extreme values. Be able to apply the concept of minimum and maximum values of functions to practical problem solving.

14. 14 Commissioned by PAMSA and German Technical Co-Operation National Certificate in Paper & Pulp Manufacturing NQF Level 2 Topic Specification Learning Outcomes On completion of this topic the learner will be able to: Demonstrate understanding the concept of extreme values of functions Demonstrate understanding of what a quadratic function is Demonstrate understanding of the concept of extreme values of quadratic functions. Express a quadratic function in a standard form. Demonstrate understanding of the concept of maximum and minimum values of quadratic functions Use graphic devices to find extreme values; Apply the concept of minimum and maximum values of functions to practical problem solving Assessment Criteria : (The assessor will observe and confirm) The learners ability to use the concept of extreme values of functions to solve relevant practical problems. Notional Hours 10 Assignment Description WHATWHAT WHYWHY HOW An individual exercise of self study and working through examples and doing exercises that are then presented in group context for correction, review and feedback. To ensure that the learners can independently apply the concept of extreme values of functions in a practical way that makes sense to them in their private and work environment. Learners do self study and work through the examples; Learners solve practical problems using data from their personal and work life and prepare these examples for presentation; Learners present the results of their work in a group discussion and receive feedback. TF209.62 Working with the extreme values of functions