# Mathematics and Statistics - annotated exemplars level 2 AS91257

## Apply graphical methods in solving problems (2.2)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 1.2MB)

This annotated exemplar is intended for teacher use only. The student work shown does not always represent a complete sample of what is required. Selected extracts are used, focused on the grade boundaries, in order to assist assessors to make judgements at the national standard.

Last updated July 2019.

There is new student evidence for the High Merit and Low Achieved exemplars for this standard.

Annotations at Low Excellence, High Merit, High Achieved, Low Achieved and High Not Achieved have been altered to better illustrate the requirements of the standard.

### Low Excellence

Commentary
Student work extract

Student 1 (PDF, 568KB)

For Excellence, the student needs to apply graphical methods, using extended abstract thinking, in solving problems.

This involves one or more of: devising a strategy to investigate a situation, identifying relevant concepts in context, developing a chain of logical reasoning, or proof, forming a generalisation, and also using correct mathematical statements, or communicating mathematical insight.

This student’s evidence is a response to the TKI task ‘Bridges’.

The student has identified relevant concepts in context and formed a generalisation for the parabolic model (1). Correct mathematical statements have been used in the response.

For a more secure Excellence, the student could complete the generalisation for the quadratic model and discuss its features/properties. The student could also consider the generalisation for the other models.

### High Merit

Commentary
Student work extract

Student 2 (PDF, 163KB)

For Merit, the student needs to apply graphical methods, using relational thinking, in solving problems.

This involves one or more of: selecting and carrying out a logical sequence of steps, connecting different concepts or representations, demonstrating understanding of concepts, forming and using a model, and also relating findings to a context, or communicating thinking using appropriate mathematical statements.

This student’s evidence is a response to the TKI task ‘Motorcycle School’.

The student has connected different concepts and representations to form the models for four sections of the course (1) and provide a possible path for the instructor (2).

To reach Excellence, the student could provide a detailed domain for the path of the instructor by considering appropriate points of intersection.

The student could also discuss the margin of safety on both sides of the path, the appropriateness of the model for the course, and provide further communication of the strategy used to determine the models.

### Low Merit

Commentary
Student work extract

Student 3 (PDF, 104KB)

For Merit, the student needs to apply graphical methods, using relational thinking, in solving problems.

This involves one or more of: selecting and carrying out a logical sequence of steps, connecting different concepts or representations, demonstrating understanding of concepts, forming and using a model, and also relating findings to a context, or communicating thinking using appropriate mathematical statements.

This student’s evidence is a response to the TKI task ‘Bridges’.

The student has formed and used models in determining appropriate functions for the first half of the bridge (1) and the whole bridge (2). The findings have been related to the context.

For a more secure Merit, the student could strengthen the communication by explaining their thinking more clearly using appropriate mathematical statements. The cubic model is not symmetric (3), and the student could also consider a piecewise model consisting of two cubic functions for the whole bridge.

### High Achieved

Commentary
Student work extract

Student 4 (PDF, 129KB)

For Achieved, the student needs to apply graphical methods in solving problems.

This involves selecting and using methods, demonstrating knowledge of the properties of functions and graphs and communicating using appropriate representations.

This student’s evidence is a response to the TKI task ‘Bridges’.

The student has selected and used the graph of a logarithmic function, its features and equation (1), the properties of the logarithmic function by considering the asymptote and domain (2) and the graph, features, equation and properties of the quadratic model (3).

The student has also demonstrated knowledge of the properties of functions and graphs and communicated using appropriate representations.

To reach Merit, the student would need to develop equations to model the complete bridge. The student could also develop the discussion on the steepness of the logarithmic model at the beginning by referring to specific points.

### Low Achieved

Commentary
Student work extract

Student 5 (PDF, 118KB)

For Achieved, the student needs to apply graphical methods in solving problems.

This involves selecting and using methods, demonstrating knowledge of the properties of functions and graphs and communicating using appropriate representations.

This student’s evidence is a response to the TKI task ‘Motorcycle School’.

The student has selected and used graphs for the first three sections of the course (1), features of the trigonometric function in order to determine two of the constants in the model (2), and a feature of the parabola (3).

The student has also demonstrated knowledge of properties of functions and graphs, and communicated using appropriate representations.

For a more secure Achieved, the student could determine the correct equations for Sections 2 and 3 and discuss the features and properties of these graphs.

### High Not Achieved

Commentary
Student work extract

Student 6 (PDF, 119KB)

For Achieved, the student needs to apply graphical methods in solving problems.

This involves selecting and using methods, demonstrating knowledge of the properties of functions and graphs and communicating using appropriate representations.

This student’s evidence is a response to the TKI task ‘Bridges’.

The student has selected and used the properties of the quadratic function by considering the domain, vertex and symmetry of the quadratic (1).

To reach Achieved, the student would need to determine the equation of the quadratic, and consider using other graphs to model the half bridge.