# Mathematics and Statistics - annotated exemplars level 2 AS91269

## Apply systems of equations in solving problems (2.14)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 306KB)

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.

### Low Excellence

Commentary
Student work extract

Student 1 (PDF, 72KB)

For Excellence, the student needs to apply systems of equations, using extended abstract thinking, in solving problems.

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

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

This student has devised a strategy to investigate a problem by finding the coordinates of the white dot (1) and the black dot (2). The student has also investigated the grey line (3) and use the discriminant to investigate when it is a tangent (4).

For a more secure Excellence, the student could communicate more clearly the reasons for the discriminant being zero.

### High Merit

Commentary
Student work extract

Student 2 (PDF, 70KB)

For Merit, the student needs to apply systems of equations, 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 and terms, forming and using a model, and also relating findings to a context or communicating thinking using appropriate mathematical statements.

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

This student has connected different concepts or representations by solving simultaneous equations to find the co-ordinates of the white dot (1), the black dot (2) and the dark grey dot (3). Appropriate mathematical statements have been used throughout the task. This student has investigated how the constant of the equation can be changed by guess and check (4).

To reach Excellence the student could use simultaneous equations and the discriminant to investigate how the constant of the equation can be changed to solve the final part of the problem.

### Low Merit

Commentary
Student work extract

Student 3 (PDF, 32KB)

For Merit, the student needs to apply systems of equations, 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 and terms, forming and using a model, and also relating findings to a context or communicating thinking using appropriate mathematical statements.

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

This student has connected different concepts and representations by using simultaneous equations to find the co-ordinates of the white dot (1) and the dark grey dot (2). Appropriate mathematical statements have been used throughout the task. This student has made an error in finding the coordinates of the black dot (3).

For a more secure Merit, the student could find the correct co-ordinates of the black dot.

### High Achieved

Commentary
Student work extract

Student 4 (PDF, 32KB)

For Achieved, the student needs to apply systems of equations in solving problems.

This involves selecting and using methods, demonstrating knowledge of concepts and terms, and communicating using appropriate representations.

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

This student has demonstrated knowledge of concepts and terms by linking the equations to the graphs (1), and interpreted the solution of a system of equations in context by finding the co-ordinates of the white dot (2). This student has made an error in finding the coordinates of the black dot.

To reach Merit, the student would need to find the correct coordinates of the black dot and/or the dark grey dot.

### Low Achieved

Commentary
Student work extract

Student 5 (PDF, 28KB)

For Achieved, the student needs to apply systems of equations in solving problems.

This involves selecting and using methods, demonstrating knowledge of concepts and terms, and communicating using appropriate representations.

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

This student has demonstrated knowledge of concepts and terms by linking the equations to the graphs (1), and interpreted the solution of a system of equations in context by finding the coordinates of the white dot (2).

For a more secure Achieved, the student could make some progress towards finding the coordinates of the black and/or grey dots.

### High Not Achieved

Commentary
Student work extract

Student 6 (PDF, 28KB)

For Achieved, the student needs to apply systems of equations in solving problems.

This involves selecting and using methods, demonstrating knowledge of concepts and terms, and communicating using appropriate representations.

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

This student has connected different representations by linking the equations to the graphs (1).

To reach Achieved the student could solve the quadratic equation and interpret the solutions in context.