# Mathematics and Statistics - annotated exemplar Level 3 AS91587

## Apply systems of simultaneous equations in solving problems (3.15)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 329KB)

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, 97KB)

For Excellence, the student needs to apply systems of simultaneous 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, forming a generalisation and also using correct mathematical statements, or communicating mathematical insight.

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

The student has found the amount of each type of food to meet the daily requirements (1) and found a general solution which satisfies the situation with 6μg of Vitamin A in the Zany product (2).

They have also identified an appropriate range of values for the amount of Zany for the new situation and given two possible solutions (3). The student has also indicated they have considered amounts other than 5 micrograms and 6 micrograms for the amount of Vitamin A in Zany (4).

For a more secure Excellence, the student would need to accurately communicate their thinking about how 6 μg of vitamin A in the Zany product relates to the general solution and develop the discussion on the dependency of the equations.

### High Merit

Commentary
Student work extract

Student 2 (PDF, 90KB)

For Merit, the student needs to apply systems of simultaneous 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, 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 ‘Roger’s rabbits’.

The student has shown evidence of relational thinking by finding the amount of each type of food required to meet the daily requirements (1).

They have also demonstrated algebraically that the system of equations is consistent and indicated that increasing the amount of vitamin A in Zany food does not provide a unique solution (2).

The student has identified a possible solution for the amount of each type of food if Zany uses 6 μg of vitamin A and made an appropriate recommendation to Roger (3).

To reach Excellence, the student would need to generalise the amount of each type of food required if Zany contains 6μg of vitamin A.

### Low Merit

Commentary
Student work extract

Student 3 (PDF, 38KB)

For Merit, the student needs to apply systems of simultaneous 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, 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 ‘Roger’s rabbits’.

The student has shown evidence of relational thinking by finding the amount of each type of food required to meet the daily requirements (1) and by identifying that the change to 6μg of vitamin in Zany food produces multiple solutions (2).

For a more secure Merit, the student could interpret the multiple solutions in the context of the problem and provide a possible solution which meets the new situation.

### High Achieved

Commentary
Student work extract

Student 4 (PDF, 35KB)

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

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

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

The student has formed a system of simultaneous equations (1), used them to find a solution, and made an appropriate recommendation regarding the amount of each type of food required (2).

To reach Merit, the student would need to consider how the amount of each type of food would change if the number of micrograms of vitamin A in the Zany food changes to 6.

### Low Achieved

Commentary
Student work extract

Student 5 (PDF, 30KB)

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

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

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

The student has formed the equations for each vitamin (1) and solved the system of equations (2).

For a more secure Achieved, the student would need to indicate what is represented by each variable and interpret the solution in context.

### High Not Achieved

Commentary
Student work extract

Student 6 (PDF, 60KB)

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

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

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

The student has formed the equation for each of the vitamins (1).

To reach Achieved, the student would need to correctly solve the system of equations.