# Mathematics and Statistics - annotated exemplar Level 3 AS91574

## Apply linear programming methods in solving problems (3.2)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 650KB)

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 June 2019.

There is new student evidence for the Low Merit exemplar for this standard.

Annotations at Low Excellence, Low Merit and High Achieved and Low Achieved have been altered to better illustrate the requirements of the standard.

### Low Excellence

Commentary
Student work extract

Student 1 (PDF, 128KB)

For Excellence, the student needs to apply linear programming methods, 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 evidence is a student’s response to the TKI task ‘Ted’s tomatoes’.

The student has determined the number of hectares of artichokes and tomatoes to maximise the current income (1), and identified relevant concepts in context by forming an equation for a possible future income (2).

The student has identified the multiple solutions, but made a transfer error when communicating the maximum income (3). They have given an explanation for the multiple solutions (4) and used correct mathematical statements in the response.

For a more secure Excellence, the student could generalise the response to the future situation further, for example by investigating a general model for the future income I = 2kx+ky.

### High Merit

Commentary
Student work extract

Student 2 (PDF, 143KB)

For Merit, the student needs to apply linear programming 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 evidence is a student’s response to the TKI task ‘Ted’s tomatoes’.

The student has connected different concepts or representations by identifying the feasible region for the inequalities (1) and identifying the number of hectares required for each vegetable to maximise the income (2).

A possible income function for future years has been provided (3). The student has identified two solutions for the new income and has selected one as the optimal value for future years (4). They have also related the findings to the context.

To reach Excellence, the student could recognise that there are multiple solutions for the situation in future years which are along the line AB, and relate this to the context.

### Low Merit

Commentary
Student work extract

Student 3 (PDF, 126KB)

For Merit, the student needs to apply linear programming 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.

The student has connected different concepts or representations by locating the feasible region for the inequalities (1) and identifying the number of rods and pillars to maximise the profit (2). They have also related the findings to the context.

Modified inequalities for the increased hours for drilling and grinding have been given (3).

For a more secure Merit, the student would need to investigate the effect of the increased hours on the feasible region and maximum profit.

### High Achieved

Commentary
Student work extract

Student 4 (PDF, 110KB)

For Achieved, the student needs to apply linear programming methods 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 ‘Ted’s tomatoes’.

The student has formed the linear inequalities for the problem (1) and found the feasible region (2). They have also made some progress towards optimising the solution by determining the income for each vertex of the feasible region (3).

To reach Merit, the student could identify the vertex which maximises the income function, in order to make a recommendation regarding the number of hectares for each vegetable.

### Low Achieved

Commentary
Student work extract

Student 5 (PDF, 122KB)

For Achieved, the student needs to apply linear programming methods 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 ‘Ted’s tomatoes’.

The student has formed the linear inequalities for the problem (1) and found the feasible region (2).

For a more secure Achieved, the student could indicate what each variable represents and make some progress towards finding the optimal solution.

### High Not Achieved

Commentary
Student work extract

Student 6 (PDF, 98KB)

For Achieved, the student needs to apply linear programming methods 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 ‘Ted’s tomatoes’.

The student has formed some of the linear inequalities (1) and used these to find a feasible region.

To reach Achieved, the student could form the equation of the inequality for the hours of labour, and use this to find the correct feasible region for the problem.