# Mathematics and Statistics - annotated exemplar Level 3 AS91574 (A)

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

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 See EXEMPLAR B for this standard TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 473KB)

The task asks students to use the constraints provided to recommend the number of hectares of artichokes and tomatoes for Ted to maximize his income in the current year and in future years.

### Low Excellence

Commentary
Student response

Student 1 page 1 (PDF, 142KB)

Student 1 page 2 (PDF, 135KB)

For Excellence the student is required to apply linear programming methods, using extended abstract thinking, in solving problems. This involves devising a strategy to investigate or solve a problem, developing a chain of logical reasoning and using correct mathematical statements or communicating mathematical insight.

There is evidence of extended and abstract thinking by providing the feasible region (1) to satisfy all the constraints and the student has made a recommendation regarding the number of hectares of artichokes and tomatoes required to maximise the income (2).

The student has identified that there are multiple solutions for this situation in future years (3).

For a more secure Excellence the student would need to accurately communicate that the multiple solutions all lie on the line formed by one of the constraints.

### High Merit

Commentary
Student response

Student 2 (PDF, 301KB)

For Merit the student is required to apply linear programming methods, using relational thinking, in solving problems. This involves selecting and carrying out a logical sequence of steps, connecting different concepts and representations, demonstrating understanding of concepts and also relating findings to a context, or communicating thinking using appropriate mathematical statements.

The student has shown evidence of relational thinking by identifying the feasible region (1) for the system of linear inequalities and identifying the number of hectares required for each vegetable to maximise the income (2).

The student has identified that there are two solutions and has selected one as the optimal value for future years, but has not provided the evidence to support this claim (3).

To be awarded Excellence the student would need to identify that there are multiple solutions for the situation in future years based on supporting evidence.

### Low Merit

Commentary
Student response

Student 3 (PDF, 161KB)

For Merit the student is required to apply linear programming methods, using relational thinking, in solving problems. This involves selecting and carrying out a logical sequence of steps, connecting different concepts and representations, demonstrating understanding of concepts and also relating findings to a context, or communicating thinking using appropriate mathematical statements.

The student has shown evidence of relational thinking by providing the feasible region for the system of linear inequalities (1) and identifying the number of hectares for each vegetable to maximise the income for the current year (2).

For a more secure Merit the student would need to consider how the payments in the future would affect the optimal area for each vegetable.

### High Achieved

Commentary
Student response

Student 4 (PDF, 131KB)

For Achieved the student is required 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.

The student has shown evidence of applying linear programming methods by forming the constraints (1) and using them to identify the feasible region (2). The student has also evaluated the income for each vertex of the feasible region (3).

To be awarded Merit the student would need to identify the vertex which maximises the income function to make a recommendation regarding the number of hectares for each vegetable.

### Low Achieved

Commentary
Student response

Student 5 (PDF, 161KB)

For Achieved the student is required 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.

The student has shown evidence of linear programming methods by providing the equation of the inequality for the hours of labour (1) and the feasible region identified by the four constraints (2).

For a more secure Achieved the student would need to indicate what is represented by each variable.

### High Not Achieved

Commentary
Student response

Student 6 (PDF, 101KB)

For Achieved the student is required 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.

The student has provided a feasible region showing at least two of the constraints (1).

To be awarded Achieved the student would need to form the equation of the inequality for the hours of labour.