# Mathematics and Statistics - annotated exemplars level 2 AS91259

## Apply trigonometric relationships in solving problems (2.4)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 693KB)

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.

All annotations have been altered to correct the title of the TKI resource and to better illustrate the requirements of the standard.

### Low Excellence

Commentary
Student work extract

Student 1 (PDF, 94KB)

For Excellence, the student needs to apply trigonometric relationships, 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 ‘School Spare Land Subdivision’.

The student has devised a strategy to investigate the situation of subdividing the land for the sale. The student has shown that the total area can be subdivided into four sections of at least 400 m2 (1).

The student has also shown how four sections can be created, not all of which are triangles that satisfy the requirement of the sale (2). Correct mathematical statements have been used throughout the response.

For a more secure Excellence, the student could improve the communication, for example by clearly explaining how subsections 3 and 4 are created from ∆ABC, and also by finding and stating clearly the dimensions of the four subsections.

### High Merit

Commentary
Student work extract

Student 2 (PDF, 114KB)

For Merit, the student needs to apply trigonometric relationships, 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 ‘School Spare Land Subdivision’.

The student has selected and carried out a logical sequence of steps to calculate the areas of the two triangles on either side of the pipeline (1), and to show that each triangle can be subdivided into two sections with an area of more than 400 m2 (2). Appropriate mathematical statements have been used throughout the response.

To reach Excellence, the student would need to provide a subdivision into four sections, not all of which are triangles.

### Low Merit

Commentary
Student work extract

Student 3 (PDF, 150KB)

For Merit, the student needs to apply trigonometric relationships, 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 ‘School Spare Land Subdivision’.

The student has selected and carried out a logical sequence of steps to connect the areas of the triangles to four sections of at least 400 m2 (1). Appropriate mathematical statements have been used.

For a more secure Merit, the student could start to investigate possible dimensions for the four triangular subdivisions on the diagram to meet the requirements that each of them is more than 400 m2.

### High Achieved

Commentary
Student work extract

Student 4 (PDF, 121KB)

For Achieved, the student needs to apply trigonometric relationships in solving problems.

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

This student’s evidence is a response to the TKI task ‘School Spare Land Subdivision’.

The student has selected and used the cosine rule to find the length of the pipeline (1), the sine rule to find an angle in a triangle (2), and the formula for the area of a triangle to find the areas of Triangle A and Triangle B (3). The student has communicated their working using appropriate representations.

To reach Merit, the student could relate the two areas of the triangles to the requirement for sections of at least 400 m2. The additional line on the diagram (4) shows the start of subdividing the land.

### Low Achieved

Commentary
Student work extract

Student 5 (PDF, 85KB)

For Achieved, the student needs to apply trigonometric relationships in solving problems.

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

This student’s evidence is a response to the TKI task ‘School Spare Land Subdivision’.

The student has selected and used the cosine rule to find the length of the pipeline (1), and the formula for the area of a triangle to find the area of section 1 (2). The student has communicated using appropriate representations.

For a more secure Achieved, the student could make progress towards finding the area of the second triangle.

### High Not Achieved

Commentary
Student work extract

Student 6 (PDF, 117KB)

For Achieved, the student needs to apply trigonometric relationships in solving problems.

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

This student’s evidence is a response to the TKI task ‘School Spare Land Subdivision’.

The student has selected and used the cosine rule to find the length of the pipeline (1).

The student has incorrectly thought that halving the angle will result in the length of the opposite side being halved and the subsequent calculations are wrong (2). This student has attempted to use the sine rule but has misinterpreted the answer as a length (3).

To reach Achieved, the student would need to select and use one more method correctly whilst making progress towards solving the problem, for example by finding the area of triangle ADC.