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Mathematics and Statistics  annotated exemplar Level 3 AS91575
Apply trigonometric methods in solving problems (3.3)
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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  

For Excellence, the student needs to apply trigonometric 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 logical chain of 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 ‘Maths End Ferris Wheels’. This student has devised a strategy to investigate a problem by finding a model for the Kiddywheel (1) and a model for the Flyinghigh wheel (2), although the latter model is incorrect and it does not simplify the problem. The student has used these to solve the problem (3). Correct mathematical statements have been used throughout the response. For a more secure Excellence, the student could have found the correct equation for the Flyinghigh wheel. The student could also consider and explain that one of the intervals is not a sensible solution, and have chosen to discard it. 
High Merit
Commentary  

For Merit, the student needs to apply trigonometric 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, or 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 ‘Exact Values’. This student has demonstrated an understanding of concepts by finding the value of the reciprocal trigonometric functions for angles found using the compound angle formulae and double angle formulae (1). The values for tan120° and cot120° are incorrect (2). In finding general solutions, the student has found all the angles that have a sine and cosine of and a tangent of 1 but has not clearly communicated what the angles represent (3). The statements for the reciprocal ratios are incorrect (4). To reach Excellence, the student could correct the errors and the generalisations need to be extended to exact values other than those for the angles in the special triangles. 
Low Merit
Commentary  

For Merit, the student needs to apply trigonometric 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, or 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 ‘Exact Values’. This student has connected different concepts in finding the value of the reciprocal trigonometric functions for the angles found using the compound angle formulae (1). For a more secure Merit, the student could use the unit circle or graphs to investigate other angles. 
High Achieved
Commentary  

For Achieved, the student needs to apply trigonometric 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 ‘Maths End Ferris Wheels’. This student has selected and used properties of trigonometric functions in identifying the correct equation of the Kiddy wheel (1) and finding the correct equation of the Flyinghigh wheel (2). This student has solved trigonometric equations to find the correct intervals for both Ferris wheels (3). To reach Merit, the student could link the solutions back to the problem to find an interval when Jade can see Manu and show a contextual understanding of the problem. 
Low Achieved
Commentary  

For Achieved, the student needs to apply trigonometric 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 ‘Maths End Ferris Wheels’. This student has selected and used properties of trigonometric functions in finding the correct equation of the Kiddywheel (1) and solved a trigonometric equation to find an interval when Jade is above 5 m (2). For a more secure Achieved, the student could complete the equation for the Flyinghigh wheel and find an interval for Manu. 
High Not Achieved
Commentary  

For Achieved, the student needs to apply trigonometric 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 ‘Exact Values’. This student has selected and used reciprocal trigonometric functions to find exact values for some of the reciprocal functions for the angles in the special triangles (1). The use of the double angle formula to find sin90° is correct but this is a known angle (2). To reach Achieved, the student could determine the exact value for sin135°. 