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Mathematics and Statistics  annotated exemplar Level 3 AS91573
Apply the geometry of conic sections in solving problems (3.1)
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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 the geometry of conic sections, 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, or forming a generalisation, and also using correct mathematical statements or communicating mathematical insight. This evidence is from a student’s response to the TKI task ‘Orbits’. This student has devised a strategy to determine the distance of the asteroid from the sun when it crosses the orbits of Earth (1), Mars (2) and the comet (3). Correct mathematical statements have been used in the response. For a more secure Excellence, the student could have presented the findings in a written report and interpreted the distances more clearly in context, for example, by relating the value y = 149.5 x 10^{6}km to the distance of the asteroid from the sun when the asteroid crosses the orbit of Earth. 
High Merit
Commentary  

For Merit, the student needs to apply the geometry of conic sections, 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 from a student’s response to the TKI task ‘Orbits’. This student has formed a model for the orbit of the comet (1) and for the orbit of Earth (2). The models for the orbits of the comet and Earth have been used to find the distances from the sun when the asteroid crosses their orbits (3). Appropriate mathematical statements have been used in the response. To reach Excellence, the student could form and use a model for the orbit of Mars. 
Low Merit
Commentary  

For Merit, the student needs to apply the geometry of conic sections, 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 from a student’s response to the TKI task ‘Orbits’. This student has formed a model for the orbit of the comet (1) and the orbit of Earth (2). The model for the comet has been used to find the distance the asteroid is from the sun when it crosses this orbit (3). For a more secure Merit, the student could use the model for the orbit of Earth to find the correct distance of the asteroid from the sun when it crosses the orbit. 
High Achieved
Commentary  

For Achieved, the student needs to apply the geometry of conic sections in solving problems. This involves selecting and using methods, demonstrating knowledge of concepts and terms and communicating using appropriate representations. This evidence is from a student’s response to the TKI task ‘Orbits’. This student has selected and used properties of an ellipse and a parabola (1) to form the equation of Earth’s orbit and the path of the comet (2). This student has communicated using appropriate representations. To reach Merit, the student could consider the path of the asteroid, and use the equations that have been formed to find the distances from the sun to where the asteroid crosses the orbit of Earth and/or the path of the comet. 
Low Achieved
Commentary  

For Achieved, the student needs to apply the geometry of conic sections in solving problems. This involves selecting and using methods, demonstrating knowledge of concepts and terms and communicating using appropriate representations. This evidence is from a student’s response to the TKI task ‘Orbits’. This student has selected and used the properties of a parabola to find the correct equation of the path of the comet (1). This student has selected and used the properties of an ellipse (2) in starting to find the equations of the ellipse for Earth and Mars, but the subsequent use of properties and equations are incorrect. This student has communicated using appropriate representations. For a more secure Achieved, the student could use the properties of an ellipse to find a correct equation for the orbit of Mars and/or Earth. 
High Not Achieved
Commentary  

For Achieved, the student needs to apply the geometry of conic sections in solving problems. This involves selecting and using methods, demonstrating knowledge of concepts and terms and communicating using appropriate representations. This evidence is from a student’s response to the TKI task ‘Orbits’. This student has selected and used the properties of a parabola (1) to find the equation of the path of the comet. To reach Achieved, the student could select and use properties of an ellipse to find the correct equation of one ellipse. 