# Mathematics - annotated exemplars level 1 AS91029

## Apply linear algebra in solving problems (1.4)

 Read the STANDARD TKI Mathematics and Statistics Assessment Resources Download all these exemplars and commentary (PDF, 887KB)

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 May 2019.

Annotations to Low Excellence and Low Achieved have been altered to better reflect the requirements of the standard.

### Low Excellence

Commentary
Student response

Student 1 (PDF, 243KB)

For Excellence, the student needs to apply linear algebra, 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 assessment resource ‘Taxi Charges’.

This student has identified relevant concepts in context to represent the three taxi companies’ charges graphically (1), recommended which taxi company to use for two destinations (2), and recommended distances for which P&G Taxis is the cheapest (3).

The student has also described and justified two options for Fred’s Taxi Company to be the cheapest company (4) (5). Correct mathematical statements have been used throughout the response.

For a more secure Excellence, the student would need to show greater depth in their thinking, for example, by discussing the reasons for selecting their options for Fred’s Taxi Company to be the cheapest company.

The student could also consider a range of distances for which the first of the suggestions for Fred’s Taxis will be cheaper than the second.

### High Merit

Commentary
Student response

Student 2 (PDF, 89KB)

For Merit, the student needs to apply linear algebra, using relational thinking, in solving problems.

This involves one or more of: selecting and carrying out a logical sequence of steps, connecting different concepts and 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 assessment resource ‘Taxi Charges’.

This student has connected different concepts and representations to represent the three taxi companies’ charges algebraically (1), recommended which taxi company to use for two destinations (2), and recommended distances for which P&G Taxis is the cheapest (3).

The student has also recommended new charge rates for Fred’s Taxis (4). Appropriate mathematical statements have been used throughout the response.

To reach Excellence, the student needs to alter option 1 so that it is always cheaper and explain their thinking about the new charges more clearly, for example with graphs.

### Low Merit

Commentary
Student response

Student 3 (PDF, 176KB)

For Merit, the student needs to apply linear algebra, using relational thinking, in solving problems.

This involves one or more of: selecting and carrying out a logical sequence of steps, connecting different concepts and 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 assessment resource ‘Taxi Charges’.

This student has connected different concepts and representations to represent the three taxi companies’ charges graphically (1), recommended which taxi company to use for one destination (2), and recommended, with justification, distances for which P&G Taxis is the cheapest (3).

The student has also described one option for Fred’s Taxi Company to be the cheapest company (4). Appropriate mathematical statements have been used.

For a more secure Merit, the student would need to recommend which company is cheaper for a second destination and investigate further possible new charges for Fred’s Taxi Company.

### High Achieved

Commentary
Student response

Student 4 (PDF, 122KB)

For Achieved, the student needs to apply linear algebra in solving problems.

This involves selecting and using a range of methods, demonstrating knowledge of algebraic concepts and terms, and communicating solutions which usually require only one or two steps.

This student’s evidence is a response to the TKI assessment resource ‘Taxi Charges’.

This student has represented the three taxi companies’ charges graphically (1), formed a linear model (2), recommended which taxi company to use for two destinations (3) and used simultaneous equations (implied by reading off the graph) to recommend distances for which P&G Taxis is the cheapest (4).

To reach Merit, the student would need to clearly communicate their thinking with appropriate mathematical statements. For example, linking their decisions to the evidence visible in the graphs.

### Low Achieved

Commentary
Student response

Student 5 (PDF, 143KB)

For Achieved, the student needs to apply linear algebra in solving problems.

This involves selecting and using a range of methods, demonstrating knowledge of algebraic concepts and terms, and communicating solutions which usually require only one or two steps.

This student’s evidence is a response to the TKI assessment resource ‘Taxi Charges’.

This student has represented the three taxi companies’ charges graphically (1), formed a linear model (2) and used formulae to recommend which taxi company to use for two destinations (3).

For a more secure Achieved, the student would need to improve the quality of the communication.

### High Not Achieved

Commentary
Student response

Student 6 (PDF, 146KB)

For Achieved, the student needs to apply linear algebra in solving problems.

This involves selecting and using a range of methods, demonstrating knowledge of algebraic concepts and terms, and communicating solutions which usually require only one or two steps.

This student’s evidence is a response to the TKI assessment resource ‘Taxi Charges’.

This student has formed a linear model (1) and correctly used formulae to recommend which taxi company to use for two destinations (2).

To reach Achieved, the student would need to provide evidence of selecting and using another method in solving the problem.