Achievement Standard Mathematics and Statistics 91029: Apply linear algebra in solving problems Resource Reference: Mathematics and Statistics 1.4A Resource Title: Taxi Charges Credits: 3 Achievement Achievement with Merit Apply linear algebra in solving problems. Apply linear algebra, using relational thinking, in solving problems. Achievement with Excellence Apply linear algebra, using extended abstract thinking, in solving problems. Student instructions Introduction At Tauranga airport, there are three different taxi companies with taxis available for hire. This assessment activity requires you to interpret and compare the charges the taxi companies use for different trips. You will be assessed on your understanding and application of linear graphs and linear equations and on your ability to communicate your solutions clearly and accurately. Task At Tauranga airport, there are three different taxi companies with taxis available for hire: Fred’s Taxi Company P and G Taxi Company Flatrate Taxi Company The attached Resource Page gives the hire charge for each company and the distances to common destinations in Tauranga from the airport. 1. Represent the three taxi companies' charges using the same representation, for example, three equations or three graphs with the same variables and scale. 2. Recommend which taxi company to use for a trip to two of the common destinations. 3. Recommend distances for which it would be cheapest to use P and G Taxi Company. 4. Fred, who owns Fred’s Taxi Company, wants to be the cheapest taxi company people can use to travel to any destination. Write and describe at least two different ways Fred could realistically change his charges to achieve this goal. Include specific examples of the rates he could use. Resource Page Taxi Company Charges: Fred’s Taxi Company P and G Taxi Company Flatrate Taxi Company Come with us! Cheap rates! Our advantage is clear! Fixed charge: $5.00 C = 0.65D + 2 Per kilometre: $0.50 Where: $25.00 flat fee to any destination up to 60 km. C is the cost in dollars, and Additional charges apply over 60 km. D is the distance in kilometres Distance to some common destinations from the airport: Destination Distance in kilometres City Centre 17 Port 24 Bethlehem 45 Internal assessment resource Mathematics and Statistics 1.4A for Achievement Standard 91029 PAGE FOR TEACHER USE Assessment schedule: Mathematics and Statistics 91029 Taxi Charges Evidence/Judgements for Achievement Evidence/Judgements for Achievement with Merit Evidence/Judgements for Achievement with Excellence Applying linear algebra will involve using a range of methods in solving problems, demonstrating knowledge of algebraic concepts and terms, and communicating solutions which would usually require only one or two steps. Student must select and use at least three different methods, for example, formulae, graphing, or simultaneous equations: Using formulae Fred’s cost to city = 0.5 x 17 + 5 = $13.50 Forming a linear model, Fred’s: y = 0.5x + 5 Graphing a linear model Fred’s graph Solving simultaneous equations, inequations or graphs PG and Fred cost the same for a trip of 20 km. Relational thinking will involve 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 relating findings to a context, or communicating thinking using appropriate mathematical statements. For example, a student might demonstrate understanding of concepts and communicate their chain of reasoning by solving Fred and PG equations simultaneously and using the solution to determine appropriate selections for different distance ranges Solving C = 0.5D + 5 and C = 0.65D + 2 gives a distance of 20 km where their charges are the same. PG has a cheaper fixed charge and larger slope so it is cheapest for distances less than 20 km. Fred is cheapest for more than 20 km as long as he’s cheaper than Flatrate at $25.00. For example, to go to the Port the costs are: Fred = 0.5 x 24 + 5 = $17 PG = 0.65 x 24 + 2 = $17.60 Flat = $25 Use Fred because it is the cheapest. Extended abstract thinking will involve one or more of: Demonstrating understanding of abstract concepts, Developing a chain of logical reasoning, or proof, Forming a generalisation, And using correct mathematical statements, or communicating mathematical insight. For example, a student might provide a chain of logical reasoning, with statements of supporting evidence: Over 40 km, Fred starts to cost more than Flatrate (0.5 x 41 + 5 = 25.50) For all distances over 40km he needs to drop to a flat rate (like Flatrate Taxi Company), as Flatrate has zero slope and doesn’t go up until 60 km. The rate needs to be cheaper than Flatrate’s $25 per journey. E.g. Rate = $24 for any length journey over 40km For any journey less than 20 km Fred needs to have a charge which is lower than PG. His charges are higher because even though his per km rate is low, his fixed charge is much higher than PG. If he undercuts PG’s fixed charge and matched their per km rate, he would always be cheaper. E.g. His charges could be C = 0.65D + 1.5 His fixed charge is $1.50 and his rate is the same as P&G. Final grades will be decided using professional judgement based on a holistic examination of the evidence provided against the criteria in the Achievement Standard. This resource is copyright © Crown 2010 Page 3 of 4 Internal assessment resource Mathematics and Statistics 1.4A for Achievement Standard 91029 PAGE FOR TEACHER USE Teacher Resource Page y 40 P and G y = 0.65x + 2 Fred y = 0.5x +5 Point of Intersection ( 35.4 , 25 ) 30 Flatrate y = 25 Point of Intersection ( 40 , 25 ) 20 Point of Intersection ( 20 , 15 ) 10 10 20 This resource is copyright © Crown 2010 30 40 50 60 Page 4 of 4 x