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
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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
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