Mathematics and Statistics Achievement Standard
91029(v1) Apply linear algebra in solving problems
Travelling Drainlayer
Credits: 3
NZAMT
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Teacher guidelines
The following guidelines are designed to ensure that teachers can carry out a valid and consistent
assessment using this internal assessment resource.
Read also:

The Achievement Standard Mathematics and Statistics 91029 explanatory notes at
http://www.nzqa.govt.nz/ncea/assessment/search.do?query=mathematics&view=achievement
s&level=01
 The senior subject guides at http://seniorsecondary.tki.org.nz, in particular:
o http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievementobjectives/Achievement-objective-NA6-5
o http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievementobjectives/Achievement-objective-NA6-7
o http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievementobjectives/Achievement-objective-NA6-8
 The assessment exemplars and moderator comments at
http://www.nzqa.govt.nz/qualifications-standards/qualifications/ncea/ncea-subjectresources/ncea-study-resource-mathematics/exemplars/
These notes contain information, definitions, and requirements that are crucial when interpreting the
standard and assessing students against it.
AS 91026 NZAMT 2013
Context/Setting
The context for this resource is plumbing work and choosing a drainlayer from a comparison of three
different companies and associated costs. Students will be able to use information given about linear
relationships in an equation form, a graphical form and in a written form and demonstrate rate of
change and simultaneaous solution concepts to achieve this standard. There is ample opportunity for
a student to demonstrate connected thinking and justify decisions for Merit and likewise introduce
different and innovative ideas for Excellence. Students need to be prepared to think about the whole
problem.
This resource could be used as part of a learning portfolio that contains evidence of a student’s
learning process from a variety of activities.
Conditions
All activities must be performed under conditions which ensure students’ own work is
assessed.
Resource requirements
Access to appropriate technology is expected.
Additional information
None.
AS 91026 NZAMT 2013
Travelling Drainlayer
NZAMT
Mathematics and Statistics 91029 (AS 1.4) v1
Achievement

Apply linear algebra in solving
problems.
Credits:3
Achievement with Merit

Apply linear algebra, using
relational thinking, in solving
problems.
Achievement with
Excellence

Apply linear algebra, using
extended abstract thinking, in
solving problems.
Student instructions sheet
PipeLine, a major plumbing business, uses the services of a local drainlayers. The company does
work in town, close to town and on rural properties more than 30 km from the city. The manager is
considering three drainlayers named A-One, BestDrains and Creative Drains.
These three drainlayers have been selected because their work standards are high and their hourly
rates are similar. However, they charge PipeLine different amounts for travelling to jobs. Your task
is to investigate the information and recommend the best plumber to use or jobs with different travel
distances.
Use the information provided on the Resource Sheet about the three drain laying businesses to
complete the task.
YOUR TASK
Write a report to the manager of PipeLine with your recommendation.
See Resource Sheet A for information about the travel rates for the drain layers.
You will be assessed on your depth of understanding and application of linear graphs. It is important
you communicate your thinking and your solutions clearly and relate your findings to the context.
Your overall grade will be determined by the quality of your solution
AS 91026 NZAMT 2013
Resource Sheet A • TRAVELLING DRAINLAYER
Creative: The cost
equation is:
A-One: The cost is $50
plus 40 cents per km.
C= 2.4d + 10
where C is the total cost
of the call-out fee in
dollars, and d is the
number of kilometres to
the job.
The drainlayers and their costs
Cost in dollars (C)
BestDrains: The costs are shown in the graph below.
Distance, d, from the PipeLine shop (km)
AS 91029 NZAMT 2012
Assessment Schedule: Mathematics and Statistics AS 1.4 Travelling Drainlayer
Evidence/Judgments for achievement
Evidence/Judgments for achievement with
merit
Evidence/Judgments for achievement with
excellence
Apply linear algebra in solving problems.
Apply linear algebra, using relational thinking, in solving
problems.
Apply linear algebra, using extended abstract
thinking, in solving problems.
Students use appropriate methods, demonstrate
knowledge of algebraic concepts and terms and
communicate solutions to problems that usually
require only one or two steps.
Students carry out a logical sequence of steps; connect
different concepts and representations; demonstrate
understanding of mathematical concepts and
communicate thinking using correct mathematical
statements.
Students devise a strategy to solve this problem and
communicate using correct mathematical statements
or communicate using mathematical insight.
Students show at least 3 different methods.
These must be relevant to solving the problem
and could be for example:-
Students have to make a recommendation of the
drainlayer to be hired by PipeLine with justification of
the choice. A strategy is clear.
Students devise a clear strategy, make a clear
recommendation with justified reasoning and
complete a report to the shareholders. Insight is
shown.
1. Using formulae eg substitution into
The correct choice may be any drainlayer with sufficient
justification. Justification should include the points of
intersection for the 3 models and the intervals for which
the models are cheapest.
Thinking must include consideration of factors that
may impact on models and hence the
recommendations.
For example
See Excellence example attached below.
formulae
2. Forming an equation. Eg A-One
drainlayers. C= 0.4d + 50
3. graphing a linear model
4. manipulating linear models
5. comparing rate of change to the gradient
eg The gradient of 0.4 represents 40
cents per km.
6. using simultaneous equations or using
intersections of straight lines when solving
the problem
Students must demonstrate an
understanding of concepts and terms and
also to communicate solutions.
AS 91029 NZAMT 2012
From the graph the Creative Drains is the
cheapest to hire for jobs under 15 km. BestDrains
becomes less costly to hire from 15km to 25 km.
After 25 km A-One becomes the cheapest to hire.
REPORT
Dear Manager
The drainlayer I would recommend for long
distance is A-One. This drainlayer is the most
expensive at a call-out fee of $50 but the
mileage is only 40 cents per kilometre. After 25
km A-One has the lowest kilometre rate and
continues to be so.
Incorrect rounding or lack of units will not be
penalised.
Note that evidence for “Achievement” may be
contained in a partial report.
For work up to 12km away then the best rate is
offered by Creative Drains, and for work which
involves travelling between 12 and 25 km then
BestDrains is the best option.
If PipeLine’s contracts gradually involve laying
drains closer to the city then it would be
advisable to consider Creative Drains as they
have a lower call-out fee and their charge per
kilometre is not that much higher than A-One.
If most of your rural work at the moment is more
than 25 km away I recommend you hire A-One.
Should you prefer to use only one drainlayer
then I would recommend BestDrains. This
plumber will be slightly more expensive than
Creative for short travel jobs and increasingly
more expensive for long distance jobs but is the
cheapest between 12.5 to 25km.
Yours faithfully
A. Student
AS 91029 NZAMT 2012
Final grades will be decided using professional judgment based on a holistic examination of the evidence provided against the criteria in the
Achievement Standard.
AS 91029 NZAMT 2012
An Excellence Solution
Strategy
Put the information in one form so a comparison can be made.
This strategy explains the thinking being used and the choice
is justified.
All into equation form
Or all into graphical form
The idea of a common form for comparison is very good
mathematical technique.
Or all into table of values form.
I am going to choose the equation form and solve the equations using
simultaneous equations.
Doing two methods, one using algebra or generalised
methods and the other using graphs or a visual method for
checking is also a very good example of abstract thinking.
For a visual check and comparison I will also graph these equations.
The table of values form would be quite hard to see exact values and in
my experience is not as useful as the previous two methods.
I will recommend the cheapest option for the longer term or more than 1
year. There may be other features.
AS 91029 NZAMT 2012
The comment about the table is based upon experience and
is insight to a better solution.
A student knowing these techniques and performing them is
forming a M or E solution.
Solution
1. A-One
Call-out charge is $50 and road charges are 40 cents per kilometre.
A
In equation form this is C  0.40d  50
Changing written information into an
equation is forming an equation.
where C is the cost and d is the distance travelled from the shop to the job.
A
10
The gradient is
= or 0.4 dollars per kilometre.
25
Interpreting the gradient formed
The constant 50 is the call-out fee.
For distances of around 30 km the cost is $12 + $50 = $62
40 km will cost $66
2. BestDrains
The y intercept is $20 and represents the call-out fee.
From the graph, the vertical intercept is $20 and the gradient is
cents per kilometre.
Thus the equation is C  1.6d  20
For a distance 30 km from the shop the cost is $68
40 km will cost $84
AS 91029 NZAMT 2012
40
or 1.6. This is 16
25
A
Calculating the gradient
A
Forming the equation
3. Creative Drains
A
C  2.4d  10 means the call-out fee is $10 and the rate per kilometre is $2.40
Interpreting gradient in context
For a distance of 30 km from the shop the cost is $82.
Interpreting intercept in context
40 km will cost $106
4. General Comment
A
A One is the cheapest for distances over 25km with a fixed cost of $50 and a rate of 40c
per km.
These comments are all observations and
relate aspects of equations to linear
relationships.
BestDrains is cheapest between 12.5km and 25km with a fixed cost of $20 and a rate of
$1.60 per km.
Creative Drains is cheapest up to 12.5km with a fixed cost of $10 and a charge out rate of
$2.40 per km.
AS 91029 NZAMT 2012
There is only analysis at this stage but a
student doing at least three different A of
the above examples clearly is
Achievement.
Graph of all three (Cost ($) vs Distance from the PipeLine shop)
Here a graphing package
has been using to draw all
the equations so a visual
comparison can be made.
A
Graphs but no justification or
comment.
M
A choice is made and a
reason for the choice made.
A One is the yellowy colour, Purple is BestDrains, Blue is Creative Drains.
AS 91029 NZAMT 2012
From the graph Creative Drains is the cheapest until about 12 km. Then BestDrains
becomes less costly to run up to 25 km. After 25 km A-One, the most expensive call-out
fee, becomes the cheapest to run. This is due to the low cost per kilometre.
M
The models are interpreted.
M
A decision is made, a recommendation
made, and justified.
Solving the equations
M/E
A-One, C = 0.4d + 50
Algebra solution, or read from a graphical
package as was done here.
BestDrains, C = 1.6d + 20
Creative Drains, C = 2.4d + 10
The rounding is of no concern, do we
round up or down?
Drainlayers A and B have the same cost at 25 km. We are not interested in the actual
cost. After that B is more expensive and continues to be so.
M
Drainlayers B and C have the same cost at 12.5 km after which B is the cheapest and
continues to be so.
Interpretation, communicated.
The best choice is A-One in the longer term.
REPORT
E
Dear Manager
A written report with a clearly
communicated and justified
recommendation.
The drainlayer I would recommend for long distance is A-One. This drainlayer is the most
expensive at a call-out fee of $50 but the mileage is only 40 cents per kilometre. After 25
km A-One has the lowest kilometre rate and continues to be so.
For work up to 12km away then the best rate is offered by Creative Drains, and for work
which involves travelling between 12 and 25 km then BestDrains is the best option.
AS 91029 NZAMT 2012
E
There is some insight here as the solution
If PipeLine’s contracts gradually involve laying drains closer to the city then it would be
advisable to consider Creative Drains as they have a lower call-out fee and their charge
per kilometre is not that much higher than A-One.
If most of your rural work at the moment is more than 25 km away I recommend you hire
A-One.
Should you prefer to use only one drainlayer then I would recommend BestDrains. This
plumber will be slightly more expensive than Creative for short travel jobs and increasingly
more expensive for long distance jobs but is the cheapest between 12.5 to 25km.
Yours faithfully
A. Student
Computer Services.
AS 91029 NZAMT 2012
is interpreted in context.
E Insight
Other information is considered.