Mathematics and Statistics Achievement Standard 91032
Apply right angled triangles in solving measurement problems
NZAMT
The Painter’s Problem • Practice
Credits: 3
Note: These are secure practice assessments for schools (not for individual tutoring) to be used
as a base from which schools can develop their own assessments and therefore schools
should:
1. Check the assessment against the standards (these are not pre-moderated)
2. Do the assessment before it is given to students and adjust to suit your school
3. Check the assessment schedule
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 91032 explanatory notes at
http://www.nzqa.govt.nz/ncea/assessment/search.do?query=mathematics&view=achievements&l
evel=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-GM6-1
o http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievementobjectives/Achievement-objective-GM6-5
o http://seniorsecondary.tki.org.nz/Mathematics-and-statistics/Achievementobjectives/Achievement-objective-GM6-6
o http://nzmaths.co.nz/elaborations-level-five-geometry-and-measurement?parent_node=
 The assessment exemplars and moderator comments at http://www.nzqa.govt.nz/qualificationsstandards/qualifications/ncea/ncea-subject-resources/ncea-study-resourcemathematics/exemplars/
These notes contain information, definitions, and requirements that are crucial when interpreting the
standard and assessing students against it.
Context/setting
This achievement standard requires students to demonstrate knowledge of measurement concepts,
trigonometric ratios and Pythagoras’ Theorem in two and three dimensions.
In this investigation students will measure several lengths and an angle in relation to a wall and use their
measurements to calculate some distances.
You will need to find a wall whereby you have access to the base of it and can stand at a distance away
from it on the level.
Pre-requisites
It is assumed that before attempting this assessment students have had experience in using clinometers
and other measurement devices in order to calculate lengths and angles using the methods of similar
triangles, trigonometry and Pythagoras’ Theorem. In addition they should be familiar with using the ‘rule
of thumb’ method for determining the required lengths for similar triangles.
AS 91032 NZAMT 2012
Conditions
This task should be completed in two sessions. Between sessions teachers will need to collect work to
ensure authenticity is maintained.
During the first session, students should be taken to an appropriate wall and work in groups of up to
three to take and record their measurements.
During the second session students should work independently to complete the task. An appropriate
length of time for this is one hour.
Students will need to use calculators to complete the task.
Where manageable, one reassessment opportunity should be available for all students.
Resource requirements
Tape measure of suitable length or similar and some method of measuring angles e.g. clinometer
AS 91032 NZAMT 2012
NZAMT
Name:_________________________________ Teacher:________________
1
AS 91032
AS 91032 (v1) Apply right angled triangles in solving measurement problems
The Painter’s Problem
Credits: 3
You should answer ALL questions in this booklet.
Show ALL working for ALL questions.
If you need more space for any answer, use the blank pages provided and carefully number the question.
YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE ASSESSMENT.
Achievement
Achievement with Merit
Apply right-angled triangles in
solving measurement problems.
Apply right-angled triangles,
using relational thinking, in
solving measurement problems.
Achievement with Excellence
Apply right-angled triangles,
using extended abstract
thinking, in solving
measurement problems.
.
Overall level of performance
AS 91032 NZAMT 2012
AS 91032 NZAMT 2012
Student instructions sheet
Introduction
The side wall of the Maths Block is to be painted. A painter is to be employed to do the job. What is the
shortest ladder that they will need in order to safely complete the painting job? As part of your task you
will be required to find the height of the wall. You will need to use at least two different methods in order
to do this.
Part One: Measuring at the wall
Working with a partner use the equipment provided to take whatever measurements you require to
enable you to calculate the height of the wall using the methods you have selected.
Take suitable measurements in order to find the height of the wall.
 Draw suitable sketches
 Take appropriate measurements
Write your measurements on your sketch(es)
Part Two: Solving the problem
A ladder is needed to enable the painter to be able to safely reach all parts of the wall.

Calculate the minimum length of ladder required if the industry safety recommendations for angle
between the ground and the ladder are between 60o and 70o. Justify your answer.
The quality of your diagrams, descriptions and explanations, and how well you link this
to the context will determine your overall grade.
AS 91032 NZAMT 2012
Assessment Schedule: Mathematics and Statistics AS 1.7 The Painter’s Problem
Evidence/Judgements for
Achievement
Evidence/Judgements for
Achievement with Merit
Evidence/Judgements for
Achievement with Excellence
Apply right-angled triangles in solving
measurement problems.
Apply right-angled triangles, using
relational thinking, in solving
measurement problems.
Apply right-angled triangles, using
extended abstract thinking, in solving
measurement problems.
Students use a range of methods in
solving measurement problems,
demonstrating knowledge of
measurement and geometric
concepts and terms, and
communicating solutions which would
usually require only one or two steps.
Students will select and carry out a
logical sequence of steps and either
relate findings to the context or
communicate thinking using
appropriate mathematical
statements.
Students devise a strategy to
investigate or solve a problem and
use correct mathematical statements.
Students will demonstrate at least 3 of the
following methods:
Students will take 3 accurate measurements
(length  0.2m. or angle  5) and demonstrate
at least two of the following methods:
Students will take 3 accurate measurements
(length  0.2m. or angle  5) and demonstrate
all of the following methods:
Method – Use Trigonometry to find the height
of the wall
Method – Calculate the height of the wall using
at least two different methods. Chose the most
appropriate measurement for the height of the
wall and justify the choice
Method - Measure at a level of precision
appropriate to the task
At least 3 measurements within the accuracy
(length  0.2m. or angle  5) with at least two
correct units.
For example
Height of student to eye 160 cm +/- 2cm
Distance from wall 5.00m +/- 0.2m
Angle from student to top of wall 380 +/- 50
Length of arm 0.52 m +/- 2cm
Method – Use Trigonometry to find the height
of the wall
5 X Tan 38 = 3.91m
Method – Use similar triangles to find the
height of the wall
Method- Use Trigonometry to find the length of
the ladder
Alternate methods for solving these problems
are acceptable.
The student has communicated a clear
strategy and applied this to solve, or largely
solve, the problem stated.
3.91 + 1.6 = 5.51m
Method – Use similar triangles to find the
height of the wall
0.3 / 0.52 x 10.75 = 6.20m
Method – Use Trigonometry to find the length
of the ladder
L=5.01/sin 70o
L = 5.33m
Mathematical communication is essential.
Diagrams should be clear and correct. The
solutions should be communicated in the
context of the question.
Students should not be penalised for incorrect
rounding, minor errors or running arithmetic.
Method- Investigate finding the minimum
length of the ladder using at least two different
options. Decide which option results in the
minimum ladder length and justify the decision
based on appropriate mathematical analysis
Clear mathematical communication is
essential: Students must use correct
mathematical statements throughout the
problem and explain their thinking. They
should provide clear diagrams and
explanations. Solutions should be
communicated in the context of the question.
Alternate methods for solving these problems
are acceptable.
Students should not be penalised for incorrect
rounding, or minor errors or running arithmetic.
Alternate methods for solving these problems
are acceptable.
Students should not be penalised for incorrect
rounding, minor errors, running arithmetic or
failing to communicate clearly.
Final grades will be decided using professional judgement based on a holistic examination of the evidence provided
against the criteria in the Achievement Standard.
AS 91032 NZAMT 2012
EXCELLENCE WORKED SOLUTION
Part One: Measuring at the Wall
Part Two: Solving the problem
Finding the height of the wall
Strategy
1. Draw sketches of the problem
2. Take accurate and appropriate measurements at the wall and record them on
the sketches
3. Calculate the height using trigonometry
4. Calculate the height using similar triangles
5. Decide which result from methods used in 3 and 4 is the most accurate and
explain the decision
Finding the minimum length of the ladder
Strategy
1. State and justify any assumptions made
2. Draw a sketch of the problem including all relevant measurements (taken at the
wall)
3. Calculate the length of the ladder using trigonometry
4. Recalculate the length of the ladder using a different angle of elevation (both
AS 91032 NZAMT 2012
A/M/E
A/M
A/M
E
A/M
angles need to be within the range of 60 – 70 degrees)
5. Compare the solutions and state the minimum ladder length with justification
Method 1 Using Trigonometry
38o
h
5m
o
Tan38 = h/5
h = 3.906m
Adding on height of student to eye level: 3.906 + 1.60 = 5.506m
Height of wall is 5.51m (to 3 sf)
Method 2 Using similar triangles
0.3m
0.52m
h
10.75m
The ratio of 0.3/0.52 is equal to the ratio of h/10.75
h = 10.75 x 0.3/0.52 = 6.201m
I decided to use the height as 5.51m as it is more accurate. With method 2 it assumed
that my eye was at ground level in order to make the right-angle triangle with the wall.
This was not the case and this resulted the height calculation being an over-estimation
of the actual height.
Finding the minimum length of the ladder
Reach 0.5m
Wall
5.51m
Length L
70o
Assumption: The ladder does not need to go to the top of the wall as it is safer for the painter to
stand below the top of the ladder and reach up to paint the top of the wall. If the top of the ladder
is placed 50 cm below the top of the wall then this will give the painter enough height to reach to
the top of the wall, when standing near the top of the ladder (measured length of arm 0.52m i.e.
approximately 0.5m).
The angle of 70o has been used as it is the maximum safe angle.
Finding the height of the ladder (using a 70o angle)
Height reached up wall by ladder = 5.51-0.5 = 5.01m
Sin 70o= 5.01/L
L = 5.01/sin 70o
L = 5.332 m
L = 5.33 m (3 sf)
Checking to see if this is the minimum length of ladder by using angle of 600
Sin 60o= 5.01/L
L = 5.01/sin 60o
L = 5.785 m
AS 91032 NZAMT 2012
E
L = 5.79 m (3 sf)
The minimum length required for the ladder is 5.33m
Insight(!)
Calculating how far the base of the ladder would be from the wall:
Using a 70o angle
5.01m
5.33m
70o
d
Distance from base of ladder to wall (d) =
5.332 – 5.012 = d2
d = 1.819m
d = 1.82 m (3 sf)
Using a 60o angle
5.01m
5.79m
60o
d
Distance from base of ladder to wall (d) =
5.792 – 5.012 = d2
d = 2.902m
d = 2.90 m (3 sf)
In addition to resulting in the minimum ladder length required, using a 70 o angle also means that
the distance between the base of the wall and bottom of the ladder is also shorter i.e. 1.82m
compared to 2.90m which means that ladder would be more stable using the 70o angle.
AS 91032 NZAMT 2012