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Department of Education
Measuring the
Width of a Road
A Multidimensional Learning & Assessment Task
Designed for a Year 9 Mathematics Program
Written by Andrew Hill, Brighton Secondary College, 30 March 2007.
This task was designed by Andrew Hill, Denver De Kretser and Lachlan Champion, of
Brighton Secondary College, in 2005. This version has been modified to suit assessment
according to the 2007 Victorian Essential Learning Standards.
Last updated: 17.05.07
© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
Measuring the Width of a Road
Guidelines for teachers
Synopsis
In teams, students apply the mathematics of scale and transformation to measure the width of
a road. In doing so, they monitor their own learning and develop their ability to work
collaboratively with others.
Rationale
The application of mathematics to real life situations stands in contrast to the simplified, neat
and tidy nature of text book mathematics. This activity is designed to assist students towards
an implicit understanding and experience of mathematics, as applied to “real-world” contexts.
Purpose
The purpose of this activity is for students to gain experience in applying mathematics to a
“real-life” situation in cooperation with others.
Assessment
This task includes three types of assessment:
•
•
•
Assessment as learning, as students formatively set their own goals within the
requirements of the task, reflect on their learning, and consider the feedback of their peers
and teachers.
Assessment for learning, as teachers adapt the task to the learning needs of particular
groups of students. This is to ensure that students do not feel overwhelmed or lost in the
complexity of the task, and that they are adequately challenged and have the opportunity
to take their learning further.
Assessment of learning, as teachers make summative judgements about student learning
based on student feedback, teacher observations, and the assessment of students’ written
reports. These judgements are made in accordance with the Victorian Essential Learning
Standards.
Victorian Essential Learning Standards (VELS)
Assessment of this task relates to the following strands, domains and dimensions of VELS:
Strand
Domain
Dimension
Physical, Personal and
Social Learning
Interpersonal Development
Building social relationships
Discipline-based
Learning
Working in teams
Personal Learning
The individual learner
Mathematics
Measurement, Chance & Data
Space
Working mathematically
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© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
Background
To succeed in this task, each team of students will require the following abilities amongst
their members:
•
•
•
•
•
The ability to conceptualise the goals of the task, and a sequence of actions that will
enable those goals to be achieved.
The ability to record measurements, calculations and conclusions in an orderly and
coherent way.
The ability to orchestrate the actions of six to eight people.
The ability to read and comprehend instructions and questions, including mathematical
terms.
A grasp of the relevant mathematics progressing towards VELS Level 5.5 as described for
the following three dimensions:
Measurement, chance and data: The estimation, measurement and conversion of metric
units of length;
Space: The mathematics of transformations and scale factors.
Working mathematically: The deduction of lengths and distances from what is already
known.
Measuring length “y”
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© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
To assist students in the development of the appropriate abilities, the following learning
activities may be completed in the lessons prior to doing this activity, Measuring the Width of
a Road:
Learning
activity
Skill based
exercises
Description
Application
and
analytical
exercises
Students read and answer descriptive questions
concerning the application of relevant mathematical
methods, and the appropriate selection or
adaptation of methods to suit particular
mathematical problems. Each student writes a
personal reflection about their progress with these
exercises.
Right angle
rope
triangles
Students complete simple questions concerning
methods involving ratios, similar triangles and
Pythagoras’ theorem.
Students make large 3:4:5 rope triangles, knotted at
the vertices. In groups of three, they use them to
check for right angles in the classroom, on sporting
courts, etc. Students complete a verbal and written
team reflection.
VELS Dimensions
Assessed
• Measurement, chance
and data.
• Space
•
•
•
•
•
•
•
•
Reading
Measurement, chance
and data.
Structure
Working
mathematically
The individual learner
Space
Working
mathematically
Working in teams
Human
Class is divided into two competing teams. The aim •
triangle
is to make similar triangles of the specified ratios
•
competition before the other team does. They make the triangles
with their bodies by lying or sitting on the floor.
•
The teacher(s) coordinate a 1-3-6-class analysis of
the actions that facilitated team success.
The individual learner
Measurement, chance
and data.
Working in teams
•
•
Working in teams
Measurement, chance
and data.
Space
Working
mathematically
Measuring
the height
of a pole by
similar
triangles.
Teams of three or four students determine the
height of a pole on the school grounds. They refer
to the shadow of the pole, the known height of one
of the students and the method of similar triangles
to complete their calculation.
•
•
Checking a right angle using a 3:4:5 rope triangle
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© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
Equipment list
The following items will equip between 24 and 32 students. Increase the quantities
proportionately for larger groups.
Quantity
Item
Purpose
1
Cricket bag
For transporting equipment.
4
Clipboards
For recording group measurements on planning sheets.
4
Tape measures
For carrying out measurements.
4
3:4:5 Rope Triangles
For checking right angles.
16
Broomsticks
For marking points on a straight line by line-of-sight.
Recommended learning activity sequence
This recommended sequence of activities can be adapted to the prevailing circumstances.
Activity Purpose
Activity
1
•
2
To introduce
the problem
and the means
by which it
will be
tackled.
To rehearse
the activity.
•
•
•
•
•
•
•
•
•
•
3
5
To take
•
measurements.
•
•
•
Teacher(s) explain the problem and its relationship to
problems dealt with by professionals such as surveyors,
draftsmen and navigators.
By instruction, demonstrate how the theory of similar
triangles is relevant to the solution of the problem.
Set a few related exercises to complete.
Provide feedback concerning the solutions to the exercises.
Jointly reflecting on the rehearsal, students assume specified
roles within their teams; and each team chooses a name.
Students are organised into teams of six to eight students.
Teacher(s) highlight the importance of the cooperation and
safe conduct of all group members.
The methods of assessment are explained, by reference to the
assessment rubrics.
The procedure is demonstrated in the classroom or school
grounds with the help of a few students.
Each team rehearses the procedure.
Students conduct a personal and group reflection concerning
the completion of the rehearsal and the mathematical method.
Teacher(s) restate expectations of courteous and safe conduct
whilst doing the activity.
Teacher(s) and students travel to the road with the equipment.
Teams take and record their measurements.
Students reflect on their performance of the activity,
individually and in groups. Teachers also provide feedback to
the teams.
© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Activity Purpose
Activity
4
•
To calculate
and compare
solutions; and
to complete
student
assessments.
•
•
•
•
•
•
Department of Education
On poster paper, each team completes the calculations with
the aim of obtaining a common understanding of the relevant
mathematics, the reliability of the measurements, and their
calculation for the width of the road.
The accuracy of the calculated road width is determined by
reference to Google Earth (using the measurement tool)
and/or a measurement provided by a road authority.
Consequently, a correct scale for the diagram on the planning
sheet is determined.
Students compare their calculated values with the values
determined by other teams and the officially correct values.
Students complete self and peer assessments using Rubrics 1
and 2.
Students write individual reports of the activity according to
specifications provided by the teacher(s).
Teacher assessments are completed.
Checking the measurement
Each team’s calculated measurement may be checked by cross-reference with other groups,
by reference to an official value such as that provided by a road authority, or by using the
measurement feature in Google Earth.
The image below shows the use of the measurement tool in Google Earth to arrive at a value
for the width of a major highway.
6
© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
Calculations
Students are provided with a diagram, similar to the one shown below, as an aid to taking
measurements and completing calculations.
E
w
y
Calculation Method No. 1
Triangle EBC is similar to triangle EAD.
Hence,
w w+ y
=
x
z
⇒ wz = x ( w + y )
⇒ wz = wx + xy
⇒ wz − wx = xy
⇒ w( z − x) = xy
⇒ w=
xy
z−x
Method 2
Triangle EBC is similar to the smaller triangle with hypotenuse DC.
Hence,
w
x
xy
=
; therefore w =
y z−x
z−x
Some students find this method easier to understand than the other.
7
© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
Rubrics
Rubric 1: Peer assessment
Following the completion of the task, students consider each individual’s contribution to the
team according to the criteria shown on Assessment Rubric 1.
Students are required to reach majority agreement about each person’s contribution whilst
also justifying their decisions in a respectful and reasonable way. Teacher supervision and
moderation is appropriate here.
The important thing here is to provide constructive comments to the other members of the
team, so they may reflect on how they have contributed and what they’ve learnt.
Rubric 2: Teacher and Student self assessment
This rubric provides a means for teacher assessment and student self assessment. Each box
corresponds to a particular standard. The relevant learning focus statements have been
interpreted according to the task at VELS Levels 3, 4, 5 and 6.
Measuring the “buffer” zone.
8
© State of Victoria, 2007
www.education.vic.gov.au/studentlearning/assessment/
Department of Education
Measuring the Width of a Road
Students’ Instruction Sheet
Introduction
A footbridge is to be built across a major road and the engineering draftsman, who will design
the bridge, requires an accurate measurement of the width of the road. The bridge will be
supported on either side of the road, with no additional supports in between. Imagine your
team is a company that has been contracted to measure the width of the road. You are
required to use a method that enables you to safely take the measurement without having to
cross the road.
Your success in completing this task will have direct bearing on your company’s future
contracts and your team is in competition with the other teams. Your aim is to demonstrate
that you can complete the task more efficiently and accurately than the other teams.
Teams
It is not possible for individuals to satisfactorily complete this task in the time available – this
is a task that must be completed in cooperation with others. You will need to form a team of
six to eight people.
Safety and behaviour requirements
•
•
Under no circumstances are you to set foot on the road surface whilst conducting
measurements. For safety reasons you are to stay on the footpaths, keeping a “buffer
zone” between yourselves and the road surface. You will need to factor a measurement of
the width of the buffer zone into your calculations. The only time you are permitted to set
foot on a road surface is when crossing a road in a safe manner (e.g. at green traffic lights)
when on route to or from the school.
Pedestrian and bicycle traffic must not be obstructed and you are to treat all members of
the public with due courtesy and respect.
Equipment
The following materials will be available to each group:
•
•
•
•
•
Planning diagram
4 x broomsticks for determining line of sight
1 x measuring tape
1 x clipboard
1 x knotted piece of rope, with which to make a 3:4:5 right-angle triangle
Assessment
Your understanding of the mathematics involved will be assessed. You will also be assessed
according to your personal learning habits, your courtesy toward others, and your cooperation
with the other group members. Each team will also be assessed according to their ability to
work together in a cooperative and productive way.
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Your company
Together, choose a name for your surveying company and agree on your individual roles
according to the information provided in the table below:
Role
Managing
Director
Communications
Manager
Public Relations
Manager
Chief Surveyor
Surveyor 1
Surveyor 1
Surveyor 1
Surveyor 1
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Responsibilities
Competency
Question
To ensure the team’s actions are
Did the group understand
coordinated.
what was needed for them to
act together in a harmonious
way and complete the task
successfully?
To keep a record of the team’s activities
Was a complete written
and give a verbal report on behalf of the
record kept? Was the report
team.
clearly communicated?
To ensure group members treat each other, Did the team make a good
other groups, and members of the public
impression on others? Did
with courtesy and respect.
the communications manager
make a good impression on
others when giving the
report?
To ensure that measurements and
Were the measurements
calculations are taken correctly. They
taken correctly, and an
coordinate the following:
accurate result obtained for
a) Place pole A on the edge of the
the width of the road and the
buffer zone directly opposite an
scale of the diagram?
object visible on the other side of
the road.
b) Line up pole A and B with the
object on the other side of the road
c) Place pole C at right angles to AB.
Verify that ∠ABC is a right angle.
d) Place pole D on the edge of the
buffer zone, then line it up
between pole C and the object on
the other side of the road. Verify
that ∠DAB is a right angle.
e) Measure distances x, y and z.
Verify those measurements.
To assist with the various physical tasks:
• Determine straight lines of sight
• Create the required right angles
• Measure distances accurately
Were the physical
measurement tasks
accurately completed in
cooperation with others?
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Department of Education
Measuring the Width of a Road
Planning Sheet
Label this diagram with the names of the people who will stand at points A, B, C and D,
check line of sight, check right angles, and measure distances x, y and z.
A pole, tree or other object visible on the other side of the road.
A
x
D
y
B
Students holding broomsticks
11
z
C
Measure lengths x, y and z after
first checking that the angles
∠DAB and ∠ABC are right-angles.
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Department of Education
Measuring the Width of a Road: Assessment Rubric 1
Peer Assessment: Interpersonal Learning, Working in Teams
Chosen company name: ___________________________________________________________________
Team members should complete this table together.
Group
member
12
Role performed
Comments made by other group members
Worked
collaboratively and
negotiated roles?
Provided strategies
to help the team
achieve the agreed
goals?
Respected, involved, Provided ways for
and built on the
team members to
ideas of others?
improve their
performance?
Reflected on and
improved his/her
personal
performance?
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Measuring the Width of a Road: Assessment Rubric 2
Individual Assessment: Multiple Dimensions
Name: _______________________________ Form: ______ Assessing Teacher: _____ Date: _________
Mathematics
Interpersonal
Development
Personal
Learning
Standards
3.00
4.00
5.00
6.00
The individual
learner
With support, you identify
your learning strengths and
weaknesses
You independently identify
your learning strengths and
weaknesses.
You identify your learning
strengths and take actions to
address your weaknesses.
You evaluate the effectiveness
of your learning habits, and
modify them according to your
reflections.
Building social
relationships
You support others and
acknowledge individual
differences.
You accept and display
empathy for the points of view
and feelings of your peers.
You recognise and describe
how peers influence your
behaviour.
Working in
teams
You cooperate with others in
teams.
You explore the ideas of others
and perform a specific role
responsibly.
Measurement,
chance and
data
You estimate and measure
using informal units of length.
You allocate tasks
cooperatively and accept
responsibility for a specific
role.
You estimate and measure
metric units of length.
You describe how local and
global values and beliefs
determine your and others’
social relationships.
You work collaboratively in a
team, negotiate roles, and
complete complex tasks on
time.
You estimate, measure and
convert metric units of length,
whilst also considering errors.
Space
You locate and identify places
on maps and diagrams.
You use scale to describe
relative location on maps and
diagrams.
You use lines and scales to
specify location and direction
on maps and diagrams.
You investigate the use of
mathematics in a “real”
situation.
You describe the simple
mathematical deductions of the
task.
You use appropriate strategies
Working
Mathematically to find solutions.
You estimate, measure and
convert metric units of length.
You use transformations,
including enlargements,
considering the effect of scale
factors.
You predict certain lengths and
distances from what you
already know.
Notes: Reference is made to the VELS standards. The descriptors are taken from the standards and the learning focus statements, and are provided as
signposts for teacher judgement.
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