Download Measurement

Measurement
Level: Secondary 4 / 6
Project aims:
The main objectives of this project are to let students:
1. Understand the importance of measurement in our daily life.
2.
3.
4.
Explore different methods for measurement.
Understand that measurements can never be exact.
Compare the accuracy of different measuring methods.
Learning objectives:
1.
2.
3.
4.
Understand the importance of measurement in science.
Gain familiarity with a variety of measuring instruments and methods.
Understand that measurements are never exact through error analysis.
Recognize the limitation of measuring instruments used.
Tasks:
In the following project tasks, students are asked to design their own methods for
measuring particular object(s), building(s) and distance(s). They have to explain how
they make the measurements, identify the possible sources of error of each method,
and compare and estimate the accuracies of their measurements.
Task 1: Measuring length
Measure the length of an object (shorter than one foot long recommended), say a
pencil, using a one feet ruler without finer scale. Students may use other tools in their
measurement when necessary. However, the tools should not contain any scale or
mark, which can directly be used to measure the length of the object.
Task 2: Estimating number
Estimate the number of small objects inside a container, say, the number of beans in a
jar without counting the number directly, or to estimate the number of tiles in a floor
which the laboratory is located by measuring the number of tiles per square meter.
Task 3: Measuring height
Measure the height of a given tall building or structure, such as the Lion Rock or the
International Finance Centre in Hong Kong.
Task 4: Measuring distance
Measure the distance between any two particular places, without the use of any
electronic equipment or professional devices
Apparatus/Materials:
Any appropriate apparatus and materials proposed by students are welcomed. Typical
examples include meter ruler (scale covered with paper), identical coins (many),
protractor, beaker, balance, measuring tape, tennis ball (as a falling object), stop watch,
string / rope with a weight, meter wheel and so on.
Pre-knowledge:
Measurement is of central importance in our daily life. It is one of the concrete ways
we deal with our world. Measurement is particularly important in science. Science is
concerned with the description and understanding of nature and measurement is one
of its most fundamental tools. One may say that science cannot exist without
measurement and we start learning physics by learning how to measure a few physical
quantities. Physics attempts to describe nature in an objective way through
measurement.
However, in every measurement, there is uncertainty associated with it – there is no
such thing as a perfectly accurate measurement. There are uncertainties arise from the
measuring instrument, the people doing the measurement, the method of making the
measurement and the characteristics of the systems being measured. We need to learn
to deal with these uncertainties in measurement through error analysis.
Possible Methods:
Task 1: Measuring length
Method 1: Mid-point cut
Cut a piece of paper of one feet long into two, four, eight…until the divided
length matches the pencil “exactly”. We can then obtain the total length of
the pencil.
Method 2: Coin measurement
Measure the length of the one meter-ruler and the pencil by comparing with
the height of a stack of coins. Record the number of coins that have the
same length as the one meter-ruler as Y and the number of coins having
same length as the pencil as X. The length of the pencil can then be
calculated by Y/X (in meter)
Method 3: Triangle rule
To form a triangle using the one
meter-ruler and the pencil, the
length of the pencil is
then x tan  (in meter), where
the angle can be measured by a
protractor
x
θ
One meter- ruler
Task 2: Estimating number
Method 1: Bisection method
Separate the objects inside the container into two parts and put them inside
two beakers respectively. Place the two beakers on the two sides on a beam
balance such that the weight balance on each side. Put aside one beaker and
separate the objects in the other one into two parts again. Repeat the above
process and record the number of measurements x until the number of
objects inside the beaker is small enough to count. Record this number as y.
The total number of objects is roughly y  2 x .
Method 2: Counting the grids
If we are to estimate the number of tiles used on a floor, we can first count
the number of tiles per square meter, and then estimate the area of the floor
in terms of meter squared.
Task 3: Measuring height
Method 1: Trigonometry
Height of the building (H)
 L tan   L tan 
(where L is the horizontal h
distance
between
the
person and the building)
θ
H
L
β
Method 2: Kinematics equation
s  ut 
1 2
at
2
where u  0, a  g  9.81ms 2
and t is the time taken for the ball to drop from the top of
the building to the ground
(assume air resistance is neglected)
Task 4: Measuring distance
Method 1: String measurement
Cut a piece of string of particular length, say 10 meters. This can be done by
measuring a 2-meter long piece of string using a measuring tape and then
fold it to make five sections so that it becomes 10 meters long. The distance
between the two places that we want to estimate can be measured segment
by segment. By counting the total number of segments, we can calculate the
total distance by multiplying 10 meters with the number of segment
counted.
Method 2: Wheel measurement
Rotate a wheel along the required distance and count the total number of
revolutions. On the other hand, measure
the radius of the wheel and hence
F
calculate its circumference. The distance
can then be estimated by the number of
B
revolutions times the circumference of
d
A
the wheel used.
Method 3: Trigonometry
To measure a distant point F, we can at two separate points A and B to
measure the bearing of F using compass. With angle BAF and ABF, and
the distance between AB, we have
sin AFB sin ABF

AB
AF
Distance of between AF can be found.
Teachers’ note:
Normally, error within 5% to 10% to the actual value is considered to be reasonable.
For more information on error analysis, please see the additional materials about
measurement and error.
Teachers’ guide:
Students are encouraged to propose other methods for measurements and to identify
the plausible sources of error in different measuring methods before they perform the
actual measurements. They should also estimate of the accuracy of their results after
the experiments. Students should learn to use error analysis methods in their error
estimation.
Teachers are recommended to encourage the students to focus on analyzing and
estimating the errors involved in each method of measurements rather than just trying
to make the measurements as “accurate” as possible.
The level of difficulty of the tasks can be adjusted by choosing different kind of
objects, buildings or distances to be measured by the students. Teachers may choose
different objects for students to perform their measurements based on their abilities.
Experience in pilot study:
A pilot study on this project was carried out in a secondary school for Form Six
students during school holiday. During a 10-day holiday, classes were made on three
non-consecutive days. Around 40 students were divided into nine groups and a list of
three projects was given to them. A rundown of the event is outlined below. The
briefing, planning and discussions were done in the first class, while the
implementations were done in the second and third class. A project presentation was
carried out after the holiday in a supplementary lesson after school.
Each group was asked to choose a project from the list. Out of the 3 projects, namely,
Measurement, Projectile motion and Acceleration in traveling object, 4 groups chose
this project. Students studied Tasks 1, 3 and 4. We shall concentrate on these groups
of students in the following.
Teacher
Stage
Students
 Tried out the project himself.
 Marked down some critical
points where student may fail in
the experiment.
 Planned how to help students
who may be stuck in the critical
points.
 Prepared handouts, reference
Preparation
materials, a supplementary
handout to calculate error, and a
learning log as an assessment
tool.
 Gave introduction and briefing
to each project.
 Facilitated students to choose
project.
DAY 1
(Afternoon)
Briefing
(1 hour)
 Observed the students.
 Went to each group to facilitate
student’s discussion.
 Answered
questions
from
students.
 Prepared apparatus after the
class.
 Went through the projects in the
class.
 Chose a project from the list.
 Discussed the procedure, setup,
apparatus, division of labor,
Planning
& Discussion
(2 hours)
precautions etc. of the project.
 Consolidated a list of apparatus
that they would like the teacher
to prepare for them.
 Got a supplementary handout
for calculating error.
 Facilitated the implementation, DAY 2 & 3  Implemented the project.
helped students to solve their (Whole days)  Took measurements.
 Modified the setup or procedure
problems faced.
 Responded
to
students’ Implementation
whenever they found necessary.
(I)
problems upon their request.
 Asked the teacher as last resort
(3 hours)
 Provide apparatus to students
when faced difficulty.
upon their request.
 Assessed students by observing
 They were not supposed to seek
Implementation
students. Ask them to do some
help from teacher.
(II)
key steps if they have done
(3 hrs+1 day)
them before teacher came.
 Assessed the presentation.
 Launched peer evaluation.
DAY 4
(2 Periods)
 Each group presented their
results to the class using digital
projector.
Peer Evaluation  Graded the presentation of their
(1 hour)
peers.
Presentation &
 Graded the in-class assessment,
learning log, presentation and
reports.
 Handed it their reports.
Assessment
The roles of teacher – facilitator and assessor
During the stages of project briefing, planning and discussion, and in the
beginning of implementation, the teacher acted like a facilitator and helped
students to solve problems. Afterwards, the teacher acted like an assessor,
questioned and graded the students. In both stages the teacher did not
proactively helped students to overcome all their problems. However, he
repeatedly reminded student about his expectation on the progress of the
project work. The aim of switching roles was to make sure that the students
had a good progress and acquired enough guidance before they were
graded.
Assessments
In the preparation, the teacher also identified a few benchmark points as
marking points. Points were given to students who demonstrated that they
have achieved these benchmarks. Since the teacher knew when he had to
grade the students, it made in-class assessment easier. The teacher may
assess the students by questioning or simply requested the students to repeat
the experiment before him if the teacher cannot asses the student during the
process.
Presentation
Students presented their project after the holiday in a supplementary lesson
after school. Each group had 5 minutes to present in front of the class. They
used PowerPoint as their media of presentation. The presentation was
graded by both the teacher and the students. Students gave their comments
and scores regarding the content and delivery skills on a peer evaluation
sheet.
Students’ work
Students had one day break after the project briefing day. On the second day
of class, students performed experiments on measuring the length, height
and distance. The following pictures show the measurement performed by
students.
Figure 1: Measuring height
Figure 2: Measuring distance
Challenges faced by teacher and students
As resources are limited, the teacher allowed students to do 3 different
projects together. This made teacher’s preparation time tripled.
In order to allow students to measure some large objects for a non-trivial
measurement in a safe and controlled environment, the teacher utilized the
whole school building. Using that, students can perform their study close to
their daily life. Also, the true value of height of the building can be easily
found out by suspending a long rope from the ceiling to the ground, this
make the student able to compare their value to the true value. Unlike a
mountain which the height is known to the public, students can only know
the true value until when the teacher find the time appropriate.
In this project the efforts should be made on the estimation and control of
error. Some students shifted their focus on repeating the measurements
many times without paying careful attention to error analysis which is an
important learning objective of this project. Hence, teachers are suggested
to remind students to spare time in explaining the estimation of error in
class so that if they have any questions or difficulties, they can seek help
and guidance before they manipulate the error independently at home.
Questions:
1.
2.
Which method of measurement is more accurate? Can you account for this?
How could you deal with the difficulty in measurement involving slope and
gradient, like staircase?
Extension:
1.
2.
3.
Measure the thickness of a thin sheet of paper that is too thin to be measured
using normal measuring instrument.
In room temperature and pressure, one mole of air molecules occupies a volume
of 24.5 liter and has a mass of 14.4g. Estimate the mass of air inside a room.
Which is the “heaviest”? The people in a room, the chairs and tables, or the air in
that room?
Let the “cleaning area” of an object be the total area of that object that you wish
to clean. (For example, the top surface of a desk and the inner surface of a
cabinet but not those outside your visibility.) Clean a small area and record the
time needed. Try estimating the time needed to clean the whole cleaning area, the
error in estimating the cleaning area and the cleaning time. Raise out the factors
we haven’t took into account if this value is outside your expectation.
Appendix:
Propagation of error in different forms
(This appendix is adopted from the draft Error Estimation - 誤差估算 released in
October 2005 by Science Education Section of Curriculum Development Institute,
EMB.)
In many cases, a particular result will be obtained by combining a number of
measurements of different quantities in some ways.
For instance, when two independent measurements ( A  A ) and ( B  B ) are
combined to give a result X, the associated error terms X will be given by:
Combinations of A and B
Error of X
X  A B
X  A B
X  A  B
X  AB
X
A B


X
A
B
A
X 
B
X  kA ( k is a constant)
X
A

or X  k A
X
A
X  An
X
A
n
X
A
X  AB n
X
A
B

n
X
A
B
The combinations involve addition and multiplication with more variables or more
terms can be resolved into combinations listed above.
Example 1: Combinations of error involving addition and multiplication
A student finds the constant speed of a moving trolley with a stopwatch. The equation
used is v  s t . The distance s is measured with a meter ruler, the time t is measured
with a stopwatch. Their values are s = (4.01 ± 0.01) m; t = (3.2 ± 0.2) s.
The speed v
= 4.01 / 3.2 ms-1
= 1.253 ms-1 (calculator display)
= 1.3 ms-1
This result is corrected to 2 significant figures since the time measured is
only known to 2 significant figures.
The estimated error in speed, v
 s t 
= v


t 
 s
 0.01 0.2 
= 1.253

 ms-1
 4.01 3.2 
= 0.081437 ms-1 (calculator display)
= 0.1 ms-1
Since v has 1 decimal place, the error term ∆v should also has 1 decimal
place, i.e. v = 0.1 ms-1.
Therefore the speed of the trolley is 1.3 ± 0.1 ms-1.
Handling the trigonometric functions of an angle
In the tasks, students may measure an angle and apply trigonometric functions of the
angle for their analysis. Here shows two examples to deal with the tangent of an
angle.
Example 2: Handling the trigonometric functions with multiple measurements
Students used a protractor to measure the angle of elevation to a building and apply
the formula h  d tan  to calculate the height of the building. If
d  (8.23  0.005) m and θ = 27º, 29º, 28º, 31º
Then the average value of height is
h
 27  29  28  31 
 8.23 tan 

4


 4.515m
Since the tangent of an angle is an increasing function, the extreme values
of the height are
hmax  8.235 tan 31 = 4.948m
hmin  8.225 tan 27 = 4.190m
hmax  h = 4.948 – 4.515 = 0.433m
h  hmin = 4.515 – 4.190 = 0.325m
So, maximum error = 0.433m
After rounding off to have the same decimal
h  (4.52  0.43) m
place
with
d,
Example 3: Handling the sine of the angle with single measurement
If a student has only one measurement for the angle, and have result
d  (8.23  0.005) m,   (29  0.5)
Then the mean and extreme values are:
h
 8.23 tan 29 = 4.561m
hmax  8.235 tan 29.5 = 4.659m
hmin  8.225 tan 28.5 = 4.466m
hmax  h = 4.659 – 4.561 = 0.098m
h  hmin = 4.561 – 4.466 = 0.095m
So, maximum error = 0.098m
After rounding off to have the same significant figure with d,
h  (4.56  0.10) m
Further Example
(This method is provided for students with strong mathematical background.)
For any function f(x), we can find the error by
d

 f  x    f x  x
 dx

f x  x 
 f x 
f x 
 f x 
f x  x
x x
x  x x x  x
we have
 f x  1 f x  x   f x  x 

x
2
x
for an infinitesimally small error, that is, x  0 ,
or, for a small value of error,
lim
 f  x  df  x 

x
dx
d

 f  x    f x  x
 dx

Example 4: Handling error by calculus
A student measures the angle of elevation of a building. He finds the angle is
  34  0.5 . Find the value of sin  and its error.
34  34 

180
0.5  0.5 
rad  0.593

180
rad  0.00872
sin   sin 0.593  0.559
 d

sin    
sin     cos 0.593  0.00872  0.00723
 d

sin   sin    0.559  0.007
* Notice that we should use radian scale because we are using calculus.