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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.