Teaching Significant Figures Using a Learning Cycle E. Park Guymon, Helen J. James,' and Spencer L. Seager Weber State College, Ogden, UT 84408 "Whatever became of significant figures?" How often do instructors echo this comment as they grade problem sets or lab reports? Can we teach the use of significant figures in a way that will he retained better by our students? The rules for using significant figures are usually found in the first chapter (or possibly the appendix) of most textbooks and seem to be forgotten quickly by everyone as soon as their discussion in class is finished. The importance of correctly recording measurements is often emphasized in class, hut few students seem to recognize that this idea should be applied to data collected in the laboratory. Another frustration occurs because students misinterpret the rules, possibly because the rules are not clearly stated in terms the students understand. This problem is indicated by recent articles that are concerned with the wording used to state the rules governing the significance of zeros in a val11e.34 ..
We have developed an exercise called "Rand R" (rectangles and rulers) which helps our students establish the rules for themselves. This exercise involves the students in making measurements which helps them relate the use of significant figures to laboratory experiences. We have used R and R in both classroom and laboratory settings depending upon the course being taught. R and R is written as a learning cycle, a type of activity developed to help students' reasoning and to give them concrete experiences with a new c o n ~ e p tA. ~learning cycle consists of three phases: exploration, invention, and application. During exploration students are encouraged t o learn through their own concrete experiences. In the second phase, students are asked to invent part or all of a concept for themselves. This procedure gives students confidence as they gain familiarity with the invented concept. During application the students are required to use the concept in a situation which gives them more understanding. Details of our experiences with the R and R learning cycle are given below. Exploration The students work in airs which encouraees them to discuss results and prod each other to achieve better results. Each team receives a packet of material containing four different colored rectangles and a ruler calibrated incentimeters. We made the rectangles and rulers from plastic index cards. They are sturdy and come in various colors. We used a paper cutter to make the rectangles of a given color as nearly as possihle identical in size to ail others-of thnt calm. The rectnngles had the folluwing dimrnsions: 12.0 X 10.5 cni, 2OUX 2.5cm. 13.5 X 0.7 rm.and 3.0 X 2.5cm. The teamsare asked to deteimiue the length, width, perimeter, and area of each rectangle. After com~letinethis task the students are rquired to repent the measurements, hut w i ~ ha ruler cnlihmttd in tenths of a centimeter. The results for a team are shown in Table 1. During this exploration many students show lots of concern about estimating values with the rulers
Table 1. Tvoical Student Data Rectangle
Length (cm)
Width (cm)
Perimeter (cm)
Area (cm2)
Ruler Calibrated to a Centimeter Black Red Green
Orange
12
45 46 11.6 28.2
126
10.4 2.42 2.55
44.8 45.24 11.46
124.8
13.22
27.8
10.5 2.6 2.6 13.4
20.4 3.2 0.7
53.04 8.32 9.38
Ruler Calibrated to Tenths of a Centimeter Black
12
Red
20.2 3.18 0.66
Green
Oranoe
48.684 8.109 8.9896
that they are using. But few show the same concern when they use the measured values in calculations. Also, notice that the data in Table 1 indicate that the team estimated readings to tenths of a centimeter, except when the measured value was perceived to be a whole number (the length of the black rectangle was recorded as 12 instead of 12.0). When these rectangles are cut, i t is important to cut one carefully enough to allow for this type of student result. lnventlon A primary goal of the invention phase is to involve the students and instructor in a discussion that encourages the students to be active learners. We begin this phase by having each team list its results on a blackboard. Table 2 shows some typical results for the hlack rectangle. Usually, discussions (and sometimes arguments) begin among the students as soon as the first results are listed. We try to center the inital discussion around the question, "Are the recorded measurements consistent?" Within a few minutes the students agree that the length of the hlack rectangle should be recorded as 12.0 cm or 12.00 cm depending upon the ruler used. They begin to recognize that for an experimental measurement the last recorded digit is an estimate that will include a reading error. Next, we lead the discussion to consider whether the calculated perimeter and area values are consistent with the correctly recorded lengths. Students often consider results
Table 2. Typical Data for the Black Rectangle Team #
Lengm (cm)
Width (cm)
Perimeter (cm)
Area (cmZ)
Ruler Calibrated to a Centimeter 1
12
2 3
12.0
4
12
12
10.5 10.4
45 44.8
10.6 10.5
45.2 45.0
126 124.8 127.2 126.0
Ruler Calibrated to Tenths of a Centimeter
Karplus. R. "Science Curriculum improvement Study"; University of California:Berkeley. CA. 1974. 786
Journal of Chemical Education
1 2 3 4
12 12.00 12.0 12.0
10.4 10.52 10.5 10.48
45
124.8 126.24 126
44.96
125.76
44.8
45.04
such as those of team number two to be correct. Two exercises usually enlighten them. First, we ask them to determine the perimeter by using the length measured with the first ruler and the width measured with the second. For Team 2 this would be 12.0 cm 12.0 cm 10.52 cm
Table 3.
Pre- and Post-Test Results
Measurement Task Score
Pretest
Calculation Task Post-test
Pre-test
Post-test
1 (2.7%) 2 (5.4%) 2 (5.4%) 32(86.5%)
0 (0.0%) 32 (86.5%) 3 (8.1%) 2 (5.4%)
0 (0.0%) 24(64.9%) 1 (2.7%) 12 (32.4%)
3 (3.2%) 16(17.2%) 1 1 1 % 73(78.5%)
7 (7.5%) 53 (57.0%) 6 (6.5%) 27 (29.0%)
1 (1.1%) 53 (57.0%) 3 (3.2%) 36(38.7%)
Rand R Students
LLykLL, 45.04 cm
Almost without exception the students agree on a value of 45.0 cm as the correct one. They explain that since only the last digit can be an estimate, the 4 must be dropped or the last two digits will both contain estimates. The students are asked to use these ideas and write a rule for the use of significant figures in addition andlor subtraction operations. In the second exercise we ask the students to calculate areas using a constant length, but a width that has values of plus or minus 0.1 cm from the recorded value. Again, Team 2 would find Recorded width + 0.1: 12.0 crn X 10.5 cm = 126.0 crn2 Recorded values: 12.0 crn X 10.4 em = 124.8 crn2 Recorded width - 0.1: 12.0 x 10.3 cm = 123.6 crn2 The students see that the results vary in the last two digits. They conclude that since only the last digit can contain any estimate, the results must he rounded, and the correct value should be written as 125 cm2. After doing several examples the students begin counting significant figures correctly and actually derive their own rules for multiplication and/or division. In order to clarifv the zero ornblem we ask them to record the length and width in meters and to calculate the area: By comparison to the earlier result (124.8 cm2) which was rounded to 3 significant figures, students see that the zero to the rieht of the decimal in the answer cannot he sienificant. ~hese and, & fact, the result must he rounded to 0.0125 i2. concrete experiences helo them recoenize the usefulness of the rules and also help them state i h e rules in their own words. Appllcatlon
The application phase varies depending upon the setting in which we use R and R. In a classroom, students are asked to calculate the volume of a single rectangle. Since it is difficult to measure the thickness of a single rectangle, the students form a stack and determine the averaee thickness. This approach brings up the relationship between significant fieures and countine numbers. In a laboratorv settine. the &dents must determine the readability limits of sue?; equipment as graduated cylinders and burets. We also ask them to decide on the maximum number of significant figures they can obtain for measurements made with these devices.
-
Comparison of R and R versus Tradltonal Approach
During the 1984-85 academic year tests were given to 37 freshmen chemistry students at the beginning of the quarter before they did the R and R activity as a 3-hour laboratory
0 1 2 3
2 (5.4%) 16(43.2%) 2 (5.4%) 17 (46.0%)
Traditional Students 0 1 2 3
9 (9.7%) 1 1 1 8 % 4 (4.3%) 69 (74.2%)
experiment, and at the end of the quarter. These students received no other formal instruction dealing with significant figures. The same end-of-quarter test was given to 97 students who received significant figure instruction in a traditional lecture setting. Circumstances prevented us from giving this group the pre-test before they received significant figure instruction in class. Two test auestions dealt with significant figures. One question was a measurement task that required the students to record a measured quantity using the correct number of significant figures. The second question required students to do a calculation and express the result using the correct number of significant figures. The questions were worked out on the test paper and were graded on a 0-3 basis as follows: 0 = no attempt made to solve problem, 1 = question done incorrectly, 2 = correct number of sienificant fieures used but incorrect results obrained, 3 = correct number of significant figures and correct r r u l r obtained. The results of the test are r i w n in'l'able 3. Both groups handled the measurement cask quite well in the post-test. The R and R group did a little better with 91.9% scoring 2 or 3 compared to 79.6% of the traditional students. Neither group did as well on the calculation task, but the traditional students did a little better than the R and R group. Of the traditional students, 41.9% scored 2 or 3, compared to 35.1% of the R and R group. Conclusions
The correct use of significant figures can be taught, hut we prefer to have students establish the rules rather than membrize them. In our experience, students who do this are more willing to accept the responsibility for using them correctly. In addition, the gentle reminder "remember R and R is often all that is needed to answer student questions about significant figures. The data collected in the pre- and posttests indicate that R and R is a t least as effective as the traditional a~wroach to teachine sienificant fieures. Trv R .. " and R for yourself, or develop your own concrete experiences that allow students to discover significant figures for themselves. You will like the results. A sample of the materials we use can he obtained for $1.50. Send requests and make checks payable to the Chemistry Department, Weber State College.
- -
Volume 63
Number 9
September 1986
787
