In the Classroom edited by
Applications and Analogies
Arthur M. Last University College of the Fraser Valley Abbotsford, BC, Canada
Teaching Significant Figures Using Age Conversions Thomas D. Crute Department of Chemistry and Physics, Augusta State University, Augusta, GA 30904;
[email protected] Understanding and properly manipulating significant figures, while normally tackled early in a chemistry course, can present a lasting challenge. Many students will likely learn the rules by rote, but not necessarily understand the rationale and meaning. Some useful approaches have appeared in this Journal to assist students with the uncertainty of measurements and the necessity of significant figures through exercises (1) and activities (2). Herein is an alternative, requiring little preparation and no equipment, that allows students to intuitively discover and discuss the uncertainty of measurements and conversion factors and the necessity of significant figures. By allowing students to study new concepts with familiar topics such as age, they will be more comfortable later applying these concepts to unfamiliar chemical measurements. The Activity In class, students are simply asked about a particular famous person and how many days old he or she is. The instructor will need to prepare for class by calculating an accurate answer. This is readily done by finding the Julian dates for the current date and the birth date in question and subtracting. The Julian Day Count is a count of days since noon of January 1, 4713 BCE and is frequently used by those who are interested in determining time spans for cyclic events. This system avoids the confusion associated with months of different numbers of days and leap years. The Julian date (JD) is available through Internet sources (3) or print materials (4). The activity starts with no given information about the age of the person in question and progresses with increasing information leading to increasingly precise age estimations. Students will recognize that a person’s age is a measured value, and this estimation is only as precise as the measuring device. They intuitively become aware of error in the estimation of the age in years as well as the conversion to age in days. As the age in years is known with increasing precision,
the calculated age in days may be expressed with increasing precision. Example and Discussion A person who is familiar to everyone must be selected to use as a class example. Using George W. Bush as an example in 2004, students might estimate based on his appearance that he is 60 years old. This immediately opens a discussion of how many significant figures are in 60. My class will generally agree that it is 1 significant figure, meaning the actual value is between 50 and 70 years. Therefore, significant figure rules allow us to report the age in days to 1 significant figure. This allows us to state that he is 20,000 days old (60 years × 365.25 days兾year = 21,915 days), or confidently between 10,000 and 30,000 days old. Most students will agree that we can not state that he is definitely between 21,914 and 21,916 days old as the calculator answer might suggest. George W. Bush was born on July 6, 1946, which was approximately Julian date 2,432,008.1 For a class discussion held September 1, 2004 (JD 2,453,250), Mr. Bush would be approximately 21,242 days old or 58.157 years. The instructor could use this value to reveal an increasingly precise age in years (i.e., one more decimal place for each successive calculation) for Mr. Bush to allow students to calculate the age in days with increasing precision as outlined in Table 1. I find that at some point the discussion will turn to the conversion factor relating days and years, thus opening up a discussion of exact versus inexact values. The typical student will recognize that a leap day every fourth year will result in an average length of year of 365.25 days. Some students may recognize that there are exceptions whereby a particular “fourth year” does not have a leap day (such as the years 1700, 1800, and 1900, which are common years and not leap years) resulting in a long term average of 365.24219 solar days per tropical year (5). Once one recognizes that every fourth year
Table 1. Age of George W. Bush on September 1, 2004 Based on Conversions from Increasingly Precise Values in Years Information Provided
Calculator Display of Age/days
6 x 101 years old (based on appearance)
60 x 365.25 = 21,915
Reported Age/days
Significant Figures
20,000
1
58 years old (born in 1946)
58 x 365.25 = 21,184.5
21,000
2
58.2 years old (born in July 1946)
58.2 x 365.25 = 21,257.55
21,300
3
58.16 years
58.16 x 365.25 = 21,242.94
21,240
4
58.157 years
58.157 x 365.25 = 21,241.84
21,242
5
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In the Classroom
of George W. Bush’s life has been a leap year, one can use 365.25 days兾year as a reasonable conversion. Analysis of all the calculations in Table 1 reveals that none of them are wrong according to the precision reported by significant figure rules. For instance a value of 58.2 years is provided to the students in the third entry of Table 1, and although Bush is not 21,300 days old, he is between 21,200 and 21,400 days as implied by a three significant figure value. Extended discussions might underscore the propagation of rounding errors such as using a value of 365 days兾year as a conversion factor. Students might be asked how to determine someone’s age more exactly, including having students speculate whether any two people can be exactly the same age. This discussion ought to lead to the conclusion that while two people might be born on the same date, they are not born at the same instant once one considers the birth time to an appropriate number of significant figures. Followup Activities A useful assignment would have students select a famous person, research the birth date, and perform their own series of calculations as was done in class. More sophisticated students might be asked to contemplate the limitations of significant figures by, for instance, calculating the percent error for the age in years versus the age in days. More specifically students can discover that the range suggested by 2 signifi-
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cant figure value in years (58 years ±1 year is a 2-year range) is a smaller range than the range suggested by a 2 significant figure value in days (21,000 days ±1000 days is a 2000 day range or greater than a 5-year range). This discussion will help highlight both the strengths and weaknesses of using significant figure rules to express uncertainty. Notes 1. Obviously greater precision of this Julian date is available from knowing the hour of birth and thereby including fractional days in the Julian date.
Literature Cited 1. Pacer, R. A. J. Chem. Educ. 2000, 77, 1435. 2. Kirksey, H. G. J. Chem. Educ. 1992, 69, 497. 3. Julian Date Converter. http://aa.usno.navy.mil/data/docs/ JulianDate.html. Welch, Doug. Calendar Date to Julian Date Calculator. http://wwwmacho.mcmaster.ca/JAVA/JD.html. Julian Date Converters. http://onlineconverters.com/calendars.html (all accessed Jun 2005). 4. The Astronomical Almanac for the Year 2003; U.S. Government Printing Office: Washington, DC, 2001; p B4. 5. CRC Handbook of Chemistry and Physics, 65th ed.; Weast, R. C., Astle, M. J., Beyer, W. H., Eds; CRC Press: Boca Raton, FL, 1984; p F-319.
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