What Is the Half-Life of Basketball Teams?Zdeslav Hrepic

Citation: The Physics Teacher 51, 415 (2013); doi: 10.1119/1.4820854 View online: http://dx.doi.org/10.1119/1.4820854 View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/51/7?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in What Makes Usain Bolt Unique as a Sprinter? Phys. Teach. 48, 365 (2010); 10.1119/1.3479707 From J. J. Thomson to FAIR, what do we learn from LargeScale Mass and HalfLife Measurements of Bareand FewElectron Ions? AIP Conf. Proc. 1224, 28 (2010); 10.1063/1.3431427 Biological Half-Life of Cardiolite® Phys. Teach. 46, 522 (2008); 10.1119/1.3023652 Mechanical Simulation of a Half-Life Phys. Teach. 46, 369 (2008); 10.1119/1.2971223 What makes bowling balls hook? Am. J. Phys. 72, 1170 (2004); 10.1119/1.1767099

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DOI: 10.1119/1.4820854 The Physics Teacher ◆ Vol. 51, OctOber 2013 415

confidence in my promise, as I was not sure myself in what way and how much of it I would be able to deliver.

However, as I started introducing the topic, other interest-ing and productive parallels became clear, beyond the simple exclusion similarity. A critical analogy, and perhaps also the most direct one, is about the half-life of each process. Half-life (t½) is the time required for a quantity to fall to half its value as compared to the beginning of the measured time period. In physics, it is typically used to describe a property of radioac-tive decay, but it is also used in other fields to describe quan-tity that follows an exponential decay.

When confronting something new, we have little choice but to try and understand it in terms of something we already know. For this reason it is possible that everything we under-stand in physics (perhaps in all of science) comes through some kind of analogy and metaphor. This paper presents a captivating way to teach radioactive half-life as students already have an interest and some intuitive understanding of one side of the analogy. But more fundamentally, it also illustrates a rich pedagogical opportunity for exploring the promises and perils of analogies in general.

To elaborate on the analogy, I use below the bracket cre-ated by President Obama for the 2012 tournament (Fig. 1). The bracket is posted on the White House website.2

What do basketball teams have in common with radioactive nuclei? It turns out, there is more here than first meets the eye. The National

Collegiate Athletic Association (NCAA) basketball tourna-ments feeds fans’ craving when NBA competitions are not in swing, and the college tournament time has been referred to as “March Madness” or the “Big Dance”1 as many fans par-ticipate in “bracketing,” i.e., predicting winners.

During one of the “Big Dances,” the lab activity for the day in my physical science course was radioactive decay and, at the onset, a group of students (playfully gathered around a chart on a paper) suggested that instead of worrying about radioactivity, we have some fun “bracketing” the NCAA basketball tournament. At the time, I was not familiar with bracketing. However, this physical science lab course targets non-science majors and my best chance in diverting students’ attention away from the activity they “proposed” was by of-fering (or at least suggesting) in exchange something fun and interesting about the originally planned activity. Still having “my” topic in mind while listening to students explain the forecasting procedure, I realized there are some basic similari-ties between the tournament progression and the radioactive decay process. Namely, both of them start with a certain num-ber of constituents (basketball teams and atoms, respectively) that are progressively ex-cluded (teams eliminated and atoms disintegrated).

So I suggested that we go on with the originally planned radioactivity lab, and I made a pretentious announcement that un-derstanding radioactiv-ity will put their seeding predictions on some sci-entific grounds. Because, I claimed, what they wanted to do is similar to what I was about to teach. Need-less to say, students did not believe they should take me seriously, nor did they expect me to deliver on this promise, but they seemed to appreciate the attempt to throw some-thing of value into the bar-gain. Likewise, I did not blame them for the lack of

Fig. 1. President Obama’s bracket for NCAA 2012 men’s basketball tournament.

What Is the Half-Life of Basketball Teams?Zdeslav Hrepic, Columbus State University, Columbus, Ga

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416 The Physics Teacher ◆ Vol. 51, OctOber 2013

They thus “decayed,” leaving N = (1/2)(N0) coins. After the next round this is reduced to N = (1/2)2(N0) = 1/4(N0). And after n rounds, the number of remaining coins is N = (1/2)n(N0).

Equivalently, the analysis is very similar if we start with a large number (N0) of single elimination tournament teams. The pairs play a set of games and losers are eliminated (with “tail side up”). Thus they “decayed,” which again leaves N = (1/2)n(N0) teams after n rounds.

We point out that the half-life (t½) is the average time re-quired for one half of the teams (i.e., nuclei) to decay. If the sets of game rounds are evenly spaced (say daily or weekly, as it is for halving of unstable nuclei), the number of rounds (n) can be expressed as a ratio of time elapsed and half-life time. So n = (t)/(t½).

Thus:t t1/2

One might object that all coins and all nuclei are created (more or less) the same and all basketball teams may not be. However, for undergraduate students, basketball teams are (clearly) more fun to play with (no pun intended), and for this benefit we may assume, at least hypothetically, that the teams start with equal chances in a tournament.

Furthermore, this distinction and similarity provide another teachable moment based on this analogy. At the be-ginning of the process, it is not possible to predict which par-ticular atoms will disintegrate at any given time. What we do know, however, is that after one half-life, half of the original nuclei will remain in the sample. Likewise, we do not know which basketball teams will win in any of the rounds (this is precisely what makes “bracketing” predictions popular fun). But after one game set, exactly half of the teams are elimi-nated and another half are still playing. And all chances are renewed again for the next round.

As I kept explaining and pinpointing these analogies to my students, I noticed a combination of an exciting understand-ing of the topic and the presented analogies combined with a simultaneous disbelief about the extent of the clear parallels between the topics of radioactivity and bracketing.

While exploring similarities between these concepts, it is necessary and worthwhile to stress the differences inherent in any analogy. And the critical one in this case is the size of the sample. Namely, statistics breaks apart with small numbers, and the number of teams in the typical tournament does not compare to the number of unstable nuclei in any practical sample. Another important limitation of this analogy is that in true radioactive decay, each atom has exactly the same probability of decaying within a given amount of time. Pair-ing teams in a tournament is the equivalent of creating a situ-ation involving two particular teams where one team will certainly “decay” and the other will certainly not during a round of the tournament. In this respect the coin tossing analogy is stronger because the model presented here (unlike the coin tossing model) pairs probabilities. The behavior of

The NCAA basketball tournaments start with 64 college and university teams. As in any single elimination tourna-ment, after the first round of games, there are exactly half of the teams left in the competition. The number is then halved again after another game set round. This is what happens also after each half-life with radioactive nuclei. So, knowing the dates of game rounds, a bracket (such as one shown in Fig. 1) can be used to determine the average half-life of a basketball team during the tournament. In the case of the NCAA 2012 men’s basketball tournament, the first round was played on March 15 and 16, starting with 64 teams. For a measure of sampling we can take three playoff rounds and notice that the original “Round of 64” (March 15/16) goes to “Elite Eight” for March 24/25. The reduction of teams from 64 to 8 accounts for three half-lives and this occurred in a nine-day period between March 15 and March 24. This makes a half-life of a basketball team on this tournament three days long, thus making basketball teams on a tournament somewhat short-lived “particles.” (Although exact dates for NCAA tournament game rounds vary in different years, the days of the “Round of 64” and “Elite Eight” are conveniently held nine days apart from year to year.)

Introductory college physics textbooks typically introduce the topic of half-life by discussing elimination of coins. Each coin has a 50% chance of landing on either side. So if the tail side is assigned as the elimination side, after one throw of a group of coins we can expect that nearly half of them will “dis-integrate.” Therefore, a toss of a large number of coins can be considered equivalent to an elimination round of games in the tournament. Both are further equivalent to the elimination of half of the nuclei during the half-life of a radioactive sample.

To take the point one step further, the coin analogy can be replaced with the tournament analogy to derive the formula for the decay of nuclei in a radioactive sample in terms of half-life.

The use of coin tossing and counting as an analogy in mathematics of radioactive decay is probably familiar3 to ma-ny. You start with a large number (N0) of coins, throw them all simultaneously, and eliminate all those that land tail side up.

Fig. 2. The plot of the number of remaining teams vs time in units of the number of rounds completed resembles an exponential decay curve.

Tournament Rounds0 1 2 3 4 5 6 7

70605040302010 0

Rem

aini

ng T

eam

s

Basketball Teams Decay Rate

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The Physics Teacher ◆ Vol. 51, OctOber 2013 417

one coin doesn’t affect the behavior of another, and after all the coins are flipped and the tails are cleared away, the num-ber of heads remaining is likely not exactly half the initial number. So it is with independent radioactive atoms after the passage of a half-life. However, there is no probability at all involved in the determination of the number of teams re-maining after a round of a tournament.

But analogical differences can be just as productive for target learning as are the similarities. All models, and espe-cially analogies, have limitations by definition. Nevertheless, they also have a great scaffolding power for teaching, in ad-dition to utilitarian value. Analogy in general is considered one of the very fundamental ways humans learn, and earlier research shows that carefully chosen and presented analo-gies can promote student learning of physics.4,5 This paper proposes a way in which students’ understanding of natural radioactive decay can be taught or improved by making use of a rich analogy with a very familiar and popular resource—a sports tournament. Through the proposed analogical scaf-folding and follow-up discussion, the instructor can explore the tournament concept on one side (with which students are very familiar) and compare it to concepts of natural radioac-tivity and half-life, which are difficult topics and also typically do not lend themselves easily to demonstration.

While using the tournament model for this purpose, the above listed limitations of the model should be kept in mind. But at the same time, they themselves can be productively utilized to deepen students’ understanding by comparing and contrasting a familiar tournament model with the tar-get radioactive decay concept from a variety of angles, thus reaching higher levels of Bloom’s taxonomy (analysis and evaluation).

References1. NCAA Turner Sports Interactive Inc., “Big Dance: Rock out at

the Final Four,” 2012.2. M. Compton, “President Obama’s 2012 NCAA Tournament

Bracket,” 2012.3. R. Knight, B. Jones, and S. Field, “Nuclear Decay and Half-

Lives,” in College Physics (Pearson, 2010), pp. 1003-1004.4. N. S. Podolefsky and N. D. Finkelstein, “Analogical scaffold-

ing and the learning of abstract ideas in physics: An example from electromagnetic waves,” Phys. Rev. ST Phys. Educ. Res. 3, 010109 (2007).

5. N. S. Podolefsky and N. D. Finkelstein, “Use of analogy in learning physics: The role of representations,” Phys. Rev. ST Phys. Educ. Res. 2, 020101 (2006).

Zdeslav Hrepic has a BA in physics and polytechnic teaching from University of Split, Croatia, and MS and PhD degrees in physics education from Kansas State University. He is an assistant professor at Columbus State University (GA), and teaches a range of introductory physics courses and science education courses. His pedagogical and research interests revolve around students’ understanding of a range of physics topics and technology-enhanced active learning environments.Department of Earth and Space Sciences, Columbus State University, Columbus, GA 31907; drz@columbusstate.edu; www.hrepic.com

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